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Ai"3B!'gY~> @endspecial 466 5066 a(c)445 5068 y Fp(\015)g Fv(1999)j Fw(Kluwer)f(A)l(c)l(ademic) h(Publishers.)56 b(Printe)l(d)29 b(in)e(the)h(Netherlands.)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.1)p eop %%Page: 2 2 2 1 bop 246 100 a Fz(2)873 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(Section)h(2)i(describ)s(es)d(three)i(algorithms)f (for)g Fr(SSa)-6 b(t)o Fz(,)27 b(one)g(based)g(on)f(the)h(Da)m(vis-)246 399 y(Putnam-Logemann-Lo)m(v)m(eland)42 b(\(DPLL\))h(algorithm)e(for)h Fr(Sa)-6 b(t)o Fz(,)42 b(a)h(v)-5 b(ariation)246 506 y(based)37 b(on)h(a)g(\\univ)m(ersal")f(searc)m(h)i(algorithm)e(for)g Fr(Sa)-6 b(t)o Fz(,)38 b(and)g(one)g(that)g(com-)246 614 y(bines)23 b(sto)s(c)m(hastic)i(sampling)e(and)h(lo)s(cal)g(searc)m (h.)i(Section)f(4)g(presen)m(ts)f(empirical)246 722 y(results)29 b(and)h(puts)f(them)h(in)m(to)h(p)s(ersp)s(ectiv)m(e.)246 940 y(1.1.)65 b Fr(Deterministic)33 b(Sa)-6 b(tisfiability)246 1136 y Fz(In)48 b(the)i(deterministic)d(satis\014abilit)m(y)g(problem,) h(or)h Fr(Sa)-6 b(t)o Fz(,)49 b(w)m(e)h(are)g(giv)m(en)f(a)246 1244 y(Bo)s(olean)f(form)m(ula)f(and)g(wish)f(to)j(determine)d(whether) i(there)g(is)e(some)j(as-)246 1352 y(signmen)m(t)40 b(to)h(the)f(v)-5 b(ariables)39 b(in)g(the)i(form)m(ula)e(that)i(results)e(in)g(the)i (form)m(ula)246 1460 y(ev)-5 b(aluating)26 b(to)j Fy(True)n Fz(.)f(This)e(problem)f(is)i(connected)h(to)g(problems)e(throughout)246 1568 y(computer)43 b(science,)h(from)f(circuit)f(design)h(and)g (complexit)m(y)g(theory)h(to)g(AI.)246 1676 y(The)35 b(last)h(sev)m(eral)g(y)m(ears)h(has)f(seen)g(tremendous)f(progress)h (in)e(our)i(abilit)m(y)e(to)246 1784 y(solv)m(e)e Fr(Sa)-6 b(t)30 b Fz(problems.)g(In)h(part)g(due)g(to)h(the)g(success)g(of)f (solving)f(satis\014abilit)m(y)246 1891 y(problems,)45 b(researc)m(hers)i(are)h(\014nding)c(e\016cien)m(t)j(w)m(a)m(ys)h(of)f (mo)s(deling)e(div)m(erse)246 1999 y(problems)29 b(within)g(the)j (satis\014abilit)m(y)d(framew)m(ork,)j(suc)m(h)f(as)h(planning)c (\(Kautz)246 2107 y(&)i(Selman,)f(1996\).)362 2215 y(A)39 b(formal)f(de\014nition)f(of)i(the)g Fr(Sa)-6 b(t)38 b Fz(decision)g(problem)f(follo)m(ws.)h(Let)i Fs(x)g Fz(=)246 2323 y Fm(h)p Fl(x)333 2337 y Fn(1)372 2323 y Fl(;)15 b(x)464 2337 y Fn(2)504 2323 y Fl(;)g(:)g(:)g(:)i(;)e(x)758 2337 y Fk(n)805 2323 y Fm(i)33 b Fz(b)s(e)g(a)g(collection)g(of)g Fl(n)f Fz(Bo)s(olean)i(\(0/1\))h(v)-5 b(ariables,)32 b(and)g Fl(\036)p Fz(\()p Fs(x)p Fz(\))246 2431 y(b)s(e)37 b(a)i Fl(k)s Fz(-CNF)g(Bo)s(olean)g(form)m(ula)e(on)h(these)h(v)-5 b(ariables)37 b(with)g Fl(m)h Fz(clauses.)g(F)-8 b(or)246 2539 y(example,)35 b(\()p Fl(x)719 2553 y Fn(1)782 2539 y Fm(_)p 865 2489 92 4 v 22 w Fl(x)917 2553 y Fn(2)980 2539 y Fm(_)23 b Fl(x)1116 2553 y Fn(4)1155 2539 y Fz(\)\()p Fl(x)1277 2553 y Fn(2)1341 2539 y Fm(_)g Fl(x)1477 2553 y Fn(3)1539 2539 y Fm(_)g Fl(x)1675 2553 y Fn(4)1715 2539 y Fz(\)\()p 1785 2489 V Fl(x)1837 2553 y Fn(1)1900 2539 y Fm(_)p 1984 2489 V 23 w Fl(x)2036 2553 y Fn(2)2098 2539 y Fm(_)p 2182 2489 V 23 w Fl(x)2234 2553 y Fn(3)2274 2539 y Fz(\))35 b(is)f(a)h(form)m(ula)f(with)246 2647 y Fl(k)51 b Fz(=)d(3)c(literals)f(p)s(er)g(clause,)h Fl(n)k Fz(=)g(4)c(v)-5 b(ariables)43 b(and)g Fl(m)48 b Fz(=)g(3)d(clauses.)f(An)246 2755 y(assignmen)m(t)36 b Fl(\013)i Fz(to)f(the)g(Bo)s(olean)g(v)-5 b(ariables)36 b Fs(x)h Fz(is)f Fj(satisfying)45 b Fz(if)36 b(it)g(mak)m(es)i(the)246 2863 y(form)m(ula)22 b Fy(True)f Fz(\()p Fl(\036)p Fz(\()p Fl(\013)p Fz(\))27 b(=)e(1\))f(and)e(unsatisfying)f(if)g(it)i(mak)m(es) h(the)f(form)m(ula)f Fy(False)246 2971 y Fz(\()p Fl(\036)p Fz(\()p Fl(\013)p Fz(\))34 b(=)e(0\).)j(The)g(decision)e(problem)g Fr(Sa)-6 b(t)33 b Fz(asks,)i(giv)m(en)g(a)g(Bo)s(olean)g(form)m(ula)246 3079 y Fl(\036)p Fz(\()p Fs(x)p Fz(\))29 b(in)d Fl(k)s Fz(-CNF,)j(do)s(es)f(it)g(ha)m(v)m(e)h(a)f(satisfying)f(assignmen)m(t?) h(Or,)f(sym)m(b)s(olically)-8 b(,)246 3187 y(w)m(e)30 b(w)m(an)m(t)h(to)f(kno)m(w)g(whether)f Fm(9)p Fl(x)1399 3201 y Fn(1)1438 3187 y Fl(;)15 b(:)g(:)g(:)i(;)e Fm(9)p Fl(x)1743 3201 y Fk(n)1790 3187 y Fz(\()p Fl(\036)p Fz(\()p Fs(x)p Fz(\))26 b(=)f(1\).)31 b(T)-8 b(able)30 b(I)f(summarizes)246 3295 y(notation.)246 3513 y(1.2.)65 b Fr(Stochastic)32 b(Sa)-6 b(tisfiability)246 3708 y Fz(P)m(apadimitriou)22 b(\(1985\))27 b(explored)d(an)h(extension)f(of)h Fr(Sa)-6 b(t)24 b Fz(in)f(whic)m(h)g(a)j(random-)246 3816 y(ized)i(quan)m (ti\014er)g(is)f(in)m(tro)s(duced.)h(The)g(sto)s(c)m(hastic)i Fr(Sa)-6 b(t)27 b Fz(\()p Fr(SSa)-6 b(t)o Fz(\))29 b(problem)f(is)f(to) 246 3924 y(ev)-5 b(aluate)24 b(a)g(Bo)s(olean)g(form)m(ula)f(con)m (taining)g(b)s(oth)g(existen)m(tial)g(and)g(randomized)246 4032 y(quan)m(ti\014ers.)29 b(F)-8 b(or)31 b(example:)788 4224 y Fm(9)p Fl(x)891 4238 y Fn(1)930 4224 y Fl(;)1029 4161 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1029 4161 a Fi(R)1088 4161 y currentpoint grestore moveto 1088 4161 a 1029 4224 a Fl(y)1074 4238 y Fn(2)1113 4224 y Fl(;)15 b Fm(9)p Fl(x)1256 4238 y Fn(3)1295 4224 y Fl(;)g(:)g(:)g(:)i(;)e Fm(9)p Fl(x)1600 4238 y Fk(n)p Fx(\000)p Fn(1)1737 4224 y Fl(;)1836 4161 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1836 4161 a Fi(R)1895 4161 y currentpoint grestore moveto 1895 4161 a 1836 4224 a Fl(y)1881 4238 y Fk(n)1928 4224 y Fz(\()p Fl(E)5 b Fz([)p Fl(\036)p Fz(\()p Fs(x)p Fz(\)])27 b Fm(\025)e Fl(\022)s Fz(\))p Fl(:)246 4417 y Fz(In)f(w)m(ords,)h(this) f(form)m(ula)g(asks)h(whether)f(there)h(is)f(a)i(v)-5 b(alue)24 b(for)h Fl(x)2467 4431 y Fn(1)2531 4417 y Fz(suc)m(h)g(that)h Fj(for)246 4525 y(r)-5 b(andom)36 b Fz(v)-5 b(alues)25 b(of)i Fl(y)989 4539 y Fn(2)1054 4525 y Fz(\(c)m(ho)s(ose)h(0)f(or)f(1) h(with)e(equal)h(probabilit)m(y\),)f(there)i(exists)246 4633 y(a)38 b(v)-5 b(alue)37 b(of)h Fl(x)734 4647 y Fn(3)773 4633 y Fz(,)g(suc)m(h)f(that...)i(the)f Fj(exp)-5 b(e)g(cte)g(d)49 b Fz(v)-5 b(alue)37 b(of)h(the)g(Bo)s(olean)g(form)m(ula)246 4741 y Fl(\036)p Fz(\()p Fs(x)p Fz(\))32 b(meets)h(or)e(exceeds)i(a)f (threshold)e(0)e Fm(\024)f Fl(\022)j Fm(\024)d Fz(1.)33 b(This)c(particular)h(example)246 4848 y(of)47 b(sto)s(c)m(hastic)h (satis\014abilit)m(y)d(consists)h(of)i(alternating)e(b)s(et)m(w)m(een)i (making)f(a)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.2)p eop %%Page: 3 3 3 2 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)853 b Fz(3)246 259 y Fv(T)-6 b(able)24 b(I.)38 b(Summary)21 b(of)j(notation)h(used)e(for)h(Bo)r(olean)i(form)n (ulae.)e(Some)f(notation)h(is)g(in)n(tro)r(duced)f(in)246 350 y(later)j(sections.)295 473 y(0)c Fh(<)f(n)569 b Fv(n)n(um)n(b)r(er)24 b(of)i(Bo)r(olean)i(v)l(ariables)295 582 y(0)22 b Fh(<)f(k)575 b Fv(n)n(um)n(b)r(er)24 b(of)i(literals)h(p)r (er)f(clause)295 691 y(0)c Fh(<)f(m)g Fp(\024)606 635 y Fg(P)687 656 y Ff(k)687 714 y(i)p Fe(=1)804 631 y Fg(\000)839 657 y Fe(2)p Ff(n)862 716 y(i)908 631 y Fg(\001)1051 691 y Fv(n)n(um)n(b)r(er)j(of)i(clauses)295 800 y Fh(b)c Fp(2)f(f)p Fv(0)p Fh(;)14 b Fv(1)p Fp(g)442 b Fv(Bo)r(olean)27 b(v)l(alue)295 909 y Fu(x)21 b Fv(=)h Fp(h)p Fh(x)519 917 y Fe(1)553 909 y Fh(;)13 b(x)631 917 y Fe(2)665 909 y Fh(;)g(:)g(:)g(:)g(;)g(x)879 917 y Ff(n)921 909 y Fp(i)100 b Fv(Bo)r(olean)27 b(v)l(ariables,)g Fh(x)1730 917 y Ff(i)1777 909 y Fp(2)22 b(f)p Fv(0)p Fh(;)14 b Fv(1)p Fp(g)295 1018 y Fh(\036)p Fv(\()p Fu(x)p Fv(\))603 b(Bo)r(olean)27 b(form)n(ula)f(in)g(CNF)295 1127 y Fh(\036)p Fp(d)375 1136 y Ff(x)410 1147 y Fd(i)436 1136 y Fe(=)p Ff(b)1051 1127 y Fv(restriction)h(\(v)l(ariable)f(assignmen)n(t\))f(of)i Fh(\036)e Fv(\(ren)n(um)n(b)r(ers)f(v)l(ariables\))295 1237 y Fh(\013)707 b Fv(assignmen)n(t)25 b(to)h Fu(x)295 1346 y Fh(Q)p Fv(\()p Fh(x)430 1354 y Ff(i)456 1346 y Fv(\))21 b Fp(2)h(f9)p Fh(;)13 b Fp(8)p Fh(;)821 1294 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 821 1294 a Fc(R)871 1294 y currentpoint grestore moveto 871 1294 a 821 1346 a Fp(g)192 b Fv(quan)n(ti\014er)25 b(for)h(v)l(ariable)g Fh(x)1846 1354 y Ff(i)1898 1346 y Fv(\(existen)n(tial,)g(univ)n(ersal,)h(or)f (randomized\))295 1455 y Fp(F)357 1423 y Ff(k)q(;n)350 1468 y(m)1051 1455 y Fv(\014xed-clause)f(distribution)295 1564 y(3)d Fp(\024)f Fh(s)g Fp(\024)g Fv(2)612 1532 y Ff(n)p Fe(+1)749 1564 y Fp(\000)c Fv(1)187 b(size)26 b(of)h(smallest)f(DPLL)g(tree)295 1673 y(0)c Fp(\024)f Fh(w)560 b Fv(size)26 b(of)h(an)f(assignmen)n(t)f(sample)295 1782 y(0)d Fp(\024)f Fh(\022)j Fp(\024)d Fv(1)436 b(sto)r(c)n(hastic)27 b(satisfaction)h(threshold)295 1891 y(0)22 b Fp(\024)f Fh(t)g Fp(\024)g Fh(n)439 b Fv(satisfaction)28 b(threshold,)e(where)g (2)2074 1859 y Ff(t)2128 1891 y Fv(is)g(the)f(n)n(um)n(b)r(er)f(of)j (assignmen)n(ts)1102 2000 y(needed)e(for)i(a)f(p)r(ositiv)n(e)g (instance)295 2109 y(0)c Fp(\024)f Fh(c)h Fp(\024)f Fh(n)433 b Fv(n)n(um)n(b)r(er)24 b(of)i(c)n(hoice)g(v)l(ariables)h(in)e Ft(E-Majsa)-5 b(t)246 2534 y Fz(c)m(hoice)41 b(of)g(v)-5 b(alue)40 b(for)h(an)f Fl(x)h Fz(v)-5 b(ariable)39 b(\(p)s(erhaps)h (based)g(on)h(v)-5 b(alues)40 b(of)h(lo)m(w)m(er-)246 2642 y(n)m(um)m(b)s(ered)28 b(v)-5 b(ariables\))29 b(and)g(a)i(c)m (hance)g(selection)f(of)g(a)g(v)-5 b(alue)29 b(for)h(a)g Fl(y)j Fz(v)-5 b(ariable.)246 2750 y(F)d(or)40 b(this)f(reason,)h(P)m (apadimitriou)e(referred)h(to)h Fr(SSa)-6 b(t)38 b Fz(as)i(a)h(\\game)g (against)246 2858 y(nature.")j(An)f(instance)h(of)g Fr(SSa)-6 b(t)42 b Fz(is)h Fj(p)-5 b(ositive)52 b Fz(if)43 b(the)h(threshold)e (is)h(met)h(or)246 2966 y(exceeded.)26 b(This)d(pap)s(er)h(also)h (considers)e(a)j(generalization)e(of)h(P)m(apadimitriou's)246 3074 y(class,)37 b(Extended)30 b Fr(SSa)-6 b(t)o Fz(,)37 b(that)i(allo)m(ws)e(univ)m(ersal,)f(existen)m(tial)h(and)g(random-)246 3182 y(ized)29 b(quan)m(ti\014ers,)h(and)f(a)i(more)f(constrained)g(v)m (ersion,)g(Alternating)f Fr(SSa)-6 b(t)o Fz(,)30 b(in)246 3290 y(whic)m(h)f(existen)m(tial)h(and)f(randomized)h(quan)m(ti\014ers) f(strictly)g(alternate.)246 3553 y(1.2.1.)65 b Fj(Semantics)34 b(of)f(Extende)-5 b(d)33 b Fr(SSa)-6 b(t)31 b Fj(and)j(De)-5 b(cision)33 b(T)-7 b(r)i(e)g(es)246 3661 y Fz(An)27 b(Extended)j Fr(SSa)-6 b(t)26 b Fz(form)m(ula)h(is)g(de\014ned)f(b)m(y)i(a)g(triple) e(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))28 b(where)f Fl(\036)h Fz(is)f(a)246 3769 y(CNF)33 b(form)m(ula)g(with)f(underlying) e(ordered)j(v)-5 b(ariables)32 b Fl(x)2238 3783 y Fn(1)2277 3769 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)2531 3783 y Fk(n)2578 3769 y Fz(,)34 b(0)c Fm(\024)g Fl(\022)j Fm(\024)d Fz(1)246 3877 y(is)g(a)i(threshold,)e(and)h Fl(Q)h Fz(is)e(a)i(mapping)e(from)h (v)-5 b(ariables)30 b(to)j(quan)m(ti\014ers)d(\(exis-)246 3985 y(ten)m(tial)36 b Fm(9)p Fz(,)g(univ)m(ersal)e Fm(8)p Fz(,)i(and)g(randomized)1889 3922 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1889 3922 a Fi(R)1947 3922 y currentpoint grestore moveto 1947 3922 a 1889 3985 a Fz(\).)h(De\014ne)f Fl(\036)2327 3952 y Fx(0)2386 3985 y Fz(=)e Fl(\036)p Fm(d)2585 4000 y Fk(x)2625 4010 y Ff(i)2652 4000 y Fn(=)p Fk(b)2741 3985 y Fz(,)j(where)246 4093 y Fl(\036)300 4060 y Fx(0)365 4093 y Fz(is)k(the)h(\()p Fl(n)28 b Fm(\000)f Fz(1\)-v)-5 b(ariable)42 b(CNF)g(form)m(ula)f(obtained)g(from)h(assigning)e(the)246 4201 y(single)27 b(v)-5 b(ariable)27 b Fl(x)888 4215 y Fk(i)944 4201 y Fz(the)h(Bo)s(olean)h(v)-5 b(alue)28 b Fl(b)g Fz(in)f(the)h Fl(n)p Fz(-v)-5 b(ariable)27 b(CNF)i(form)m(ula) e Fl(\036)246 4309 y Fz(and)h(simplifying)c(the)29 b(result)e (\(including)f(an)m(y)j(necessary)g(v)-5 b(ariable)27 b(ren)m(um)m(b)s(er-)246 4417 y(ing\).)k(V)-8 b(ariables)31 b(are)h(n)m(um)m(b)s(ered)e(so)i(that)g Fl(x)1794 4431 y Fn(1)1865 4417 y Fz(corresp)s(onds)e(to)j(the)e(outermost)246 4525 y(\(leftmost\))g(quan)m(ti\014er)e(and)h Fl(x)1306 4539 y Fk(n)1383 4525 y Fz(to)h(the)g(innermost.)362 4633 y(The)k(v)-5 b(alue)35 b(of)g Fl(\036)h Fz(\(under)e(quan)m (ti\014er)h(order)f Fl(Q)p Fz(\),)i Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\),)37 b(is)d(de\014ned)g(b)m(y)246 4741 y(induction)20 b(on)j(the)g(n)m(um)m(b)s(er)f(of)h(quan)m (ti\014ers)f(as)h(follo)m(ws.)g(Let)g Fl(x)2396 4755 y Fn(1)2459 4741 y Fz(b)s(e)f(the)h(v)-5 b(ariable)246 4848 y(asso)s(ciated)39 b(with)e(the)i(outermost)h(quan)m(ti\014er:)e (\(1\))i(if)d Fl(\036)i Fz(con)m(tains)g(an)g(empt)m(y)1819 5847 y Fo(paper.tex;)j(1/09/1999;)f(0:14;)g(p.3)p eop %%Page: 4 4 4 3 bop 246 100 a Fz(4)873 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(clause,)i(then)f Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))26 b(=)f(0;)j(\(2\))g(if)e Fl(\036)h Fz(con)m(tains)g(no)f(clauses)h(then)g Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b(=)246 399 y(1;)21 b(\(3\))g(if)e Fl(Q)p Fz(\()p Fl(x)705 413 y Fn(1)745 399 y Fz(\))25 b(=)g Fm(9)p Fz(,)20 b(then)g Fl(v)s(al)r Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b(=)e(max\()p Fj(val)q Fz(\()p Fl(\036)p Fm(d)2122 413 y Fk(x)2162 422 y Fe(1)2197 413 y Fn(=0)2292 399 y Fl(;)15 b(Q)p Fz(\))p Fl(;)g Fj(val)q Fz(\()p Fl(\036)p Fm(d)2720 413 y Fk(x)2760 422 y Fe(1)2795 413 y Fn(=1)2890 399 y Fl(;)g(Q)p Fz(\)\);)246 506 y(\(4\))33 b(if)e Fl(Q)p Fz(\()p Fl(x)638 520 y Fn(1)678 506 y Fz(\))d(=)g Fm(8)p Fz(,)k(then)f Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))29 b(=)f(min)n(\()p Fj(val)q Fz(\()p Fl(\036)p Fm(d)2059 520 y Fk(x)2099 529 y Fe(1)2135 520 y Fn(=0)2229 506 y Fl(;)15 b(Q)p Fz(\))p Fl(;)g Fj(val)q Fz(\()p Fl(\036)p Fm(d)2657 520 y Fk(x)2697 529 y Fe(1)2733 520 y Fn(=1)2827 506 y Fl(;)g(Q)p Fz(\)\);)246 614 y(\(5\))31 b(if)e Fl(Q)p Fz(\()p Fl(x)634 628 y Fn(1)674 614 y Fz(\))d(=)889 551 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 889 551 a Fi(R)948 551 y currentpoint grestore moveto 948 551 a 889 614 a Fz(,)31 b(then)f Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b(=)e(\()p Fj(val)q Fz(\()p Fl(\036)p Fm(d)1898 628 y Fk(x)1938 637 y Fe(1)1973 628 y Fn(=0)2067 614 y Fl(;)15 b(Q)p Fz(\))21 b(+)f Fj(val)q Fz(\()p Fl(\036)p Fm(d)2567 628 y Fk(x)2607 637 y Fe(1)2642 628 y Fn(=1)2736 614 y Fl(;)15 b(Q)p Fz(\)\))p Fl(=)p Fz(2.)362 722 y(F)-8 b(or)31 b(example,)f(the)h(form)m(ula)859 834 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 859 834 a Fi(R)918 834 y currentpoint grestore moveto 918 834 a 859 897 a Fl(x;)15 b Fm(9)p Fl(y)s(;)1149 834 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1149 834 a Fi(R)1208 834 y currentpoint grestore moveto 1208 834 a 1149 897 a Fl(z)t Fz(\()p Fl(E)5 b Fz([\()p Fl(x)22 b Fm(_)e Fl(y)s Fz(\)\()p Fl(y)j Fm(_)p 1784 847 47 4 v 20 w Fl(z)t Fz(\)\()p 1900 847 52 4 v Fl(x)e Fm(_)p 2054 847 48 4 v 20 w Fl(y)s Fz(\)])26 b Fm(\025)e Fz(1)p Fl(=)p Fz(2\))p Fl(:)246 1071 y Fz(has)31 b(v)-5 b(alue)32 b(3)p Fl(=)p Fz(4.)h(No)m(w,)g(giv)m(en)f Fl(\036)p Fz(,)g Fl(Q)p Fz(,)g(and)g(a)g(threshold)e Fl(\022)s Fz(,)i(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))32 b(is)f Fy(True)g Fz(if)246 1179 y(and)e(only)h(if)f Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))26 b Fm(\025)f Fl(\022)s Fz(.)362 1287 y(Note)35 b(that)g(this)d(coincides)h(with)g(the)h(usual)e(de\014nition)g(of)i (the)g(quan)m(ti\014ed)246 1395 y(Bo)s(olean)42 b(form)m(ula)g(problem) e(\()p Fr(QBF)p Fz(\),)j(where)e(all)h(of)g(the)g(quan)m(ti\014ers)f (are)i Fm(8)246 1503 y Fz(or)34 b Fm(9)p Fz(.)f(The)h(usual)e (de\014nition)g(of)i(truth)f(for)h Fl(\036)g Fz(on)g(an)g(instance)f (of)h Fr(QBF)g Fz(is)f(as)246 1611 y(follo)m(ws.)25 b(If)h Fl(\036)g Fz(con)m(tains)g(an)g(empt)m(y)g(clause,)g(then)g Fl(\036)g Fz(is)f Fy(False)o Fz(;)h(if)f Fl(\036)h Fz(con)m(tains)g(no) 246 1719 y(clauses,)32 b(then)g Fl(\036)h Fz(is)e Fy(True)o Fz(;)i(otherwise,)f(if)f Fl(\036)e Fz(=)f Fm(9)p Fl(x)2017 1733 y Fn(1)2056 1719 y Fl(\036)2110 1686 y Fx(0)2133 1719 y Fz(,)33 b(then)f Fl(\036)h Fz(is)e Fy(True)g Fz(if)h(and)246 1826 y(only)25 b(if)h Fl(\036)575 1793 y Fx(0)598 1826 y Fm(d)638 1840 y Fk(x)678 1849 y Fe(1)713 1840 y Fn(=0)833 1826 y Fz(is)g Fy(True)f Fz(or)i Fl(\036)1300 1793 y Fx(0)1323 1826 y Fm(d)1363 1840 y Fk(x)1403 1849 y Fe(1)1438 1840 y Fn(=1)1559 1826 y Fz(is)e Fy(True)o Fz(,)i(and)e(\014nally)g(if) g Fl(\036)h Fz(=)e Fm(8)p Fl(x)2690 1840 y Fn(1)2729 1826 y Fl(\036)2783 1793 y Fx(0)2807 1826 y Fz(,)i(then)246 1934 y Fl(\036)33 b Fz(is)g Fy(True)f Fz(if)h(and)g(only)f(if)h(b)s (oth)g Fl(\036)1480 1901 y Fx(0)1503 1934 y Fm(d)1543 1948 y Fk(x)1583 1957 y Fe(1)1618 1948 y Fn(=0)1746 1934 y Fz(and)g Fl(\036)1980 1901 y Fx(0)2003 1934 y Fm(d)2043 1948 y Fk(x)2083 1957 y Fe(1)2118 1948 y Fn(=1)2246 1934 y Fz(are)h Fy(True)n Fz(.)g(Using)f(the)246 2042 y(de\014nition)22 b(of)i(v)-5 b(alue,)24 b Fl(\036)h Fz(is)e Fy(True)g Fz(if)g(and)h(only)g(if)f Fl(v)s(al)r Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b Fl(>)e Fz(0.)g(The)e(base)i(case,)246 2150 y(when)j Fl(\036)i Fz(con)m(tains)g(an)g(empt)m(y)g(clause)f(or)h (no)f(clauses)h(clearly)f(coincide.)g(If)g Fl(\036)d Fz(=)246 2258 y Fm(9)p Fl(x)349 2272 y Fn(1)388 2258 y Fl(\036)442 2225 y Fx(0)465 2258 y Fz(,)31 b(then)f Fl(\036)h Fz(is)e Fy(True)h Fz(if)f(and)h(only)g(if)f Fl(\036)1723 2225 y Fx(0)1747 2258 y Fm(d)1787 2272 y Fk(x)1827 2281 y Fe(1)1861 2272 y Fn(=0)1986 2258 y Fz(is)h Fy(True)f Fz(or)i Fl(\036)2465 2225 y Fx(0)2488 2258 y Fm(d)2528 2272 y Fk(x)2568 2281 y Fe(1)2603 2272 y Fn(=1)2727 2258 y Fz(is)f Fy(True)o Fz(,)246 2366 y(whic)m(h)g(is)g (the)h(case)h(if)f(and)f(only)g(if)h(max\()p Fj(val)q Fz(\()p Fl(\036)1901 2333 y Fx(0)1924 2366 y Fm(d)1964 2380 y Fk(x)2004 2389 y Fe(1)2039 2380 y Fn(=0)2134 2366 y Fl(;)15 b(Q)p Fz(\))p Fl(;)g Fj(val)q Fz(\()p Fl(\036)2522 2333 y Fx(0)2546 2366 y Fm(d)2586 2380 y Fk(x)2626 2389 y Fe(1)2661 2380 y Fn(=1)2755 2366 y Fl(;)g(Q)p Fz(\)\))28 b Fl(>)246 2474 y Fz(0,)j(as)f(desired.)f(The)h(de\014nitions)e (similarly)f(coincide)j(when)f Fl(\036)c Fz(=)g Fm(8)p Fl(x)2655 2488 y Fn(1)2694 2474 y Fl(\036)2748 2441 y Fx(0)2772 2474 y Fz(.)362 2582 y(It)g(will)e(b)s(e)h(useful)f(to)j (visualize)d(the)j(ab)s(o)m(v)m(e)g(inductiv)m(e)e(pro)s(cedure)g(as)h (a)g(deci-)246 2690 y(sion)d(tree.)h(A)g Fj(de)-5 b(cision)27 b(tr)-5 b(e)g(e)31 b Fl(T)k Fz(o)m(v)m(er)24 b(v)-5 b(ariables)22 b Fl(x)1928 2704 y Fn(1)1967 2690 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)2221 2704 y Fk(n)2291 2690 y Fz(is)22 b(a)h(binary)f(decision)246 2798 y(tree)31 b(where)g(eac)m(h)h(in)m(ternal)e(no)s(de)g(in)g(the)h (tree)h(is)e(lab)s(eled)f(with)h(a)h(v)-5 b(ariable)30 b Fl(x)2982 2812 y Fk(i)3010 2798 y Fz(,)246 2906 y(and)g(the)h(t)m(w)m (o)i(outgoing)e(edges)h(are)f(lab)s(eled)e(with)h(the)h(t)m(w)m(o)i (di\013eren)m(t)d(settings)246 3014 y(of)c Fl(x)397 3028 y Fk(i)425 3014 y Fz(:)g Fl(x)528 3028 y Fk(i)581 3014 y Fz(=)f(0)i(and)e Fl(x)973 3028 y Fk(i)1026 3014 y Fz(=)g(1.)i(Eac)m (h)f(leaf)g(of)g(the)g(tree)h(is)e(lab)s(eled)f(with)g(either)i(0)g(or) 246 3122 y(1.)f(Note)h(that)f(an)m(y)g(total)g(truth)f(assignmen)m(t)g Fl(\013)h Fz(to)h(the)e(underlying)e(v)-5 b(ariables)23 b(is)246 3230 y(consisten)m(t)i(with)f(exactly)h(one)h(path)e(of)i Fl(T)13 b Fz(.)25 b(A)g(decision)e(tree)j Fl(T)38 b Fz(o)m(v)m(er)26 b Fl(x)2695 3244 y Fn(1)2734 3230 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)2988 3244 y Fk(n)246 3337 y Fz(represen)m(ts)23 b(a)g(CNF)h(form)m(ula)e Fl(\036)p Fz(\()p Fl(x)1419 3351 y Fn(1)1459 3337 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)1713 3351 y Fk(n)1760 3337 y Fz(\))24 b(if)e(for)h(ev)m(ery)h(truth)e(assignmen)m(t)h Fl(\013)p Fz(,)246 3445 y(the)g(v)-5 b(alue)22 b(of)h(the)f(leaf)h(no)s(de)f(on)g (the)h(path)g(consisten)m(t)g(with)e Fl(\013)i Fz(is)f(equal)g(to)i Fl(\036)p Fz(\()p Fl(\013)p Fz(\).)246 3553 y(A)k(decision)e(tree)i Fl(T)41 b Fz(for)27 b Fl(\036)h Fz(is)f Fj(c)-5 b(omplete)29 b Fz(if)e(it)g(is)g(a)h(full)d(binary)h(tree)j(of)f(heigh)m(t)f Fl(n)p Fz(;)246 3661 y(notice)k(that)h(a)g(giv)m(en)f Fl(\036)g Fz(will)e(ha)m(v)m(e)k(man)m(y)e(decision)f(trees)i(that)g (can)f(represen)m(t)246 3769 y(it;)38 b(some)h(ma)m(y)g(b)s(e)f(v)m (ery)g(small)f(while)g(others)h(\(suc)m(h)h(as)g(the)f(complete)h (tree\))246 3877 y(can)29 b(ha)m(v)m(e)g(exp)s(onen)m(tial)f(size)g(in) g Fl(n)p Fz(.)g(Figure)g(1)h(giv)m(es)g(an)g(example)f(of)h(a)g (decision)246 3985 y(tree)i(for)f(\()p Fl(x)20 b Fm(_)g Fl(y)s Fz(\)\()p Fl(y)k Fm(_)p 1023 3935 47 4 v 20 w Fl(z)t Fz(\)\()p 1139 3935 52 4 v Fl(x)d Fm(_)p 1293 3935 48 4 v 20 w Fl(y)r Fz(\).)362 4093 y(A)27 b Fj(c)-5 b(anonic)g(al)28 b Fz(decision)d(tree)i(for)g(an)f(Extended)k Fr(SSa)-6 b(t)25 b Fz(instance)h(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))27 b(is)246 4201 y(a)j(decision)e(tree)i Fl(T)42 b Fz(where)30 b(the)f(ordering)g(of)g(the)h(v)-5 b(ariables)29 b(in)f Fl(T)42 b Fz(is)29 b(consisten)m(t)246 4309 y(with)g(the)h(quan) m(ti\014er)g(order.)362 4417 y(Giv)m(en)24 b(a)h(canonical)f(tree)h Fl(T)37 b Fz(for)25 b(an)f(Extended)30 b Fr(SSa)-6 b(t)23 b Fz(form)m(ula)g Fl(\036)p Fz(,)i(the)g(v)-5 b(alue)246 4525 y(of)32 b Fl(\036)p Fz(,)g Fl(v)s(al)r Fz(\()p Fl(\036;)15 b(Q)p Fz(\),)34 b(can)e(b)s(e)f(computed)h(from)f Fl(T)45 b Fz(b)m(y)32 b(lab)s(eling)d(the)j(in)m(termediate)246 4633 y(no)s(des)g(of)i Fl(T)46 b Fz(in)32 b(a)i(b)s(ottom-up)f (fashion:)f(Inductiv)m(ely)-8 b(,)32 b(if)h Fl(v)j Fz(is)d(unlab)s (eled,)e(and)246 4741 y(has)42 b(c)m(hildren)f Fl(v)826 4708 y Fx(0)849 4741 y Fz(,)i Fl(v)964 4708 y Fx(00)1050 4741 y Fz(with)e(lab)s(els)g Fl(L)p Fz(\()p Fl(v)1681 4708 y Fx(0)1705 4741 y Fz(\),)i Fl(L)p Fz(\()p Fl(v)1952 4708 y Fx(00)1995 4741 y Fz(\))g(resp)s(ectiv)m(ely)-8 b(,)43 b(then:)g(\(i\))f(if)246 4848 y Fl(Q)p Fz(\()p Fl(v)s Fz(\))26 b(=)f Fm(9)p Fz(,)g(then)h Fl(L)p Fz(\()p Fl(v)s Fz(\))g(=)f(max\()p Fl(L)p Fz(\()p Fl(v)1510 4815 y Fx(0)1534 4848 y Fz(\))p Fl(;)15 b(L)p Fz(\()p Fl(v)1753 4815 y Fx(00)1797 4848 y Fz(\)\);)27 b(\(ii\))e(if)f Fl(Q)p Fz(\()p Fl(v)s Fz(\))i(=)f Fm(8)p Fz(,)h(then)f Fl(L)p Fz(\()p Fl(v)s Fz(\))i(=)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.4)p eop %%Page: 5 5 5 4 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)853 b Fz(5)425 208 y gsave currentpoint currentpoint translate 270 neg rotate neg exch neg exch translate 425 208 a @beginspecial 97 @llx 164 @lly 419 @urx 686 @ury 1800 @rwi @setspecial %%BeginDocument: /u/majercik/papers/SAT2000/fmcharts/dpll.ps % % Frame ps_prolog 5.5, for use with Adobe Unix Frame 5.5 products % % This ps_prolog file is Copyright (c) 1986-1996 Adobe Systems, Incoporated. % All rights reserved. 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This has no effect on bitmap images. /FMPrintAllColorsAsBlack false def % % Frame products can either set their own line screens or use a printer's % default settings. Three flags below control this separately for no % separations, spot separations and process separations. If a flag % is true, then the default printer settings will not be changed. If it is % false, Frame products will use their own settings from a table based on % the printer's resolution. /FMUseDefaultNoSeparationScreen true def /FMUseDefaultSpotSeparationScreen true def /FMUseDefaultProcessSeparationScreen false def % % For any given PostScript printer resolution, Frame products have two sets of % screen angles and frequencies for printing process separations, which are % recomended by Adobe. The following variable chooses the higher frequencies % when set to true or the lower frequencies when set to false. This is only % effective if the appropriate FMUseDefault...SeparationScreen flag is false. /FMUseHighFrequencyScreens true def % % The following is a set of predefined optimal frequencies and angles for various % common dpi settings. This is taken from "Advances in Color Separation Using % PostScript Software Technology," from Adobe Systems (3/13/89 P.N. LPS 0043) % and corrolated with information which is in various PPD (4.0) files. % % The "dpiranges" figure is the minimum dots per inch device resolution which % can support this setting. The "low" and "high" values are controlled by the % setting of the FMUseHighFrequencyScreens flag above. The "TDot" flags control % the use of the "Yellow Triple Dot" feature whereby the frequency id divided by % three, but the dot function is "trippled" giving a block of 3x3 dots per cell. % % PatFreq is a compromise pattern frequency for ps Level 2 printers which is close % to the ideal WYSIWYG pattern frequency of 9 repetitions/inch but does not beat % (too badly) against the screen frequencies of any separations for that DPI. /dpiranges [ 2540 2400 1693 1270 1200 635 600 0 ] def /CMLowFreqs [ 100.402 94.8683 89.2289 100.402 94.8683 66.9349 63.2456 47.4342 ] def /YLowFreqs [ 95.25 90.0 84.65 95.25 90.0 70.5556 66.6667 50.0 ] def /KLowFreqs [ 89.8026 84.8528 79.8088 89.8026 84.8528 74.8355 70.7107 53.033 ] def /CLowAngles [ 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 ] def /MLowAngles [ 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 ] def /YLowTDot [ true true false true true false false false ] def /CMHighFreqs [ 133.87 126.491 133.843 108.503 102.523 100.402 94.8683 63.2456 ] def /YHighFreqs [ 127.0 120.0 126.975 115.455 109.091 95.25 90.0 60.0 ] def /KHighFreqs [ 119.737 113.137 119.713 128.289 121.218 89.8026 84.8528 63.6395 ] def /CHighAngles [ 71.5651 71.5651 71.5651 70.0169 70.0169 71.5651 71.5651 71.5651 ] def /MHighAngles [ 18.4349 18.4349 18.4349 19.9831 19.9831 18.4349 18.4349 18.4349 ] def /YHighTDot [ false false true false false true true false ] def /PatFreq [ 10.5833 10.0 9.4055 10.5833 10.0 10.5833 10.0 9.375 ] def % % PostScript Level 2 printers contain an "Accurate Screens" feature which can % improve process separation rendering at the expense of compute time. This % flag is ignored by PostScript Level 1 printers. /FMUseAcccurateScreens true def % % The following PostScript procedure defines the spot function that Frame % products will use for process separations. You may un-comment-out one of % the alternative functions below, or use your own. % % Dot function /FMSpotFunction {abs exch abs 2 copy add 1 gt {1 sub dup mul exch 1 sub dup mul add 1 sub } {dup mul exch dup mul add 1 exch sub }ifelse } def % % Line function % /FMSpotFunction { pop } def % % Elipse function % /FMSpotFunction { dup 5 mul 8 div mul exch dup mul exch add % sqrt 1 exch sub } def % % /FMversion (5.5) def /fMLevel1 /languagelevel where {pop languagelevel} {1} ifelse 2 lt def /FMPColor fMLevel1 { false /colorimage where {pop pop true} if } { true } ifelse def /FrameDict 400 dict def systemdict /errordict known not {/errordict 10 dict def errordict /rangecheck {stop} put} if % The readline in PS 23.0 doesn't recognize cr's as nl's on AppleTalk FrameDict /tmprangecheck errordict /rangecheck get put errordict /rangecheck {FrameDict /bug true put} put FrameDict /bug false put mark % Some PS machines read past the CR, so keep the following 3 lines together! currentfile 5 string readline 00 0000000000 cleartomark errordict /rangecheck FrameDict /tmprangecheck get put FrameDict /bug get { /readline { /gstring exch def /gfile exch def /gindex 0 def { gfile read pop dup 10 eq {exit} if dup 13 eq {exit} if gstring exch gindex exch put /gindex gindex 1 add def } loop pop gstring 0 gindex getinterval true } bind def } if /FMshowpage /showpage load def /FMquit /quit load def /FMFAILURE { 2 copy exch = = flush FMshowpage /Helvetica findfont 12 scalefont setfont 72 200 moveto show 72 220 moveto show FMshowpage FMquit } def /FMVERSION { FMversion ne { (Adobe Frame product version does not match ps_prolog! Check installation;) (also check ~/fminit and ./fminit for old versions) FMFAILURE } if } def /fmConcatProcs { /proc2 exch cvlit def/proc1 exch cvlit def/newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }def FrameDict begin [ /ALDsave /FMdicttop /FMoptop /FMpointsize /FMsetsize /FMsaveobject /b /bitmapsave /blut /bpside /bs /bstring /bwidth /c /cf /cs /cynu /depth /edown /fh /fillvals /fw /fx /fy /g /gfile /gindex /grnt /gryt /gstring /height /hh /i /im /indx /is /k /kk /landscape /lb /len /llx /lly /m /magu /manualfeed /n /offbits /onbits /organgle /orgbangle /orgbfreq /orgbproc /orgbxfer /orgfreq /orggangle /orggfreq /orggproc /orggxfer /orghalftone /orgmatrix /orgproc /orgrangle /orgrfreq /orgrproc /orgrxfer /orgxfer /pagesave /paperheight /papersizedict /paperwidth /pos /pwid /r /rad /redt /sl /str /tran /u /urx /ury /val /width /width /ws /ww /x /x1 /x2 /xindex /xpoint /xscale /xx /y /y1 /y2 /yelu /yindex /ypoint /yscale /yy /tintGray ] { 0 def } forall /FmBD {bind def} bind def systemdict /pdfmark known systemdict /currentdistillerparams known and { /fMAcrobat true def /FmPD /pdfmark load def /FmPT /show load def currentdistillerparams /CoreDistVersion get 2000 ge { /FmPD2 /pdfmark load def /FmPA { mark exch /Dest exch 5 3 roll /View [ /XYZ null 6 -2 roll FmDC exch pop null] /DEST FmPD }FmBD } { /FmPD2 /cleartomark load def /FmPA {pop pop pop}FmBD } ifelse } { /fMAcrobat false def /FmPD /cleartomark load def /FmPD2 /cleartomark load def /FmPT /pop load def /FmPA {pop pop pop}FmBD } ifelse /FmDC { transform fMDefaultMatrix defaultmatrix itransform cvi exch cvi exch }FmBD /FmBx { dup 3 index lt {3 1 roll exch} if 1 index 4 index lt {4 -1 roll 3 1 roll exch 4 1 roll} if }FmBD /FMnone 0 def /FMcyan 1 def /FMmagenta 2 def /FMyellow 3 def /FMblack 4 def /FMcustom 5 def /fMNegative false def /FrameSepIs FMnone def /FrameSepBlack 0 def /FrameSepYellow 0 def /FrameSepMagenta 0 def /FrameSepCyan 0 def /FrameSepRed 1 def /FrameSepGreen 1 def /FrameSepBlue 1 def /FrameCurGray 1 def /FrameCurPat null def /FrameCurColors [ 0 0 0 1 0 0 0 1] def /FrameColorEpsilon .001 def /eqepsilon { sub dup 0 lt {neg} if FrameColorEpsilon le } bind def /FrameCmpColorsCMYK { 2 copy 0 get exch 0 get eqepsilon { 2 copy 1 get exch 1 get eqepsilon { 2 copy 2 get exch 2 get eqepsilon { 3 get exch 3 get eqepsilon } {pop pop false} ifelse }{pop pop false} ifelse } {pop pop false} ifelse } bind def /FrameCmpColorsRGB { 2 copy 4 get exch 0 get eqepsilon { 2 copy 5 get exch 1 get eqepsilon { 6 get exch 2 get eqepsilon }{pop pop false} ifelse } {pop pop false} ifelse } bind def /RGBtoCMYK { 1 exch sub 3 1 roll 1 exch sub 3 1 roll 1 exch sub 3 1 roll 3 copy 2 copy le { pop } { exch pop } ifelse 2 copy le { pop } { exch pop } ifelse dup dup dup 6 1 roll 4 1 roll 7 1 roll sub 6 1 roll sub 5 1 roll sub 4 1 roll } bind def /CMYKtoRGB { dup dup 4 -1 roll add 5 1 roll 3 -1 roll add 4 1 roll add 1 exch sub dup 0 lt {pop 0} if 3 1 roll 1 exch sub dup 0 lt {pop 0} if exch 1 exch sub dup 0 lt {pop 0} if exch } bind def /FrameSepInit { 1.0 RealSetgray } bind def /FrameSetSepColor { /FrameSepBlue exch def /FrameSepGreen exch def /FrameSepRed exch def /FrameSepBlack exch def /FrameSepYellow exch def /FrameSepMagenta exch def /FrameSepCyan exch def /FrameSepIs FMcustom def setCurrentScreen } bind def /FrameSetCyan { /FrameSepBlue 1.0 def /FrameSepGreen 1.0 def /FrameSepRed 0.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 1.0 def /FrameSepIs FMcyan def setCurrentScreen } bind def /FrameSetMagenta { /FrameSepBlue 1.0 def /FrameSepGreen 0.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 1.0 def /FrameSepCyan 0.0 def /FrameSepIs FMmagenta def setCurrentScreen } bind def /FrameSetYellow { /FrameSepBlue 0.0 def /FrameSepGreen 1.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 1.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMyellow def setCurrentScreen } bind def /FrameSetBlack { /FrameSepBlue 0.0 def /FrameSepGreen 0.0 def /FrameSepRed 0.0 def /FrameSepBlack 1.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMblack def setCurrentScreen } bind def /FrameNoSep { /FrameSepIs FMnone def setCurrentScreen } bind def /FrameSetSepColors { FrameDict begin [ exch 1 add 1 roll ] /FrameSepColors exch def end } bind def /FrameColorInSepListCMYK { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsCMYK { pop true exit } if } forall dup true ne {pop false} if } bind def /FrameColorInSepListRGB { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsRGB { pop true exit } if } forall dup true ne {pop false} if } bind def /RealSetgray /setgray load def /RealSetrgbcolor /setrgbcolor load def /RealSethsbcolor /sethsbcolor load def end /setgray { FrameDict begin FrameSepIs FMnone eq { RealSetgray } { FrameSepIs FMblack eq { RealSetgray } { FrameSepIs FMcustom eq FrameSepRed 0 eq and FrameSepGreen 0 eq and FrameSepBlue 0 eq and { RealSetgray } { 1 RealSetgray pop } ifelse } ifelse } ifelse end } bind def /setrgbcolor { FrameDict begin FrameSepIs FMnone eq { RealSetrgbcolor } { 3 copy [ 4 1 roll ] FrameColorInSepListRGB { FrameSepBlue eq exch FrameSepGreen eq and exch FrameSepRed eq and { 0 } { 1 } ifelse } { FMPColor { RealSetrgbcolor currentcmykcolor } { RGBtoCMYK } ifelse FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind def /sethsbcolor { FrameDict begin FrameSepIs FMnone eq { RealSethsbcolor } { RealSethsbcolor currentrgbcolor setrgbcolor } ifelse end } bind def FrameDict begin /setcmykcolor where { pop /RealSetcmykcolor /setcmykcolor load def } { /RealSetcmykcolor { 4 1 roll 3 { 3 index add 0 max 1 min 1 exch sub 3 1 roll} repeat RealSetrgbcolor pop } bind def } ifelse userdict /setcmykcolor { FrameDict begin FrameSepIs FMnone eq { RealSetcmykcolor } { 4 copy [ 5 1 roll ] FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and { 0 } { 1 } ifelse } { FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind put fMLevel1 { /patScreenDict 7 dict dup begin <0f1e3c78f0e1c387> [ 45 { pop } {exch pop} .5 2 sqrt] FmBD <0f87c3e1f0783c1e> [ 135 { pop } {exch pop} .5 2 sqrt] FmBD [ 0 { pop } dup .5 2 ] FmBD [ 90 { pop } dup .5 2 ] FmBD <8142241818244281> [ 45 { 2 copy lt {exch} if pop} dup .75 2 sqrt] FmBD <03060c183060c081> [ 45 { pop } {exch pop} .875 2 sqrt] FmBD <8040201008040201> [ 135 { pop } {exch pop} .875 2 sqrt] FmBD end def } { /patProcDict 5 dict dup begin <0f1e3c78f0e1c387> { 3 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <0f87c3e1f0783c1e> { 3 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def <8142241818244281> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -1 -1 moveto 9 9 lineto stroke } bind def <03060c183060c081> { 1 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <8040201008040201> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def end def /patDict 15 dict dup begin /PatternType 1 def /PaintType 2 def /TilingType 3 def /BBox [ 0 0 8 8 ] def /XStep 8 def /YStep 8 def /PaintProc { begin patProcDict bstring known { patProcDict bstring get exec } { 8 8 true [1 0 0 -1 0 8] bstring imagemask } ifelse end } bind def end def } ifelse /tintCMYK { 1 tintGray sub FrameCurColors 0 4 getinterval aload pop 4 index mul 5 1 roll 3 index mul 5 1 roll 2 index mul 5 1 roll mul 4 1 roll }bind def /tintRGB { 1 tintGray sub FrameCurColors 4 3 getinterval aload pop 1 exch sub 3 index mul 1 exch sub 4 1 roll 1 exch sub 2 index mul 1 exch sub 4 1 roll 1 exch sub mul 1 exch sub 3 1 roll }bind def /combineColor { /tintGray 1 1 FrameCurGray sub FrameCurColors 7 get mul sub def FrameSepIs FMnone eq { graymode fMLevel1 or not { [/Pattern [/DeviceCMYK]] setcolorspace tintCMYK FrameCurPat setcolor } { FrameCurColors 3 get 1.0 ge { tintGray RealSetgray } { fMAcrobat not FMPColor graymode and and { tintCMYK RealSetcmykcolor } { tintRGB RealSetrgbcolor } ifelse } ifelse } ifelse } { FrameCurColors 0 4 getinterval aload FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and FrameSepIs FMcustom eq and { tintGray } { 1 } ifelse } { FrameSepIs FMblack eq {tintGray 1.0 exch sub mul 1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop tintGray 1.0 exch sub mul 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop tintGray 1.0 exch sub mul 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop tintGray 1.0 exch sub mul 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse graymode fMLevel1 or not { [/Pattern [/DeviceGray]] setcolorspace FrameCurPat setcolor } { graymode not fMLevel1 and { dup 1 lt {pop FrameCurGray} if } if RealSetgray } ifelse } ifelse } bind def /savematrix { orgmatrix currentmatrix pop } bind def /restorematrix { orgmatrix setmatrix } bind def /fMDefaultMatrix matrix def /fMatrix2 matrix def /dpi 72 0 fMDefaultMatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /freq dpi dup 72 div round dup 0 eq {pop 1} if 8 mul div def /sangle 1 0 fMDefaultMatrix defaultmatrix dtransform exch atan def sangle fMatrix2 rotate fMDefaultMatrix defaultmatrix fMatrix2 concatmatrix dup 0 get /sflipx exch def 3 get /sflipy exch def /screenIndex { 0 1 dpiranges length 1 sub { dup dpiranges exch get 1 sub dpi le {exit} {pop} ifelse } for } bind def /getCyanScreen { FMUseHighFrequencyScreens { CHighAngles CMHighFreqs} {CLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getMagentaScreen { FMUseHighFrequencyScreens { MHighAngles CMHighFreqs } {MLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getYellowScreen { FMUseHighFrequencyScreens { YHighTDot YHighFreqs} { YLowTDot YLowFreqs } ifelse screenIndex dup 3 1 roll get 3 1 roll get { 3 div {2 { 1 add 2 div 3 mul dup floor sub 2 mul 1 sub exch} repeat FMSpotFunction } } {/FMSpotFunction load } ifelse 0.0 exch } bind def /getBlackScreen { FMUseHighFrequencyScreens { KHighFreqs } { KLowFreqs } ifelse screenIndex get 45.0 /FMSpotFunction load } bind def /getSpotScreen { getBlackScreen } bind def /getCompositeScreen { getBlackScreen } bind def /FMSetScreen fMLevel1 { /setscreen load }{ { 8 dict begin /HalftoneType 1 def /SpotFunction exch def /Angle exch def /Frequency exch def /AccurateScreens FMUseAcccurateScreens def currentdict end sethalftone } bind } ifelse def /setDefaultScreen { fMLevel1 { FMPColor { orgrxfer cvx orggxfer cvx orgbxfer cvx orgxfer cvx setcolortransfer } { orgxfer cvx settransfer } ifelse orgfreq organgle orgproc cvx setscreen } { orghalftone sethalftone }ifelse } bind def /setCurrentScreen { FrameSepIs FMnone eq { FMUseDefaultNoSeparationScreen { setDefaultScreen } { getCompositeScreen FMSetScreen } ifelse } { FrameSepIs FMcustom eq { FMUseDefaultSpotSeparationScreen { setDefaultScreen } { getSpotScreen FMSetScreen } ifelse } { FMUseDefaultProcessSeparationScreen { setDefaultScreen } { FrameSepIs FMcyan eq { getCyanScreen FMSetScreen } { FrameSepIs FMmagenta eq { getMagentaScreen FMSetScreen } { FrameSepIs FMyellow eq { getYellowScreen FMSetScreen } { getBlackScreen FMSetScreen } ifelse } ifelse } ifelse } ifelse } ifelse } ifelse } bind def end /FMDOCUMENT { array /FMfonts exch def dup 1 gt {/#copies exch def} {pop} ifelse FrameDict begin 0 ne /manualfeed exch def /paperheight exch def /paperwidth exch def 0 ne /fMNegative exch def 0 ne /edown exch def /yscale exch def /xscale exch def fMLevel1 not { /orghalftone currenthalftone def }if FMPColor { currentcolorscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def cvlit /orgbproc exch def /orgbangle exch def /orgbfreq exch def cvlit /orggproc exch def /orggangle exch def /orggfreq exch def cvlit /orgrproc exch def /orgrangle exch def /orgrfreq exch def currentcolortransfer fMNegative { 1 1 4 { pop { 1 exch sub } fmConcatProcs 4 1 roll } for 4 copy setcolortransfer } if cvlit /orgxfer exch def cvlit /orgbxfer exch def cvlit /orggxfer exch def cvlit /orgrxfer exch def } { currentscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def currenttransfer fMNegative { { 1 exch sub } fmConcatProcs dup settransfer } if cvlit /orgxfer exch def } ifelse end } def /FMENDDOCUMENT { FMDocSave restore } def /FMBEGINPAGE { FrameDict begin /pagesave save def 3.86 setmiterlimit /landscape exch 0 ne def landscape { 90 rotate 0 exch dup /pwid exch def neg translate pop }{ pop /pwid exch def } ifelse edown { [-1 0 0 1 pwid 0] concat } if xscale yscale scale /orgmatrix matrix def gsave } def /FMENDPAGE { grestore pagesave restore end showpage } def /FMFONTDEFINE { FrameDict begin findfont ReEncode 1 index exch definefont FMfonts 3 1 roll put end } def /FMFILLS { FrameDict begin dup array /fillvals exch def dict /patCache exch def end } def /FMFILL { FrameDict begin fillvals 3 1 roll put end } def /FMNORMALIZEGRAPHICS { newpath 1 setlinewidth 0 setlinecap 0 0 0 sethsbcolor 0 setgray } bind def /FMBEGINEPSF { end /FMEPSF save def /showpage {} def FMNORMALIZEGRAPHICS [/fy /fx /fh /fw /ury /urx /lly /llx] {exch def} forall fx fw 2 div add fy fh 2 div add translate rotate fw 2 div neg fh 2 div neg translate fw urx llx sub div fh ury lly sub div scale llx neg lly neg translate /FMdicttop countdictstack 1 add def /FMoptop count def } bind def /FMENDEPSF { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMEPSF restore FrameDict begin } bind def FrameDict begin /setmanualfeed { statusdict /manualfeed true put } bind def /max {2 copy lt {exch} if pop} bind def /min {2 copy gt {exch} if pop} bind def /inch {72 mul} def /pagedimen { paperheight sub abs 16 lt exch paperwidth sub abs 16 lt and {/papername exch def} {pop} ifelse } bind def /setpapername { /papersizedict 14 dict def papersizedict begin /papername /unknown def /Letter 8.5 inch 11.0 inch pagedimen /LetterSmall 7.68 inch 10.16 inch pagedimen /Tabloid 11.0 inch 17.0 inch pagedimen /Ledger 17.0 inch 11.0 inch pagedimen /Legal 8.5 inch 14.0 inch pagedimen /Statement 5.5 inch 8.5 inch pagedimen /Executive 7.5 inch 10.0 inch pagedimen /A3 11.69 inch 16.5 inch pagedimen /A4 8.26 inch 11.69 inch pagedimen /A4Small 7.47 inch 10.85 inch pagedimen /B4 10.125 inch 14.33 inch pagedimen /B5 7.16 inch 10.125 inch pagedimen end } bind def /papersize { papersizedict begin /Letter {lettertray letter} def /LetterSmall {lettertray lettersmall} def /Tabloid {11x17tray 11x17} def /Ledger {ledgertray ledger} def /Legal {legaltray legal} def /Statement {statementtray statement} def /Executive {executivetray executive} def /A3 {a3tray a3} def /A4 {a4tray a4} def /A4Small {a4tray a4small} def /B4 {b4tray b4} def /B5 {b5tray b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end statusdict begin stopped end } bind def /manualpapersize { papersizedict begin /Letter {letter} def /LetterSmall {lettersmall} def /Tabloid {11x17} def /Ledger {ledger} def /Legal {legal} def /Statement {statement} def /Executive {executive} def /A3 {a3} def /A4 {a4} def /A4Small {a4small} def /B4 {b4} def /B5 {b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end stopped } bind def /desperatepapersize { mark statusdict begin /setpageparams where { pop paperwidth paperheight 0 1 {setpageparams} stopped } { true } ifelse { /setpagedevice where { pop 1 dict dup begin /PageSize [ paperwidth paperheight ] def end {setpagedevice} stopped } { true } ifelse } { false } ifelse end {cleartomark true}{cleartomark false}ifelse } bind def /papersizefailure { FMAllowPaperSizeMismatch not { (The requested paper size is not available in any currently-installed tray) (Edit the PS file to "FMAllowPaperSizeMismatch true" to use default tray) FMFAILURE } if } def /DiacriticEncoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /grave /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /.notdef /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /.notdef /AE /Oslash /.notdef /.notdef /.notdef /.notdef /yen /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ordfeminine /ordmasculine /.notdef /ae /oslash /questiondown /exclamdown /logicalnot /.notdef /florin /.notdef /.notdef /guillemotleft /guillemotright /ellipsis /.notdef /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /.notdef /.notdef /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /.notdef /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron ] def /ReEncode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall 0 eq {/Encoding DiacriticEncoding def} if currentdict end } bind def FMPColor { /BEGINBITMAPCOLOR { BITMAPCOLOR} def /BEGINBITMAPCOLORc { BITMAPCOLORc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUECOLOR } def /BEGINBITMAPTRUECOLORc { BITMAPTRUECOLORc } def /BEGINBITMAPCMYK { BITMAPCMYK } def /BEGINBITMAPCMYKc { BITMAPCMYKc } def } { /BEGINBITMAPCOLOR { BITMAPGRAY} def /BEGINBITMAPCOLORc { BITMAPGRAYc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUEGRAY } def /BEGINBITMAPTRUECOLORc { BITMAPTRUEGRAYc } def /BEGINBITMAPCMYK { BITMAPCMYKGRAY } def /BEGINBITMAPCMYKc { BITMAPCMYKGRAYc } def } ifelse /K { FMPrintAllColorsAsBlack { 8 1 roll dup 1 eq 2 index 1 eq and 3 index 1 eq and not {7 {pop} repeat 0 0 0 1 0 0 0} if 8 -1 roll } if FrameCurColors astore pop combineColor } bind def /graymode true def fMLevel1 { /fmGetFlip { fMatrix2 exch get mul 0 lt { -1 } { 1 } ifelse } FmBD } if /setPatternMode { fMLevel1 { 2 index patScreenDict exch known { pop pop patScreenDict exch get aload pop freq mul 5 2 roll fMatrix2 currentmatrix 1 get 0 ne { 3 -1 roll 90 add 3 1 roll sflipx 1 fmGetFlip sflipy 2 fmGetFlip neg mul } { sflipx 0 fmGetFlip sflipy 3 fmGetFlip mul } ifelse 0 lt {exch pop} {pop} ifelse fMNegative { {neg} fmConcatProcs } if bind systemdict /setscreen get exec /FrameCurGray exch def } { /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq sangle landscape {90 add} if {/ypoint exch def /xpoint exch def /xindex xpoint 1 add 2 div bpside mul cvi def /yindex ypoint 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get 1 7 xindex 8 mod sub bitshift and 0 ne fMNegative {not} if {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen offbits offbits onbits add dup 0 ne {div} {pop pop .5} ifelse fMNegative {1.0 exch sub} if /FrameCurGray exch def } ifelse } { pop pop dup patCache exch known { patCache exch get } { dup patDict /bstring 3 -1 roll put patDict 9 PatFreq screenIndex get div dup matrix scale makepattern dup patCache 4 -1 roll 3 -1 roll put } ifelse /FrameCurGray 0 def /FrameCurPat exch def } ifelse /graymode false def combineColor } bind def /setGrayScaleMode { graymode not { /graymode true def fMLevel1 { setCurrentScreen } if } if /FrameCurGray exch def combineColor } bind def /normalize { transform round exch round exch itransform } bind def /dnormalize { dtransform round exch round exch idtransform } bind def /lnormalize { 0 dtransform exch cvi 2 idiv 2 mul 1 add exch idtransform pop } bind def /H { lnormalize setlinewidth } bind def /Z { setlinecap } bind def /PFill { graymode fMLevel1 or not { gsave 1 setgray eofill grestore } if } bind def /PStroke { graymode fMLevel1 or not { gsave 1 setgray stroke grestore } if stroke } bind def /X { fillvals exch get dup type /stringtype eq {8 1 setPatternMode} {setGrayScaleMode} ifelse } bind def /V { PFill gsave eofill grestore } bind def /Vclip { clip } bind def /Vstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /N { PStroke } bind def /Nclip { strokepath clip newpath } bind def /Nstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /M {newpath moveto} bind def /E {lineto} bind def /D {curveto} bind def /O {closepath} bind def /L { /n exch def newpath normalize moveto 2 1 n {pop normalize lineto} for } bind def /Y { L closepath } bind def /R { /y2 exch def /x2 exch def /y1 exch def /x1 exch def x1 y1 x2 y1 x2 y2 x1 y2 4 Y } bind def /rarc {rad arcto } bind def /RR { /rad exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def mark newpath { x1 y1 rad add moveto x1 y2 x2 y2 rarc x2 y2 x2 y1 rarc x2 y1 x1 y1 rarc x1 y1 x1 y2 rarc closepath } stopped {x1 y1 x2 y2 R} if cleartomark } bind def /RRR { /rad exch def normalize /y4 exch def /x4 exch def normalize /y3 exch def /x3 exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def newpath normalize moveto mark { x2 y2 x3 y3 rarc x3 y3 x4 y4 rarc x4 y4 x1 y1 rarc x1 y1 x2 y2 rarc closepath } stopped {x1 y1 x2 y2 x3 y3 x4 y4 newpath moveto lineto lineto lineto closepath} if cleartomark } bind def /C { grestore gsave R clip setCurrentScreen } bind def /CP { grestore gsave Y clip setCurrentScreen } bind def /F { FMfonts exch get [FMsetsize 0 0 FMpointsize 0 0] makefont setfont } bind def /Q { /FMpointsize exch def /FMsetsize FMpointsize def F } bind def /QQ { /FMsetsize exch def /FMpointsize exch def F } bind def /T { moveto show } bind def /RF { rotate 0 ne {-1 1 scale} if } bind def /TF { gsave moveto RF show grestore } bind def /P { moveto 0 32 3 2 roll widthshow } bind def /PF { gsave moveto RF 0 32 3 2 roll widthshow grestore } bind def /S { moveto 0 exch ashow } bind def /SF { gsave moveto RF 0 exch ashow grestore } bind def /B { moveto 0 32 4 2 roll 0 exch awidthshow } bind def /BF { gsave moveto RF 0 32 4 2 roll 0 exch awidthshow grestore } bind def /G { gsave newpath normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /Gstrk { savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 0.0 0.0 1.0 5 3 roll arc restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /Gclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GG { gsave newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /GGclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GGstrk { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /A { gsave savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 2 copy 0.0 0.0 1.0 5 3 roll arc round cvi 360 mod exch round cvi 360 mod eq {closepath} if restorematrix PStroke grestore } bind def /Aclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /Astrk { Gstrk } bind def /AA { gsave savematrix newpath 3 index 2 div add exch 4 index 2 div sub exch normalize 3 index 2 div sub exch 4 index 2 div add exch translate rotate scale 0.0 0.0 1.0 5 3 roll arc restorematrix PStroke grestore } bind def /AAclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /AAstrk { GGstrk } bind def /BEGINPRINTCODE { /FMdicttop countdictstack 1 add def /FMoptop count 7 sub def /FMsaveobject save def userdict begin /showpage {} def FMNORMALIZEGRAPHICS 3 index neg 3 index neg translate } bind def /ENDPRINTCODE { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMsaveobject restore } bind def /gn { 0 { 46 mul cf read pop 32 sub dup 46 lt {exit} if 46 sub add } loop add } bind def /cfs { /str sl string def 0 1 sl 1 sub {str exch val put} for str def } bind def /ic [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 {0 hx} {1 hx} {2 hx} {3 hx} {4 hx} {5 hx} {6 hx} {7 hx} {8 hx} {9 hx} {10 hx} {11 hx} {12 hx} {13 hx} {14 hx} {15 hx} {16 hx} {17 hx} {18 hx} {19 hx} {gn hx} {0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14} {15} {16} {17} {18} {19} {gn} {0 wh} {1 wh} {2 wh} {3 wh} {4 wh} {5 wh} {6 wh} {7 wh} {8 wh} {9 wh} {10 wh} {11 wh} {12 wh} {13 wh} {14 wh} {gn wh} {0 bl} {1 bl} {2 bl} {3 bl} {4 bl} {5 bl} {6 bl} {7 bl} {8 bl} {9 bl} {10 bl} {11 bl} {12 bl} {13 bl} {14 bl} {gn bl} {0 fl} {1 fl} {2 fl} {3 fl} {4 fl} {5 fl} {6 fl} {7 fl} {8 fl} {9 fl} {10 fl} {11 fl} {12 fl} {13 fl} {14 fl} {gn fl} ] def /ms { /sl exch def /val 255 def /ws cfs /im cfs /val 0 def /bs cfs /cs cfs } bind def 400 ms /ip { is 0 cf cs readline pop { ic exch get exec add } forall pop } bind def /rip { bis ris copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop ris gis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop gis bis copy pop dup add is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /rip4 { kis cis copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop cis mis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop mis yis copy pop dup dup add is exch cf cs readline pop { ic exch get exec add } forall pop pop yis kis copy pop 3 mul is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /wh { /len exch def /pos exch def ws 0 len getinterval im pos len getinterval copy pop pos len } bind def /bl { /len exch def /pos exch def bs 0 len getinterval im pos len getinterval copy pop pos len } bind def /s1 1 string def /fl { /len exch def /pos exch def /val cf s1 readhexstring pop 0 get def pos 1 pos len add 1 sub {im exch val put} for pos len } bind def /hx { 3 copy getinterval cf exch readhexstring pop pop } bind def /wbytes { dup dup 8 gt { pop 8 idiv mul } { 8 eq {pop} {1 eq {7 add 8 idiv} {3 add 4 idiv} ifelse} ifelse } ifelse } bind def /BEGINBITMAPBWc { 1 {} COMMONBITMAPc } bind def /BEGINBITMAPGRAYc { 8 {} COMMONBITMAPc } bind def /BEGINBITMAP2BITc { 2 {} COMMONBITMAPc } bind def /COMMONBITMAPc { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def cvtProc /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} image bitmapsave restore grestore } bind def /BEGINBITMAPBW { 1 {} COMMONBITMAP } bind def /BEGINBITMAPGRAY { 8 {} COMMONBITMAP } bind def /BEGINBITMAP2BIT { 2 {} COMMONBITMAP } bind def /COMMONBITMAP { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def cvtProc /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} image bitmapsave restore grestore } bind def /ngrayt 256 array def /nredt 256 array def /nbluet 256 array def /ngreent 256 array def fMLevel1 { /colorsetup { currentcolortransfer /gryt exch def /blut exch def /grnt exch def /redt exch def 0 1 255 { /indx exch def /cynu 1 red indx get 255 div sub def /magu 1 green indx get 255 div sub def /yelu 1 blue indx get 255 div sub def /kk cynu magu min yelu min def /u kk currentundercolorremoval exec def % /u 0 def nredt indx 1 0 cynu u sub max sub redt exec put ngreent indx 1 0 magu u sub max sub grnt exec put nbluet indx 1 0 yelu u sub max sub blut exec put ngrayt indx 1 kk currentblackgeneration exec sub gryt exec put } for {255 mul cvi nredt exch get} {255 mul cvi ngreent exch get} {255 mul cvi nbluet exch get} {255 mul cvi ngrayt exch get} setcolortransfer {pop 0} setundercolorremoval {} setblackgeneration } bind def } { /colorSetup2 { [ /Indexed /DeviceRGB 255 {dup red exch get 255 div exch dup green exch get 255 div exch blue exch get 255 div} ] setcolorspace } bind def } ifelse /fakecolorsetup { /tran 256 string def 0 1 255 {/indx exch def tran indx red indx get 77 mul green indx get 151 mul blue indx get 28 mul add add 256 idiv put} for currenttransfer {255 mul cvi tran exch get 255.0 div} exch fmConcatProcs settransfer } bind def /BITMAPCOLOR { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def fMLevel1 { colorsetup /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} {is} {is} true 3 colorimage } { colorSetup2 /is width depth wbytes string def /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {cf is readhexstring pop} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPCOLORc { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def fMLevel1 { colorsetup /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} {is} {is} true 3 colorimage } { colorSetup2 /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {ip} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPTRUECOLORc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris} {gis} {bis} true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYKc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip4 pop cis} {mis} {yis} {kis} true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUECOLOR { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf gis readhexstring pop } { cf bis readhexstring pop } true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYK { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /mis width string def /yis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf mis readhexstring pop } { cf yis readhexstring pop } { cf kis readhexstring pop } true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUEGRAYc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris gis bis width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAYc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop cis mis yis kis width cgray} image bitmapsave restore grestore } bind def /cgray { /ww exch def /k exch def /y exch def /m exch def /c exch def 0 1 ww 1 sub { /i exch def c i get m i get y i get k i get CMYKtoRGB .144 mul 3 1 roll .587 mul 3 1 roll .299 mul add add c i 3 -1 roll floor cvi put } for c } bind def /gray { /ww exch def /b exch def /g exch def /r exch def 0 1 ww 1 sub { /i exch def r i get .299 mul g i get .587 mul b i get .114 mul add add r i 3 -1 roll floor cvi put } for r } bind def /BITMAPTRUEGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf gis readhexstring pop cf bis readhexstring pop width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /yis width string def /mis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf mis readhexstring pop cf yis readhexstring pop cf kis readhexstring pop width cgray} image bitmapsave restore grestore } bind def /BITMAPGRAY { 8 {fakecolorsetup} COMMONBITMAP } bind def /BITMAPGRAYc { 8 {fakecolorsetup} COMMONBITMAPc } bind def /ENDBITMAP { } bind def end /ALDmatrix matrix def ALDmatrix currentmatrix pop /StartALD { /ALDsave save def savematrix ALDmatrix setmatrix } bind def /InALD { restorematrix } bind def /DoneALD { ALDsave restore } bind def /I { setdash } bind def /J { [] 0 setdash } bind def (5.5) FMVERSION 1 1 0 0 612 792 0 1 3 FMDOCUMENT 0 0 /Times-Roman FMFONTDEFINE 1 0 /Times-Italic FMFONTDEFINE 32 FMFILLS 0 0 FMFILL 1 0.1 FMFILL 2 0.3 FMFILL 3 0.5 FMFILL 4 0.7 FMFILL 5 0.9 FMFILL 6 0.97 FMFILL 7 1 FMFILL 8 <0f1e3c78f0e1c387> FMFILL 9 <0f87c3e1f0783c1e> FMFILL 10 FMFILL 11 FMFILL 12 <8142241818244281> FMFILL 13 <03060c183060c081> FMFILL 14 <8040201008040201> FMFILL 16 1 FMFILL 17 0.9 FMFILL 18 0.7 FMFILL 19 0.5 FMFILL 20 0.3 FMFILL 21 0.1 FMFILL 22 0.03 FMFILL 23 0 FMFILL 24 FMFILL 25 FMFILL 26 <3333333333333333> FMFILL 27 <0000ffff0000ffff> FMFILL 28 <7ebddbe7e7dbbd7e> FMFILL 29 FMFILL 30 <7fbfdfeff7fbfdfe> FMFILL 612 792 1 FMBEGINPAGE 0 FrameSetSepColors FrameNoSep 0 0 0 1 0 0 0 1 K J 27 251 779 719 C 27 332 779 707 C 0 0 0 1 0 0 0 1 K 0 X 90 450 15.74 16.52 187.99 507.91 G 0.5 H 2 Z 7 X 90 450 15.74 16.52 187.99 507.91 A 0 X 90 450 15.74 16.52 587.87 599.45 A 90 450 15.74 16.52 426.43 678.04 A 90 450 15.74 16.52 266.63 600.67 A 265.56 583.66 187.86 523.43 2 L N 266.25 584.25 338 525.5 2 L N 425.75 661.5 587.75 616 2 L N 425.5 661.5 266.43 617 2 L N 90 450 15.74 16.52 292.94 412.52 G 90 450 15.74 16.52 292.94 412.52 A 90 450 15.74 16.52 338.52 508.91 A 338.25 492.5 292.75 429 2 L N 338.75 492.5 377 429 2 L N 90 450 15.74 16.52 378.24 412.52 G 90 450 15.74 16.52 378.24 412.52 A 90 450 15.74 16.52 659.42 506.05 G 90 450 15.74 16.52 659.42 506.05 A 0 18 Q (0) 655.2 468.5 T (1) 503.94 468.5 T 1 24 Q (z) 334.66 505.76 T 0 18 Q (0) 183.46 468.5 T 1 24 Q (x) 421.39 674.1 T 0 18 Q (1) 373.08 373.16 T 1 24 Q (y) 585.5 596.3 T (y) 262.7 596.73 T 0 18 Q (0) 287.78 373.16 T 0 24 Q 1 F (x) 482.41 657.1 T 0 18 Q (0) 530.66 657.1 T 0 24 Q (=) 505.13 657.1 T 1 F (y) 165.41 564.73 T 0 18 Q (1) 204.21 564.73 T (=) 185.06 564.73 T 1 24 Q (y) 494.98 558.3 T 0 18 Q (1) 533.78 558.3 T (=) 514.63 558.3 T 1 24 Q (y) 303.41 564.73 T 0 18 Q (0) 342.21 564.73 T (=) 323.06 564.73 T 1 24 Q (y) 637.98 558.3 T 0 18 Q (0) 676.78 558.3 T (=) 657.63 558.3 T 1 24 Q (z) 259.81 462.11 T 0 18 Q (1) 297.29 462.11 T (=) 278.14 462.11 T 1 24 Q (z) 367.5 462.11 T 0 18 Q (0) 404.98 462.11 T (=) 385.83 462.11 T 1 24 Q (x) 315.89 657.5 T 0 18 Q (1) 354.76 657.5 T (=) 335.61 657.5 T 90 450 15.74 16.52 509.57 505.62 G 7 X 90 450 15.74 16.52 509.57 505.62 A 587.75 583 509.43 521.14 2 L 0 X N 588.25 583 659.5 522.75 2 L N 27 251 779 719 C 0 180 792 792 C 0 0 0 1 0 0 0 1 K FMENDPAGE FMENDDOCUMENT %%EndDocument @endspecial 1925 208 a currentpoint grestore moveto 1925 208 a 246 1813 a Fw(Figur)l(e)42 b(1.)c Fv(A)j(decision)h(tree)g(for)g(the)f(form)n(ula)g(\()p Fh(x)27 b Fp(_)h Fh(y)s Fv(\)\()p Fh(y)h Fp(_)p 2181 1772 39 4 v 27 w Fh(z)s Fv(\)\()p 2280 1772 44 4 v Fh(x)e Fp(_)p 2430 1772 41 4 v 28 w Fh(y)r Fv(\))41 b(enco)r(des)h(whic)n(h) 246 1904 y(assignmen)n(ts)25 b(are)i(satisfying)g(and)e(whic)n(h)h(are) g(not.)246 2169 y Fz(min)n(\()p Fl(L)p Fz(\()p Fl(v)576 2136 y Fx(0)600 2169 y Fz(\))p Fl(;)15 b(L)p Fz(\()p Fl(v)819 2136 y Fx(00)863 2169 y Fz(\)\);)27 b(and)e(\(iii\))e(if)i Fl(Q)p Fz(\()p Fl(v)s Fz(\))h(=)1777 2106 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1777 2106 a Fi(R)1836 2106 y currentpoint grestore moveto 1836 2106 a 1777 2169 a Fz(,)g(then)f Fl(L)p Fz(\()p Fl(v)s Fz(\))h(=)f(\()p Fl(L)p Fz(\()p Fl(v)2510 2136 y Fx(0)2534 2169 y Fz(\))11 b(+)g Fl(L)p Fz(\()p Fl(v)2806 2136 y Fx(00)2848 2169 y Fz(\)\))p Fl(=)p Fz(2.)246 2277 y(The)32 b(v)-5 b(alue)33 b(of)g Fl(\036)h Fz(is)e(the)h(v)-5 b(alue)33 b(of)g(the)h(ro)s(ot,)f(and)g Fl(\036)g Fz(is)g(a)g(p)s(ositiv)m(e)f(instance)h(if)246 2385 y(and)c(only)h(if)f(that)i(v)-5 b(alue)30 b(is)f(greater)j(than)e (or)h(equal)e(to)j Fl(\022)s Fz(.)362 2493 y(Note)24 b(that,)g(while)d(not)i(explored)f(in)f(detail)h(in)g(this)f(pap)s(er,) h(the)h(de\014nition)e(of)246 2601 y(the)32 b(randomized)f(quan)m (ti\014er)1366 2538 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1366 2538 a Fi(R)1424 2538 y currentpoint grestore moveto 1424 2538 a 1398 2601 a Fz(can)h(b)s(e)f(extended)i(to)f(allo)m(w)g (arbitrary)f(ratio-)246 2709 y(nal)f(probabilit)m(y)f(distributions)f (o)m(v)m(er)33 b(the)f(set)g Fm(f)p Fz(0)p Fl(;)15 b Fz(1)p Fm(g)33 b Fz(instead)e(of)h(the)f(uniform)246 2817 y(distribution)22 b(considered)j(in)g(this)h(pap)s(er.)f(The)h (resulting)e(generalization)i(do)s(es)246 2925 y(not)42 b(c)m(hange)h(an)m(y)f(of)h(the)f(complexit)m(y)g(results)e(describ)s (ed)g(in)h(the)h(next)g(sec-)246 3032 y(tion,)26 b(and)g(is)g(k)m(ey)i (to)f(making)f(a)i(connection)f(b)s(et)m(w)m(een)g Fr(SSa)-6 b(t)o Fz(,)27 b(planning)d(under)246 3140 y(uncertain)m(t)m(y)-8 b(,)31 b(and)f(b)s(elief)e(net)m(w)m(ork)j(inference.)246 3358 y(1.3.)65 b Fr(Sa)-6 b(tisfiability)33 b(and)h(Uncer)-6 b(t)g(ainty)246 3553 y Fz(As)35 b(AI)h(tec)m(hniques)f(are)i(used)e (more)g(and)h(more)g(to)g(attac)m(k)i(real-w)m(orld)d(prob-)246 3661 y(lems,)e(man)m(y)h(researc)m(hers)g(ha)m(v)m(e)h(em)m(braced)f (probabilit)m(y)d(theory)j(as)g(a)g(w)m(a)m(y)h(of)246 3769 y(represen)m(ting)23 b(the)h(p)s(erv)-5 b(asiv)m(e)23 b(uncertain)m(t)m(y)h(they)g(\014nd.)f(Tw)m(o)h(sp)s(eci\014c)f (examples)246 3877 y(of)f(this)f(trend)g(are)h(the)g(increasing)f(use)g (of)h(Mark)m(o)m(v)i(decision)c(pro)s(cess)i(mo)s(dels)e(in)246 3985 y(planning)k(and)i(learning,)f(and)h(b)s(elief)f(net)m(w)m(orks)i (in)f(reasoning)g(and)g(kno)m(wledge)246 4093 y(represen)m(tation;)34 b(ho)m(w)m(ev)m(er,)h(the)f(in\015uence)f(of)h(probabilistic)c(mo)s (dels)j(in)f(AI)i(is)246 4201 y(felt)c(quite)g(broadly)-8 b(.)362 4309 y(Whereas)42 b(man)m(y)g(basic)f(deterministic)e(problems) h(are)i(complete)g(for)f(the)246 4417 y(w)m(ell-kno)m(wn)27 b(complexit)m(y)h(class)h(NP,)g(and)f(are)h(therefore)g(formally)e (equiv)-5 b(alen)m(t)246 4525 y(to)27 b Fr(Sa)-6 b(t)o Fz(,)27 b(man)m(y)g(planning)e(and)h(reasoning)g(problems)f(in)h (probabilistic)d(mo)s(dels)246 4633 y(lie)f(in)h(other)h(complexit)m(y) g(classes,)g(suc)m(h)f(as)h(#P)g(or)g(PP)f(\(Roth,)i(1996;)g(Littman,) 246 4741 y(Goldsmith,)k(&)h(Mundhenk,)f(1998\).)k(This)28 b(means)j(that,)g(with)e(resp)s(ect)i(to)g(the)246 4848 y(curren)m(t)j(state)i(of)f(complexit)m(y)g(theory)-8 b(,)35 b(these)h(problems)d(cannot)i(b)s(e)f(p)s(olyno-)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.5)p eop %%Page: 6 6 6 5 bop 246 100 a Fz(6)873 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(mially)34 b(reduced)j(to)g Fr(Sa)-6 b(t)o Fz(.)37 b(Ho)m(w)m(ev)m(er,)j(man)m(y)d(standard)f(uncertain)g (reasoning)246 399 y(problems)d(can)i(b)s(e)f(reduced)g(to)i(sp)s (ecial)d(cases)j(of)f Fr(SSa)-6 b(t)n Fz(,)35 b(as)g(describ)s(ed)e (next.)246 506 y(The)d(probabilistic)d(problems)i(discussed)f(are)j (complete)g(for)g(NP,)f(PP,)h(NP)2910 471 y Fn(PP)3010 506 y Fz(,)246 614 y(and)e(PSP)-8 b(A)m(CE.)362 722 y(The)39 b(NP-complete)h(problem)e Fr(Sa)-6 b(t)39 b Fz(is)f(the)i Fr(SSa)-6 b(t)38 b Fz(problem)g(obtained)h(b)m(y)246 830 y(using)19 b(only)h(existen)m(tial)g(quan)m(ti\014ers)f(and)i (setting)f Fl(\022)28 b Fz(=)d(1.)c(The)f(problem)g(of)g(\014nd-)246 938 y(ing)k(the)h(most)g(probable)f(explanation)g(\(MPE\))i(in)d(a)j(b) s(elief)d(net)m(w)m(ork)j(\(Dec)m(h)m(ter,)246 1046 y(1996\))48 b(is)d(\(p)s(olynomially\))e(equiv)-5 b(alen)m(t)45 b(to)i Fr(Sa)-6 b(t)o Fz(.)46 b(A)g(related)f(problem)g(from)246 1154 y(planning)21 b(under)h(uncertain)m(t)m(y)h(is)g(determining)e (whether)i(there)h(is)e(some)i(c)m(hoice)246 1262 y(of)c(actions)h(suc) m(h)g(that)g(the)g(probabilit)m(y)d(of)j(the)g(most)g(lik)m(ely)e(tra)5 b(jectory)22 b(through)246 1370 y(state)31 b(space)g(to)g(the)g(goal)g (exceeds)g(a)g(giv)m(en)f(probabilit)m(y)e(threshold.)362 1478 y(The)38 b(class)g(PP,)g(or)h(probabilistic)c(p)s(olynomial)g (time,)k(can)g(b)s(e)e(de\014ned)g(in)246 1586 y(sev)m(eral)i(w)m(a)m (ys.)h(F)-8 b(or)39 b(the)g(purp)s(oses)e(of)i(this)f(pap)s(er,)g(it)g (is)g(useful)f(to)j(de\014ne)e(it)246 1694 y(b)m(y)28 b(one)g(of)g(its)g(complete)h(problems,)d Fr(Majsa)-6 b(t)o Fz(.)28 b(In)f(its)h(standard)f(form)m(ulation,)246 1802 y Fr(Majsa)-6 b(t)29 b Fz(asks)i(whether,)f(giv)m(en)h(a)g(Bo)s (olean)h(form)m(ula)e Fl(\036)p Fz(\()p Fs(x)p Fz(\))h(in)f(CNF,)h(at)h (least)246 1910 y(half)e(of)h(its)f(assignmen)m(ts)h(are)g(satisfying.) f(Th)m(us,)g Fr(Majsa)-6 b(t)29 b Fz(is)i(concerned,)g(not)246 2017 y(just)j(with)g(the)i(existence)f(of)h(a)f(satisfying)f(assignmen) m(t,)i(but)e(the)h Fj(numb)-5 b(er)46 b Fz(of)246 2125 y(satisfying)29 b(assignmen)m(ts.)362 2233 y(The)40 b Fr(Majsa)-6 b(t)39 b Fz(problem)h(connects)i(satis\014abilit)m(y)c(to)k (probabilit)m(y)c(in)i(the)246 2341 y(sense)21 b(that,)h(if)e(w)m(e)i (imagine)e(that)i(all)e(assignmen)m(ts)h(are)h(equally)d(lik)m(ely)-8 b(,)21 b Fr(Majsa)-6 b(t)246 2449 y Fz(asks)30 b(whether)f(the)h (probabilit)m(y)d(of)j(a)h(satisfying)d(assignmen)m(t)i(is)f(at)h (least)h(1)p Fl(=)p Fz(2.)246 2557 y(Th)m(us,)e Fr(Majsa)-6 b(t)29 b Fz(can)h(b)s(e)g(expressed)g(as)h(an)f(instance)g(of)h Fr(SSa)-6 b(t)n Fz(:)1129 2722 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1129 2722 a Fi(R)1187 2722 y currentpoint grestore moveto 1187 2722 a 1129 2785 a Fl(x)1181 2799 y Fn(1)1220 2785 y Fl(;)15 b(:)g(:)g(:)i(;)1481 2722 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1481 2722 a Fi(R)1540 2722 y currentpoint grestore moveto 1540 2722 a 1481 2785 a Fl(x)1533 2799 y Fk(n)1580 2785 y Fz(\()p Fl(E)5 b Fz([)p Fl(\036)p Fz(\()p Fs(x)p Fz(\)])27 b Fm(\025)e Fz(1)p Fl(=)p Fz(2\))246 3014 y(\(or)32 b Fl(\022)s Fz(,)g(more)h(generally;)e(P)m(apadimitriou) f(\(1985\))35 b(refers)d(to)h(this)e(as)i(\\Thresh-)246 3122 y(old)40 b Fr(Sa)-6 b(t)o Fz(",)41 b(but)f(this)g(pap)s(er)f(uses) h(simply)f(\\)p Fr(Majsa)-6 b(t)n Fz(")42 b(for)e(this)g(generaliza-) 246 3230 y(tion\).)28 b(Th)m(us,)f Fr(Majsa)-6 b(t)27 b Fz(is)g(obtained)h(from)f Fr(SSa)-6 b(t)27 b Fz(b)m(y)h(using)f(only) g(randomized)246 3337 y(quan)m(ti\014ers.)362 3445 y(As)22 b(with)e(man)m(y)i(decision)e(problems,)g Fr(Majsa)-6 b(t)20 b Fz(is)g(p)s(olynomially)f(equiv)-5 b(alen)m(t)246 3553 y(to)40 b(the)g(problem)e(of)i(actually)f(computing)g(the)h (probabilit)m(y)d(of)j(a)g(satisfying)246 3661 y(assignmen)m(t,)24 b(since)f(binary)g(searc)m(h)h(on)g Fl(\022)i Fz(can)f(b)s(e)e(used)g (to)i(\014nd)e(the)h(exact)i(v)-5 b(alue)246 3769 y(of)31 b Fl(\022)i Fz(for)e(whic)m(h)f(the)i(probabilit)m(y)c(is)j(at)h(least) f(as)h(big,)e(but)h(no)g(bigger)g(than,)g Fl(\022)s Fz(.)246 3877 y(Th)m(us,)36 b(the)h(class)g(PP)g(can)g(b)s(e)g(view)m(ed)f(as)i (the)f(decision-problem)d(v)m(ersion)j(of)246 3985 y(#P)o(,)31 b(whic)m(h)e(actually)h Fj(c)-5 b(ounts)39 b Fz(the)30 b(n)m(um)m(b)s(er)f(of)i(satisfying)e(assignmen)m(ts.)362 4093 y(The)42 b(problem)g(of)h(b)s(elief)e(net)m(w)m(ork)j (inference|giv)m(en)e(a)i(b)s(elief)d(net)m(w)m(ork,)246 4201 y(v)-5 b(alues)32 b(for)h(its)g(evidence)g(no)s(des,)f(and)h(the)g (v)-5 b(alue)33 b(for)g(a)h(query)e(no)s(de,)h(what)g(is)246 4309 y(the)c(probabilit)m(y)f(that)i(the)g(query)f(no)s(de)f(tak)m(es)j (on)f(the)g(giv)m(en)f(v)-5 b(alue)29 b(giv)m(en)h(the)246 4417 y(evidence?|is)35 b(#P)o(-complete)i(\(Roth,)g(1996\).)i(In)c (addition,)g(an)m(y)h(b)s(elief)f(net-)246 4525 y(w)m(ork)c(\(with)g (rational)g(conditional)f(probabilit)m(y)f(tables\))i(can)h(b)s(e)f (represen)m(ted)246 4633 y(as)37 b(a)g(Bo)s(olean)g(form)m(ula.)f(Th)m (us,)g(b)s(elief)f(net)m(w)m(ork)i(inference)f(is)g(equiv)-5 b(alen)m(t)36 b(to)246 4741 y Fr(Majsa)-6 b(t)n Fz(.)48 b(A)g(PP-complete)h(planning)c(problem)i(is)g(plan)f(ev)-5 b(aluation)48 b(in)f(a)246 4848 y(probabilistic)27 b(domain)i (\(Littman,)h(Goldsmith,)f(&)h(Mundhenk,)f(1998\).)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.6)p eop %%Page: 7 7 7 6 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)853 b Fz(7)362 291 y(The)40 b(complexit)m(y)h (class)f(NP)1385 255 y Fn(PP)1526 291 y Fz(is)g(formed)g(b)m(y)h(com)m (bining)e(NP)i(and)f(PP.)246 399 y(It)e(is)f(the)h(class)g(of)h (problems)d(that)j(can)f(b)s(e)g(solv)m(ed)g(b)m(y)g(guessing)f(a)i (solution)246 506 y(\(NP\))34 b(and)e(then)h(p)s(erforming)e(a)j(PP)f (calculation)f(for)h(v)m(eri\014cation.)g(A)h(satis\014-)246 614 y(abilit)m(y)g(problem)g(that)i(is)e(complete)i(for)f(this)g(class) g(is)g Fr(E-Majsa)-6 b(t)34 b Fz(\(\\exists")246 722 y Fr(Majsa)-6 b(t)n Fz(\))35 b(\(Littman,)g(Goldsmith,)f(&)h(Mundhenk,) e(1998\),)38 b(whic)m(h)c(com)m(bines)246 830 y(elemen)m(ts)e(of)h Fr(Sa)-6 b(t)31 b Fz(and)h Fr(Majsa)-6 b(t)o Fz(.)32 b(An)g Fr(E-Majsa)-6 b(t)32 b Fz(instance)g(is)f(de\014ned)g(b)m(y)i(a) 246 938 y(CNF)24 b(Bo)s(olean)g(form)m(ula)f Fl(\036)p Fz(\()p Fs(x)p Fz(\))i(on)f Fl(n)f Fz(Bo)s(olean)h(v)-5 b(ariables,)23 b(a)h(threshold)e(v)-5 b(alue)24 b Fl(\022)s Fz(,)246 1046 y(and)31 b(a)i(n)m(um)m(b)s(er)e(0)f Fm(\024)e Fl(c)h Fm(\024)f Fl(n)p Fz(.)33 b(The)e(v)-5 b(ariables)32 b Fl(x)1909 1060 y Fk(c)p Fn(+1)2033 1046 y Fl(;)15 b(:)g(:)g(:)i(;)e (x)2287 1060 y Fk(n)2367 1046 y Fz(are)33 b(the)f(\\c)m(hance")246 1154 y(v)-5 b(ariables,)39 b(and)h(the)g(decision)f(problem)g(is)g(to)i (rep)s(ort)f(whether)f(there)i(is)e(an)246 1262 y(assignmen)m(t)30 b(to)i(the)f(\\c)m(hoice")i(v)-5 b(ariables)30 b Fl(x)1775 1276 y Fn(1)1814 1262 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)2068 1276 y Fk(c)2134 1262 y Fz(so)31 b(that)g(the)h(probabilit)m(y)246 1370 y(that)g(they)f(com)m(bine)g(with)f(the)i(c)m(hance)g(v)-5 b(ariables)30 b(to)i(constitute)g(a)f(satisfying)246 1478 y(assignmen)m(t)43 b(of)h Fl(\036)p Fz(\()p Fs(x)p Fz(\))h(is)e(at)h(least)g Fl(\022)s Fz(.)f(Th)m(us,)g Fr(E-Majsa)-6 b(t)43 b Fz(is)f(also)i(an)g Fr(SSa)-6 b(t)246 1586 y Fz(problem:)799 1694 y Fm(9)p Fl(x)902 1708 y Fn(1)941 1694 y Fl(;)15 b(:)g(:)g(:)i(;)e Fm(9)p Fl(x)1246 1708 y Fk(c)1281 1694 y Fl(;)1380 1631 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1380 1631 a Fi(R)1438 1631 y currentpoint grestore moveto 1438 1631 a 1380 1694 a Fl(x)1432 1708 y Fk(c)p Fn(+1)1556 1694 y Fl(;)g(:)g(:)g(:)i(;)1817 1631 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1817 1631 a Fi(R)1876 1631 y currentpoint grestore moveto 1876 1631 a 1817 1694 a Fl(x)1869 1708 y Fk(n)1916 1694 y Fz(\()p Fl(E)5 b Fz([)p Fl(\036)p Fz(\()p Fs(x)p Fz(\)])28 b Fm(\025)d Fl(\022)s Fz(\))p Fl(:)362 1826 y Fz(Note)34 b(that,)f(if)f Fl(c)d Fz(=)f Fl(n)p Fz(,)33 b Fr(E-Majsa)-6 b(t)31 b Fz(is)h(simply)e Fr(Sa)-6 b(t)31 b Fz(\(for)i Fl(\022)e Fz(=)d(1\).)34 b(If)e Fl(c)d Fz(=)g(0,)246 1934 y Fr(E-Majsa)-6 b(t)35 b Fz(is)h(exactly)h Fr(Majsa)-6 b(t)n Fz(.)37 b(In)f(terms)g(of)h(complexit)m(y)f(classes,)h(NP)f Fm(\022)246 2042 y Fz(PP)k Fm(\022)g Fz(NP)651 2007 y Fn(PP)751 2042 y Fz(;)g Fr(E-Majsa)-6 b(t)38 b Fz(is)g(at)j(least)e(as)h(hard)e(as)i Fr(Majsa)-6 b(t)n Fz(,)40 b(whic)m(h)e(is)g(at)246 2150 y(least)43 b(as)g(hard)f(as)h Fr(Sa)-6 b(t)o Fz(.)43 b(In)f(some)h(sense,)g(then,)g(the)g(p)s(eak)g(complexit)m(y)f(for)246 2258 y Fr(E-Majsa)-6 b(t)29 b Fz(is)g(ac)m(hiev)m(ed)j(with)d(in)m (termediate)h(v)-5 b(alues)29 b(of)i Fl(c)p Fz(.)362 2366 y(Other)44 b(problems)f(ha)m(v)m(e)j(b)s(een)e(sho)m(wn)h(to)g(b)s (e)f(NP)2180 2330 y Fn(PP)2281 2366 y Fz(-complete,)h(suc)m(h)g(as)246 2474 y(\014nding)f(optimal)j(size-b)s(ounded)e(plans)g(in)h(uncertain)g (domains)g(\(Littman,)246 2582 y(Goldsmith,)31 b(&)i(Mundhenk,)e (1998\))k(and)d(generating)h(\\explanations")g(in)e(b)s(e-)246 2690 y(lief)f(net)m(w)m(orks)h(\(P)m(eot,)j(1998\).)f(In)e(addition,)f (the)h(b)s(elief)e(net)m(w)m(ork)j(problems)e(of)246 2798 y(calculating)g(a)h(maxim)m(um)e(a)i(p)s(osteriori)e(\(MAP\))j(h)m (yp)s(othesis)d(or)i(a)g(maxim)m(um)246 2906 y(exp)s(ected)g(utilit)m (y)f(\(MEU\))i(solution)e(\(Dec)m(h)m(ter,)k(1996\))f(are)f(also)f (complete)g(for)246 3014 y(NP)376 2978 y Fn(PP)476 3014 y Fz(.)j(In)f(these)i(problems,)d(the)i(c)m(hoice)h(v)-5 b(ariables)33 b(corresp)s(ond)g(to)h(the)g(plan)246 3122 y(or)c(explanation)f(and)h(the)h(c)m(hance)g(v)-5 b(ariables)29 b(to)i(the)g(uncertain)m(t)m(y)-8 b(.)362 3230 y(The)34 b(class)g(PSP)-8 b(A)m(CE)34 b(consists)g(of)h(the)g(problems)e(solv)-5 b(able)33 b(using)g(a)i(p)s(oly-)246 3337 y(nomial)43 b(amoun)m(t)i(of)f(space.)h(All)e(the)i(previously)d(men)m(tioned)i (classes)h(\(NP,)246 3445 y(PP)o(,)50 b(NP)574 3410 y Fn(PP)674 3445 y Fz(\))f(can)h(b)s(e)e(solv)m(ed)h(in)f(p)s(olynomial)e (space)k(b)m(y)f(en)m(umerating)f(all)246 3553 y(assignmen)m(ts)42 b(and)h(com)m(bining)f(the)h(results)f(in)f(the)j(appropriate)e(w)m(a)m (y)-8 b(.)45 b(The)246 3661 y(sto)s(c)m(hastic)d Fr(Sa)-6 b(t)40 b Fz(problem)g Fr(SSa)-6 b(t)40 b Fz(and)h(the)h(deterministic)d Fr(QBF)i Fz(are)h(satis-)246 3769 y(\014abilit)m(y)c(problems)g(that)j (are)g(PSP)-8 b(A)m(CE-complete.)41 b(As)f(men)m(tioned)g(earlier,)246 3877 y(eac)m(h)46 b(existen)m(tial)f(quan)m(ti\014er)f(in)g(an)h Fr(SSa)-6 b(t)44 b Fz(problem)g(can)i(b)s(e)e(view)m(ed)h(as)h(a)246 3985 y(simple)31 b(t)m(yp)s(e)i(of)g(maximization)e(op)s(erator;)j(the) f(problem)e(then)i(b)s(ecomes)g(one)246 4093 y(of)27 b(maximizing)f(the)i(probabilit)m(y)d(that)j Fl(\036)p Fz(\()p Fs(x)p Fz(\))g(is)f(satis\014ed,)f(giv)m(en)i(that)g(some)g(of) 246 4201 y(the)j(v)-5 b(ariables)31 b(are)h(under)e(the)h(con)m(trol)h (of)g(\\nature.")h(This)c(is)i(closely)g(related)246 4309 y(to)c(solving)e(a)i(\014nite-horizon)e(partially)g(observ)-5 b(able)26 b(Mark)m(o)m(v)i(decision)e(pro)s(cess)246 4417 y(\()p Fr(pomdp)p Fz(\),)d(whic)m(h)f(is)h(also)g(PSP)-8 b(A)m(CE-complete)25 b(\(P)m(apadimitriou)c(&)i(Tsitsiklis,)246 4525 y(1987\).)43 b(The)d(problem)g(remains)g(PSP)-8 b(A)m(CE)o(-complete)42 b(when)e(the)h(domain)f(is)246 4633 y(sp)s(eci\014ed)25 b(compactly)j(via)f(probabilistic)c Fr(strips)j Fz(op)s(erators)i(or)f(an)g(equiv)-5 b(alen)m(t)246 4741 y(represen)m(tation)39 b(\(Mundhenk)f Fj(et)j(al.)p Fz(,)g(1997\),)h(ev)m(en)e(if)e(the)i(domain)e(is)h(\\fully)246 4848 y(observ)-5 b(able")29 b(\(Littman,)h(1997\).)j(The)c(connection)h (b)s(et)m(w)m(een)h Fr(SSa)-6 b(t)28 b Fz(and)i(these)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.7)p eop %%Page: 8 8 8 7 bop 246 100 a Fz(8)873 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 259 y Fv(T)-6 b(able)34 b(I)r(I.)k(Di\013eren)n(t)33 b(arrangemen)n(ts)h(of)g(quan)n(ti\014ers)g(result)g(in)f Ft(SSa)-5 b(t)33 b Fv(problems)g(complete)h(for)g(di\013eren)n(t)246 350 y(complexit)n(y)24 b(classes)k(and)d(corresp)r(ond)h(to)g(basic)g (problems)f(in)h(uncertain)g(reasoning)h(and)e(planning.)p 25 428 3231 4 v 23 671 4 219 v 75 581 a(class)504 526 y(satis\014abilit)n(y)h(problem)504 635 y(Bo)r(olean)h(form)n(ula)2061 526 y(b)r(elief)g(net)n(w)n(ork)f(problem)2061 635 y(planning)g (problem)p 3254 671 V 25 700 3231 4 v 25 753 V 23 996 4 219 v 75 906 a(NP)504 850 y Ft(Sa)-5 b(t)504 960 y Fp(9)p Fh(x)591 968 y Fe(1)624 960 y Fh(;)14 b(:)f(:)g(:)g(;)g Fp(9)p Fh(x)882 968 y Ff(n)924 960 y Fv(\()p Fh(E)t Fv([)p Fh(\036)p Fv(\()p Fu(x)p Fv(\)])20 b Fp(\025)i Fh(\022)r Fv(\))2061 850 y(most)j(probable)h(explanation)2061 960 y(b)r(est)g(tra)t(jectory)p 3254 996 V 25 1024 3231 4 v 23 1267 4 219 v 75 1177 a(PP)504 1122 y Ft(Majsa)-5 b(t)553 1179 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 553 1179 a Fc(R)603 1179 y currentpoint grestore moveto 603 1179 a 553 1231 a Fh(x)597 1239 y Fe(1)632 1231 y Fh(;)13 b(:)g(:)g(:)g(;)852 1179 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 852 1179 a Fc(R)902 1179 y currentpoint grestore moveto 902 1179 a 852 1231 a Fh(x)896 1239 y Ff(n)938 1231 y Fv(\()p Fh(E)t Fv([)p Fh(\036)p Fv(\()p Fu(x)p Fv(\)])21 b Fp(\025)g Fh(\022)r Fv(\))2061 1122 y(b)r(elief)27 b(up)r(dating)f(\(inference\))2061 1231 y(plan)g(ev)l(aluation)p 3254 1267 4 219 v 25 1295 3231 4 v 23 1539 4 219 v 75 1448 a(NP)185 1416 y Fe(PP)504 1393 y Ft(E-Majsa)-5 b(t)504 1502 y Fp(9)p Fh(x)591 1510 y Fe(1)624 1502 y Fh(;)14 b(:)f(:)g(:)g(;)g Fp(9)p Fh(x)882 1510 y Ff(c)913 1502 y Fh(;)997 1450 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 997 1450 a Fc(R)1047 1450 y currentpoint grestore moveto 1047 1450 a 997 1502 a Fh(x)1041 1510 y Ff(c)p Fe(+1)1150 1502 y Fh(;)g(:)g(:)g(:)g(;)1370 1450 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1370 1450 a Fc(R)1420 1450 y currentpoint grestore moveto 1420 1450 a 1370 1502 a Fh(x)1414 1510 y Ff(n)1456 1502 y Fv(\()p Fh(E)t Fv([)p Fh(\036)p Fv(\()p Fu(x)p Fv(\)])21 b Fp(\025)g Fh(\022)r Fv(\))2061 1393 y(maxim)n(um)i(a)j(p)r(osteriori)h(h)n(yp)r(othesis)2061 1502 y(b)r(est)f(p)r(olynomial-size)g(plan)p 3254 1539 4 219 v 25 1567 3231 4 v 23 1810 4 219 v 75 1719 a(PSP)-6 b(A)n(CE)504 1664 y(Alternating)26 b Ft(SSa)-5 b(t)504 1774 y Fp(9)p Fh(x)591 1782 y Fe(1)624 1774 y Fh(;)708 1722 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 708 1722 a Fc(R)758 1722 y currentpoint grestore moveto 758 1722 a 708 1774 a Fh(x)752 1782 y Fe(2)786 1774 y Fh(;)14 b(:)f(:)g(:)g(;)g Fp(9)p Fh(x)1044 1782 y Ff(n)p Fq(\000)p Fe(1)1164 1774 y Fh(;)1248 1722 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1248 1722 a Fc(R)1298 1722 y currentpoint grestore moveto 1298 1722 a 1248 1774 a Fh(x)1292 1782 y Ff(n)1334 1774 y Fv(\()p Fh(E)t Fv([)p Fh(\036)p Fv(\()p Fu(x)p Fv(\)])21 b Fp(\025)g Fh(\022)r Fv(\))2061 1664 y(in\015uence)k(diagrams)2061 1774 y(b)r(est)h(p)r(olynomial-horizon)g(plan)p 3254 1810 4 219 v 25 1838 3231 4 v 246 2235 a Fz(planning)e(problems)h(has)h (b)s(een)g(exploited)g(to)i(create)g(a)f(comp)s(etitiv)m(e)g(planning) 246 2343 y(algorithm)33 b(\(Ma)5 b(jercik)34 b(&)f(Littman,)h(1999\).)i (In\015uence)d(diagrams)g(\(Shac)m(h)m(ter,)246 2451 y(1986\))50 b(are)f(a)g(b)s(elief-net)m(w)m(ork-lik)m(e)f(represen)m (tation)g(for)h(the)f(same)h(t)m(yp)s(e)g(of)246 2559 y(problem.)362 2667 y(T)-8 b(able)37 b(I)s(I)f(summarizes)g(the)h (relations)f(b)s(et)m(w)m(een)i(the)f(complexit)m(y)g(classes,)246 2775 y(sto)s(c)m(hastic)j(satis\014abilit)m(y)c(problems,)i(b)s(elief)f (net)m(w)m(ork)j(problems,)e(and)g(plan-)246 2883 y(ning)29 b(problems)f(discussed.)246 3122 y(1.3.1.)65 b Fj(Appr)-5 b(oximability)36 b(of)c Fr(SSa)-6 b(t)246 3230 y Fz(F)e(or)26 b(applications)d(in)h(planning)e(and)j(decision)f(making,)g(the)i (underlying)c(prob-)246 3337 y(lem)37 b(is)h(often)h(the)g(related)f (optimization)f(problem)g(asso)s(ciated)i(with)e Fr(SSa)-6 b(t)o Fz(:)246 3445 y(Giv)m(en)41 b(an)h(instance)f(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))43 b(output)e Fl(v)s(al)r Fz(\()p Fl(\036)p Fz(\).)i(Moreo)m(v)m(er,)h(an)e(e\016cien)m(t)g(al-) 246 3553 y(gorithm)35 b(that)i(will)c(return)i(a)i(close)f(appro)m (ximation)f(to)i(the)f(actual)h(v)-5 b(alue)35 b(is)246 3661 y(nearly)30 b(as)h(go)s(o)s(d)g(as)g(an)g(exact)i(solution.)d (Belo)m(w)h(is)g(a)g(brief)f(review)g(of)h(what)g(is)246 3769 y(kno)m(wn)f(ab)s(out)g(the)g(appro)m(ximabilit)m(y)e(of)j Fr(SSa)-6 b(t)n Fz(.)362 3877 y(Let)27 b Fj(Opt)h Fz(b)s(e)e(an)h (optimization)f(problem;)f(that)j(is,)e Fj(Opt)p Fz(\()p Fl(x)p Fz(\))i(is)e(a)h(real-v)-5 b(alued)246 3985 y(function)27 b(with)h(domain)g Fl(D)g Fm(\022)d(f)p Fz(0)p Fl(;)15 b Fz(1)p Fm(g)1547 3952 y Fx(\003)1617 3985 y Fz(\(bit)28 b(strings\).)h(Let)g Fl(A)g Fz(b)s(e)f(an)h(algorithm)246 4093 y(that)23 b(appro)m(ximates)f Fj(Opt)p Fz(\()p Fl(x)p Fz(\):)h(on)f(input)f Fl(x)k Fm(2)g(f)p Fz(0)p Fl(;)15 b Fz(1)p Fm(g)2040 4060 y Fx(\003)2081 4093 y Fz(,)23 b(the)f(algorithm)f(pro)s(duces)246 4201 y(an)j(output)g Fl(A)p Fz(\()p Fl(x)p Fz(\).)h(Let)g Fl(\017)f Fz(b)s(e)g(a)g(function) f(from)h(in)m(tegers)h(\(input)d(string)i(lengths\))246 4309 y(to)d(real)g(n)m(um)m(b)s(ers)e(b)s(et)m(w)m(een)j(0)f(and)g(1.)g (W)-8 b(e)22 b(sa)m(y)g(that)g(algorithm)e Fl(A)h Fj(appr)-5 b(oximates)246 4417 y(Opt)22 b Fz(to)h(within)d(ratio)i(0)k Fl(<)f(\017)p Fz(\()p Fm(\001)p Fz(\))h Fl(<)f Fz(1)e(if)e(for)h(all)f Fl(x)26 b Fm(2)e Fl(D)s Fz(,)f Fl(\017)p Fz(\()p Fm(j)p Fl(x)p Fm(j)p Fz(\))j Fm(\024)f Fl(A)p Fz(\()p Fl(x)p Fz(\))p Fl(=)p Fj(Opt)r Fz(\()p Fl(x)p Fz(\))h Fm(\024)246 4525 y Fz(1)p Fl(=)p Fz(\()p Fl(\017)p Fz(\()p Fm(j)p Fl(x)p Fm(j)p Fz(\)\).)31 b(In)c(other)h(w)m(ords,)g(for)g(all)f (inputs)f Fl(x)p Fz(,)i Fl(A)p Fz(\()p Fl(x)p Fz(\))h(and)e Fj(Opt)p Fz(\()p Fl(x)p Fz(\))i(di\013er)e(b)m(y)h(a)246 4633 y(factor)h(of)g(at)h(most)f Fl(\017)p Fz(\()p Fm(j)p Fl(x)p Fm(j)p Fz(\),)h(where)e Fm(j)p Fl(x)p Fm(j)h Fz(is)f(the)h (length)f(of)h(the)g(input)e(string)h Fl(x)p Fz(.)h(If)246 4741 y(suc)m(h)d(an)g Fl(A)h Fz(exists)f(for)g Fj(Opt)p Fz(,)h(then)f Fj(Opt)h Fz(can)f(b)s(e)g Fj(appr)-5 b(oximate)g(d)40 b Fz(to)27 b(within)d(ratio)246 4848 y Fl(\017)p Fz(\()p Fm(\001)p Fz(\).)34 b(Condon)f Fj(et)i(al.)f Fz(\(1997\))i(observ)m(ed) d(that)i(there)e(exists)g(a)h(constan)m(t)h Fl(c)c(>)f Fz(0)1819 5847 y Fo(paper.tex;)42 b(1/09/1999;)f(0:14;)g(p.8)p eop %%Page: 9 9 9 8 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)853 b Fz(9)246 291 y(suc)m(h)33 b(that)h(appro)m(ximating)e Fr(SSa)-6 b(t)32 b Fz(to)i(within)c(ratio)k (2)2176 258 y Fx(\000)p Fk(n)2274 234 y Ff(c)2343 291 y Fz(is)f(PSP)-8 b(A)m(CE-hard.)246 399 y(Th)m(us,)25 b(the)h(general)g(problem,)e(with)h(un)m(b)s(ounded)e(alternations)i (of)h(quan)m(ti\014ers,)246 506 y(is)44 b(v)m(ery)h(hard)f(to)i(appro)m (ximate.)g(Ho)m(w)m(ev)m(er,)h(what)e(ab)s(out)g(sp)s(ecial)f(cases)i (of)246 614 y Fr(SSa)-6 b(t)n Fz(,)26 b(suc)m(h)g(as)g Fr(SSa)-6 b(t)24 b Fz(with)g Fl(i)j Fz(alternations)e(of)h(quan)m (ti\014ers)e(where)i Fl(i)g Fz(is)e(a)j(small)246 722 y(constan)m(t,)45 b(indep)s(enden)m(t)c(of)j Fl(n)p Fz(?)f(The)g (argumen)m(t)h(of)f(Condon)g Fj(et)i(al.)f Fz(\(1997\))246 830 y(breaks)39 b(do)m(wn,)g(and)f(it)h(is)g(op)s(en)f(whether)h(or)g (not)h(the)f(problem)f(remains)g(as)246 938 y(hard)29 b(to)i(appro)m(ximate)g(as)f(the)h(general)f(v)m(ersion.)362 1046 y(Lusena,)43 b(Goldsmith,)f(&)h(Mundhenk)f(\(1998\))k(found)c (that)i(appro)m(ximat-)246 1154 y(ing)f(solutions)f(to)j(Mark)m(o)m(v)h (decision)c(pro)s(cesses)i(to)h(within)c(ratio)j Fl(\016)k Fz(is)43 b(also)246 1262 y(in)m(tractable.)246 1492 y(1.4.)65 b Fr(Random)33 b(F)m(ormulae)246 1696 y Fz(T)-8 b(o)29 b(empirically)d(ev)-5 b(aluate)29 b(algorithms)f(for)h(satis\014abilit) m(y)d(and)j(sto)s(c)m(hastic)g(sat-)246 1804 y(is\014abilit)m(y)-8 b(,)23 b(it)i(is)g(useful)f(to)j(ha)m(v)m(e)g(a)f(go)s(o)s(d)f(testb)s (ed)h(of)g(hard)e(form)m(ulae.)i(This)e(is)h(a)246 1912 y(non)m(trivial)i(problem:)h(sp)s(eci\014c)g(hand-co)s(ded)g(instances) h(of)g Fr(Sa)-6 b(t)28 b Fz(are)i(kno)m(wn)e(to)246 2020 y(b)s(e)c(satis\014able)f(or)i(not,)g(and)f(usually)f(for)h(a)h(clear)g (\(easily)f(computable\))h(reason.)246 2128 y(What)g(is)e(sough)m(t)i (is)e(a)i(natural)e(distribution)d(o)m(v)m(er)26 b Fr(Sa)-6 b(t)23 b Fz(instances)h(that)h(is)e(hard)246 2236 y(on)g(a)m(v)m (erage.)k(This)21 b(has)j(led)e(man)m(y)i(researc)m(hers)g(to)g(study)f (randomly)f(generated)246 2343 y(form)m(ulae)37 b(\(although)g(this)g (distribution)c(is)k(not)h(kno)m(wn)f(to)h(b)s(e)f(a)m(v)m(erage-case) 246 2451 y(hard\).)44 b(The)g(usual)g(\014xed-clause)g(random)g Fl(k)s Fz(-CNF)h(mo)s(del)f(is)g(used)g(in)f(this)246 2559 y(pap)s(er,)34 b(whic)m(h)g(is)h(de\014ned)f(in)g(terms)h(of)h (three)g(in)m(teger)f(parameters)h Fl(n;)15 b(k)39 b Fz(and)246 2667 y Fl(m)p Fz(;)21 b(a)i(form)m(ula)e(is)g(generated)h(b) m(y)g(selecting)g Fl(m)f Fz(clauses)h(of)g(size)g Fl(k)i Fz(indep)s(enden)m(tly)-8 b(.)246 2775 y(A)27 b(clause)g(is)g (generated)h(b)m(y)g(randomly)d(selecting)j(one)f(of)h Fl(n)f Fz(v)-5 b(ariables)26 b Fl(k)k Fz(times,)246 2883 y(rejecting)e(duplicate)f(selections,)i(and)f(randomly)e(negating)j(it) f(with)f(probabil-)246 2999 y(it)m(y)d(1)p Fl(=)p Fz(2.)h(This)e (distribution)d(is)j(denoted)i Fm(F)1749 2955 y Fk(k)r(;n)1740 3011 y(m)1854 2999 y Fz(.)f(In)g(the)g(exp)s(erimen)m(ts)f(describ)s (ed)246 3107 y(in)32 b(Section)h(3,)g(a)h(program)f(called)f Fy(makewff)p Fz(,)g(a)m(v)-5 b(ailable)32 b(from)h(A)-8 b(T&T)33 b(Labs)g({)246 3215 y(Researc)m(h,)24 b(w)m(as)f(used)f(to)i (generate)g(random)f(form)m(ulae)f(from)h(this)f(distribution.)362 3323 y(The)27 b(random)f Fl(k)s Fz(-CNF)i(mo)s(del)e(has)h(b)s(een)f (widely)f(studied)h(for)h(sev)m(eral)g(go)s(o)s(d)246 3431 y(reasons.)d(First,)g(it)f(is)g(an)h(in)m(trinsically)d(natural)i (mo)s(del,)g(analogous)h(to)h(the)f(ran-)246 3539 y(dom)c(graph)h(mo)s (del,)f(that)h(sheds)f(ligh)m(t)g(on)h(fundamen)m(tal)f(structural)g (prop)s(erties)246 3647 y(of)29 b(the)g(satis\014abilit)m(y)e(problem.) h(As)h(with)f Fr(Sa)-6 b(t)o Fz(,)29 b(it)g(is)f(p)s(ossible)f(to)j (gain)f(insigh)m(t)246 3755 y(in)m(to)39 b(the)g(mathematical)g (structure)g(of)g Fr(SSa)-6 b(t)37 b Fz(b)m(y)i(lo)s(oking)f(at)i(the)f (b)s(eha)m(vior)246 3863 y(of)c(randomly)e(generated)k(form)m(ulae.)e (Second,)g(using)e(an)i(appropriate)f(c)m(hoice)246 3971 y(of)j(parameters,)h(randomly)e(c)m(hosen)i(form)m(ulae)f(are)h (empirically)d(di\016cult)g(for)246 4079 y(satis\014abilit)m(y)44 b(algorithms,)i(and)g(are)h(a)g(commonly)f(used)f(b)s(enc)m(hmark)h (\(Gu)246 4187 y Fj(et)38 b(al.)p Fz(,)f(1997\).)h(The)e(next)h (section)f(brie\015y)f(reviews)g(some)i(analytical)e(results)246 4295 y(concerning)29 b(random)h Fr(Sa)-6 b(t)29 b Fz(and)h Fr(Majsa)-6 b(t)29 b Fz(instances.)246 4525 y(1.4.1.)65 b Fj(R)-5 b(andom)35 b Fr(Sa)-6 b(t)246 4633 y Fz(An)40 b(in)m(teresting)f(prop)s(ert)m(y)h(of)h(random)e Fl(k)s Fz(-CNF)i(form)m(ulae)f(is)g(that)h(when)e Fl(m)246 4741 y Fz(is)k(small)g(relativ)m(e)i(to)g Fl(n)p Fz(,)f(almost)h(all)e(form) m(ulae)h(are)h(satis\014able,)f(and)g(most)246 4848 y(algorithms)g (\014nd)g(it)h(easy)h(to)g(sho)m(w)f(this)f(satis\014abilit)m(y)-8 b(.)44 b(When)h Fl(m)h Fz(is)e(large)1819 5847 y Fo(paper.tex;)e (1/09/1999;)f(0:14;)g(p.9)p eop %%Page: 10 10 10 9 bop 246 100 a Fz(10)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(relativ)m(e)32 b(to)i Fl(n)p Fz(,)e(almost)h(all) e(form)m(ulae)h(are)h(unsatis\014able)d(and)i(unsatis\014abilit)m(y)246 399 y(is)i(relativ)m(ely)g(easy)h(to)h(sho)m(w.)f(F)-8 b(or)36 b(in)m(termediate)e(v)-5 b(alues)35 b(of)g Fl(m)f Fz(\(around)g(4)p Fl(:)p Fz(2)p Fl(n)246 506 y Fz(for)e Fl(k)g Fz(=)c(3\),)34 b(appro)m(ximately)d(half)h(of)g(the)h(form)m (ulae)f(are)h(satis\014able)e(and)h(it)g(is)246 614 y(v)m(ery)37 b(di\016cult)e(to)j(sho)m(w)f(satis\014abilit)m(y)-8 b(.)36 b(Th)m(us,)g(with)g(resp)s(ect)h(to)h Fl(m)p Fz(,)f(random)246 722 y Fr(Sa)-6 b(t)30 b Fz(instances)h(exhibit)f(an)i (\\easy-hardest-hard")g(pattern.)g(This)e(pattern)i(is)246 830 y(consisten)m(t)i(for)f(v)-5 b(arious)33 b(v)-5 b(alues)33 b(of)h Fl(n)g Fz(but)f(b)s(ecomes)h(more)g(pronounced)e(with)246 938 y(larger)e Fl(n)p Fz(.)g(In)g(addition,)e(the)j(hard)e(region)h(is) g(v)m(ery)g(large)h(for)f(large)h Fl(n)p Fz(.)362 1046 y(A)37 b(k)m(ey)h(conjecture)f(ab)s(out)g(the)g(random)f Fl(k)s Fz(-CNF)h Fr(Sa)-6 b(t)36 b Fz(form)m(ula)g(mo)s(del)g(is)246 1154 y(that)41 b(there)g(exists)g(a)g(constan)m(t)i Fl(\013)1480 1169 y Fk(k)1563 1154 y Fz(suc)m(h)e(that)h(a)f(random)f Fl(k)s Fz(-CNF)i(form)m(ula)246 1262 y(from)24 b Fm(F)529 1218 y Fk(k)r(;n)520 1274 y(m)659 1262 y Fz(is)g(almost)h(certainly)f (satis\014able)g(for)h Fl(m=n)g(<)g(\013)2312 1277 y Fk(k)2380 1262 y Fz(\(as)g Fl(n)g Fz(gets)g(large\),)246 1370 y(and)34 b(almost)h(certainly)f(unsatis\014able)f(if)h Fl(m=n)f(>)f(\013)2094 1385 y Fk(k)2137 1370 y Fz(.)j(As)g(men)m (tioned)g(ab)s(o)m(v)m(e,)246 1478 y(empirical)26 b(evidence)j (\(Selman,)f(Mitc)m(hell,)h(&)f(Lev)m(esque,)i(1996\))h(suggests)f (that)246 1586 y(this)j(0{1)i(threshold)e(v)-5 b(alue)34 b(for)g Fl(k)h Fz(=)c(3)k(o)s(ccurs)f(at)h(around)e(4)p Fl(:)p Fz(2,)j(and)d(a)i(recen)m(t)246 1694 y(theoretical)e(result)f(b) m(y)h(F)-8 b(riedgut)32 b(\(1997\))k(sho)m(ws)d(that)g(for)g(eac)m(h)h Fl(n)p Fz(,)f(there)g(is)f(a)246 1802 y(sharp)25 b(0{1)j(threshold)d Fl(\013)1107 1817 y Fk(k)1150 1802 y Fz(\()p Fl(n)p Fz(\))i(that)g(p)s (ossibly)c(v)-5 b(aries)26 b(with)g Fl(n)p Fz(.)g(It)h(is)e(kno)m(wn)i (that)246 1910 y Fl(\013)304 1924 y Fn(2)384 1910 y Fz(=)41 b(1)g(\(Ch)m(v\023)-45 b(atal)40 b(&)g(Reed,)g(1992;)i(Go)s(erdt,)e (1996\),)j(and)c(that)h(for)g Fl(k)45 b Fz(=)c(3,)246 2017 y(3)p Fl(:)p Fz(003)27 b Fm(\024)e Fl(\013)632 2032 y Fk(k)675 2017 y Fz(\()p Fl(n)p Fz(\))g Fm(\024)g Fz(4)p Fl(:)p Fz(601)33 b(\(F)-8 b(rieze)31 b(&)f(Suen,)g(1996;)i(Kirousis)c Fj(et)k(al.)p Fz(,)f(1998\).)362 2125 y(An)40 b(in)m(teresting)h (feature)g(of)g(the)g(threshold)e(is)h(that)h(the)g(dual)f(problems)246 2233 y(of)e(b)s(ounding)c(the)k(threshold)f(v)-5 b(alue)37 b(from)g(ab)s(o)m(v)m(e)i(and)e(b)s(elo)m(w)g(app)s(ear)g(to)i(b)s(e) 246 2341 y(asymmetric.)20 b(A)m(t)h(sligh)m(tly)e(b)s(elo)m(w)h(the)g (0{1)i(threshold,)d(there)h(are)h(simple)d(linear-)246 2449 y(time)46 b(algorithms)g(for)h(\014nding)e(a)i(satisfying)f (assignmen)m(t,)h(but)f(at)i(sligh)m(tly)246 2557 y(ab)s(o)m(v)m(e)37 b(the)f(0{1)h(threshold,)e(there)h(are)g(no)g(kno)m(wn)f(algorithms)g (that)h(quic)m(kly)246 2665 y(pro)s(duce)24 b(pro)s(ofs)h(of)i (unsatis\014abilit)m(y)-8 b(.)23 b(Moreo)m(v)m(er,)28 b(it)e(has)f(b)s(een)g(pro)m(v)m(en)i(that)f(no)246 2773 y(Resolution-based)37 b(metho)s(d)g(\(including)e(the)j(DPLL)g(pro)s (cedure)f(that)i(forms)246 2881 y(the)49 b(basis)g(of)g(sev)m(eral)h (algorithms)e(in)g(this)h(pap)s(er\))f(will)f(succeed)j(in)e(sub-)246 2989 y(exp)s(onen)m(tial)42 b(time)i(on)f(a)h(broad)f(range)h(of)g (randomly)e(generated)j(form)m(ulae)246 3097 y(\(Ch)m(v\023)-45 b(atal)37 b(&)f(Reed,)g(1992;)j(Beame)f(&)e(Pitassi,)f(1996;)k(Beame)e Fj(et)i(al.)p Fz(,)e(1998\).)246 3205 y(In)46 b(particular,)f(Beame)j Fj(et)f(al.)g Fz(\(1998\))i(sho)m(w)e(that)g(for)f Fl(k)56 b Fz(=)c(3)46 b(and)g Fl(m)53 b Fm(\025)246 3313 y Fl(n)301 3280 y Fn(2)340 3313 y Fl(=)15 b Fz(log)i Fl(n)p Fz(,)j(almost)h (certainly)f(a)h(random)f(form)m(ula)g(has)g(a)h(p)s(olynomial-size)d (deci-)246 3421 y(sion)23 b(tree,)j(but)e(for)h Fl(m)g(<)g(n)1180 3388 y Fn(6)p Fk(=)p Fn(5)1289 3421 y Fz(,)g(almost)g(certainly)f(a)h (random)f(form)m(ula)g(requires)246 3528 y(a)g(sup)s(erp)s (olynomial-size)19 b(decision)j(tree.)j(This)d(suggests)i(that)g(DPLL)g (will)d(p)s(er-)246 3636 y(form)29 b(v)m(ery)i(badly)d (\(asymptotically)i(in)f Fl(n)p Fz(\))h(ev)m(en)h(for)e(ranges)i(of)f Fl(m)g Fz(w)m(a)m(y)h(ab)s(o)m(v)m(e)246 3744 y(the)d(0{1)h(threshold)d (\(up)h(to)h Fl(n)1288 3711 y Fn(6)p Fk(=)p Fn(5)1398 3744 y Fz(\),)g(and)f(that)h(the)g(easy-hardest-hard)g(pattern)246 3852 y(observ)m(ed)k(empirically)e(ma)m(y)i(actually)g(b)s(e)g (misleading|for)e(large)i(enough)g Fl(n)p Fz(,)246 3960 y(the)44 b(exp)s(ected)g(pattern)h(is)e(more)h(lik)m(e)f (easy-hardest-...-hardest-easy)-8 b(.)48 b(This)246 4068 y(p)s(oin)m(ts)43 b(out)i(that)g(when)f(analyzing)f(algorithms)h(for)g (random)g(instances)g(of)246 4176 y Fr(Sa)-6 b(t)34 b Fz(it)h(is)g(imp)s(ortan)m(t)g(to)h(kno)m(w)g(whether)f(the)h (algorithm)f(is)f(applied)g(ab)s(o)m(v)m(e,)246 4284 y(b)s(elo)m(w,)29 b(or)i(at)g(the)g(threshold.)246 4525 y(1.4.2.)65 b Fj(R)-5 b(andom)35 b Fr(Majsa)-6 b(t)246 4633 y Fz(A)27 b(random)f 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b Fl(f)10 b Fz(\()p Fl(k)s(;)15 b(n;)g(m;)g(t)p Fz(\))24 b(b)s(e)f(the)h(probabilit)m(y)d(that)j(a)g(random)e Fl(k)s Fz(-CNF)j(form)m(ula)d(on)246 730 y Fl(n)27 b Fz(v)-5 b(ariables)27 b(with)f Fl(m)i Fz(clauses)g(has)f(at)i(least)f (2)1840 697 y Fk(t)1898 730 y Fz(satisfying)f(truth)g(assignmen)m(ts.) 246 838 y(Giv)m(en)d Fl(m)h Fz(and)f(a)h(function)e Fl(t)p Fz(\()p Fl(n)p Fz(\))i(of)g Fl(n)p Fz(,)f(what)h(can)g(b)s(e)f(said)f (ab)s(out)i(the)g(limit)d(of)j Fl(f)246 945 y Fz(as)g Fl(n)g Fz(gets)h(large?)f(Surprisingly)-8 b(,)21 b(in)j(spite)h(of)g(a) h(n)m(um)m(b)s(er)e(of)h(articles)g(concerning)246 1053 y(the)40 b(coun)m(ting)g(of)h(satisfying)e(assignmen)m(ts,)h(there)g (are)h(v)m(ery)g(few)f(analytical)246 1161 y(results)29 b(kno)m(wn)i(ab)s(out)g(this)f(function)f Fl(f)10 b Fz(,)31 b(and)f(its)g(0{1)j(threshold)c(b)s(eha)m(vior)h(is)246 1269 y(far)g(from)g(determined.)f(Some)h(simple)f(results)g(are)i (summarized)d(next.)362 1377 y(The)c(exp)s(ected)h(n)m(um)m(b)s(er)e 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b(e)32 b(w)m(an)m(t)h(to)f (calculate)g(for)g(what)f(v)-5 b(alues)31 b(of)h Fl(m)f Fz(do)s(es)246 2046 y(\(2\(7)p Fl(=)p Fz(8\))531 2013 y Fk(m=n)678 2046 y Fz(\))713 2013 y Fk(n)761 2046 y Fz(2)806 2013 y Fx(\000)p Fk(t)937 2046 y Fz(go)48 b(to)g(zero)f(as)h Fl(n)e Fz(go)s(es)h(to)h(in\014nit)m(y)-8 b(.)45 b(This)h(will)e(happ)s (en)246 2154 y(whenev)m(er)30 b(2\(7)p Fl(=)p Fz(8\))895 2121 y Fk(m=n)1067 2154 y Fl(<)25 b Fz(2)1208 2121 y Fk(t=n)1317 2154 y Fz(.)30 b(Solving)f(for)h Fl(m)p Fz(,)g(whenev)m(er) 937 2314 y Fl(m)25 b Fm(\025)g Fz(\(log)1291 2336 y Fn(8)p Fk(=)p Fn(7)1416 2314 y Fz(2\)\()p Fl(n)c Fm(\000)f Fl(t)p Fz(\))26 b Fm(\031)e Fz(5)p Fl(:)p Fz(19\()p Fl(n)e Fm(\000)e Fl(t)p Fz(\))p Fl(;)576 b Fz(\(1\))246 2474 y(a)35 b(random)f(3-CNF)h (form)m(ula)f(with)f Fl(m)i Fz(clauses)f(and)g Fl(n)g Fz(v)-5 b(ariables)34 b(almost)g(cer-)246 2582 y(tainly)g(has)i(few)m (er)g(than)f(2)1184 2549 y Fk(t)1250 2582 y Fz(satisfying)f(assignmen)m (ts.)i(As)f(exp)s(ected,)i(as)f Fl(t)f Fz(in-)246 2690 y(creases,)d(few)m(er)f(random)g(clauses)g(are)g(needed)g(to)h (generate)h(an)e(instance)g(that)246 2798 y(is)d(not)i(p)s(ositiv)m(e.) f(Empirical)e(estimates)j(of)g(the)g(0{1)h(threshold)d(in)h(Section)g (3.1)246 2906 y(con\014rm)g(the)i(form)f(of)g(Equation)g(1.)362 3014 y(While)20 b(the)i(ab)s(o)m(v)m(e)h(calculation)e(giv)m(es)h(an)f (upp)s(er)f(b)s(ound)g(on)h Fl(m)h Fz(as)f(a)h(function)246 3122 y(of)35 b Fl(t)g Fz(and)f Fl(n)p Fz(,)h(this)f(upp)s(er)f(b)s (ound)g(is)h(lik)m(ely)g(larger)h(than)f(the)i(actual)f(n)m(um)m(b)s (er.)246 3230 y(In)i(sev)m(eral)i(pap)s(ers)e(\(Kamath)i Fj(et)h(al.)p Fz(,)f(1995;)h(Kirousis)c Fj(et)k(al.)p Fz(,)f(1998\),)i(it)c(has)246 3337 y(b)s(een)28 b(demonstrated)i(that)g (most)g(form)m(ulae)f(will)e(ha)m(v)m(e)k(a)f(m)m(uc)m(h)g(smaller)e(n) m(um-)246 3445 y(b)s(er)36 b(of)h(satisfying)f(assignmen)m(ts)h(than)g (the)g(exp)s(ected)g(n)m(um)m(b)s(er)f(of)h(satisfying)246 3553 y(assignmen)m(ts,)27 b(b)s(ecause)g(for)g(those)h(form)m(ulae)f (that)h(are)f(satis\014able,)g(they)g(actu-)246 3661 y(ally)34 b(ha)m(v)m(e)j(man)m(y)f(satisfying)f(assignmen)m(ts)g(\(th)m (us)h(sk)m(ewing)f(the)h(a)m(v)m(erage\).)j(In)246 3769 y(particular,)24 b(Kamath)j Fj(et)h(al.)f Fz(\(1995\))i(sho)m(w)d(that) g(if)f(a)i(random)e Fl(k)s Fz(-CNF)i(form)m(ula)246 3877 y(is)33 b(satis\014able,)g(then)h(with)f(exp)s(onen)m(tially)f(high)h (probabilit)m(y)e(\(probabilit)m(y)h(at)246 3985 y(least)c(1)16 b Fm(\000)f Fz(2)648 3952 y Fx(\000)p Fk(\017n)779 3985 y Fz(\))28 b(it)f(has)h(an)g(exp)s(onen)m(tial)f(n)m(um)m(b)s(er)g(of)h (satisfying)e(truth)i(assign-)246 4093 y(men)m(ts.)h(Let)h Fl(Z)36 b Fz(b)s(e)29 b(the)g(n)m(um)m(b)s(er)f(of)h(v)-5 b(ariables)28 b(that)i(are)g(not)f(men)m(tioned)g(at)h(all)246 4201 y(when)38 b Fl(m)h Fz(clauses)f(are)i(c)m(hosen)f(randomly)-8 b(.)38 b(F)-8 b(or)40 b Fl(k)j Fz(=)c(3,)h(the)f(exp)s(ectation)h(of) 246 4309 y Fl(Z)7 b Fz(,)28 b Fl(E)5 b Fz([)p Fl(Z)i Fz(],)30 b(is)d Fl(n)p Fz(\(1)18 b Fm(\000)e Fz(1)p Fl(=n)p Fz(\))1123 4276 y Fn(3)p Fk(m)1226 4309 y Fz(,)29 b(whic)m(h)f(is)f (appro)m(ximately)i Fl(ne)2323 4276 y Fx(\000)p Fn(3)p Fk(m=n)2557 4309 y Fz(.)g(Kamath)g Fj(et)246 4417 y(al.)35 b Fz(\(1995\))j(sho)m(w)d(amazingly)f(tigh)m(t)i(tail)e(b)s(ounds)f(on) i(this)f(quan)m(tit)m(y)-8 b(.)36 b(In)e(fact,)246 4525 y(the)i(w)m(eak)m(est)i(of)e(their)f(results)g(sho)m(ws)g(that)i(the)f (probabilit)m(y)d(of)j(b)s(eing)f(more)246 4633 y(than)22 b(a)i(constan)m(t)g(factor)g(a)m(w)m(a)m(y)h(from)e(the)g(exp)s (ectation)g(is)f(exp)s(onen)m(tially)g(small)246 4741 y(in)29 b Fl(n)p Fz(.)i(This)e(implies)g(that)i(if)f Fl(\036)h Fz(is)f(a)i(random)e(form)m(ula)g(where)h Fl(m)g Fz(is)f(b)s(elo)m(w)g(the)246 4848 y(0{1)37 b(threshold,)e(then)h Fl(\036)h Fz(almost)f(certainly)g(will)e(b)s(e)h(satis\014able)h(and,)g (b)m(y)g(the)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.11)p eop %%Page: 12 12 12 11 bop 246 100 a Fz(12)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 301 y Fz(ab)s(o)m(v)m(e)31 b(result,)f(almost)g(certainly) f(will)f(ha)m(v)m(e)j(at)h(least)e(2)2201 268 y Fk(ne)2277 245 y Fq(\000)p Fe(3)p Ff(m=n)2519 301 y Fz(satisfying)f(as-)246 409 y(signmen)m(ts,)d(since)f(eac)m(h)j(of)e(the)h(v)-5 b(ariables)25 b(that)i(are)f(not)h(men)m(tioned)f(at)h(all)e(can)246 517 y(b)s(e)31 b(set)i(in)e(an)m(y)h(p)s(ossible)e(w)m(a)m(y)-8 b(.)33 b(Th)m(us,)f(almost)g(certainly)-8 b(,)32 b(whenev)m(er)g Fl(m)g Fz(is)f(less)246 625 y(than)36 b(the)h(0{1)i(threshold,)c(a)i (random)g Fr(Majsa)-6 b(t)35 b Fz(instance)h(with)g(parameters)246 733 y(\(3)p Fl(;)15 b(m;)g(n;)g(t)p Fz(\))32 b(is)d(p)s(ositiv)m(e)g (whenev)m(er)i Fl(t)25 b Fm(\024)g Fl(ne)1759 700 y Fx(\000)p Fn(3)p Fk(m=n)1993 733 y Fz(.)362 841 y(The)31 b(ab)s(o)m(v)m(e)i (argumen)m(t)f(states)h(that)f(while)d(the)j(exp)s(ected)g(n)m(um)m(b)s (er)e(of)i(sat-)246 949 y(isfying)41 b(assignmen)m(ts)i(is)g(\(2\(7)p Fl(=)p Fz(8\))1454 916 y Fk(m=n)1601 949 y Fz(\))1636 916 y Fk(n)1684 949 y Fz(,)g(almost)h(certainly)e(the)i(n)m(um)m(b)s (er)e(of)246 1057 y(satisfying)32 b(assignmen)m(ts)h(will)e(b)s(e)i (substan)m(tially)f(b)s(elo)m(w)h(the)g(exp)s(ected)h(v)-5 b(alue.)246 1165 y(The)25 b(b)s(eautifully)d(simple)h(argumen)m(t)j(of) f(Kirousis)e Fj(et)28 b(al.)e Fz(\(1998\))i(demonstrates)246 1273 y(that)38 b(this)e(is)h(not)g(the)h(only)f(source)g(of)h(w)m (astefulness)f(in)f(these)i(calculations.)246 1381 y(\(See)31 b(the)f(thesis)g(of)g(Ac)m(hlioptas)g(\(1999\))j(for)d(more)h (details.\))362 1489 y(What)36 b(can)f(b)s(e)f(said)g(of)h(a)h(lo)m(w)m (er)f(b)s(ound)e(on)i Fl(m)f Fz(as)i(a)f(function)f(of)h Fl(t)p Fz(?)g(Here)246 1596 y(again,)41 b(v)m(ery)h(little)d(seems)j (to)g(b)s(e)e(kno)m(wn.)h(Assume)f(that)i(almost)f(certainly)-8 b(,)246 1704 y(there)24 b(are)h Fl(l)i Fz(disjoin)m(t)c(clauses,)h Fl(C)1378 1718 y Fn(1)1418 1704 y Fl(;)15 b(C)1523 1718 y Fn(2)1563 1704 y Fl(;)g(:)g(:)g(:)h(;)f(C)1829 1719 y Fk(l)1880 1704 y Fz(in)23 b(a)i(random)f Fr(Majsa)-6 b(t)23 b Fz(form)m(ula)246 1812 y(with)29 b(parameters)h Fl(k)s(;)15 b(m;)g(n;)g(t)p Fz(.)362 1920 y(Let)30 b Fl(k)e Fz(=)d(3.)30 b(There)f(are)g(2)1252 1887 y Fn(3)p Fk(l)1343 1920 y Fz(total)h(assignmen)m(ts)f(for)g(the)h Fl(l)h Fz(disjoin)m(t)c(clauses.)246 2028 y(There)i(are)h(7)g(w)m(a)m (ys)h(to)f(mak)m(e)h(eac)m(h)g(clause)e Fy(True)g Fz(for)g(3-CNF)i (form)m(ulae.)e(Since)246 2136 y(the)22 b(clauses)g(are)h(disjoin)m(t,) e(there)i(are)f(7)1588 2103 y Fk(l)1637 2136 y Fz(w)m(a)m(ys)h(to)g (mak)m(e)h(all)d(the)h(clauses)g(true,)h(so)246 2244 y(the)28 b(exact)h(n)m(um)m(b)s(er)e(of)g(assignmen)m(ts)h(that)g(mak)m (e)h(one)f(or)g(more)g(clauses)g Fy(False)246 2352 y Fz(is)k(2)385 2319 y Fn(3)p Fk(l)469 2352 y Fm(\000)21 b Fz(7)606 2319 y Fk(l)633 2352 y Fz(,)33 b(whic)m(h)f(is)g(b)s(et)m(w) m(een)i(2)1447 2319 y Fn(3\()p Fk(l)q Fx(\000)p Fn(1\))1686 2352 y Fz(and)f(2)1911 2319 y Fn(3)p Fk(l)1973 2352 y Fz(.)g(This)e(giv)m(es)i(a)h(lo)m(w)m(er)f(b)s(ound)246 2460 y(on)38 b Fl(m)h Fz(as)g(a)g(function)e(of)i Fl(t)p Fz(,)g(the)g(n)m(um)m(b)s(er)e(of)i(satisfying)e(assignmen)m(ts,)i (since)246 2568 y Fl(l)f Fz(can)f(b)s(e)f(c)m(hosen)i(as)e(a)i (function)d(of)i Fl(m)f Fz(and)g Fl(n)h Fz(suc)m(h)f(that)h(almost)g (certainly)246 2676 y(a)c(3-CNF)i(random)d(form)m(ula)h(with)f Fl(m)h Fz(clauses)g(o)m(v)m(er)h Fl(n)f Fz(v)-5 b(ariables)32 b(con)m(tains)h(at)246 2784 y(least)d Fl(l)j Fz(v)-5 b(ariable-disjoin)m(t)27 b(clauses.)362 2892 y(The)f(ab)s(o)m(v)m(e)h (giv)m(es)g(v)m(ery)g(coarse)g(upp)s(er)d(and)i(lo)m(w)m(er)g(b)s (ounds)f(on)h(the)g(v)-5 b(alue)26 b(of)246 3000 y Fl(m)e Fz(for)g(whic)m(h)f(a)i(random)f Fr(Majsa)-6 b(t)23 b Fz(form)m(ula)g(is)h Fy(False)o Fz(/)p Fy(True)f Fz(almost)i(certainly) 246 3107 y(as)40 b(a)g(function)f(of)h Fl(t)p Fz(.)g(Do)g(w)m(e)h(kno)m (w)e(that)i Fl(f)10 b Fz(\()p Fl(k)s(;)15 b(m;)g(n;)g(t)p Fz(\))40 b(actually)g(exhibits)e(a)246 3215 y(sharp)f(0{1)i(threshold)e (b)s(eha)m(vior?)g(As)h(men)m(tioned)g(earlier,)g(F)-8 b(riedgut)38 b(\(1997\))246 3323 y(has)32 b(pro)m(v)m(en)i(that)f (random)g Fr(Sa)-6 b(t)32 b Fz(has)g(a)i(sharp)e(0{1)i(threshold,)e (although)g(the)246 3431 y(0{1)h(threshold)d(v)-5 b(alue)31 b Fl(\013)1109 3446 y Fk(k)1184 3431 y Fz(ma)m(y)i(b)s(e)e(a)h (function)f(of)h Fl(n)p Fz(.)f(The)h(ideas)f(in)f(his)h(pro)s(of)246 3539 y(are)f(general)g(enough)g(that)h(they)f(extend)g(to)h Fr(Majsa)-6 b(t)n Fz(.)30 b(In)g(other)g(w)m(ords,)g(from)246 3647 y(F)-8 b(riedgut)29 b(\(1997\),)i(it)e(follo)m(ws)f(that)i(there)f (exists)g(a)h(function)e Fl(\013)2468 3662 y Fk(k)2510 3647 y Fz(\()p Fl(n)p Fz(\))i(suc)m(h)f(that)246 3755 y(for)h(an)m(y)i Fl(\017)26 b(>)g Fz(0,)32 b(there)f(exists)g Fl(t)26 b(>)g Fz(1)31 b(suc)m(h)g(that)h(a)f(random)f Fr(Majsa)-6 b(t)29 b Fz(instance)246 3863 y(with)i(parameters)h(\()p Fl(k)s(;)15 b(m;)g(n;)g(t)p Fz(\),)34 b(where)e Fl(m)d Fz(=)f(\()p Fl(\013)1960 3878 y Fk(k)2003 3863 y Fz(\()p Fl(n)p Fz(\))22 b Fm(\000)f Fl(\017)p Fz(\))p Fl(n)p Fz(,)33 b(is)e(p)s(ositiv)m(e)g(with)246 3971 y(high)f(probabilit)m(y) -8 b(,)30 b(while)g(a)j(random)e(form)m(ula)g(with)f(parameters)j(\()p Fl(k)s(;)15 b(m;)g(n;)g(t)p Fz(\),)246 4079 y(where)39 b Fl(m)i Fz(=)g(\()p Fl(\013)844 4094 y Fk(k)887 4079 y Fz(\()p Fl(n)p Fz(\))27 b(+)f Fl(\017)p Fz(\))p Fl(n)p Fz(,)40 b(is)f(not)h(a)g(p)s(ositiv)m(e)f(instance)h(with)e(high)h (prob-)246 4187 y(abilit)m(y)-8 b(,)34 b(and)g(moreo)m(v)m(er)i(has)f (zero)h(satisfying)d(truth)h(assignmen)m(ts)h(with)e(high)246 4295 y(probabilit)m(y)-8 b(.)25 b(Therefore,)i(although)f(this)g(line)g (of)h(argumen)m(t)h(do)s(esn't)e(lead)h(to)h(a)246 4403 y(prediction)h(of)i(where)f(the)h(0{1)h(threshold)d(is,)i(or)f(ev)m(en) i(whether)e(it)h(is)e(a)j(linear)246 4511 y(function)d(of)h Fl(n)p Fz(,)g(it)g(do)s(es)g(imply)e(that)j(it)f(exists.)362 4618 y(F)-8 b(or)21 b(random)f(instances)g(of)g(more)h(general)g Fr(SSa)-6 b(t)18 b Fz(problems,)h(there)i(are)g(more)246 4726 y(parameters,)k(since)f(the)h(v)-5 b(alidit)m(y)23 b(of)h(the)h(form)m(ula)f(is)f(highly)g(dep)s(enden)m(t)g(on)i(the)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.12)p eop %%Page: 13 13 13 12 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(13)246 291 y(exact)33 b(sequence)g(of)f (quan)m(ti\014ers.)g(T)-8 b(o)32 b(date,)i(there)e(are)h(no)f (theoretical)g(results)246 399 y(concerning)21 b(the)h(threshold)e(b)s (eha)m(vior)h(of)h(random)f(instances)h(of)g(general)f Fr(SSa)-6 b(t)o Fz(.)676 742 y Fs(2.)74 b(Algorithms)34 b(for)h(Sto)s(c)m(hastic)h(Satis\014abilit)m(y)246 968 y Fz(The)28 b Fr(SSa)-6 b(t)28 b Fz(problem)f(is)h(PSP)-8 b(A)m(CE)29 b(complete,)g(and)g(therefore)g(w)m(ould)f(seem)h(to)246 1076 y(require)h(a)i(di\013eren)m(t)f(set)h(of)g(algorithmic)e(approac) m(hes)j(from)e(those)h(dev)m(elop)s(ed)246 1184 y(for)k(the)g(NP)q (-complete)h Fr(Sa)-6 b(t)o Fz(.)36 b(In)g(fact,)i(this)d(section)h (sho)m(ws)h(that)g(the)f(funda-)246 1292 y(men)m(tal)c(algorithmic)f (ideas)g(dev)m(elop)s(ed)h(in)f(the)h Fr(Sa)-6 b(t)31 b Fz(comm)m(unit)m(y)h(pro)m(vide)f(an)246 1400 y(excellen)m(t)26 b(starting)f(p)s(oin)m(t)g(for)h Fr(SSa)-6 b(t)24 b Fz(algorithms|the)h (gap)h(b)s(et)m(w)m(een)g(NP)g(and)246 1508 y(PSP)-8 b(A)m(CE)30 b(ma)m(y)h(not)f(b)s(e)g(as)h(large)f(as)h(it)f(seems.)362 1616 y(This)d(section)i(describ)s(es)e(three)i(algorithms)e(for)i Fr(SSa)-6 b(t)o Fz(.)28 b(Section)h(2.1)h(dev)m(el-)246 1724 y(ops)i(a)h(DPLL-based)f(algorithm.)g(It)g(sho)m(ws)h(that)g(the)f (basic)g(structure)g(of)h(the)246 1831 y Fr(Sa)-6 b(t)29 b Fz(DPLL)h(algorithm)g(is)f(applicable)f(to)j(Extended)f Fr(SSa)-6 b(t)o Fz(.)362 1939 y(Section)24 b(2.2)i(describ)s(es)d (another)i(algorithm)f(that)h(is)f(similar)e(to)j(the)g(DPLL-)246 2047 y(based)c(algorithm,)g(but)g(that)i(has)e(a)i(guaran)m(tee)g(of)f (b)s(eing)e(e\016cien)m(t)j(with)d(resp)s(ect)246 2155 y(to)30 b(other)h(decision-tree-based)e(algorithms.)g(This)f(t)m(yp)s (e)i(of)h(algorithm)e(can)h(b)s(e)246 2263 y(view)m(ed)i(as)h(a)g (\\univ)m(ersal")e(searc)m(h)j(algorithm)d(for)i(DPLL-based)f(pro)s (cedures.)246 2371 y(The)k(algorithm)g(is)f(guaran)m(teed)j(to)f(solv)m (e)g(the)g(Extended)30 b Fr(SSa)-6 b(t)35 b Fz(instance)h(in)246 2479 y(time)g(that)i(is)e(e\016cien)m(t)i(in)d(the)i(size)g(of)g(the)h (smallest)e(decision)f(tree.)j(As)f(dis-)246 2587 y(cussed)e(in)g (Section)h(4,)h(although)f(the)g(theoretical)g(p)s(erformance)g(of)g (the)h(uni-)246 2695 y(v)m(ersal)e(algorithm)g(is)g(sup)s(erior)e(in)i (the)h(w)m(orst)g(case,)h(it)e(do)s(es)h(not)g(p)s(erform)e(as)246 2803 y(w)m(ell)29 b(on)h(random)g(form)m(ulae)g(as)g(the)h(DPLL)f (algorithm.)362 2911 y(The)k(algorithms)g(in)f(Section)i(2.1)h(and)e (Section)g(2.2)i(are)f(b)s(oth)f(sound)g(and)246 3019 y(complete)20 b(algorithms:)g(giv)m(en)g(an)g(arbitrary)f(Extended)30 b Fr(SSa)-6 b(t)19 b Fz(instance)h(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\),)246 3127 y(b)s(oth)32 b(of)h(these)g(algorithms)e(are)i(guaran)m (teed)h(to)g(return)d(the)i(correct)h(answ)m(er,)246 3235 y(although)21 b(the)i(running)d(time)i(can)g(b)s(e)g(exp)s(onen)m (tial.)g(As)g(with)f Fr(Sa)-6 b(t)o Fz(,)23 b(incomplete)246 3342 y(or)j(appro)m(ximate)g(algorithms)f(can)h(often)g(solv)m(e)h (large)f(problems)e(more)i(quic)m(kly)246 3450 y(than)e(complete)h (algorithms)f(lik)m(e)f(DPLL.)i(F)-8 b(or)26 b Fr(Sa)-6 b(t)n Fz(,)25 b(randomized)f(lo)s(cal)g(searc)m(h)246 3558 y(pro)s(cedures)37 b(lik)m(e)h Fr(w)-8 b(alksa)i(t)36 b Fz(ha)m(v)m(e)k(b)s(een)e(sho)m(wn)g(to)h(b)s(e)f(successful)g(for)g (large)246 3666 y(problem)28 b(instances.)i(Similarly)-8 b(,)27 b Fr(Majsa)-6 b(t)28 b Fz(instances)i(are)g(v)m(ery)h(naturally) d(ap-)246 3774 y(pro)m(ximated)i(using)e(random)i(sampling.)e(Section)i (2.3)i(describ)s(es)c(a)j(new)f Fr(SSa)-6 b(t)246 3882 y Fz(algorithm)20 b(that)j(com)m(bines)e(lo)s(cal)g(searc)m(h)h(with)e (random)h(sampling)e(to)k(compute)246 3990 y(appro)m(ximate)41 b(answ)m(ers)g(to)g Fr(SSa)-6 b(t)40 b Fz(problems)f(\(existen)m(tial)i (and)f(randomized)246 4098 y(quan)m(ti\014ers)29 b(only\).)246 4324 y(2.1.)65 b Fr(A)34 b(D)m(a)-8 b(vis-Putnam-Logemann-Lo)n(veland) 32 b(Algorithm)246 4525 y Fz(The)20 b(Da)m(vis-Putnam-Logemann-Lo)m(v)m (eland)j(\(DPLL\))f(algorithm)e(for)h(Bo)s(olean)246 4633 y(satis\014abilit)m(y)32 b(\(Da)m(vis,)j(Logemann,)h(&)e(Lo)m(v)m (eland,)h(1962\))h(w)m(orks)f(b)m(y)f(en)m(umer-)246 4741 y(ating)29 b(partial)g(assignmen)m(ts)h(and)f(monitoring)f(for)i (opp)s(ortunities)d(to)k(simplify)246 4848 y(the)36 b(form)m(ula.)g (These)h(simpli\014cations,)c(or)k(pruning)d(rules,)h(mak)m(e)i(it)g(p) s(ossible)1780 5847 y Fo(paper.tex;)k(1/09/1999;)h(0:14;)e(p.13)p eop %%Page: 14 14 14 13 bop 246 100 a Fz(14)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(to)39 b(solv)m(e)g(problems)d(whose)j(en)m(tire)f (set)h(of)f(assignmen)m(ts)h(could)e(not)i(b)s(e)e(fully)246 399 y(en)m(umerated.)362 506 y(DPLL)d(is)g(designed)f(to)i(solv)m(e)g Fr(Sa)-6 b(t)33 b Fz(problems,)g(and,)h(th)m(us,)g(only)f(needs)h(to) 246 614 y(deal)21 b(with)g(existen)m(tial)h(quan)m(ti\014ers.)f(The)g (algorithm)g(describ)s(ed)f(in)h(this)g(section)246 722 y(can)f(b)s(e)g(view)m(ed)g(as)g(an)g(extension)g(of)h(the)f(DPLL)g (algorithm)g(to)h(Extended)30 b Fr(SSa)-6 b(t)246 830 y Fz(b)m(y)39 b(pro)m(viding)e(pruning)g(rules)h(for)h(univ)m(ersal)e (and)i(randomized)f(quan)m(ti\014ers.)246 938 y(Cadoli,)26 b(Gio)m(v)-5 b(anardi,)28 b(&)f(Sc)m(haerf)h(\(1998\))j(pro)m(vide)c(a) h(set)h(of)f(pruning)d(rules)h(for)246 1046 y(univ)m(ersal)i(quan)m (ti\014ers)h(\(for)i(solving)d Fr(QBF)i Fz(problems\);)f(the)i (puri\014cation)d(and)246 1154 y(unit)i(propagation)h(pruning)d(rules)i (describ)s(ed)f(here)j(are)f(in)m(tro)s(duced)f(and)h(jus-)246 1262 y(ti\014ed)24 b(there.)h(Birn)m(baum)e(&)i(Lozinskii)e(\(1999\))k (describ)s(e)c(a)j(similar)c(adaptation)246 1370 y(to)31 b(DPLL)f(for)g(coun)m(ting)g(satisfying)f(assignmen)m(ts.)362 1478 y(The)43 b Fi(evalssat)f Fz(algorithm)g(in)g(Figure)g(2)i(tak)m (es)h(form)m(ula)d Fl(\036)h Fz(and)g(lo)m(w)g(and)246 1586 y(high)30 b(thresholds)g Fl(\022)929 1601 y Fk(l)986 1586 y Fz(and)h Fl(\022)1207 1601 y Fk(h)1251 1586 y Fz(.)h(It)g(returns)e(a)i(v)-5 b(alue)31 b(less)g(than)h Fl(\022)2462 1601 y Fk(l)2519 1586 y Fz(if)e(and)h(only)g(if)246 1694 y(the)d(v)-5 b(alue)27 b(of)g(the)h(Extended)i Fr(SSa)-6 b(t)26 b Fz(form)m(ula)h(is)g(less)g(than)g Fl(\022)2373 1709 y Fk(l)2399 1694 y Fz(,)h(a)g(v)-5 b(alue)27 b(greater)246 1802 y(than)c Fl(\022)494 1817 y Fk(h)562 1802 y Fz(if)f(and)h(only)f (if)h(the)g(v)-5 b(alue)23 b(of)h(the)f(Extended)30 b Fr(SSa)-6 b(t)22 b Fz(form)m(ula)g(is)h(greater)246 1910 y(than)28 b Fl(\022)499 1925 y Fk(h)543 1910 y Fz(,)h(and)f(otherwise)f (the)i(exact)h(v)-5 b(alue)28 b(of)g(the)h(Extended)h Fr(SSa)-6 b(t)27 b Fz(form)m(ula.)246 2017 y(Th)m(us,)43 b(it)g(can)h(b)s(e)f(used)g(to)i(solv)m(e)f(the)g(Extended)30 b Fr(SSa)-6 b(t)42 b Fz(decision)h(problem)246 2125 y(b)m(y)h(setting)g Fl(\022)741 2140 y Fk(l)814 2125 y Fz(=)k Fl(\022)976 2140 y Fk(h)1068 2125 y Fz(=)g Fl(\022)s Fz(.)43 b(It)i(can)f(also)g(b) s(e)f(used)g(to)i(compute)f(the)h(exact)246 2233 y(v)-5 b(alue)39 b(of)g(the)h(form)m(ula)f(b)m(y)g(setting)h Fl(\022)1597 2248 y Fk(l)1663 2233 y Fz(=)g(0)g(and)f Fl(\022)2088 2248 y Fk(h)2173 2233 y Fz(=)h(1.)g(The)f(algorithm's)246 2341 y(basic)29 b(structure)g(is)g(to)h(compute)g(the)g(v)-5 b(alue)29 b(of)h(the)g(Extended)g Fr(SSa)-6 b(t)28 b Fz(form)m(ula)246 2449 y(from)41 b(its)h(de\014nition)e(\(Section)i (1.2.1\);)j(this)c(tak)m(es)j(place)e(in)f(the)i(section)f(of)246 2557 y(the)28 b(pseudo)s(co)s(de)f(lab)s(eled)f(\\)p Fb(Splitting)p Fz(",)h(whic)m(h)g(en)m(umerates)h(all)f(assignmen)m (ts,)246 2665 y(applying)33 b(op)s(erators)i(recursiv)m(ely)e(from)i (left)g(to)g(righ)m(t.)g(Ho)m(w)m(ev)m(er,)i(it)e(is)f(made)246 2773 y(more)29 b(complex)h(\(and)f(e\016cien)m(t\))i(b)m(y)e(a)h(set)g (of)g(pruning)d(rules,)h(describ)s(ed)g(next.)246 3014 y(2.1.1.)65 b Fj(Unit)33 b(R)-5 b(esolution)246 3122 y Fz(When)24 b(a)i(Bo)s(olean)f(form)m(ula)f Fl(\036)h Fz(is)f(ev)-5 b(aluated)26 b(that)f(con)m(tains)g(a)h(v)-5 b(ariable)23 b Fl(x)2815 3136 y Fk(i)2869 3122 y Fz(that)246 3230 y(app)s(ears)i(alone)g(in)g(a)h(clause)f(in)g Fl(\036)h Fz(with)e Fj(sign)33 b Fl(b)25 b Fz(\(0)i(if)p 2071 3180 81 4 v 25 w Fl(x)2123 3244 y Fk(i)2176 3230 y Fz(is)e(in)g(the)g (clause,)h(1)g(if)f Fl(x)3007 3244 y Fk(i)246 3337 y Fz(is)32 b(in)g(the)i(clause\),)f(the)h(normal)e(left-to-righ)m(t)i(ev) -5 b(aluation)33 b(of)h(quan)m(ti\014ers)e(can)246 3445 y(b)s(e)26 b(in)m(terrupted)g(to)j(deal)e(with)f(this)g(v)-5 b(ariable.)27 b(This)e(is)i(called)f Fj(unit)k(r)-5 b(esolution)7 b Fz(,)246 3553 y(b)m(y)30 b(analogy)h(with)e(DPLL.)362 3661 y(If)21 b(the)g(quan)m(ti\014er)f(asso)s(ciated)i(with)e Fl(x)1665 3675 y Fk(i)1714 3661 y Fz(is)h(existen)m(tial,)g Fl(x)2300 3675 y Fk(i)2349 3661 y Fz(can)g(b)s(e)g(eliminated)246 3769 y(from)35 b(the)i(form)m(ula)e(b)m(y)i(assigning)d(it)i(v)-5 b(alue)36 b Fl(b)g Fz(and)g(recursing.)f(As)h(in)f(DPLL,)246 3877 y(this)29 b(is)g(b)s(ecause)i(assigning)e Fl(x)1293 3891 y Fk(i)1346 3877 y Fz(=)c(1)c Fm(\000)f Fl(b)30 b Fz(is)f(guaran)m(teed)j(to)f(mak)m(e)g Fl(\036)g Fy(False)n Fz(,)g(so)246 3985 y Fl(x)298 3999 y Fk(i)357 3985 y Fz(=)g Fl(b)j Fz(can)h(b)s(e)e(no)h(w)m(orse.)h(Similarly)-8 b(,)31 b(if)i(the)h(quan)m(ti\014er)f(asso)s(ciated)h(with)f Fl(x)3007 3999 y Fk(i)246 4093 y Fz(is)h(randomized,)g(it)g(is)g(the)h (case)h(that)g(one)f(branc)m(h)g(of)g(the)g(computation)g(will)246 4201 y(return)g(a)h(zero,)i(so)e Fl(x)1004 4215 y Fk(i)1068 4201 y Fz(can)h(b)s(e)f(eliminated)e(from)i(the)g(form)m(ula)f(b)m(y)i (assigning)246 4309 y(it)29 b(v)-5 b(alue)30 b Fl(b)g Fz(and)g(con)m(tin)m(uing)f(recursiv)m(ely)-8 b(.)30 b(The)f(resulting)g(v)-5 b(alue)29 b(is)h(divided)d(b)m(y)246 4417 y(t)m(w)m(o)35 b(\(or)g(m)m(ultiplied)c(b)m(y)j(the)g(probabilit)m (y)e(asso)s(ciated)j(with)e(the)h(randomized)246 4525 y(quan)m(ti\014er,)40 b(more)h(generally\),)g(since)g(it)f(represen)m (ts)h(the)h(v)-5 b(alue)40 b(of)i(only)e(one)246 4633 y(branc)m(h.)362 4741 y(On)30 b(the)h(other)g(hand,)f(if)g(the)h(quan)m (ti\014er)e(asso)s(ciated)j(with)d Fl(x)2505 4755 y Fk(i)2564 4741 y Fz(is)h(univ)m(ersal,)246 4848 y(it)d(is)g(immediate)g(that)h (the)g(en)m(tire)f(form)m(ula)g(is)g Fy(False)f Fz(since)i(it)f(will)e (not)j(b)s(e)f(the)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e (p.14)p eop %%Page: 15 15 15 14 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(15)246 507 y Fi(evalssat)o Fz(\()p Fl(\036;)15 b(Q;)g(\022)824 522 y Fk(l)850 507 y Fl(;)g(\022)933 522 y Fk(h)978 507 y Fz(\))26 b(:=)f Fm(f)342 615 y Fz(if)k Fl(\036)h Fz(is)g(the)g(empt)m(y)h(set,)g (return)f(1)342 723 y(if)f Fl(\036)h Fz(con)m(tains)h(an)f(empt)m(y)h (clause,)f(return)g(0)342 831 y(/*)h Fj(Unit)h(R)-5 b(esolution)32 b Fz(*/)342 939 y(if)d Fl(x)477 953 y Fk(i)535 939 y Fz(is)h(a)h(unit)d(v)-5 b(ariable)30 b(with)f(sign)g Fl(b)h Fz(and)g Fl(Q)p Fz(\()p Fl(x)2035 953 y Fk(i)2064 939 y Fz(\))25 b(=)g Fm(9)p Fz(,)438 1047 y(return)k Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1139 1062 y Fk(x)1179 1072 y Ff(i)1206 1062 y Fn(=)p Fk(b)1295 1047 y Fl(;)15 b(Q;)g(\022)1490 1062 y Fk(l)1516 1047 y Fl(;)g(\022)1599 1062 y Fk(h)1644 1047 y Fz(\))342 1155 y(if)29 b Fl(x)477 1169 y Fk(i)535 1155 y Fz(is)h(a)h(unit)d(v)-5 b(ariable)30 b(and)f Fl(Q)p Fz(\()p Fl(x)1571 1169 y Fk(i)1600 1155 y Fz(\))c(=)g Fm(8)p Fz(,)438 1263 y(return)k(0)342 1371 y(if)g Fl(x)477 1385 y Fk(i)535 1371 y Fz(is)h(a)h(unit)d(v)-5 b(ariable)30 b(with)f(sign)g Fl(b)h Fz(and)g Fl(Q)p Fz(\()p Fl(x)2035 1385 y Fk(i)2064 1371 y Fz(\))25 b(=)2279 1308 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 2279 1308 a Fi(R)2338 1308 y currentpoint grestore moveto 2338 1308 a 2279 1371 a Fz(,)438 1479 y(return)k Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1139 1494 y Fk(x)1179 1504 y Ff(i)1206 1494 y Fn(=)p Fk(b)1295 1479 y Fl(;)15 b(Q;)g Fz(2)p Fl(\022)1535 1494 y Fk(l)1562 1479 y Fl(;)g Fz(2)p Fl(\022)1690 1494 y Fk(h)1735 1479 y Fz(\))p Fl(=)p Fz(2)342 1587 y(/*)31 b Fj(Puri\014c)-5 b(ation)31 b Fz(*/)342 1695 y(if)e Fl(x)477 1709 y Fk(i)535 1695 y Fz(is)h(a)h(pure)e(v)-5 b(ariable)29 b(with)g(sign)h Fl(b)g Fz(and)g Fl(Q)p Fz(\()p Fl(x)2051 1709 y Fk(i)2079 1695 y Fz(\))c(=)f Fm(9)p Fz(,)438 1803 y(return)k Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1139 1818 y Fk(x)1179 1828 y Ff(i)1206 1818 y Fn(=)p Fk(b)1295 1803 y Fl(;)15 b(Q;)g(\022)1490 1818 y Fk(l)1516 1803 y Fl(;)g(\022)1599 1818 y Fk(h)1644 1803 y Fz(\))342 1911 y(if)29 b Fl(x)477 1925 y Fk(i)535 1911 y Fz(is)h(a)h(pure)e(v)-5 b(ariable)29 b(with)g(sign)h Fl(b)g Fz(and)g Fl(Q)p Fz(\()p Fl(x)2051 1925 y Fk(i)2079 1911 y Fz(\))c(=)f Fm(8)p Fz(,)438 2018 y(return)k Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1139 2033 y Fk(x)1179 2043 y Ff(i)1206 2033 y Fn(=1)p Fx(\000)p Fk(b)1385 2018 y Fl(;)15 b(Q;)g(\022)1580 2033 y Fk(l)1607 2018 y Fl(;)g(\022)1690 2033 y Fk(h)1734 2018 y Fz(\))342 2126 y(/*)31 b Fj(Splitting)g Fz(*/)342 2234 y(if)e Fl(Q)p Fz(\()p Fl(x)584 2248 y Fn(1)624 2234 y Fz(\))c(=)g Fm(9)p Fz(,)30 b Fm(f)438 2342 y Fl(v)482 2356 y Fn(0)546 2342 y Fz(=)25 b Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 2356 y Fk(x)1105 2365 y Fe(1)1140 2356 y Fn(=0)1235 2342 y Fl(;)15 b(Q;)g(\022)1430 2357 y Fk(l)1456 2342 y Fl(;)g(\022)1539 2357 y Fk(h)1584 2342 y Fz(\))438 2450 y(if)29 b Fl(v)565 2464 y Fn(0)630 2450 y Fm(\025)c Fl(\022)769 2465 y Fk(h)813 2450 y Fz(,)31 b(return)e Fl(v)1191 2464 y Fn(0)438 2558 y Fl(v)482 2572 y Fn(1)546 2558 y Fz(=)c Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 2572 y Fk(x)1105 2581 y Fe(1)1140 2572 y Fn(=1)1235 2558 y Fl(;)15 b(Q;)g Fz(max)q(\()p Fl(\022)1635 2573 y Fk(l)1661 2558 y Fl(;)g(v)1745 2572 y Fn(0)1785 2558 y Fz(\))p Fl(;)g(\022)1903 2573 y Fk(h)1948 2558 y Fz(\))438 2666 y(return)29 b(max\()p Fl(v)964 2680 y Fn(0)1004 2666 y Fl(;)15 b(v)1088 2680 y Fn(1)1128 2666 y Fz(\))342 2774 y Fm(g)342 2882 y Fz(if)29 b Fl(Q)p Fz(\()p Fl(x)584 2896 y Fn(1)624 2882 y Fz(\))c(=)g Fm(8)p Fz(,)30 b Fm(f)438 2990 y Fl(v)482 3004 y Fn(0)546 2990 y Fz(=)25 b Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 3004 y Fk(x)1105 3013 y Fe(1)1140 3004 y Fn(=0)1235 2990 y Fl(;)15 b(Q;)g(\022)1430 3005 y Fk(l)1456 2990 y Fl(;)g(\022)1539 3005 y Fk(h)1584 2990 y Fz(\))438 3098 y(if)29 b Fl(v)565 3112 y Fn(0)630 3098 y Fl(<)c(\022)769 3113 y Fk(l)794 3098 y Fz(,)31 b(return)e Fl(v)1172 3112 y Fn(0)438 3206 y Fl(v)482 3220 y Fn(1)546 3206 y Fz(=)c Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 3220 y Fk(x)1105 3229 y Fe(1)1140 3220 y Fn(=1)1235 3206 y Fl(;)15 b(Q;)g(\022)1430 3221 y Fk(l)1456 3206 y Fl(;)g Fz(min)o(\()p Fl(v)1727 3220 y Fn(0)1767 3206 y Fl(;)g(\022)1850 3221 y Fk(h)1895 3206 y Fz(\)\))438 3314 y(return)29 b(min)n(\()p Fl(v)946 3328 y Fn(0)986 3314 y Fl(;)15 b(v)1070 3328 y Fn(1)1110 3314 y Fz(\))342 3422 y Fm(g)342 3529 y Fz(if)29 b Fl(Q)p Fz(\()p Fl(x)584 3543 y Fn(1)624 3529 y Fz(\))c(=)839 3466 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 839 3466 a Fi(R)898 3466 y currentpoint grestore moveto 898 3466 a 839 3529 a Fz(,)30 b Fm(f)438 3637 y Fl(v)482 3651 y Fn(0)546 3637 y Fz(=)25 b Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 3651 y Fk(x)1105 3660 y Fe(1)1140 3651 y Fn(=0)1235 3637 y Fl(;)15 b(Q;)g Fz(2)p Fl(\022)1475 3652 y Fk(l)1522 3637 y Fm(\000)20 b Fz(1)p Fl(;)15 b Fz(2)p Fl(\022)1786 3652 y Fk(h)1832 3637 y Fz(\))438 3745 y(if)29 b(\()p Fl(v)600 3759 y Fn(0)660 3745 y Fz(+)20 b(1\))p Fl(=)p Fz(2)27 b Fl(<)e(\022)1087 3760 y Fk(l)1112 3745 y Fz(,)31 b(return)e Fl(v)1490 3759 y Fn(0)1530 3745 y Fl(=)p Fz(2)438 3853 y(if)g Fl(v)565 3867 y Fn(0)604 3853 y Fl(=)p Fz(2)e Fm(\025)e Fl(\022)860 3868 y Fk(h)904 3853 y Fz(,)31 b(return)e Fl(v)1282 3867 y Fn(0)1321 3853 y Fl(=)p Fz(2)438 3961 y Fl(v)482 3975 y Fn(1)546 3961 y Fz(=)c Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1065 3975 y Fk(x)1105 3984 y Fe(1)1140 3975 y Fn(=1)1235 3961 y Fl(;)15 b(Q;)g Fz(2)p Fl(\022)1475 3976 y Fk(l)1522 3961 y Fm(\000)20 b Fl(v)1657 3975 y Fn(0)1696 3961 y Fl(;)15 b Fz(2)p Fl(\022)1824 3976 y Fk(h)1890 3961 y Fm(\000)20 b Fl(v)2025 3975 y Fn(0)2064 3961 y Fz(\))438 4069 y(return)29 b(\()p Fl(v)795 4083 y Fn(0)855 4069 y Fz(+)20 b Fl(v)990 4083 y Fn(1)1029 4069 y Fz(\))p Fl(=)p Fz(2)342 4177 y Fm(g)246 4285 y(g)246 4519 y Fw(Figur)l(e)32 b(2.)38 b Fv(The)31 b Fc(evalssat)g Fv(algorithm)f(generalizes)j(the)d (DPLL)g(algorithm)h(for)g(satis\014abilit)n(y)246 4610 y(to)26 b(solv)n(e)g(Extended)e Ft(SSa)-5 b(t)25 b Fv(problems.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.15)p eop %%Page: 16 16 16 15 bop 246 100 a Fz(16)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(case)38 b(that)f(all)f(v)-5 b(alues)36 b(of)h Fl(x)1220 305 y Fk(i)1285 291 y Fz(mak)m(e)h(the)f(form)m(ula)f Fy(True)o Fz(.)h(The)f(algorithm)g(can)246 399 y(return)f(a)h(zero)h (in)e(this)g(case.)j(This)c(is)h(stronger)i(than)f(the)g(pruning)e(p)s (ossible)246 506 y(for)c(either)g(existen)m(tial)f(or)i(randomized)e(v) -5 b(ariables.)246 855 y(2.1.2.)65 b Fj(Puri\014c)-5 b(ation)34 b(and)g(Thr)-5 b(esholding)246 963 y Fz(The)34 b Fj(puri\014c)-5 b(ation)44 b Fz(pruning)32 b(rules)i(apply)g(when)g (there)h(is)f(a)i(v)-5 b(ariable)33 b Fl(x)2805 977 y Fk(i)2869 963 y Fz(that)246 1071 y(app)s(ears)e(only)g(with)f(one)i (sign)f Fl(b)g Fz(in)g Fl(\036)p Fz(.)h(If)f Fl(Q)p Fz(\()p Fl(x)1890 1085 y Fk(i)1919 1071 y Fz(\))d(=)f Fm(9)p Fz(,)k(the)h(algorithm)f(assigns)246 1179 y Fl(x)298 1193 y Fk(i)363 1179 y Fz(=)37 b Fl(b)h Fz(and)f(recurses.)h(This)e(is) g(v)-5 b(alid)37 b(b)s(ecause)g(an)m(y)h(clause)g(satis\014ed)f(b)m(y)g (an)246 1287 y(assignmen)m(t)25 b(with)g Fl(x)964 1301 y Fk(i)1017 1287 y Fz(=)g(1)11 b Fm(\000)g Fl(b)26 b Fz(will)d(also)j(b)s(e)f(satis\014ed)g(b)m(y)g(assigning)g Fl(x)2684 1301 y Fk(i)2737 1287 y Fz(=)g Fl(b)p Fz(.)h(By)246 1395 y(similar)31 b(reasoning,)i(if)g Fl(Q)p Fz(\()p Fl(x)1227 1409 y Fk(i)1255 1395 y Fz(\))e(=)f Fm(8)p Fz(,)k(the)g(algorithm)e(recurses)h(after)i(assigning)246 1503 y Fl(x)298 1517 y Fk(i)351 1503 y Fz(=)25 b(1)19 b Fm(\000)e Fl(b)p Fz(.)30 b(In)m(terestingly)-8 b(,)29 b(puri\014cation)e(pruning)f(do)s(es)j(not)h(app)s(ear)f(p)s(ossible) 246 1611 y(for)35 b(randomized)f(v)-5 b(ariables.)34 b(Both)i(assignmen)m(ts)f(to)h(a)g(randomized)e(v)-5 b(ariable)246 1719 y(giv)m(e)23 b Fj(some)31 b Fz(con)m(tribution)22 b(to)h(the)h(v)-5 b(alue)22 b(of)i(the)f(Extended)30 b Fr(SSa)-6 b(t)21 b Fz(form)m(ula,)i(and)246 1826 y(m)m(ust)30 b(b)s(e)g(considered)f(indep)s(enden)m(tly)-8 b(.)362 1934 y(Another)28 b(useful)f(class)h(of)h(pruning)c(rules)i(concerns)i (the)f Fj(thr)-5 b(eshold)41 b Fz(param-)246 2042 y(eters)g Fl(\022)517 2057 y Fk(l)583 2042 y Fz(and)g Fl(\022)814 2057 y Fk(h)858 2042 y Fz(.)g(While)f(some)h(care)h(m)m(ust)e(b)s(e)g (tak)m(en)i(to)g(pass)e(meaningful)246 2150 y(thresholds)j(when)g (applying)g(unit)g(resolution,)g(threshold)g(pruning)f(mainly)246 2258 y(comes)36 b(in)m(to)e(pla)m(y)h(when)f(v)-5 b(ariables)34 b(are)h(split)e(to)j(try)f(to)h(prev)m(en)m(t)f(recursiv)m(ely)246 2366 y(computing)29 b(b)s(oth)h(assignmen)m(ts)g(to)h Fl(x)1577 2380 y Fn(1)1616 2366 y Fz(,)g(the)f(outermost)i(quan)m (ti\014ed)d(v)-5 b(ariable.)362 2474 y(If)34 b Fl(Q)p Fz(\()p Fl(x)616 2488 y Fn(1)656 2474 y Fz(\))e(=)g Fm(9)p Fz(,)i(after)h(the)g(\014rst)f(recursiv)m(e)g(call)g(computing)g Fl(v)2561 2488 y Fn(0)2635 2474 y Fz(\(the)h(v)-5 b(alue)246 2582 y(of)30 b(the)g(curren)m(t)g(form)m(ula)g(with)f Fl(x)1414 2596 y Fn(1)1483 2582 y Fz(set)i(to)g Fy(False)o Fz(\),)f(it)g(is)f(p)s(ossible)f(that)j Fl(\022)2829 2597 y Fk(h)2903 2582 y Fz(has)246 2690 y(already)36 b(b)s(een)g(exceeded.)i(In)d(this)h(case,)i(the)f(algorithm)e(can)i (simply)d(return)246 2798 y Fl(v)290 2812 y Fn(0)329 2798 y Fz(,)h(without)e(ev)m(er)i(computing)f Fl(v)1425 2812 y Fn(1)1498 2798 y Fz(\(the)h(v)-5 b(alue)34 b(of)g(the)h(curren)m (t)f(form)m(ula)f(with)246 2906 y Fl(x)298 2920 y Fn(1)370 2906 y Fz(set)h(to)h Fy(True)n Fz(\).)f(In)f(particular,)f(it)h(is)g(p) s(ossible)d(that)35 b Fl(v)2257 2920 y Fn(1)2326 2906 y Fl(>)30 b(v)2471 2920 y Fn(0)2511 2906 y Fz(,)k(but)e(all)h(that)246 3014 y(is)d(signi\014can)m(t)g(is)g(whether)g(the)i(larger)e(of)i(the)f (t)m(w)m(o)h(exceeds)g Fl(\022)2432 3029 y Fk(h)2477 3014 y Fz(.)f(If)g Fl(v)2669 3028 y Fn(0)2739 3014 y Fz(exceeds)246 3122 y Fl(\022)289 3137 y Fk(l)342 3122 y Fz(but)c(falls)f(short)h(of)h Fl(\022)1067 3137 y Fk(h)1111 3122 y Fz(,)g(this)e(can)i(b)s(e)f(used)g(to)h(increase)f(the)h(lo)m(w) m(er)g(threshold)246 3230 y(for)i(the)g(recursiv)m(e)g(computation)g (of)h Fl(v)1599 3244 y Fn(1)1638 3230 y Fz(;)g(since)e(the)i(algorithm) e(m)m(ust)h(tak)m(e)i(the)246 3337 y(larger)e(of)i Fl(v)653 3351 y Fn(0)723 3337 y Fz(and)f Fl(v)945 3351 y Fn(1)984 3337 y Fz(,)g(the)h(precise)e(v)-5 b(alue)31 b(of)g Fl(v)1881 3351 y Fn(1)1951 3337 y Fz(is)f(not)i(needed)e(if)g(it)h(less)f(than) 246 3445 y Fl(v)290 3459 y Fn(0)329 3445 y Fz(.)362 3553 y(The)c(complemen)m(t)h(of)g(this)e(argumen)m(t)i(pro)m(vides)f(a)h (threshold)e(pruning)e(rule)246 3661 y(for)30 b(univ)m(ersal)e(quan)m (ti\014ers.)362 3769 y(Threshold)42 b(pruning)g(is)i(not)h(as)g(strong) g(for)f(randomized)g(v)-5 b(ariables,)43 b(al-)246 3877 y(though)32 b(it)f(can)i(b)s(e)f(done.)g(There)g(are)g(t)m(w)m(o)i(t)m (yp)s(es)e(of)g(threshold)f(pruning)f(that)246 3985 y(apply)-8 b(.)31 b(First,)g(if)g(assigning)f(0)i(to)g Fl(x)1480 3999 y Fn(1)1552 3985 y Fz(is)e(su\016cien)m(t)h(to)i(meet)f(the)g (threshold)e Fl(\022)2965 4000 y Fk(h)3010 3985 y Fz(,)246 4093 y(then)k(the)h(algorithm)e(need)h(not)h(recurse)f(on)h(assigning)e (the)h(v)-5 b(ariable)34 b(to)h(1:)g(if)246 4201 y Fl(v)290 4215 y Fn(0)329 4201 y Fl(=)p Fz(2)26 b Fm(\025)f Fl(\022)584 4216 y Fk(h)629 4201 y Fz(,)30 b(return)g Fl(v)1007 4215 y Fn(0)1046 4201 y Fl(=)p Fz(2.)362 4309 y(If)39 b(the)h(\014rst)f(v)-5 b(alue)39 b Fl(v)1113 4323 y Fn(0)1192 4309 y Fz(is)f(so)i(lo)m(w)f (that,)i(ev)m(en)f(if)e Fl(v)2170 4323 y Fn(1)2251 4309 y Fz(=)i(1,)g(\()p Fl(v)2551 4323 y Fn(0)2617 4309 y Fz(+)26 b Fl(v)2758 4323 y Fn(1)2798 4309 y Fz(\))p Fl(=)p Fz(2)42 b Fl(<)246 4417 y(\022)289 4432 y Fk(l)314 4417 y Fz(,)d(then)f(again)g(the)h(algorithm)e(need)h(not)h(compute)g Fl(v)2241 4431 y Fn(1)2280 4417 y Fz(.)g(If)e(b)s(oth)h(tests)h(fail,) 246 4525 y(the)48 b(algorithm)g(needs)g(to)h(compute)g Fl(v)1679 4539 y Fn(1)1718 4525 y Fz(,)g(but)f(can)g(adjust)g(the)h (thresholds)246 4633 y(accordingly)-8 b(.)362 4741 y(All)27 b(of)i(these)g(pruning)d(rules)i(are)h(incorp)s(orated)f(in)m(to)g(the) h(algorithm)f(giv)m(en)246 4848 y(in)h(Figure)h(2.)h(App)s(endix)c(A)k (giv)m(es)f(a)h(correctness)g(pro)s(of)f(for)g(the)h(algorithm.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.16)p eop %%Page: 17 17 17 16 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(17)246 291 y(2.1.3.)65 b Fj(Complexity)35 b(of)e Fi(evalssat)246 399 y Fz(Not)c(surprisingly) -8 b(,)25 b(the)j(w)m(orst-case)i(running)c(time)i(of)g Fi(evalssat)g Fz(is)f(exp)s(onen)m(tial)246 506 y(in)g(the)i(size)f(of) g(the)h(form)m(ula.)f(The)g(reason)h(is)e(that)i(on)g(ev)m(en)g(simple) d(instances,)246 614 y(often)46 b(the)h(underlying)c(decision)i(tree)i (is)e(alw)m(a)m(ys)i(large,)f(and)g(in)f(addition,)246 722 y(pruning)23 b(do)s(es)i(not)h(help.)f(A)h(particular)e(example)i (of)g(this)f(is)g(applying)e Fi(evalssat)246 830 y Fz(to)f (unsatis\014able)d Fr(Sa)-6 b(t)21 b Fz(form)m(ulae)g(that)h(are)g(kno) m(wn)e(to)j(require)d(exp)s(onen)m(tial)g(size)246 938 y(Resolution)25 b(pro)s(ofs.)h(Since)f(the)i(algorithm)f(\(in)f(this)h (sp)s(ecial)f(case\))j(pro)s(duces)d(a)246 1046 y(Resolution)20 b(refutation,)i(the)g(Resolution)e(lo)m(w)m(er)i(b)s(ounds)e(for)h(sp)s (eci\014c)g(form)m(ulae)246 1154 y(\(suc)m(h)j(as)h(the)g(prop)s (ositional)d(pigeonhole)i(principle\))d(imply)i(exp)s(onen)m(tial)g (time)246 1262 y(lo)m(w)m(er)30 b(b)s(ounds)e(on)j(the)f(running)e (time)i(of)h Fi(evalssat)e Fz(on)h(these)h(examples.)362 1370 y(It)39 b(is)f(also)h(in)m(teresting)f(to)h(understand)e(the)i(a)m (v)m(erage-case)k(running)36 b(time)246 1478 y(of)d Fi(evalssat)48 b Fz(and)32 b(its)h(p)s(erformance)g(on)g(more)h(general)f(form)m (ulae,)g(suc)m(h)h(as)f(in-)246 1586 y(stances)25 b(of)g Fr(Majsa)-6 b(t)22 b Fz(or)j(Alternating)30 b Fr(SSa)-6 b(t)n Fz(.)25 b(This)e(question)g(can)i(b)s(e)f(partially)246 1694 y(answ)m(ered)39 b(in)f(the)i(con)m(text)i(of)e(random)e(form)m (ulae,)i(yielding)d(results)h(on)i(the)246 1802 y(complexit)m(y)29 b(of)h(the)f(DPLL)h(algorithm)e(assuming)g(no)i(pruning)c(rules,)j (that)h(is,)246 1910 y(assuming)e(that)j(the)f(algorithm)f(needs)h(to)g (generate)i(the)e(en)m(tire)g(decision)e(tree)246 2017 y(b)s(efore)36 b(terminating.)g(This)f(is)h(in)m(teresting,)h(since)f (for)h(man)m(y)g(hard)f(form)m(ulae)246 2125 y(the)21 b(en)m(tire)g(decision)f(tree)i(m)m(ust)f(b)s(e)g(generated)h(in)e (order)g(to)i(determine)f(truth)f(or)246 2233 y(falsit)m(y)26 b(of)i(the)g(form)m(ula,)f(and,)g(th)m(us,)g(lo)m(w)m(er)h(b)s(ounds)d (on)j(the)f(decision-tree)g(size)246 2341 y(yield)d(lo)m(w)m(er)h(b)s (ounds)f(on)h(the)h(running)d(time)i(of)h(the)g(algorithm.)f(The)g (follo)m(wing)246 2449 y(theorem)30 b(follo)m(ws)g(from)g(the)g (results)g(of)g(Beame)i Fj(et)g(al.)f Fz(\(1998\).)246 2664 y(THEOREM)f(1.)61 b Fj(L)-5 b(et)28 b Fl(\036)g Fj(b)-5 b(e)27 b(a)h(3-CNF)f(formula)i(chosen)f(uniformly)h(at)f(r)-5 b(andom)246 2781 y(fr)g(om)37 b Fm(F)538 2737 y Fk(k)r(;n)529 2793 y(m)643 2781 y Fj(.)f(F)-7 b(or)37 b(al)5 b(l)36 b Fl(n)g Fj(su\016ciently)g(lar)-5 b(ge,)36 b(for)h(al)5 b(l)36 b Fl(m)p Fj(,)g Fl(n)30 b Fm(\024)h Fl(m)g Fm(\024)g Fl(n)2697 2748 y Fn(5)p Fk(=)p Fn(4)2807 2781 y Fj(,)k(with)246 2889 y(high)e(pr)-5 b(ob)g(ability)35 b(any)e(de)-5 b(cision)33 b(tr)-5 b(e)g(e)34 b(for)f Fl(\036)g Fj(r)-5 b(e)g(quir)g(es)33 b(exp)-5 b(onential)35 b(size)d(in)h Fl(n)p Fj(.)362 3104 y Fz(This)c(theorem)i(sho)m(ws)f(that)i(for)e(random)g(form)m (ulae)g(in)g(a)h(large)g(range,)g(the)246 3212 y(decision-tree)24 b(complexit)m(y)h(of)g(the)g(underlying)d(3-CNF)k(form)m(ula)e(will)e (b)s(e)j(h)m(uge,)246 3320 y(and)41 b(therefore)h(the)g(running)d(time) i(of)h(an)m(y)g(decision-tree-based)f(algorithm)246 3428 y(without)21 b(pruning)f(will)g(b)s(e)i(exp)s(onen)m(tial.)g(T)-8 b(o)23 b(mak)m(e)g(matters)g(w)m(orse,)h(the)e(DPLL)246 3536 y(algorithm)j(ab)s(o)m(v)m(e)i(searc)m(hes)h(for)e(a)g(decision)f (tree)i(in)e(a)h(v)m(ery)h(sp)s(eci\014c)e(w)m(a)m(y;)j(that)246 3644 y(is,)22 b(it)h(is)f(not)h(guaran)m(teed)h(to)g(ev)m(en)g(searc)m (h)g(for)f(the)g(smallest)f(tree.)j(This)c(leads)h(to)246 3752 y(the)g(follo)m(wing)g(in)m(teresting)g(question:)g(Giv)m(en)g(a)h (CNF)g(form)m(ula)f Fl(\036)p Fz(,)h(is)e(it)i(p)s(ossible)246 3859 y(to)35 b(e\016cien)m(tly)f(pro)s(duce)f(a)i(decision)e(tree)i (for)f Fl(\036)g Fz(that)h(is)e(close)i(to)g(an)f(optimal)246 3967 y(one?)i(An)g(a\016rmativ)m(e)h(answ)m(er)g(to)g(this)e(question)h (suggests)h(a)g(more)g(e\016cien)m(t)246 4075 y(algorithm)21 b(for)h Fr(Sa)-6 b(t)o Fz(,)22 b(and)g(moreo)m(v)m(er)i(suggests)e (that)h(if)e(one)i(is)e(using)g(a)h(decision-)246 4183 y(tree-based)g(metho)s(d,)e(that)i(an)f(\\optimal")g(algorithm)g (within)d(this)i(class)h(exists.)246 4291 y(The)32 b(next)g(section)h (describ)s(es)e(suc)m(h)h(an)g(algorithm:)g(it)g(is)g(guaran)m(teed)h (to)h(\014nd)246 4399 y(a)c(close-to-optimal)i(size)e(tree)h(e\016cien) m(tly)-8 b(.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.17)p eop %%Page: 18 18 18 17 bop 246 100 a Fz(18)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(2.2.)65 b Fr(A)34 b(Universal)f(Sear)n(ch)f (Algorithm)i(f)n(or)g(Extended)f(SSa)-6 b(t)246 539 y Fz(As)22 b(indicated)f(earlier,)h(it)g(ma)m(y)i(b)s(e)e(p)s(ossible)e (to)j(determine)f(whether)g(a)h Fr(Majsa)-6 b(t)246 647 y Fz(instance)38 b(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))40 b(is)e(p)s(ositiv)m(e)f(without)h(actually)h(constructing)f (the)h(en)m(tire)246 755 y(decision)31 b(tree,)j(via)e(pruning.)e(On)i (the)h(other)g(hand,)f(to)h(determine)f(the)h(exact)246 863 y(v)-5 b(alue)38 b(of)h Fl(\036)p Fz(,)h(the)f(en)m(tire)g (decision)e(tree)j(m)m(ust)f(b)s(e)f(examined,)g(and)h(therefore)246 971 y(a)33 b(lo)m(w)m(er)h(b)s(ound)d(on)i(the)h(b)s(est)f(p)s(ossible) d(running)h(time)i(of)g(the)h(algorithm)e(de-)246 1079 y(scrib)s(ed)24 b(here)i(will)d(b)s(e)j(a)g(linear)f(function)g(of)h (the)h(size)f(of)g(the)g(smallest)g(decision)246 1187 y(tree)k(for)g Fl(\036)p Fz(.)g(Th)m(us,)f(it)h(is)f(useful)f(to)j (understand)d(ho)m(w)i(hard)f(it)g(is)g(to)i(searc)m(h)f(for)246 1295 y(the)g(smallest)g(suc)m(h)g(decision)f(tree.)362 1403 y(Let)c(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))25 b(b)s(e)f(an)g(instance)g(of)h(Extended)30 b Fr(SSa)-6 b(t)n Fz(.)25 b(A)f Fj(blo)-5 b(ck)35 b Fz(is)23 b(a)i(maximal)246 1510 y(set)49 b(of)f(v)-5 b(ariables)47 b(that)i(app)s(ear)f(in)f(the)i (form)m(ula)f(with)f(the)i(same)f(quan)m(ti-)246 1618 y(\014er)40 b(and)g(are)h(adjacen)m(t)h(in)e(the)g(quan)m(ti\014er)g (ordering.)g(Note)i(that,)g(within)c(a)246 1726 y(blo)s(c)m(k,)30 b(quan)m(ti\014ers)f(comm)m(ute.)j(In)e(what)h(follo)m(ws,)f(assume)g (that)h(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))31 b(has)246 1834 y Fl(\024)f Fz(consecutiv)m(e)i(blo)s(c)m(ks)d(of)i(v)-5 b(ariables)29 b Fl(X)1647 1801 y Fn(1)1687 1834 y Fl(;)15 b(:)g(:)g(:)h(;)f(X)1970 1801 y Fk(\024)2016 1834 y Fz(.)362 1942 y(If)42 b Fl(\036)g Fz(is)f(an)i(Extended)29 b Fr(SSa)-6 b(t)41 b Fz(form)m(ula,)h(then)g Fl(\036)g Fz(can)h(b)s(e)e(determined) g(to)246 2050 y(b)s(e)46 b Fy(True)f Fz(or)i Fy(False)e Fz(b)m(y)i(examining)e(a)i(canonical)f(decision)f(tree)j(for)e Fl(\036)h Fz(as)246 2158 y(describ)s(ed)38 b(in)h(Section)h(1.2.1.)i (Giv)m(en)e(an)g(Extended)30 b Fr(SSa)-6 b(t)39 b Fz(instance)g Fl(\036)p Fz(,)i(the)246 2266 y Fi(evalssat)25 b Fz(algorithm)g (describ)s(ed)g(in)g(Section)h(2.1)i(can)e(b)s(e)g(generalized)g(to)i (searc)m(h)246 2374 y(for)33 b(a)h(decision)e(tree)i(that)g(is)e(more)i (general)f(than)h(a)f(canonical)g(decision)f(tree)246 2482 y(\(splits)42 b(need)j(not)f(follo)m(w)g(the)g(precise)g(quan)m (ti\014er)f(ordering\),)h(but)g(has)g(the)246 2590 y(follo)m(wing)38 b(prop)s(erties:)g(for)i(ev)m(ery)g Fl(i;)15 b(j)46 b(i)41 b(<)g(j)5 b Fz(,)40 b(ev)m(ery)h(v)-5 b(ariable)38 b(in)h(the)h(blo)s (c)m(k)246 2698 y Fl(X)328 2665 y Fk(i)387 2698 y Fz(is)31 b(split)e(up)s(on)g(in)h(the)h(decision)f(tree)i(b)s(efore)f(ev)m(ery)h (v)-5 b(ariable)29 b(in)h(the)i(blo)s(c)m(k)246 2806 y Fl(X)328 2773 y Fk(j)365 2806 y Fz(,)26 b(with)f(the)h(p)s(ossible)e (exception)i(of)g(v)-5 b(ariables)25 b(that)i(app)s(ear)e(in)g(unit)f (clauses)246 2914 y(or)37 b(pure)g(v)-5 b(ariables.)37 b(A)g(decision)g(tree)h(for)f Fl(\036)h Fz(with)f(these)h(prop)s (erties)e(will)f(b)s(e)246 3021 y(called)27 b(a)h Fj(DPLL)i(tr)-5 b(e)g(e)35 b Fz(for)28 b Fl(\036)p Fz(.)g(The)f(ev)-5 b(aluation)28 b(of)g(a)g(DPLL)g(tree)h(for)e Fl(\036)h Fz(is)f(again)246 3129 y(obtained)20 b(b)m(y)g(ev)-5 b(aluating)21 b(eac)m(h)g(v)m(ertex,)i(starting)d(at)i(the)e(lea)m(v)m (es,)j(in)c(the)i(manner)246 3237 y(describ)s(ed)35 b(in)h(Section)h (1.2.1.)i(T)-8 b(o)38 b(summarize,)f(a)g(decision)f(tree)i(ignores)f (the)246 3345 y(quan)m(ti\014er)27 b(ordering,)h(a)h(canonical)f(tree)h (follo)m(ws)f(it)g(precisely)-8 b(,)28 b(and)g(the)g(DPLL)246 3453 y(tree)j(can)f(consider)g(v)-5 b(ariables)29 b(in)g(a)i(blo)s(c)m (k)e(in)g(an)m(y)i(order.)362 3561 y(It)j(w)m(as)g(sho)m(wn)f (implicitly)e(b)m(y)i(Clegg,)i(Edmonds,)d(&)i(Impagliazzo)g(\(1996\)) 246 3669 y(and)46 b(more)g(explicitly)f(b)m(y)h(Beame)i(&)e(Pitassi)g (\(1996\))j(and)d(Ben-Sasson)h(&)246 3777 y(Wigderson)22 b(\(1999\))j(that)f(there)f(is)f(a)h(relativ)m(ely)g(e\016cien)m(t)g (deterministic)e(pro)s(ce-)246 3885 y(dure)28 b(for)h(\014nding)f(a)i (small)e(tree-lik)m(e)i(Resolution)e(pro)s(of.)h(F)-8 b(ollo)m(wing)29 b(the)h(pro)s(of)246 3993 y(of)25 b(their)f(theorem,)h (it)f(can)i(b)s(e)e(sho)m(wn)g(that)h(there)g(is)f(an)h(e\016cien)m(t)g (deterministic)246 4101 y(pro)s(cedure)k(for)h(\014nding)e(a)j(small)e (DPLL)h(tree.)246 4417 y(THEOREM)g(2.)68 b Fj(Ther)-5 b(e)35 b(is)f(a)g(deterministic)h(algorithm)h(that)f(takes)f(as)h (input)246 4525 y(an)25 b(Extende)-5 b(d)34 b Fr(SSa)-6 b(t)23 b Fj(formula)j Fl(\036)g Fj(with)f Fl(m)g Fj(clauses)g(and)h Fl(n)f Fj(underlying)g(variables)246 4633 y(and)i(outputs)h(a)g(DPLL)e (de)-5 b(cision)27 b(tr)-5 b(e)g(e,)28 b Fl(T)40 b Fj(for)27 b Fl(\036)p Fj(.)g(Mor)-5 b(e)g(over,)28 b(the)f(running)g(time)246 4741 y(of)42 b(the)h(algorithm)h(is)f(at)g(most)g Fl(n)1452 4708 y Fk(O)r Fn(\(log)13 b Fk(s)p Fn(\))1702 4741 y Fl(O)s Fz(\()p Fl(m)p Fz(\))p Fj(,)42 b(wher)-5 b(e)44 b Fl(s)e Fj(is)g(the)h(size)f(of)h(the)246 4848 y(smal)5 b(lest)33 b(DPLL)f(de)-5 b(cision)34 b(tr)-5 b(e)g(e)33 b(for)g Fl(\036)p Fj(.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h (0:14;)e(p.18)p eop %%Page: 19 19 19 18 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(19)246 291 y Fs(Pro)s(of:)79 b Fz(The)37 b(pro)s(of)f(follo)m(ws)f(Beame)j Fj(et)h(al.)e Fz(\(1998\),)i(with)d(small)f(mo)s(di\014ca-)246 399 y(tions.)c(As)i(ab)s(o)m(v)m(e,)g(let)g(\()p Fl(\036;)15 b(\022)s(;)g(Q)p Fz(\))32 b(b)s(e)g(an)g(instance)g(of)g(Extended)e Fr(SSa)-6 b(t)31 b Fz(with)g Fl(\024)246 506 y Fz(blo)s(c)m(ks)c(of)h (consecutiv)m(e)h(v)-5 b(ariables)27 b Fl(X)1555 473 y Fn(1)1595 506 y Fl(;)15 b(:)g(:)g(:)i(;)e(X)1879 473 y Fk(\024)1924 506 y Fz(.)29 b(Assume)e(that)i Fl(\036)f Fz(has)g(a)h(size)f Fl(s)246 614 y Fz(DPLL)23 b(decision)e(tree,)j(and) f(assume)g(without)f(loss)g(of)i(generalit)m(y)f(that)h(the)f(ro)s(ot) 246 722 y(no)s(de)h(is)f(lab)s(eled)g(b)m(y)i(the)f(v)-5 b(ariable)24 b Fl(x)p Fz(.)h(It)f(follo)m(ws)g(that)h(one)g(of)g(the)g (t)m(w)m(o)h(subtrees)246 830 y(has)34 b(size)g(at)g(most)h Fl(s=)p Fz(2.)g(Therefore,)f(the)g(algorithm)g(can)g(searc)m(h)h (through)e(all)246 938 y(literals)38 b Fl(l)k Fz(o)s(ccurring)d(in)g (the)i(\014rst)e(blo)s(c)m(k)h(un)m(til)f(it)g(\014nds)g(one,)i Fl(l)2549 905 y Fx(\003)2588 938 y Fz(,)g(suc)m(h)f(that)246 1046 y Fl(\036)p Fm(d)340 1061 y Fk(l)362 1042 y Fq(\003)398 1061 y Fn(=1)525 1046 y Fz(has)32 b(a)g(decision)f(tree)i(of)f(size)h (at)g(most)f Fl(s=)p Fz(2.)h(The)f(tree)h Fl(T)45 b Fz(constructed)246 1154 y(for)35 b Fl(\036)h Fz(will,)d(th)m(us,)i(ha)m(v)m(e)i(ro)s(ot)f (lab)s(eled)e(with)g Fl(l)1884 1121 y Fx(\003)1959 1154 y Fz(\(not)i(necessarily)e Fl(x)p Fz(,)i(but)f(not)246 1262 y(m)m(uc)m(h)30 b(w)m(orse,)h(as)g(indicated)e(b)m(y)h(the)g(o)m (v)m(erall)h(running)d(time\).)362 1370 y(The)35 b(main)g(algorithm,)g Fs(DPLLsearc)m(h)p Fz(,)j(tak)m(es)f(as)f(input)e(a)j(Bo)s(olean)f (for-)246 1478 y(m)m(ula)47 b Fl(\036)p Fz(.)g(If)g(there)h(is)f(a)h (literal)e Fl(l)j Fz(o)s(ccurring)d(in)g(a)i(unit)e(clause,)i(then)f (the)246 1586 y(output)29 b(of)i Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fz(\))g(consists)f(of)g(a)h(DPLL)f(tree)h(for)f Fl(\036)p Fm(d)2568 1601 y Fk(l)q Fn(=1)2715 1586 y Fz(follo)m(w)m(ed) 246 1694 y(b)m(y)35 b(a)h(DPLL)f(tree)h(for)f Fl(\036)p Fm(d)1163 1709 y Fk(l)q Fn(=0)1280 1694 y Fz(,)g(and)g(then)g(these)h (t)m(w)m(o)g(trees)g(are)g(com)m(bined)e(b)m(y)246 1802 y(adding)k(a)j(top)f(no)s(de)g(and)f(splitting)f(on)i Fl(l)r Fz(.)h(Otherwise)d(\(if)i(there)g(is)g(no)g(unit)246 1910 y(clause\),)f(then)f(for)h(eac)m(h)h(of)f(the)g(literals)e Fl(l)k Fz(o)s(ccurring)c(in)h(the)h(curren)m(t)g(blo)s(c)m(k)246 2017 y(\(the)31 b(leftmost)f(blo)s(c)m(k)g(of)g(v)-5 b(ariables)29 b(that)i(ha)m(v)m(e)h(not)e(already)g(b)s(een)g(set\),)i (apply)246 2125 y Fs(DPLLsearc)m(h)37 b Fz(to)g(the)g(form)m(ula)e Fl(\036)p Fm(d)1557 2140 y Fk(l)q Fn(=1)1710 2125 y Fz(to)i(iden)m (tify)e(the)h(literal)f Fl(l)j Fz(for)e(whic)m(h)246 2233 y Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fm(d)937 2248 y Fk(l)q Fn(=1)1054 2233 y Fz(\))30 b(terminates)h(fastest.)362 2341 y(These)h(calls)g(to)h Fs(DPLLsearc)m(h)g Fz(are)g(executed)h(in)d (a)i(sequence)f(of)h(parallel)246 2449 y(rounds)26 b(so)i(that)h(in)d (round)h Fl(i)h Fz(the)g Fl(i)p Fz(th)g(step)g(of)g(eac)m(h)h(of)f(the) g(calls)f(is)g(p)s(erformed.)246 2557 y(In)i(other)h(w)m(ords,)g(if)f Fl(l)992 2571 y Fn(1)1032 2557 y Fl(;)15 b(:)g(:)g(:)i(;)e(l)1261 2571 y Fk(p)1331 2557 y Fz(are)30 b(the)h(v)-5 b(ariables)28 b(in)h(the)i(curren)m(t)f(blo)s(c)m(k,)f(then)246 2665 y(the)g(algorithm)e(will)f(execute)k(the)f Fl(p)g Fz(recursiv)m(e)f (calls,)g Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fm(d)2832 2680 y Fk(l)2853 2690 y Ff(i)2880 2680 y Fn(=1)2975 2665 y Fz(\),)246 2773 y Fl(i)d Fz(=)g(1)p Fl(;)15 b(:)g(:)g(:)i(;)e(p)p Fz(,)26 b(in)d(a)j(do)m(v)m(etail)f(fashion,)f(executing)h(eac)m(h)i (for)d(one)i(time)e(step,)i(and)246 2881 y(then)21 b(eac)m(h)i(for)f(t) m(w)m(o)h(time)e(steps,)h(etc.)h(The)f(\014rst)f(time)g(that)i(one)f (of)g(the)g(recursiv)m(e)246 2989 y(calls)40 b(terminates,)h(sa)m(y)g (for)g(literal)e Fl(l)1564 2956 y Fx(\003)1604 2989 y Fz(,)i Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fm(d)2361 3004 y Fk(l)2383 2985 y Fq(\003)2420 3004 y Fn(=1)2514 2989 y Fz(\))g(terminates,)246 3097 y(then)36 b(w)m(e)h(ab)s(ort)g(all) e(other)i(calls,)f(except)i(for)e(the)h(call)f(corresp)s(onding)e(to)p 2942 3023 69 4 v 38 w Fl(l)2971 3070 y Fx(\003)3010 3097 y Fz(,)246 3205 y(whic)m(h)29 b(is)g(run)g(to)i(completion.)362 3313 y(The)g(output)g(of)g Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fz(\))i(consists)e(of)g(a)h(DPLL)f(tree)h(for)f Fl(\036)p Fm(d)2882 3328 y Fk(l)2904 3308 y Fq(\003)2941 3328 y Fn(=1)246 3421 y Fz(follo)m(w)m(ed)47 b(b)m(y)h(a)g(DPLL)g(tree) h(for)e Fl(\036)p Fm(d)1594 3436 y Fk(l)1616 3416 y Fq(\003)1653 3436 y Fn(=0)1747 3421 y Fz(,)h(and)f(then)h(these)g(t)m(w)m(o)h(trees) g(are)246 3528 y(com)m(bined)29 b(b)m(y)h(adding)f(a)i(top)g(no)s(de)f (and)f(splitting)f(on)j Fl(l)2225 3495 y Fx(\003)2264 3528 y Fz(.)362 3636 y(The)36 b(analysis)e(of)j(the)f(algorithm)g (follo)m(ws)f(that)i(of)f(Beame)i Fj(et)g(al.)e Fz(\(1998\).)246 3744 y(Let)26 b Fl(T)457 3758 y Fn(1)497 3744 y Fz(\()p Fl(n;)15 b(s)p Fz(;)g Fl(m)p Fz(\))26 b(denote)h(the)f(maxim)m(um)f (running)e(time)j(of)g Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fz(\))246 3852 y(o)m(v)m(er)33 b(all)e(form)m(ulae)g Fl(\036)h Fz(with)f(at)h(most)g Fl(n)g Fz(v)-5 b(ariables,)31 b Fl(m)g Fz(clauses)h(and)f(suc)m(h)h(that)246 3960 y(there)f(exists)f (a)h(decision)e(tree)i(for)g Fl(\036)g Fz(of)f(size)h(at)g(most)g Fl(s)p Fz(.)g(Consider)e(a)i(decision)246 4068 y(tree)37 b(of)g(size)g(at)g(most)h Fl(s)e Fz(and)g(let)h Fl(x)g Fz(b)s(e)f(the)h(splitting)d(v)-5 b(ariable)36 b(at)i(the)f(ro)s(ot.) 246 4176 y(The)i(left)g(and)g(righ)m(t)g(branc)m(hes)h(of)f(the)h(tree) h(giv)m(e)f(decision)e(trees)i(for)g Fl(\036)p Fm(d)2901 4190 y Fk(x)p Fn(=0)246 4284 y Fz(and)35 b Fl(\036)p Fm(d)522 4298 y Fk(x)p Fn(=1)657 4284 y Fz(,)h(and)f(the)h(smaller)f (of)h(these)h(is)e(of)h(size)g(at)g(most)h Fl(s=)p Fz(2.)f(Hence,)h(at) 246 4392 y(least)23 b(one)h(of)f(the)h(recursiv)m(e)e(calls)h (terminates)g(after)h(at)g(most)f Fl(T)2448 4406 y Fn(1)2488 4392 y Fz(\()p Fl(n)6 b Fm(\000)g Fz(1)p Fl(;)15 b(s=)p Fz(2;)g Fl(m)p Fz(\))246 4500 y(steps,)24 b(and)f(so)h(the)g(literal)f Fl(l)1204 4467 y Fx(\003)1267 4500 y Fz(is)g(found)f(after)j(at)f(most) g(that)h(n)m(um)m(b)s(er)d(of)j(rounds.)246 4608 y(The)33 b(time)g(for)h(eac)m(h)g(round)f(is)f(at)j(most)f Fl(C)7 b(n)32 b Fz(for)i(some)g(constan)m(t)h Fl(C)7 b Fz(.)33 b(Once)h Fl(l)2996 4575 y Fx(\003)246 4716 y Fz(is)d(found,)g(it)h(tak) m(es)i(at)f(most)f Fl(T)1339 4730 y Fn(1)1379 4716 y Fz(\()p Fl(n)21 b Fm(\000)g Fz(1)p Fl(;)15 b(s)p Fz(;)g Fl(m)p Fz(\))34 b(steps)e(to)h(complete)f(the)h(call)e(to)246 4824 y Fs(DPLLsearc)m(h)q Fz(\()p Fl(\036)p Fm(d)937 4839 y Fk(l)959 4819 y Fq(\003)995 4839 y Fn(=0)1090 4824 y Fz(\))39 b(\(the)h(smallest)e(tree)i(for)f Fl(l)2091 4791 y Fx(\003)2170 4824 y Fz(=)g(0)h(can)f(b)s(e)f(no)h(larger)1780 5847 y Fo(paper.tex;)i(1/09/1999;)h(0:14;)e(p.19)p eop %%Page: 20 20 20 19 bop 246 100 a Fz(20)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(than)43 b(the)g(smallest)f(tree\).)j(Th)m(us,)d (for)h(\014xed)g Fl(m)p Fz(,)g(the)h(follo)m(wing)d(recurrence)246 399 y(relation)23 b(is)g(obtained:)h Fl(T)1104 413 y Fn(1)1144 399 y Fz(\()p Fl(n;)15 b(s)p Fz(\))25 b Fm(\024)g Fl(C)7 b(nT)1653 413 y Fn(1)1692 399 y Fz(\()p Fl(n)h Fm(\000)g Fz(1)p Fl(;)15 b(s=)p Fz(2\))8 b(+)g Fl(T)2262 413 y Fn(1)2302 399 y Fz(\()p Fl(n)g Fm(\000)g Fz(1)p Fl(;)15 b(s)p Fz(\))24 b(whenev)m(er)246 506 y Fl(n)46 b Fm(\025)h Fz(1)c(and)g Fl(s)k(>)f Fz(1;)e Fl(T)1117 520 y Fn(1)1157 506 y Fz(\(0)p Fl(;)15 b(s)p Fz(\))48 b(=)e(1)e(and)f Fl(T)1852 520 y Fn(1)1891 506 y Fz(\()p Fl(n;)15 b Fz(1\))48 b(=)f(time\()p Fl(O)s Fz(\()p Fl(m)p Fz(\)\),)d(whic)m(h)246 614 y(is)c(the)h(time)f(to)i(compute)f(the)g(v) -5 b(alue)40 b(of)h(a)g(CNF)g(form)m(ula)f(with)g Fl(m)g Fz(clauses)246 722 y(under)g(a)i(giv)m(en)g(truth)f(assignmen)m(t.)h (It)g(can)g(b)s(e)f(sho)m(wn)g(b)m(y)h(induction)e(that)246 830 y Fl(T)299 844 y Fn(1)338 830 y Fz(\()p Fl(n;)15 b(s)p Fz(\))26 b Fm(\024)f Fl(O)s Fz(\()p Fl(m)p Fz(\))p Fl(n)945 797 y Fk(O)r Fn(\(log)13 b Fk(s)p Fn(\))1195 830 y Fz(.)p 2958 830 48 48 v 362 1028 a(F)-8 b(or)37 b Fr(Sa)-6 b(t)36 b Fz(and)g Fr(Majsa)-6 b(t)34 b Fz(form)m(ulae,)j(an) m(y)g(decision)e(tree)i(is)f(a)h(DPLL)f(tree,)246 1136 y(but)30 b(for)h(more)h(complicated)f(Extended)f Fr(SSa)-6 b(t)29 b Fz(instances,)i(the)h(DPLL)f(tree)h(is)246 1244 y(more)27 b(restricted)h(since)f(it)g(m)m(ust)g(split)f(on)i(all)e(the) i(v)-5 b(ariables)26 b(in)h(the)g(outermost)246 1352 y(blo)s(c)m(k)20 b(b)s(efore)g(splitting)e(on)i(an)m(y)h(v)-5 b(ariables)19 b(that)i(app)s(ear)f(later)g(in)g(the)g(quan)m(ti\014er) 246 1460 y(ordering)g(\(with)g(the)i(exception)g(of)g(unit)e (resolution)g(and)h(puri\014cation\).)f(In)g(par-)246 1568 y(ticular,)26 b(it)h(migh)m(t)g(b)s(e)g(that)h(for)f(a)g(giv)m(en) h(form)m(ula)e Fl(\036)p Fz(,)i(there)g(is)e(a)i(small)e(decision)246 1676 y(tree)35 b(for)f Fl(\036)h Fz(but)f(not)h(a)g(small)e(DPLL)i (tree)g(for)g Fl(\036)p Fz(,)g(b)s(ecause)f(of)h(the)g(quan)m(ti\014er) 246 1784 y(ordering.)i(Unfortunately)-8 b(,)38 b(the)h(follo)m(wing)d (theorem)j(due)f(to)h(Umans)f(\(1999\))246 1892 y(sho)m(ws)h(that)h(an) f(arbitrary)f(decision)g(tree)i(cannot)h(b)s(e)d(used)h(to)h(ev)-5 b(aluate)40 b(an)246 2000 y(Extended)29 b Fr(SSa)-6 b(t)29 b Fz(form)m(ula)h(e\016cien)m(tly)-8 b(.)246 2228 y(THEOREM)30 b(3.)h(\(Umans)f(\(1999\)\).)54 b Fj(L)-5 b(et)23 b Fr(QBF)p Fz(\()p Fl(i)p Fz(\))h Fj(denote)g(the)g(c)-5 b(omplexity)25 b(class)246 2336 y(of)37 b(de)-5 b(ciding)38 b(an)f(Extende)-5 b(d)34 b Fr(SSa)-6 b(t)35 b Fj(formula)k(with)f(only)g Fm(9)e Fj(and)i Fm(8)f Fj(quanti\014ers,)246 2444 y(and)d(with)g Fl(i)f Fj(quanti\014er)h(alternations)h(\(blo)-5 b(cks\).)34 b(L)-5 b(et)33 b Fr(QBF)p Fj(-tr)-5 b(e)g(e)p Fz(\()p Fl(i)p Fz(\))35 b Fj(b)-5 b(e)33 b(the)g(fol-)246 2552 y(lowing)25 b(pr)-5 b(oblem.)25 b(The)f(input)h(is)f(a)g(p)-5 b(air)26 b Fz(\()p Fl(\036;)15 b(T)e Fz(\))p Fj(,)24 b(wher)-5 b(e)26 b Fl(\036)e Fj(is)g(a)g Fr(QBF)p Fz(\()p Fl(i)p Fz(\))h Fj(formula)246 2660 y(and)37 b(wher)-5 b(e)37 b Fl(T)49 b Fj(is)36 b(a)g(de)-5 b(cision)37 b(tr)-5 b(e)g(e)37 b(that)h(c)-5 b(aptur)g(es)38 b(the)e(CNF)g(p)-5 b(art)38 b(of)e(the)h(for-)246 2768 y(mula)h Fl(\036)f Fj(\(ignoring)g(quanti\014ers\).)h(Then,)f Fr(QBF)p Fj(-tr)-5 b(e)g(e)p Fz(\()p Fl(i)p Fz(\))38 b Fj(is)g(in)f Fz(P)f Fj(if)h(and)h(only)246 2876 y(if)32 b Fr(QBF)p Fz(\()p Fl(i)20 b Fm(\000)g Fz(1\))26 b(=)f(P)p Fj(.)362 3104 y Fz(In)34 b(other)h(w)m(ords,)g(the)g(ab)s(o)m(v)m(e)i(theorem)e (tells)f(us)h(that)g(it)g(is)f(imp)s(ossible)d(to)246 3212 y(determine)f(in)f(p)s(olynomial)g(time)h(the)h(v)-5 b(alue)31 b(of)g(an)g(Extended)f Fr(SSa)-6 b(t)29 b Fz(form)m(ula)246 3320 y(giv)m(en)23 b(an)h Fj(arbitr)-5 b(ary)34 b Fz(p)s (olynomial-size)21 b(decision)h(tree)j(for)e(the)h(CNF)g(part)f(of)h (the)246 3428 y(form)m(ula,)f(unless)f(a)i(surprising)c(collapse)j(of)h (complexit)m(y)g(classes)f(o)s(ccurs.)h(Th)m(us,)246 3536 y(the)39 b(restriction)g(to)h(searc)m(hing)f(for)g(DPLL)g(trees,)i (whic)m(h)d(resp)s(ect)h(quan)m(ti\014er)246 3644 y(order,)h(is)g (necessary)-8 b(,)42 b(in)d(a)i(sense.)g(Nonetheless,)h(there)f(are)g (somewhat)g(less)246 3752 y(restricted)30 b(decision)f(trees)i(than)f (DPLL)g(trees)h(that)g(could)f(b)s(e)g(used)f(to)i(ev)-5 b(alu-)246 3859 y(ate)30 b(an)f(Extended)g Fr(SSa)-6 b(t)28 b Fz(form)m(ula.)g(F)-8 b(or)30 b(instance,)f(v)-5 b(ariables)27 b(can)i(b)s(e)g(skipp)s(ed)246 3967 y(altogether)j(and)f (still)f(the)h(v)-5 b(alues)31 b(of)h(the)f(in)m(ternal)g(no)s(des)f (can)i(b)s(e)f(calculated,)246 4075 y(as)43 b(long)f(as)h(the)g(v)-5 b(ariables)42 b(that)h(are)h(men)m(tioned)e(are)h(consisten)m(t)g(with) f(the)246 4183 y(quan)m(ti\014cation)34 b(ordering)f(\(again,)j(with)d (the)j(exception)f(of)g(unit)e(resolution\).)246 4291 y(Ho)m(w)m(ev)m(er,)j(at)f(this)e(time,)h(there)h(is)e(no)h(metho)s(d)g (kno)m(wn)g(to)h(e\016cien)m(tly)f(searc)m(h)246 4399 y(for)c(these)h(more)f(general)h(t)m(yp)s(es)f(of)g(trees.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.20)p eop %%Page: 21 21 21 20 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(21)246 291 y(2.3.)65 b Fr(A)34 b(Stochastic)e(Sampling)i(Algorithm)246 483 y Fz(Randomized)c(lo)s(cal)h(searc)m(h)h(algorithms,)f(lik)m(e)g Fr(w)-8 b(alksa)i(t)30 b Fz(\(Kautz)i(&)f(Selman,)246 591 y(1996\),)46 b(ha)m(v)m(e)f(b)s(een)f(extremely)g(successful)f(in)f (solving)h(di\016cult)f(satis\014abil-)246 699 y(it)m(y)g(problems.)g Fr(w)-8 b(alksa)i(t)40 b Fz(initially)f(mak)m(es)44 b(a)f(random)f (assignmen)m(t)h(to)g(the)246 807 y(v)-5 b(ariables)32 b(in)h(the)i Fr(Sa)-6 b(t)33 b Fz(form)m(ula.)g(If)h(this)f(is)h(not)g (a)g(satisfying)f(assignmen)m(t,)i(it)246 915 y(randomly)e(selects)j (an)g(unsatis\014ed)d(clause.)j(If)e(it)h(can)h(satisfy)f(this)f (clause)h(b)m(y)246 1023 y(\015ipping)e(a)j(v)-5 b(ariable)35 b(without)h(unsatisfying)e(an)m(y)i(other)h(clauses,)f(it)g(do)s(es)g (so.)246 1131 y(If)28 b(not,)i(with)e(probabilit)m(y)f(1/2,)k(it)e (randomly)e(\015ips)h(that)h(v)-5 b(ariable)29 b(that)g(w)m(ould)246 1239 y(unsatisfy)35 b(the)i(few)m(est)g(clauses.)g(This)e(pro)s(cess)h (con)m(tin)m(ues)h(un)m(til)d(a)j(satisfying)246 1347 y(assignmen)m(t)30 b(is)f(found)h(or)g(un)m(til)f(a)i(sp)s(eci\014ed)d (maxim)m(um)i(n)m(um)m(b)s(er)f(of)h(v)-5 b(ariables)246 1455 y(ha)m(v)m(e)42 b(b)s(een)e(\015ipp)s(ed.)e(This)h(en)m(tire)i (pro)s(cess)f(can)h(also)g(b)s(e)f(rep)s(eated)h(a)g(sp)s(eci-)246 1562 y(\014ed)30 b(n)m(um)m(b)s(er)f(of)h(times)g(starting)h(with)e(a)i (new)f(random)f(assignmen)m(t.)i(Clearly)-8 b(,)246 1670 y Fr(w)g(alksa)i(t)22 b Fz(is)i(not)h(complete;)g(it)f(ma)m(y)i(not)e (\014nd)f(a)i(satisfying)f(assignmen)m(t)g(when)246 1778 y(one)j(exists.)g(F)-8 b(urther,)27 b(it)g(cannot)h(determine)e(when)g (no)h(satisfying)f(assignmen)m(t)246 1886 y(exists.)k(But,)i(compared)f (to)g(existing)f(systematic)h(solv)m(ers,)g(randomized)f(lo)s(cal)246 1994 y(searc)m(h)46 b(algorithms)e(can)i(solv)m(e)g(random)f (satis\014abilit)m(y)e(problems)h(that)i(are)246 2102 y(orders)29 b(of)i(magnitude)e(larger)i(\(Selman,)e(Kautz,)i(&)f (Cohen,)g(1996\).)362 2210 y(The)39 b(presence)h(of)g(randomly)e(quan)m (ti\014ed)h(v)-5 b(ariables)38 b(in)g Fr(SSa)-6 b(t)39 b Fz(problems,)246 2318 y(ho)m(w)m(ev)m(er,)33 b(mak)m(es)f(it)f(imp)s (ossible)e(to)j(apply)e(lo)s(cal)h(searc)m(h)h(tec)m(hniques)f (directly)246 2426 y(to)41 b Fr(SSa)-6 b(t)39 b Fz(problems.)h(A)h(p)s (ossible)d(approac)m(h)j(is)f(suggested)h(b)m(y)g(an)g(appro)m(xi-)246 2534 y(mation)d(tec)m(hnique)h(for)g Fr(Majsa)-6 b(t)37 b Fz(problems:)g(sto)s(c)m(hastic)j(sampling.)d(In)h(this)246 2642 y(approac)m(h,)k(some)h(n)m(um)m(b)s(er)e(of)h(assignmen)m(ts)g (to)h(the)f(random)g(v)-5 b(ariables)41 b(are)246 2750 y(generated)22 b(according)f(to)i(their)d(probabilities,)e(and)j(then)g (the)h(probabilit)m(y)d(that)246 2858 y(the)36 b(form)m(ula)g(is)f (satis\014able)h(is)f(estimated)i(from)f(that)h(sample.)e(This)g (section)246 2966 y(dev)m(elops)23 b(an)h(appro)m(ximation)e(algorithm) h(for)g Fr(SSa)-6 b(t)22 b Fz(problems)g(that)j(com)m(bines)246 3073 y(sto)s(c)m(hastic)38 b(sampling)d(to)j(reduce)e(the)i(size)f(of)g (the)g(problem)f(\(Section)h(2.3.2\))246 3181 y(and)g(randomized)g(lo)s (cal)h(searc)m(h)h(to)f(solv)m(e)h(the)f(reduced)g(problem)e(e\016cien) m(tly)246 3289 y(\(Section)30 b(2.3.3\).)246 3504 y(2.3.1.)65 b Fj(Policy)33 b(T)-7 b(r)i(e)g(es)246 3611 y Fz(The)36 b(starting)h(p)s(oin)m(t)f(for)h(the)g(sto)s(c)m(hastic)h(sampling)d (algorithm)h(for)h Fr(SSa)-6 b(t)35 b Fz(is)246 3719 y(the)24 b(p)s(olicy-tree)f(represen)m(tation.)h(Figure)f(3)h(pro)m (vides)f(an)g(example)g(p)s(olicy)f(tree)246 3827 y(for)30 b(the)g Fr(SSa)-6 b(t)29 b Fz(instance)246 4014 y Fm(9)p Fl(x)349 4028 y Fn(1)388 4014 y Fl(;)15 b Fm(9)p Fl(x)531 4028 y Fn(2)570 4014 y Fl(;)669 3951 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 669 3951 a Fi(R)728 3951 y currentpoint grestore moveto 728 3951 a 669 4014 a Fl(y)714 4028 y Fn(1)753 4014 y Fl(;)852 3951 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 852 3951 a Fi(R)911 3951 y currentpoint grestore moveto 911 3951 a 852 4014 a Fl(y)897 4028 y Fn(2)936 4014 y Fl(;)g Fm(9)p Fl(x)1079 4028 y Fn(3)1118 4014 y Fl(;)g Fm(9)p Fl(x)1261 4028 y Fn(4)1301 4014 y Fl(;)1400 3951 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1400 3951 a Fi(R)1459 3951 y currentpoint grestore moveto 1459 3951 a 1400 4014 a Fl(y)1445 4028 y Fn(3)1484 4014 y Fl(;)1583 3951 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1583 3951 a Fi(R)1642 3951 y currentpoint grestore moveto 1642 3951 a 1583 4014 a Fl(y)1628 4028 y Fn(4)1667 4014 y Fl(;)g(E)5 b Fz([\()p 1839 3964 92 4 v Fl(x)1891 4028 y Fn(1)1932 4014 y Fm(_)p Fl(x)2045 4028 y Fn(3)2084 4014 y Fm(_)p 2145 3964 85 4 v Fl(y)2190 4028 y Fn(3)2228 4014 y Fz(\)\()p 2298 3964 92 4 v Fl(x)2350 4028 y Fn(2)2390 4014 y Fm(_)p 2451 3964 V Fl(x)2503 4028 y Fn(4)2542 4014 y Fm(_)p Fl(y)2648 4028 y Fn(2)2687 4014 y Fz(\)\()p Fl(x)2809 4028 y Fn(3)2849 4014 y Fm(_)p Fl(y)2955 4028 y Fn(1)2994 4014 y Fm(_)p 3055 3964 85 4 v Fl(y)3100 4028 y Fn(4)3138 4014 y Fz(\)])26 b Fm(\025)f Fl(\022)s(:)362 4201 y Fz(Eac)m(h)31 b(existen)m(tial)g(v)-5 b(ariable)29 b(can)i(tak)m(e)i(on)e(a)g(di\013eren)m(t)f(Bo)s(olean)h (v)-5 b(alue)31 b(as)g(a)246 4309 y(function)40 b(of)i(the)g(assignmen) m(t)g(to)g(the)g(randomized)f(v)-5 b(ariables)40 b(that)j(app)s(ear)246 4417 y(to)k(its)f(left)h(in)e(the)i(quan)m(ti\014er)f(ordering.)g(The)g Fj(p)-5 b(olicy-tr)g(e)g(e)55 b Fz(represen)m(tation)246 4525 y(mak)m(es)47 b(this)f(eviden)m(t)g(b)m(y)g(including)e(a)j(cop)m (y)g(of)g(the)f(v)-5 b(ariable)46 b(for)g(eac)m(h)i(of)246 4633 y(these)26 b(assignmen)m(ts.)h(Existen)m(tial)e(v)-5 b(ariables,)25 b(and)h(their)f(copies,)i(can)f(b)s(e)g(called)246 4741 y Fj(de)-5 b(cision)31 b(variables)e Fz(and)f(are)h(represen)m (ted)f(as)h(rectangular)f Fj(de)-5 b(cision)31 b(no)-5 b(des)37 b Fz(in)246 4848 y(the)c(p)s(olicy)d(tree.)k(If)e Fl(x)1029 4862 y Fk(i)1090 4848 y Fz(is)g(an)g(existen)m(tial)g(v)-5 b(ariable)32 b(suc)m(h)g(that)h Fl(r)i Fz(randomized)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.21)p eop %%Page: 22 22 22 21 bop 246 100 a Fz(22)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)440 2487 y @beginspecial 37 @llx 140 @lly 575 @urx 651 @ury 2880 @rwi @setspecial %%BeginDocument: ftree1.ps % % Frame ps_prolog 5.5, for use with Adobe Unix Frame 5.5 products % % This ps_prolog file is Copyright (c) 1986-1996 Adobe Systems, Incoporated. % All rights reserved. This ps_prolog file may be freely copied and % distributed in conjunction with documents created using FrameMaker, % FrameMaker+SGML, FrameReader, and FrameViewer as long as this % copyright notice is preserved. /FMDocSave save def % % FrameMaker users specify the proper paper size for each print job in the % "Print" dialog's "Printer Paper Size" "Width" and "Height~ fields. If the % printer that the PS file is sent to does not support the requested paper % size, or if there is no paper tray of the proper size currently installed, % then the job will not be printed. The following flag, if set to true, will % cause the job to print on the default paper in such cases. /FMAllowPaperSizeMismatch false def % % Frame products normally print colors as their true color on a color printer % or as shades of gray, based on luminance, on a black-and white printer. The % following flag, if set to true, forces all non-white colors to print as pure % black. This has no effect on bitmap images. /FMPrintAllColorsAsBlack false def % % Frame products can either set their own line screens or use a printer's % default settings. Three flags below control this separately for no % separations, spot separations and process separations. If a flag % is true, then the default printer settings will not be changed. If it is % false, Frame products will use their own settings from a table based on % the printer's resolution. /FMUseDefaultNoSeparationScreen true def /FMUseDefaultSpotSeparationScreen true def /FMUseDefaultProcessSeparationScreen false def % % For any given PostScript printer resolution, Frame products have two sets of % screen angles and frequencies for printing process separations, which are % recomended by Adobe. The following variable chooses the higher frequencies % when set to true or the lower frequencies when set to false. This is only % effective if the appropriate FMUseDefault...SeparationScreen flag is false. /FMUseHighFrequencyScreens true def % % The following is a set of predefined optimal frequencies and angles for various % common dpi settings. This is taken from "Advances in Color Separation Using % PostScript Software Technology," from Adobe Systems (3/13/89 P.N. LPS 0043) % and corrolated with information which is in various PPD (4.0) files. % % The "dpiranges" figure is the minimum dots per inch device resolution which % can support this setting. The "low" and "high" values are controlled by the % setting of the FMUseHighFrequencyScreens flag above. The "TDot" flags control % the use of the "Yellow Triple Dot" feature whereby the frequency id divided by % three, but the dot function is "trippled" giving a block of 3x3 dots per cell. % % PatFreq is a compromise pattern frequency for ps Level 2 printers which is close % to the ideal WYSIWYG pattern frequency of 9 repetitions/inch but does not beat % (too badly) against the screen frequencies of any separations for that DPI. /dpiranges [ 2540 2400 1693 1270 1200 635 600 0 ] def /CMLowFreqs [ 100.402 94.8683 89.2289 100.402 94.8683 66.9349 63.2456 47.4342 ] def /YLowFreqs [ 95.25 90.0 84.65 95.25 90.0 70.5556 66.6667 50.0 ] def /KLowFreqs [ 89.8026 84.8528 79.8088 89.8026 84.8528 74.8355 70.7107 53.033 ] def /CLowAngles [ 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 ] def /MLowAngles [ 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 ] def /YLowTDot [ true true false true true false false false ] def /CMHighFreqs [ 133.87 126.491 133.843 108.503 102.523 100.402 94.8683 63.2456 ] def /YHighFreqs [ 127.0 120.0 126.975 115.455 109.091 95.25 90.0 60.0 ] def /KHighFreqs [ 119.737 113.137 119.713 128.289 121.218 89.8026 84.8528 63.6395 ] def /CHighAngles [ 71.5651 71.5651 71.5651 70.0169 70.0169 71.5651 71.5651 71.5651 ] def /MHighAngles [ 18.4349 18.4349 18.4349 19.9831 19.9831 18.4349 18.4349 18.4349 ] def /YHighTDot [ false false true false false true true false ] def /PatFreq [ 10.5833 10.0 9.4055 10.5833 10.0 10.5833 10.0 9.375 ] def % % PostScript Level 2 printers contain an "Accurate Screens" feature which can % improve process separation rendering at the expense of compute time. This % flag is ignored by PostScript Level 1 printers. /FMUseAcccurateScreens true def % % The following PostScript procedure defines the spot function that Frame % products will use for process separations. You may un-comment-out one of % the alternative functions below, or use your own. % % Dot function /FMSpotFunction {abs exch abs 2 copy add 1 gt {1 sub dup mul exch 1 sub dup mul add 1 sub } {dup mul exch dup mul add 1 exch sub }ifelse } def % % Line function % /FMSpotFunction { pop } def % % Elipse function % /FMSpotFunction { dup 5 mul 8 div mul exch dup mul exch add % sqrt 1 exch sub } def % % /FMversion (5.5) def /fMLevel1 /languagelevel where {pop languagelevel} {1} ifelse 2 lt def /FMPColor fMLevel1 { false /colorimage where {pop pop true} if } { true } ifelse def /FrameDict 400 dict def systemdict /errordict known not {/errordict 10 dict def errordict /rangecheck {stop} put} if % The readline in PS 23.0 doesn't recognize cr's as nl's on AppleTalk FrameDict /tmprangecheck errordict /rangecheck get put errordict /rangecheck {FrameDict /bug true put} put FrameDict /bug false put mark % Some PS machines read past the CR, so keep the following 3 lines together! currentfile 5 string readline 00 0000000000 cleartomark errordict /rangecheck FrameDict /tmprangecheck get put FrameDict /bug get { /readline { /gstring exch def /gfile exch def /gindex 0 def { gfile read pop dup 10 eq {exit} if dup 13 eq {exit} if gstring exch gindex exch put /gindex gindex 1 add def } loop pop gstring 0 gindex getinterval true } bind def } if /FMshowpage /showpage load def /FMquit /quit load def /FMFAILURE { 2 copy exch = = flush FMshowpage /Helvetica findfont 12 scalefont setfont 72 200 moveto show 72 220 moveto show FMshowpage FMquit } def /FMVERSION { FMversion ne { (Adobe Frame product version does not match ps_prolog! Check installation;) (also check ~/fminit and ./fminit for old versions) FMFAILURE } if } def /fmConcatProcs { /proc2 exch cvlit def/proc1 exch cvlit def/newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }def FrameDict begin [ /ALDsave /FMdicttop /FMoptop /FMpointsize /FMsetsize /FMsaveobject /b /bitmapsave /blut /bpside /bs /bstring /bwidth /c /cf /cs /cynu /depth /edown /fh /fillvals /fw /fx /fy /g /gfile /gindex /grnt /gryt /gstring /height /hh /i /im /indx /is /k /kk /landscape /lb /len /llx /lly /m /magu /manualfeed /n /offbits /onbits /organgle /orgbangle /orgbfreq /orgbproc /orgbxfer /orgfreq /orggangle /orggfreq /orggproc /orggxfer /orghalftone /orgmatrix /orgproc /orgrangle /orgrfreq /orgrproc /orgrxfer /orgxfer /pagesave /paperheight /papersizedict /paperwidth /pos /pwid /r /rad /redt /sl /str /tran /u /urx /ury /val /width /width /ws /ww /x /x1 /x2 /xindex /xpoint /xscale /xx /y /y1 /y2 /yelu /yindex /ypoint /yscale /yy /tintGray ] { 0 def } forall /FmBD {bind def} bind def systemdict /pdfmark known systemdict /currentdistillerparams known and { /fMAcrobat true def /FmPD /pdfmark load def /FmPT /show load def currentdistillerparams /CoreDistVersion get 2000 ge { /FmPD2 /pdfmark load def /FmPA { mark exch /Dest exch 5 3 roll /View [ /XYZ null 6 -2 roll FmDC exch pop null] /DEST FmPD }FmBD } { /FmPD2 /cleartomark load def /FmPA {pop pop pop}FmBD } ifelse } { /fMAcrobat false def /FmPD /cleartomark load def /FmPD2 /cleartomark load def /FmPT /pop load def /FmPA {pop pop pop}FmBD } ifelse /FmDC { transform fMDefaultMatrix defaultmatrix itransform cvi exch cvi exch }FmBD /FmBx { dup 3 index lt {3 1 roll exch} if 1 index 4 index lt {4 -1 roll 3 1 roll exch 4 1 roll} if }FmBD /FMnone 0 def /FMcyan 1 def /FMmagenta 2 def /FMyellow 3 def /FMblack 4 def /FMcustom 5 def /fMNegative false def /FrameSepIs FMnone def /FrameSepBlack 0 def /FrameSepYellow 0 def /FrameSepMagenta 0 def /FrameSepCyan 0 def /FrameSepRed 1 def /FrameSepGreen 1 def /FrameSepBlue 1 def /FrameCurGray 1 def /FrameCurPat null def /FrameCurColors [ 0 0 0 1 0 0 0 1] def /FrameColorEpsilon .001 def /eqepsilon { sub dup 0 lt {neg} if FrameColorEpsilon le } bind def /FrameCmpColorsCMYK { 2 copy 0 get exch 0 get eqepsilon { 2 copy 1 get exch 1 get eqepsilon { 2 copy 2 get exch 2 get eqepsilon { 3 get exch 3 get eqepsilon } {pop pop false} ifelse }{pop pop false} ifelse } {pop pop false} ifelse } bind def /FrameCmpColorsRGB { 2 copy 4 get exch 0 get eqepsilon { 2 copy 5 get exch 1 get eqepsilon { 6 get exch 2 get eqepsilon }{pop pop false} ifelse } {pop pop false} ifelse } bind def /RGBtoCMYK { 1 exch sub 3 1 roll 1 exch sub 3 1 roll 1 exch sub 3 1 roll 3 copy 2 copy le { pop } { exch pop } ifelse 2 copy le { pop } { exch pop } ifelse dup dup dup 6 1 roll 4 1 roll 7 1 roll sub 6 1 roll sub 5 1 roll sub 4 1 roll } bind def /CMYKtoRGB { dup dup 4 -1 roll add 5 1 roll 3 -1 roll add 4 1 roll add 1 exch sub dup 0 lt {pop 0} if 3 1 roll 1 exch sub dup 0 lt {pop 0} if exch 1 exch sub dup 0 lt {pop 0} if exch } bind def /FrameSepInit { 1.0 RealSetgray } bind def /FrameSetSepColor { /FrameSepBlue exch def /FrameSepGreen exch def /FrameSepRed exch def /FrameSepBlack exch def /FrameSepYellow exch def /FrameSepMagenta exch def /FrameSepCyan exch def /FrameSepIs FMcustom def setCurrentScreen } bind def /FrameSetCyan { /FrameSepBlue 1.0 def /FrameSepGreen 1.0 def /FrameSepRed 0.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 1.0 def /FrameSepIs FMcyan def setCurrentScreen } bind def /FrameSetMagenta { /FrameSepBlue 1.0 def /FrameSepGreen 0.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 1.0 def /FrameSepCyan 0.0 def /FrameSepIs FMmagenta def setCurrentScreen } bind def /FrameSetYellow { /FrameSepBlue 0.0 def /FrameSepGreen 1.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 1.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMyellow def setCurrentScreen } bind def /FrameSetBlack { /FrameSepBlue 0.0 def /FrameSepGreen 0.0 def /FrameSepRed 0.0 def /FrameSepBlack 1.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMblack def setCurrentScreen } bind def /FrameNoSep { /FrameSepIs FMnone def setCurrentScreen } bind def /FrameSetSepColors { FrameDict begin [ exch 1 add 1 roll ] /FrameSepColors exch def end } bind def /FrameColorInSepListCMYK { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsCMYK { pop true exit } if } forall dup true ne {pop false} if } bind def /FrameColorInSepListRGB { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsRGB { pop true exit } if } forall dup true ne {pop false} if } bind def /RealSetgray /setgray load def /RealSetrgbcolor /setrgbcolor load def /RealSethsbcolor /sethsbcolor load def end /setgray { FrameDict begin FrameSepIs FMnone eq { RealSetgray } { FrameSepIs FMblack eq { RealSetgray } { FrameSepIs FMcustom eq FrameSepRed 0 eq and FrameSepGreen 0 eq and FrameSepBlue 0 eq and { RealSetgray } { 1 RealSetgray pop } ifelse } ifelse } ifelse end } bind def /setrgbcolor { FrameDict begin FrameSepIs FMnone eq { RealSetrgbcolor } { 3 copy [ 4 1 roll ] FrameColorInSepListRGB { FrameSepBlue eq exch FrameSepGreen eq and exch FrameSepRed eq and { 0 } { 1 } ifelse } { FMPColor { RealSetrgbcolor currentcmykcolor } { RGBtoCMYK } ifelse FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind def /sethsbcolor { FrameDict begin FrameSepIs FMnone eq { RealSethsbcolor } { RealSethsbcolor currentrgbcolor setrgbcolor } ifelse end } bind def FrameDict begin /setcmykcolor where { pop /RealSetcmykcolor /setcmykcolor load def } { /RealSetcmykcolor { 4 1 roll 3 { 3 index add 0 max 1 min 1 exch sub 3 1 roll} repeat RealSetrgbcolor pop } bind def } ifelse userdict /setcmykcolor { FrameDict begin FrameSepIs FMnone eq { RealSetcmykcolor } { 4 copy [ 5 1 roll ] FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and { 0 } { 1 } ifelse } { FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind put fMLevel1 { /patScreenDict 7 dict dup begin <0f1e3c78f0e1c387> [ 45 { pop } {exch pop} .5 2 sqrt] FmBD <0f87c3e1f0783c1e> [ 135 { pop } {exch pop} .5 2 sqrt] FmBD [ 0 { pop } dup .5 2 ] FmBD [ 90 { pop } dup .5 2 ] FmBD <8142241818244281> [ 45 { 2 copy lt {exch} if pop} dup .75 2 sqrt] FmBD <03060c183060c081> [ 45 { pop } {exch pop} .875 2 sqrt] FmBD <8040201008040201> [ 135 { pop } {exch pop} .875 2 sqrt] FmBD end def } { /patProcDict 5 dict dup begin <0f1e3c78f0e1c387> { 3 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <0f87c3e1f0783c1e> { 3 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def <8142241818244281> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -1 -1 moveto 9 9 lineto stroke } bind def <03060c183060c081> { 1 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <8040201008040201> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def end def /patDict 15 dict dup begin /PatternType 1 def /PaintType 2 def /TilingType 3 def /BBox [ 0 0 8 8 ] def /XStep 8 def /YStep 8 def /PaintProc { begin patProcDict bstring known { patProcDict bstring get exec } { 8 8 true [1 0 0 -1 0 8] bstring imagemask } ifelse end } bind def end def } ifelse /tintCMYK { 1 tintGray sub FrameCurColors 0 4 getinterval aload pop 4 index mul 5 1 roll 3 index mul 5 1 roll 2 index mul 5 1 roll mul 4 1 roll }bind def /tintRGB { 1 tintGray sub FrameCurColors 4 3 getinterval aload pop 1 exch sub 3 index mul 1 exch sub 4 1 roll 1 exch sub 2 index mul 1 exch sub 4 1 roll 1 exch sub mul 1 exch sub 3 1 roll }bind def /combineColor { /tintGray 1 1 FrameCurGray sub FrameCurColors 7 get mul sub def FrameSepIs FMnone eq { graymode fMLevel1 or not { [/Pattern [/DeviceCMYK]] setcolorspace tintCMYK FrameCurPat setcolor } { FrameCurColors 3 get 1.0 ge { tintGray RealSetgray } { fMAcrobat not FMPColor graymode and and { tintCMYK RealSetcmykcolor } { tintRGB RealSetrgbcolor } ifelse } ifelse } ifelse } { FrameCurColors 0 4 getinterval aload FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and FrameSepIs FMcustom eq and { tintGray } { 1 } ifelse } { FrameSepIs FMblack eq {tintGray 1.0 exch sub mul 1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop tintGray 1.0 exch sub mul 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop tintGray 1.0 exch sub mul 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop tintGray 1.0 exch sub mul 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse graymode fMLevel1 or not { [/Pattern [/DeviceGray]] setcolorspace FrameCurPat setcolor } { graymode not fMLevel1 and { dup 1 lt {pop FrameCurGray} if } if RealSetgray } ifelse } ifelse } bind def /savematrix { orgmatrix currentmatrix pop } bind def /restorematrix { orgmatrix setmatrix } bind def /fMDefaultMatrix matrix def /fMatrix2 matrix def /dpi 72 0 fMDefaultMatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /freq dpi dup 72 div round dup 0 eq {pop 1} if 8 mul div def /sangle 1 0 fMDefaultMatrix defaultmatrix dtransform exch atan def sangle fMatrix2 rotate fMDefaultMatrix defaultmatrix fMatrix2 concatmatrix dup 0 get /sflipx exch def 3 get /sflipy exch def /screenIndex { 0 1 dpiranges length 1 sub { dup dpiranges exch get 1 sub dpi le {exit} {pop} ifelse } for } bind def /getCyanScreen { FMUseHighFrequencyScreens { CHighAngles CMHighFreqs} {CLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getMagentaScreen { FMUseHighFrequencyScreens { MHighAngles CMHighFreqs } {MLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getYellowScreen { FMUseHighFrequencyScreens { YHighTDot YHighFreqs} { YLowTDot YLowFreqs } ifelse screenIndex dup 3 1 roll get 3 1 roll get { 3 div {2 { 1 add 2 div 3 mul dup floor sub 2 mul 1 sub exch} repeat FMSpotFunction } } {/FMSpotFunction load } ifelse 0.0 exch } bind def /getBlackScreen { FMUseHighFrequencyScreens { KHighFreqs } { KLowFreqs } ifelse screenIndex get 45.0 /FMSpotFunction load } bind def /getSpotScreen { getBlackScreen } bind def /getCompositeScreen { getBlackScreen } bind def /FMSetScreen fMLevel1 { /setscreen load }{ { 8 dict begin /HalftoneType 1 def /SpotFunction exch def /Angle exch def /Frequency exch def /AccurateScreens FMUseAcccurateScreens def currentdict end sethalftone } bind } ifelse def /setDefaultScreen { fMLevel1 { FMPColor { orgrxfer cvx orggxfer cvx orgbxfer cvx orgxfer cvx setcolortransfer } { orgxfer cvx settransfer } ifelse orgfreq organgle orgproc cvx setscreen } { orghalftone sethalftone }ifelse } bind def /setCurrentScreen { FrameSepIs FMnone eq { FMUseDefaultNoSeparationScreen { setDefaultScreen } { getCompositeScreen FMSetScreen } ifelse } { FrameSepIs FMcustom eq { FMUseDefaultSpotSeparationScreen { setDefaultScreen } { getSpotScreen FMSetScreen } ifelse } { FMUseDefaultProcessSeparationScreen { setDefaultScreen } { FrameSepIs FMcyan eq { getCyanScreen FMSetScreen } { FrameSepIs FMmagenta eq { getMagentaScreen FMSetScreen } { FrameSepIs FMyellow eq { getYellowScreen FMSetScreen } { getBlackScreen FMSetScreen } ifelse } ifelse } ifelse } ifelse } ifelse } ifelse } bind def end /FMDOCUMENT { array /FMfonts exch def dup 1 gt {/#copies exch def} {pop} ifelse FrameDict begin 0 ne /manualfeed exch def /paperheight exch def /paperwidth exch def 0 ne /fMNegative exch def 0 ne /edown exch def /yscale exch def /xscale exch def fMLevel1 not { /orghalftone currenthalftone def }if FMPColor { currentcolorscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def cvlit /orgbproc exch def /orgbangle exch def /orgbfreq exch def cvlit /orggproc exch def /orggangle exch def /orggfreq exch def cvlit /orgrproc exch def /orgrangle exch def /orgrfreq exch def currentcolortransfer fMNegative { 1 1 4 { pop { 1 exch sub } fmConcatProcs 4 1 roll } for 4 copy setcolortransfer } if cvlit /orgxfer exch def cvlit /orgbxfer exch def cvlit /orggxfer exch def cvlit /orgrxfer exch def } { currentscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def currenttransfer fMNegative { { 1 exch sub } fmConcatProcs dup settransfer } if cvlit /orgxfer exch def } ifelse end } def /FMENDDOCUMENT { FMDocSave restore } def /FMBEGINPAGE { FrameDict begin /pagesave save def 3.86 setmiterlimit /landscape exch 0 ne def landscape { 90 rotate 0 exch dup /pwid exch def neg translate pop }{ pop /pwid exch def } ifelse edown { [-1 0 0 1 pwid 0] concat } if xscale yscale scale /orgmatrix matrix def gsave } def /FMENDPAGE { grestore pagesave restore end showpage } def /FMFONTDEFINE { FrameDict begin findfont ReEncode 1 index exch definefont FMfonts 3 1 roll put end } def /FMFILLS { FrameDict begin dup array /fillvals exch def dict /patCache exch def end } def /FMFILL { FrameDict begin fillvals 3 1 roll put end } def /FMNORMALIZEGRAPHICS { newpath 1 setlinewidth 0 setlinecap 0 0 0 sethsbcolor 0 setgray } bind def /FMBEGINEPSF { end /FMEPSF save def /showpage {} def FMNORMALIZEGRAPHICS [/fy /fx /fh /fw /ury /urx /lly /llx] {exch def} forall fx fw 2 div add fy fh 2 div add translate rotate fw 2 div neg fh 2 div neg translate fw urx llx sub div fh ury lly sub div scale llx neg lly neg translate /FMdicttop countdictstack 1 add def /FMoptop count def } bind def /FMENDEPSF { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMEPSF restore FrameDict begin } bind def FrameDict begin /setmanualfeed { statusdict /manualfeed true put } bind def /max {2 copy lt {exch} if pop} bind def /min {2 copy gt {exch} if pop} bind def /inch {72 mul} def /pagedimen { paperheight sub abs 16 lt exch paperwidth sub abs 16 lt and {/papername exch def} {pop} ifelse } bind def /setpapername { /papersizedict 14 dict def papersizedict begin /papername /unknown def /Letter 8.5 inch 11.0 inch pagedimen /LetterSmall 7.68 inch 10.16 inch pagedimen /Tabloid 11.0 inch 17.0 inch pagedimen /Ledger 17.0 inch 11.0 inch pagedimen /Legal 8.5 inch 14.0 inch pagedimen /Statement 5.5 inch 8.5 inch pagedimen /Executive 7.5 inch 10.0 inch pagedimen /A3 11.69 inch 16.5 inch pagedimen /A4 8.26 inch 11.69 inch pagedimen /A4Small 7.47 inch 10.85 inch pagedimen /B4 10.125 inch 14.33 inch pagedimen /B5 7.16 inch 10.125 inch pagedimen end } bind def /papersize { papersizedict begin /Letter {lettertray letter} def /LetterSmall {lettertray lettersmall} def /Tabloid {11x17tray 11x17} def /Ledger {ledgertray ledger} def /Legal {legaltray legal} def /Statement {statementtray statement} def /Executive {executivetray executive} def /A3 {a3tray a3} def /A4 {a4tray a4} def /A4Small {a4tray a4small} def /B4 {b4tray b4} def /B5 {b5tray b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end statusdict begin stopped end } bind def /manualpapersize { papersizedict begin /Letter {letter} def /LetterSmall {lettersmall} def /Tabloid {11x17} def /Ledger {ledger} def /Legal {legal} def /Statement {statement} def /Executive {executive} def /A3 {a3} def /A4 {a4} def /A4Small {a4small} def /B4 {b4} def /B5 {b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end stopped } bind def /desperatepapersize { mark statusdict begin /setpageparams where { pop paperwidth paperheight 0 1 {setpageparams} stopped } { true } ifelse { /setpagedevice where { pop 1 dict dup begin /PageSize [ paperwidth paperheight ] def end {setpagedevice} stopped } { true } ifelse } { false } ifelse end {cleartomark true}{cleartomark false}ifelse } bind def /papersizefailure { FMAllowPaperSizeMismatch not { (The requested paper size is not available in any currently-installed tray) (Edit the PS file to "FMAllowPaperSizeMismatch true" to use default tray) FMFAILURE } if } def /DiacriticEncoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /grave /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /.notdef /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /.notdef /AE /Oslash /.notdef /.notdef /.notdef /.notdef /yen /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ordfeminine /ordmasculine /.notdef /ae /oslash /questiondown /exclamdown /logicalnot /.notdef /florin /.notdef /.notdef /guillemotleft /guillemotright /ellipsis /.notdef /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /.notdef /.notdef /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /.notdef /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron ] def /ReEncode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall 0 eq {/Encoding DiacriticEncoding def} if currentdict end } bind def FMPColor { /BEGINBITMAPCOLOR { BITMAPCOLOR} def /BEGINBITMAPCOLORc { BITMAPCOLORc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUECOLOR } def /BEGINBITMAPTRUECOLORc { BITMAPTRUECOLORc } def /BEGINBITMAPCMYK { BITMAPCMYK } def /BEGINBITMAPCMYKc { BITMAPCMYKc } def } { /BEGINBITMAPCOLOR { BITMAPGRAY} def /BEGINBITMAPCOLORc { BITMAPGRAYc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUEGRAY } def /BEGINBITMAPTRUECOLORc { BITMAPTRUEGRAYc } def /BEGINBITMAPCMYK { BITMAPCMYKGRAY } def /BEGINBITMAPCMYKc { BITMAPCMYKGRAYc } def } ifelse /K { FMPrintAllColorsAsBlack { 8 1 roll dup 1 eq 2 index 1 eq and 3 index 1 eq and not {7 {pop} repeat 0 0 0 1 0 0 0} if 8 -1 roll } if FrameCurColors astore pop combineColor } bind def /graymode true def fMLevel1 { /fmGetFlip { fMatrix2 exch get mul 0 lt { -1 } { 1 } ifelse } FmBD } if /setPatternMode { fMLevel1 { 2 index patScreenDict exch known { pop pop patScreenDict exch get aload pop freq mul 5 2 roll fMatrix2 currentmatrix 1 get 0 ne { 3 -1 roll 90 add 3 1 roll sflipx 1 fmGetFlip sflipy 2 fmGetFlip neg mul } { sflipx 0 fmGetFlip sflipy 3 fmGetFlip mul } ifelse 0 lt {exch pop} {pop} ifelse fMNegative { {neg} fmConcatProcs } if bind systemdict /setscreen get exec /FrameCurGray exch def } { /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq sangle landscape {90 add} if {/ypoint exch def /xpoint exch def /xindex xpoint 1 add 2 div bpside mul cvi def /yindex ypoint 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get 1 7 xindex 8 mod sub bitshift and 0 ne fMNegative {not} if {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen offbits offbits onbits add dup 0 ne {div} {pop pop .5} ifelse fMNegative {1.0 exch sub} if /FrameCurGray exch def } ifelse } { pop pop dup patCache exch known { patCache exch get } { dup patDict /bstring 3 -1 roll put patDict 9 PatFreq screenIndex get div dup matrix scale makepattern dup patCache 4 -1 roll 3 -1 roll put } ifelse /FrameCurGray 0 def /FrameCurPat exch def } ifelse /graymode false def combineColor } bind def /setGrayScaleMode { graymode not { /graymode true def fMLevel1 { setCurrentScreen } if } if /FrameCurGray exch def combineColor } bind def /normalize { transform round exch round exch itransform } bind def /dnormalize { dtransform round exch round exch idtransform } bind def /lnormalize { 0 dtransform exch cvi 2 idiv 2 mul 1 add exch idtransform pop } bind def /H { lnormalize setlinewidth } bind def /Z { setlinecap } bind def /PFill { graymode fMLevel1 or not { gsave 1 setgray eofill grestore } if } bind def /PStroke { graymode fMLevel1 or not { gsave 1 setgray stroke grestore } if stroke } bind def /X { fillvals exch get dup type /stringtype eq {8 1 setPatternMode} {setGrayScaleMode} ifelse } bind def /V { PFill gsave eofill grestore } bind def /Vclip { clip } bind def /Vstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /N { PStroke } bind def /Nclip { strokepath clip newpath } bind def /Nstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /M {newpath moveto} bind def /E {lineto} bind def /D {curveto} bind def /O {closepath} bind def /L { /n exch def newpath normalize moveto 2 1 n {pop normalize lineto} for } bind def /Y { L closepath } bind def /R { /y2 exch def /x2 exch def /y1 exch def /x1 exch def x1 y1 x2 y1 x2 y2 x1 y2 4 Y } bind def /rarc {rad arcto } bind def /RR { /rad exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def mark newpath { x1 y1 rad add moveto x1 y2 x2 y2 rarc x2 y2 x2 y1 rarc x2 y1 x1 y1 rarc x1 y1 x1 y2 rarc closepath } stopped {x1 y1 x2 y2 R} if cleartomark } bind def /RRR { /rad exch def normalize /y4 exch def /x4 exch def normalize /y3 exch def /x3 exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def newpath normalize moveto mark { x2 y2 x3 y3 rarc x3 y3 x4 y4 rarc x4 y4 x1 y1 rarc x1 y1 x2 y2 rarc closepath } stopped {x1 y1 x2 y2 x3 y3 x4 y4 newpath moveto lineto lineto lineto closepath} if cleartomark } bind def /C { grestore gsave R clip setCurrentScreen } bind def /CP { grestore gsave Y clip setCurrentScreen } bind def /F { FMfonts exch get [FMsetsize 0 0 FMpointsize 0 0] makefont setfont } bind def /Q { /FMpointsize exch def /FMsetsize FMpointsize def F } bind def /QQ { /FMsetsize exch def /FMpointsize exch def F } bind def /T { moveto show } bind def /RF { rotate 0 ne {-1 1 scale} if } bind def /TF { gsave moveto RF show grestore } bind def /P { moveto 0 32 3 2 roll widthshow } bind def /PF { gsave moveto RF 0 32 3 2 roll widthshow grestore } bind def /S { moveto 0 exch ashow } bind def /SF { gsave moveto RF 0 exch ashow grestore } bind def /B { moveto 0 32 4 2 roll 0 exch awidthshow } bind def /BF { gsave moveto RF 0 32 4 2 roll 0 exch awidthshow grestore } bind def /G { gsave newpath normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /Gstrk { savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 0.0 0.0 1.0 5 3 roll arc restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /Gclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GG { gsave newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /GGclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GGstrk { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /A { gsave savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 2 copy 0.0 0.0 1.0 5 3 roll arc round cvi 360 mod exch round cvi 360 mod eq {closepath} if restorematrix PStroke grestore } bind def /Aclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /Astrk { Gstrk } bind def /AA { gsave savematrix newpath 3 index 2 div add exch 4 index 2 div sub exch normalize 3 index 2 div sub exch 4 index 2 div add exch translate rotate scale 0.0 0.0 1.0 5 3 roll arc restorematrix PStroke grestore } bind def /AAclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /AAstrk { GGstrk } bind def /BEGINPRINTCODE { /FMdicttop countdictstack 1 add def /FMoptop count 7 sub def /FMsaveobject save def userdict begin /showpage {} def FMNORMALIZEGRAPHICS 3 index neg 3 index neg translate } bind def /ENDPRINTCODE { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMsaveobject restore } bind def /gn { 0 { 46 mul cf read pop 32 sub dup 46 lt {exit} if 46 sub add } loop add } bind def /cfs { /str sl string def 0 1 sl 1 sub {str exch val put} for str def } bind def /ic [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 {0 hx} {1 hx} {2 hx} {3 hx} {4 hx} {5 hx} {6 hx} {7 hx} {8 hx} {9 hx} {10 hx} {11 hx} {12 hx} {13 hx} {14 hx} {15 hx} {16 hx} {17 hx} {18 hx} {19 hx} {gn hx} {0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14} {15} {16} {17} {18} {19} {gn} {0 wh} {1 wh} {2 wh} {3 wh} {4 wh} {5 wh} {6 wh} {7 wh} {8 wh} {9 wh} {10 wh} {11 wh} {12 wh} {13 wh} {14 wh} {gn wh} {0 bl} {1 bl} {2 bl} {3 bl} {4 bl} {5 bl} {6 bl} {7 bl} {8 bl} {9 bl} {10 bl} {11 bl} {12 bl} {13 bl} {14 bl} {gn bl} {0 fl} {1 fl} {2 fl} {3 fl} {4 fl} {5 fl} {6 fl} {7 fl} {8 fl} {9 fl} {10 fl} {11 fl} {12 fl} {13 fl} {14 fl} {gn fl} ] def /ms { /sl exch def /val 255 def /ws cfs /im cfs /val 0 def /bs cfs /cs cfs } bind def 400 ms /ip { is 0 cf cs readline pop { ic exch get exec add } forall pop } bind def /rip { bis ris copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop ris gis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop gis bis copy pop dup add is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /rip4 { kis cis copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop cis mis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop mis yis copy pop dup dup add is exch cf cs readline pop { ic exch get exec add } forall pop pop yis kis copy pop 3 mul is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /wh { /len exch def /pos exch def ws 0 len getinterval im pos len getinterval copy pop pos len } bind def /bl { /len exch def /pos exch def bs 0 len getinterval im pos len getinterval copy pop pos len } bind def /s1 1 string def /fl { /len exch def /pos exch def /val cf s1 readhexstring pop 0 get def pos 1 pos len add 1 sub {im exch val put} for pos len } bind def /hx { 3 copy getinterval cf exch readhexstring pop pop } bind def /wbytes { dup dup 8 gt { pop 8 idiv mul } { 8 eq {pop} {1 eq {7 add 8 idiv} {3 add 4 idiv} ifelse} ifelse } ifelse } bind def /BEGINBITMAPBWc { 1 {} COMMONBITMAPc } bind def /BEGINBITMAPGRAYc { 8 {} COMMONBITMAPc } bind def /BEGINBITMAP2BITc { 2 {} COMMONBITMAPc } bind def /COMMONBITMAPc { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def cvtProc /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} image bitmapsave restore grestore } bind def /BEGINBITMAPBW { 1 {} COMMONBITMAP } bind def /BEGINBITMAPGRAY { 8 {} COMMONBITMAP } bind def /BEGINBITMAP2BIT { 2 {} COMMONBITMAP } bind def /COMMONBITMAP { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def cvtProc /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} image bitmapsave restore grestore } bind def /ngrayt 256 array def /nredt 256 array def /nbluet 256 array def /ngreent 256 array def fMLevel1 { /colorsetup { currentcolortransfer /gryt exch def /blut exch def /grnt exch def /redt exch def 0 1 255 { /indx exch def /cynu 1 red indx get 255 div sub def /magu 1 green indx get 255 div sub def /yelu 1 blue indx get 255 div sub def /kk cynu magu min yelu min def /u kk currentundercolorremoval exec def % /u 0 def nredt indx 1 0 cynu u sub max sub redt exec put ngreent indx 1 0 magu u sub max sub grnt exec put nbluet indx 1 0 yelu u sub max sub blut exec put ngrayt indx 1 kk currentblackgeneration exec sub gryt exec put } for {255 mul cvi nredt exch get} {255 mul cvi ngreent exch get} {255 mul cvi nbluet exch get} {255 mul cvi ngrayt exch get} setcolortransfer {pop 0} setundercolorremoval {} setblackgeneration } bind def } { /colorSetup2 { [ /Indexed /DeviceRGB 255 {dup red exch get 255 div exch dup green exch get 255 div exch blue exch get 255 div} ] setcolorspace } bind def } ifelse /fakecolorsetup { /tran 256 string def 0 1 255 {/indx exch def tran indx red indx get 77 mul green indx get 151 mul blue indx get 28 mul add add 256 idiv put} for currenttransfer {255 mul cvi tran exch get 255.0 div} exch fmConcatProcs settransfer } bind def /BITMAPCOLOR { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def fMLevel1 { colorsetup /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} {is} {is} true 3 colorimage } { colorSetup2 /is width depth wbytes string def /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {cf is readhexstring pop} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPCOLORc { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def fMLevel1 { colorsetup /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} {is} {is} true 3 colorimage } { colorSetup2 /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {ip} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPTRUECOLORc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris} {gis} {bis} true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYKc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip4 pop cis} {mis} {yis} {kis} true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUECOLOR { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf gis readhexstring pop } { cf bis readhexstring pop } true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYK { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /mis width string def /yis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf mis readhexstring pop } { cf yis readhexstring pop } { cf kis readhexstring pop } true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUEGRAYc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris gis bis width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAYc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop cis mis yis kis width cgray} image bitmapsave restore grestore } bind def /cgray { /ww exch def /k exch def /y exch def /m exch def /c exch def 0 1 ww 1 sub { /i exch def c i get m i get y i get k i get CMYKtoRGB .144 mul 3 1 roll .587 mul 3 1 roll .299 mul add add c i 3 -1 roll floor cvi put } for c } bind def /gray { /ww exch def /b exch def /g exch def /r exch def 0 1 ww 1 sub { /i exch def r i get .299 mul g i get .587 mul b i get .114 mul add add r i 3 -1 roll floor cvi put } for r } bind def /BITMAPTRUEGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf gis readhexstring pop cf bis readhexstring pop width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /yis width string def /mis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf mis readhexstring pop cf yis readhexstring pop cf kis readhexstring pop width cgray} image bitmapsave restore grestore } bind def /BITMAPGRAY { 8 {fakecolorsetup} COMMONBITMAP } bind def /BITMAPGRAYc { 8 {fakecolorsetup} COMMONBITMAPc } bind def /ENDBITMAP { } bind def end /ALDmatrix matrix def ALDmatrix currentmatrix pop /StartALD { /ALDsave save def savematrix ALDmatrix setmatrix } bind def /InALD { restorematrix } bind def /DoneALD { ALDsave restore } bind def /I { setdash } bind def /J { [] 0 setdash } bind def (5.5) FMVERSION 1 1 0 0 612 792 0 1 6 FMDOCUMENT 0 0 /Times-Italic FMFONTDEFINE 1 0 /Times-Roman FMFONTDEFINE 2 1 /Symbol FMFONTDEFINE 32 FMFILLS 0 0 FMFILL 1 0.1 FMFILL 2 0.3 FMFILL 3 0.5 FMFILL 4 0.7 FMFILL 5 0.9 FMFILL 6 0.97 FMFILL 7 1 FMFILL 8 <0f1e3c78f0e1c387> FMFILL 9 <0f87c3e1f0783c1e> FMFILL 10 FMFILL 11 FMFILL 12 <8142241818244281> FMFILL 13 <03060c183060c081> FMFILL 14 <8040201008040201> FMFILL 16 1 FMFILL 17 0.9 FMFILL 18 0.7 FMFILL 19 0.5 FMFILL 20 0.3 FMFILL 21 0.1 FMFILL 22 0.03 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317.16 564.09 T (\051) 328.86 564.09 T 315.09 614.82 315.09 590.82 2 L 7 X V 0 X N 1 18.88 Q ( 1) 40.99 149.59 T FMENDPAGE FMENDDOCUMENT %%EndDocument @endspecial 1468 x @beginspecial 31 @llx 58 @lly 572 @urx 360 @ury 2880 @rwi @setspecial %%BeginDocument: /home/lynx1/majercik/papers/SAT2000/fmcharts/ftree2.ps % % Frame ps_prolog 5.5, for use with Adobe Unix Frame 5.5 products % % This ps_prolog file is Copyright (c) 1986-1996 Adobe Systems, Incoporated. % All rights reserved. This ps_prolog file may be freely copied and % distributed in conjunction with documents created using FrameMaker, % FrameMaker+SGML, FrameReader, and FrameViewer as long as this % copyright notice is preserved. /FMDocSave save def % % FrameMaker users specify the proper paper size for each print job in the % "Print" dialog's "Printer Paper Size" "Width" and "Height~ fields. If the % printer that the PS file is sent to does not support the requested paper % size, or if there is no paper tray of the proper size currently installed, % then the job will not be printed. The following flag, if set to true, will % cause the job to print on the default paper in such cases. /FMAllowPaperSizeMismatch false def % % Frame products normally print colors as their true color on a color printer % or as shades of gray, based on luminance, on a black-and white printer. The % following flag, if set to true, forces all non-white colors to print as pure % black. This has no effect on bitmap images. /FMPrintAllColorsAsBlack false def % % Frame products can either set their own line screens or use a printer's % default settings. Three flags below control this separately for no % separations, spot separations and process separations. If a flag % is true, then the default printer settings will not be changed. If it is % false, Frame products will use their own settings from a table based on % the printer's resolution. /FMUseDefaultNoSeparationScreen true def /FMUseDefaultSpotSeparationScreen true def /FMUseDefaultProcessSeparationScreen false def % % For any given PostScript printer resolution, Frame products have two sets of % screen angles and frequencies for printing process separations, which are % recomended by Adobe. The following variable chooses the higher frequencies % when set to true or the lower frequencies when set to false. This is only % effective if the appropriate FMUseDefault...SeparationScreen flag is false. /FMUseHighFrequencyScreens true def % % The following is a set of predefined optimal frequencies and angles for various % common dpi settings. This is taken from "Advances in Color Separation Using % PostScript Software Technology," from Adobe Systems (3/13/89 P.N. LPS 0043) % and corrolated with information which is in various PPD (4.0) files. % % The "dpiranges" figure is the minimum dots per inch device resolution which % can support this setting. The "low" and "high" values are controlled by the % setting of the FMUseHighFrequencyScreens flag above. The "TDot" flags control % the use of the "Yellow Triple Dot" feature whereby the frequency id divided by % three, but the dot function is "trippled" giving a block of 3x3 dots per cell. % % PatFreq is a compromise pattern frequency for ps Level 2 printers which is close % to the ideal WYSIWYG pattern frequency of 9 repetitions/inch but does not beat % (too badly) against the screen frequencies of any separations for that DPI. /dpiranges [ 2540 2400 1693 1270 1200 635 600 0 ] def /CMLowFreqs [ 100.402 94.8683 89.2289 100.402 94.8683 66.9349 63.2456 47.4342 ] def /YLowFreqs [ 95.25 90.0 84.65 95.25 90.0 70.5556 66.6667 50.0 ] def /KLowFreqs [ 89.8026 84.8528 79.8088 89.8026 84.8528 74.8355 70.7107 53.033 ] def /CLowAngles [ 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 ] def /MLowAngles [ 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 ] def /YLowTDot [ true true false true true false false false ] def /CMHighFreqs [ 133.87 126.491 133.843 108.503 102.523 100.402 94.8683 63.2456 ] def /YHighFreqs [ 127.0 120.0 126.975 115.455 109.091 95.25 90.0 60.0 ] def /KHighFreqs [ 119.737 113.137 119.713 128.289 121.218 89.8026 84.8528 63.6395 ] def /CHighAngles [ 71.5651 71.5651 71.5651 70.0169 70.0169 71.5651 71.5651 71.5651 ] def /MHighAngles [ 18.4349 18.4349 18.4349 19.9831 19.9831 18.4349 18.4349 18.4349 ] def /YHighTDot [ false false true false false true true false ] def /PatFreq [ 10.5833 10.0 9.4055 10.5833 10.0 10.5833 10.0 9.375 ] def % % PostScript Level 2 printers contain an "Accurate Screens" feature which can % improve process separation rendering at the expense of compute time. This % flag is ignored by PostScript Level 1 printers. /FMUseAcccurateScreens true def % % The following PostScript procedure defines the spot function that Frame % products will use for process separations. You may un-comment-out one of % the alternative functions below, or use your own. % % Dot function /FMSpotFunction {abs exch abs 2 copy add 1 gt {1 sub dup mul exch 1 sub dup mul add 1 sub } {dup mul exch dup mul add 1 exch sub }ifelse } def % % Line function % /FMSpotFunction { pop } def % % Elipse function % /FMSpotFunction { dup 5 mul 8 div mul exch dup mul exch add % sqrt 1 exch sub } def % % /FMversion (5.5) def /fMLevel1 /languagelevel where {pop languagelevel} {1} ifelse 2 lt def /FMPColor fMLevel1 { false /colorimage where {pop pop true} if } { true } ifelse def /FrameDict 400 dict def systemdict /errordict known not {/errordict 10 dict def errordict /rangecheck {stop} put} if % The readline in PS 23.0 doesn't recognize cr's as nl's on AppleTalk FrameDict /tmprangecheck errordict /rangecheck get put errordict /rangecheck {FrameDict /bug true put} put FrameDict /bug false put mark % Some PS machines read past the CR, so keep the following 3 lines together! currentfile 5 string readline 00 0000000000 cleartomark errordict /rangecheck FrameDict /tmprangecheck get put FrameDict /bug get { /readline { /gstring exch def /gfile exch def /gindex 0 def { gfile read pop dup 10 eq {exit} if dup 13 eq {exit} if gstring exch gindex exch put /gindex gindex 1 add def } loop pop gstring 0 gindex getinterval true } bind def } if /FMshowpage /showpage load def /FMquit /quit load def /FMFAILURE { 2 copy exch = = flush FMshowpage /Helvetica findfont 12 scalefont setfont 72 200 moveto show 72 220 moveto show FMshowpage FMquit } def /FMVERSION { FMversion ne { (Adobe Frame product version does not match ps_prolog! Check installation;) (also check ~/fminit and ./fminit for old versions) FMFAILURE } if } def /fmConcatProcs { /proc2 exch cvlit def/proc1 exch cvlit def/newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }def FrameDict begin [ /ALDsave /FMdicttop /FMoptop /FMpointsize /FMsetsize /FMsaveobject /b /bitmapsave /blut /bpside /bs /bstring /bwidth /c /cf /cs /cynu /depth /edown /fh /fillvals /fw /fx /fy /g /gfile /gindex /grnt /gryt /gstring /height /hh /i /im /indx /is /k /kk /landscape /lb /len /llx /lly /m /magu /manualfeed /n /offbits /onbits /organgle /orgbangle /orgbfreq /orgbproc /orgbxfer /orgfreq /orggangle /orggfreq /orggproc /orggxfer /orghalftone /orgmatrix /orgproc /orgrangle /orgrfreq /orgrproc /orgrxfer /orgxfer /pagesave /paperheight /papersizedict /paperwidth /pos /pwid /r /rad /redt /sl /str /tran /u /urx /ury /val /width /width /ws /ww /x /x1 /x2 /xindex /xpoint /xscale /xx /y /y1 /y2 /yelu /yindex /ypoint /yscale /yy /tintGray ] { 0 def } forall /FmBD {bind def} bind def systemdict /pdfmark known systemdict /currentdistillerparams known and { /fMAcrobat true def /FmPD /pdfmark load def /FmPT /show load def currentdistillerparams /CoreDistVersion get 2000 ge { /FmPD2 /pdfmark load def /FmPA { mark exch /Dest exch 5 3 roll /View [ /XYZ null 6 -2 roll FmDC exch pop null] /DEST FmPD }FmBD } { /FmPD2 /cleartomark load def /FmPA {pop pop pop}FmBD } ifelse } { /fMAcrobat false def /FmPD /cleartomark load def /FmPD2 /cleartomark load def /FmPT /pop load def /FmPA {pop pop pop}FmBD } ifelse /FmDC { transform fMDefaultMatrix defaultmatrix itransform cvi exch cvi exch }FmBD /FmBx { dup 3 index lt {3 1 roll exch} if 1 index 4 index lt {4 -1 roll 3 1 roll exch 4 1 roll} if }FmBD /FMnone 0 def /FMcyan 1 def /FMmagenta 2 def /FMyellow 3 def /FMblack 4 def /FMcustom 5 def /fMNegative false def /FrameSepIs FMnone def /FrameSepBlack 0 def /FrameSepYellow 0 def /FrameSepMagenta 0 def /FrameSepCyan 0 def /FrameSepRed 1 def /FrameSepGreen 1 def /FrameSepBlue 1 def /FrameCurGray 1 def /FrameCurPat null def /FrameCurColors [ 0 0 0 1 0 0 0 1] def /FrameColorEpsilon .001 def /eqepsilon { sub dup 0 lt {neg} if FrameColorEpsilon le } bind def /FrameCmpColorsCMYK { 2 copy 0 get exch 0 get eqepsilon { 2 copy 1 get exch 1 get eqepsilon { 2 copy 2 get exch 2 get eqepsilon { 3 get exch 3 get eqepsilon } {pop pop false} ifelse }{pop pop false} ifelse } {pop pop false} ifelse } bind def /FrameCmpColorsRGB { 2 copy 4 get exch 0 get eqepsilon { 2 copy 5 get exch 1 get eqepsilon { 6 get exch 2 get eqepsilon }{pop pop false} ifelse } {pop pop false} ifelse } bind def /RGBtoCMYK { 1 exch sub 3 1 roll 1 exch sub 3 1 roll 1 exch sub 3 1 roll 3 copy 2 copy le { pop } { exch pop } ifelse 2 copy le { pop } { exch pop } ifelse dup dup dup 6 1 roll 4 1 roll 7 1 roll sub 6 1 roll sub 5 1 roll sub 4 1 roll } bind def /CMYKtoRGB { dup dup 4 -1 roll add 5 1 roll 3 -1 roll add 4 1 roll add 1 exch sub dup 0 lt {pop 0} if 3 1 roll 1 exch sub dup 0 lt {pop 0} if exch 1 exch sub dup 0 lt {pop 0} if exch } bind def /FrameSepInit { 1.0 RealSetgray } bind def /FrameSetSepColor { /FrameSepBlue exch def /FrameSepGreen exch def /FrameSepRed exch def /FrameSepBlack exch def /FrameSepYellow exch def /FrameSepMagenta exch def /FrameSepCyan exch def /FrameSepIs FMcustom def setCurrentScreen } bind def /FrameSetCyan { /FrameSepBlue 1.0 def /FrameSepGreen 1.0 def /FrameSepRed 0.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 1.0 def /FrameSepIs FMcyan def setCurrentScreen } bind def /FrameSetMagenta { /FrameSepBlue 1.0 def /FrameSepGreen 0.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 1.0 def /FrameSepCyan 0.0 def /FrameSepIs FMmagenta def setCurrentScreen } bind def /FrameSetYellow { /FrameSepBlue 0.0 def /FrameSepGreen 1.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 1.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMyellow def setCurrentScreen } bind def /FrameSetBlack { /FrameSepBlue 0.0 def /FrameSepGreen 0.0 def /FrameSepRed 0.0 def /FrameSepBlack 1.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMblack def setCurrentScreen } bind def /FrameNoSep { /FrameSepIs FMnone def setCurrentScreen } bind def /FrameSetSepColors { FrameDict begin [ exch 1 add 1 roll ] /FrameSepColors exch def end } bind def /FrameColorInSepListCMYK { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsCMYK { pop true exit } if } forall dup true ne {pop false} if } bind def /FrameColorInSepListRGB { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsRGB { pop true exit } if } forall dup true ne {pop false} if } bind def /RealSetgray /setgray load def /RealSetrgbcolor /setrgbcolor load def /RealSethsbcolor /sethsbcolor load def end /setgray { FrameDict begin FrameSepIs FMnone eq { RealSetgray } { FrameSepIs FMblack eq { RealSetgray } { FrameSepIs FMcustom eq FrameSepRed 0 eq and FrameSepGreen 0 eq and FrameSepBlue 0 eq and { RealSetgray } { 1 RealSetgray pop } ifelse } ifelse } ifelse end } bind def /setrgbcolor { FrameDict begin FrameSepIs FMnone eq { RealSetrgbcolor } { 3 copy [ 4 1 roll ] FrameColorInSepListRGB { FrameSepBlue eq exch FrameSepGreen eq and exch FrameSepRed eq and { 0 } { 1 } ifelse } { FMPColor { RealSetrgbcolor currentcmykcolor } { RGBtoCMYK } ifelse FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind def /sethsbcolor { FrameDict begin FrameSepIs FMnone eq { RealSethsbcolor } { RealSethsbcolor currentrgbcolor setrgbcolor } ifelse end } bind def FrameDict begin /setcmykcolor where { pop /RealSetcmykcolor /setcmykcolor load def } { /RealSetcmykcolor { 4 1 roll 3 { 3 index add 0 max 1 min 1 exch sub 3 1 roll} repeat RealSetrgbcolor pop } bind def } ifelse userdict /setcmykcolor { FrameDict begin FrameSepIs FMnone eq { RealSetcmykcolor } { 4 copy [ 5 1 roll ] FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and { 0 } { 1 } ifelse } { FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind put fMLevel1 { /patScreenDict 7 dict dup begin <0f1e3c78f0e1c387> [ 45 { pop } {exch pop} .5 2 sqrt] FmBD <0f87c3e1f0783c1e> [ 135 { pop } {exch pop} .5 2 sqrt] FmBD [ 0 { pop } dup .5 2 ] FmBD [ 90 { pop } dup .5 2 ] FmBD <8142241818244281> [ 45 { 2 copy lt {exch} if pop} dup .75 2 sqrt] FmBD <03060c183060c081> [ 45 { pop } {exch pop} .875 2 sqrt] FmBD <8040201008040201> [ 135 { pop } {exch pop} .875 2 sqrt] FmBD end def } { /patProcDict 5 dict dup begin <0f1e3c78f0e1c387> { 3 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <0f87c3e1f0783c1e> { 3 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def <8142241818244281> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -1 -1 moveto 9 9 lineto stroke } bind def <03060c183060c081> { 1 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <8040201008040201> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def end def /patDict 15 dict dup begin /PatternType 1 def /PaintType 2 def /TilingType 3 def /BBox [ 0 0 8 8 ] def /XStep 8 def /YStep 8 def /PaintProc { begin patProcDict bstring known { patProcDict bstring get exec } { 8 8 true [1 0 0 -1 0 8] bstring imagemask } ifelse end } bind def end def } ifelse /tintCMYK { 1 tintGray sub FrameCurColors 0 4 getinterval aload pop 4 index mul 5 1 roll 3 index mul 5 1 roll 2 index mul 5 1 roll mul 4 1 roll }bind def /tintRGB { 1 tintGray sub FrameCurColors 4 3 getinterval aload pop 1 exch sub 3 index mul 1 exch sub 4 1 roll 1 exch sub 2 index mul 1 exch sub 4 1 roll 1 exch sub mul 1 exch sub 3 1 roll }bind def /combineColor { /tintGray 1 1 FrameCurGray sub FrameCurColors 7 get mul sub def FrameSepIs FMnone eq { graymode fMLevel1 or not { [/Pattern [/DeviceCMYK]] setcolorspace tintCMYK FrameCurPat setcolor } { FrameCurColors 3 get 1.0 ge { tintGray RealSetgray } { fMAcrobat not FMPColor graymode and and { tintCMYK RealSetcmykcolor } { tintRGB RealSetrgbcolor } ifelse } ifelse } ifelse } { FrameCurColors 0 4 getinterval aload FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and FrameSepIs FMcustom eq and { tintGray } { 1 } ifelse } { FrameSepIs FMblack eq {tintGray 1.0 exch sub mul 1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop tintGray 1.0 exch sub mul 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop tintGray 1.0 exch sub mul 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop tintGray 1.0 exch sub mul 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse graymode fMLevel1 or not { [/Pattern [/DeviceGray]] setcolorspace FrameCurPat setcolor } { graymode not fMLevel1 and { dup 1 lt {pop FrameCurGray} if } if RealSetgray } ifelse } ifelse } bind def /savematrix { orgmatrix currentmatrix pop } bind def /restorematrix { orgmatrix setmatrix } bind def /fMDefaultMatrix matrix def /fMatrix2 matrix def /dpi 72 0 fMDefaultMatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /freq dpi dup 72 div round dup 0 eq {pop 1} if 8 mul div def /sangle 1 0 fMDefaultMatrix defaultmatrix dtransform exch atan def sangle fMatrix2 rotate fMDefaultMatrix defaultmatrix fMatrix2 concatmatrix dup 0 get /sflipx exch def 3 get /sflipy exch def /screenIndex { 0 1 dpiranges length 1 sub { dup dpiranges exch get 1 sub dpi le {exit} {pop} ifelse } for } bind def /getCyanScreen { FMUseHighFrequencyScreens { CHighAngles CMHighFreqs} {CLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getMagentaScreen { FMUseHighFrequencyScreens { MHighAngles CMHighFreqs } {MLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getYellowScreen { FMUseHighFrequencyScreens { YHighTDot YHighFreqs} { YLowTDot YLowFreqs } ifelse screenIndex dup 3 1 roll get 3 1 roll get { 3 div {2 { 1 add 2 div 3 mul dup floor sub 2 mul 1 sub exch} repeat FMSpotFunction } } {/FMSpotFunction load } ifelse 0.0 exch } bind def /getBlackScreen { FMUseHighFrequencyScreens { KHighFreqs } { KLowFreqs } ifelse screenIndex get 45.0 /FMSpotFunction load } bind def /getSpotScreen { getBlackScreen } bind def /getCompositeScreen { getBlackScreen } bind def /FMSetScreen fMLevel1 { /setscreen load }{ { 8 dict begin /HalftoneType 1 def /SpotFunction exch def /Angle exch def /Frequency exch def /AccurateScreens FMUseAcccurateScreens def currentdict end sethalftone } bind } ifelse def /setDefaultScreen { fMLevel1 { FMPColor { orgrxfer cvx orggxfer cvx orgbxfer cvx orgxfer cvx setcolortransfer } { orgxfer cvx settransfer } ifelse orgfreq organgle orgproc cvx setscreen } { orghalftone sethalftone }ifelse } bind def /setCurrentScreen { FrameSepIs FMnone eq { FMUseDefaultNoSeparationScreen { setDefaultScreen } { getCompositeScreen FMSetScreen } ifelse } { FrameSepIs FMcustom eq { FMUseDefaultSpotSeparationScreen { setDefaultScreen } { getSpotScreen FMSetScreen } ifelse } { FMUseDefaultProcessSeparationScreen { setDefaultScreen } { FrameSepIs FMcyan eq { getCyanScreen FMSetScreen } { FrameSepIs FMmagenta eq { getMagentaScreen FMSetScreen } { FrameSepIs FMyellow eq { getYellowScreen FMSetScreen } { getBlackScreen FMSetScreen } ifelse } ifelse } ifelse } ifelse } ifelse } ifelse } bind def end /FMDOCUMENT { array /FMfonts exch def dup 1 gt {/#copies exch def} {pop} ifelse FrameDict begin 0 ne /manualfeed exch def /paperheight exch def /paperwidth exch def 0 ne /fMNegative exch def 0 ne /edown exch def /yscale exch def /xscale exch def fMLevel1 not { /orghalftone currenthalftone def }if FMPColor { currentcolorscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def cvlit /orgbproc exch def /orgbangle exch def /orgbfreq exch def cvlit /orggproc exch def /orggangle exch def /orggfreq exch def cvlit /orgrproc exch def /orgrangle exch def /orgrfreq exch def currentcolortransfer fMNegative { 1 1 4 { pop { 1 exch sub } fmConcatProcs 4 1 roll } for 4 copy setcolortransfer } if cvlit /orgxfer exch def cvlit /orgbxfer exch def cvlit /orggxfer exch def cvlit /orgrxfer exch def } { currentscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def currenttransfer fMNegative { { 1 exch sub } fmConcatProcs dup settransfer } if cvlit /orgxfer exch def } ifelse end } def /FMENDDOCUMENT { FMDocSave restore } def /FMBEGINPAGE { FrameDict begin /pagesave save def 3.86 setmiterlimit /landscape exch 0 ne def landscape { 90 rotate 0 exch dup /pwid exch def neg translate pop }{ pop /pwid exch def } ifelse edown { [-1 0 0 1 pwid 0] concat } if xscale yscale scale /orgmatrix matrix def gsave } def /FMENDPAGE { grestore pagesave restore end showpage } def /FMFONTDEFINE { FrameDict begin findfont ReEncode 1 index exch definefont FMfonts 3 1 roll put end } def /FMFILLS { FrameDict begin dup array /fillvals exch def dict /patCache exch def end } def /FMFILL { FrameDict begin fillvals 3 1 roll put end } def /FMNORMALIZEGRAPHICS { newpath 1 setlinewidth 0 setlinecap 0 0 0 sethsbcolor 0 setgray } bind def /FMBEGINEPSF { end /FMEPSF save def /showpage {} def FMNORMALIZEGRAPHICS [/fy /fx /fh /fw /ury /urx /lly /llx] {exch def} forall fx fw 2 div add fy fh 2 div add translate rotate fw 2 div neg fh 2 div neg translate fw urx llx sub div fh ury lly sub div scale llx neg lly neg translate /FMdicttop countdictstack 1 add def /FMoptop count def } bind def /FMENDEPSF { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMEPSF restore FrameDict begin } bind def FrameDict begin /setmanualfeed { statusdict /manualfeed true put } bind def /max {2 copy lt {exch} if pop} bind def /min {2 copy gt {exch} if pop} bind def /inch {72 mul} def /pagedimen { paperheight sub abs 16 lt exch paperwidth sub abs 16 lt and {/papername exch def} {pop} ifelse } bind def /setpapername { /papersizedict 14 dict def papersizedict begin /papername /unknown def /Letter 8.5 inch 11.0 inch pagedimen /LetterSmall 7.68 inch 10.16 inch pagedimen /Tabloid 11.0 inch 17.0 inch pagedimen /Ledger 17.0 inch 11.0 inch pagedimen /Legal 8.5 inch 14.0 inch pagedimen /Statement 5.5 inch 8.5 inch pagedimen /Executive 7.5 inch 10.0 inch pagedimen /A3 11.69 inch 16.5 inch pagedimen /A4 8.26 inch 11.69 inch pagedimen /A4Small 7.47 inch 10.85 inch pagedimen /B4 10.125 inch 14.33 inch pagedimen /B5 7.16 inch 10.125 inch pagedimen end } bind def /papersize { papersizedict begin /Letter {lettertray letter} def /LetterSmall {lettertray lettersmall} def /Tabloid {11x17tray 11x17} def /Ledger {ledgertray ledger} def /Legal {legaltray legal} def /Statement {statementtray statement} def /Executive {executivetray executive} def /A3 {a3tray a3} def /A4 {a4tray a4} def /A4Small {a4tray a4small} def /B4 {b4tray b4} def /B5 {b5tray b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end statusdict begin stopped end } bind def /manualpapersize { papersizedict begin /Letter {letter} def /LetterSmall {lettersmall} def /Tabloid {11x17} def /Ledger {ledger} def /Legal {legal} def /Statement {statement} def /Executive {executive} def /A3 {a3} def /A4 {a4} def /A4Small {a4small} def /B4 {b4} def /B5 {b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end stopped } bind def /desperatepapersize { mark statusdict begin /setpageparams where { pop paperwidth paperheight 0 1 {setpageparams} stopped } { true } ifelse { /setpagedevice where { pop 1 dict dup begin /PageSize [ paperwidth paperheight ] def end {setpagedevice} stopped } { true } ifelse } { false } ifelse end {cleartomark true}{cleartomark false}ifelse } bind def /papersizefailure { FMAllowPaperSizeMismatch not { (The requested paper size is not available in any currently-installed tray) (Edit the PS file to "FMAllowPaperSizeMismatch true" to use default tray) FMFAILURE } if } def /DiacriticEncoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /grave /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /.notdef /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /.notdef /AE /Oslash /.notdef /.notdef /.notdef /.notdef /yen /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ordfeminine /ordmasculine /.notdef /ae /oslash /questiondown /exclamdown /logicalnot /.notdef /florin /.notdef /.notdef /guillemotleft /guillemotright /ellipsis /.notdef /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /.notdef /.notdef /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /.notdef /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron ] def /ReEncode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall 0 eq {/Encoding DiacriticEncoding def} if currentdict end } bind def FMPColor { /BEGINBITMAPCOLOR { BITMAPCOLOR} def /BEGINBITMAPCOLORc { BITMAPCOLORc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUECOLOR } def /BEGINBITMAPTRUECOLORc { BITMAPTRUECOLORc } def /BEGINBITMAPCMYK { BITMAPCMYK } def /BEGINBITMAPCMYKc { BITMAPCMYKc } def } { /BEGINBITMAPCOLOR { BITMAPGRAY} def /BEGINBITMAPCOLORc { BITMAPGRAYc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUEGRAY } def /BEGINBITMAPTRUECOLORc { BITMAPTRUEGRAYc } def /BEGINBITMAPCMYK { BITMAPCMYKGRAY } def /BEGINBITMAPCMYKc { BITMAPCMYKGRAYc } def } ifelse /K { FMPrintAllColorsAsBlack { 8 1 roll dup 1 eq 2 index 1 eq and 3 index 1 eq and not {7 {pop} repeat 0 0 0 1 0 0 0} if 8 -1 roll } if FrameCurColors astore pop combineColor } bind def /graymode true def fMLevel1 { /fmGetFlip { fMatrix2 exch get mul 0 lt { -1 } { 1 } ifelse } FmBD } if /setPatternMode { fMLevel1 { 2 index patScreenDict exch known { pop pop patScreenDict exch get aload pop freq mul 5 2 roll fMatrix2 currentmatrix 1 get 0 ne { 3 -1 roll 90 add 3 1 roll sflipx 1 fmGetFlip sflipy 2 fmGetFlip neg mul } { sflipx 0 fmGetFlip sflipy 3 fmGetFlip mul } ifelse 0 lt {exch pop} {pop} ifelse fMNegative { {neg} fmConcatProcs } if bind systemdict /setscreen get exec /FrameCurGray exch def } { /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq sangle landscape {90 add} if {/ypoint exch def /xpoint exch def /xindex xpoint 1 add 2 div bpside mul cvi def /yindex ypoint 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get 1 7 xindex 8 mod sub bitshift and 0 ne fMNegative {not} if {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen offbits offbits onbits add dup 0 ne {div} {pop pop .5} ifelse fMNegative {1.0 exch sub} if /FrameCurGray exch def } ifelse } { pop pop dup patCache exch known { patCache exch get } { dup patDict /bstring 3 -1 roll put patDict 9 PatFreq screenIndex get div dup matrix scale makepattern dup patCache 4 -1 roll 3 -1 roll put } ifelse /FrameCurGray 0 def /FrameCurPat exch def } ifelse /graymode false def combineColor } bind def /setGrayScaleMode { graymode not { /graymode true def fMLevel1 { setCurrentScreen } if } if /FrameCurGray exch def combineColor } bind def /normalize { transform round exch round exch itransform } bind def /dnormalize { dtransform round exch round exch idtransform } bind def /lnormalize { 0 dtransform exch cvi 2 idiv 2 mul 1 add exch idtransform pop } bind def /H { lnormalize setlinewidth } bind def /Z { setlinecap } bind def /PFill { graymode fMLevel1 or not { gsave 1 setgray eofill grestore } if } bind def /PStroke { graymode fMLevel1 or not { gsave 1 setgray stroke grestore } if stroke } bind def /X { fillvals exch get dup type /stringtype eq {8 1 setPatternMode} {setGrayScaleMode} ifelse } bind def /V { PFill gsave eofill grestore } bind def /Vclip { clip } bind def /Vstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /N { PStroke } bind def /Nclip { strokepath clip newpath } bind def /Nstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /M {newpath moveto} bind def /E {lineto} bind def /D {curveto} bind def /O {closepath} bind def /L { /n exch def newpath normalize moveto 2 1 n {pop normalize lineto} for } bind def /Y { L closepath } bind def /R { /y2 exch def /x2 exch def /y1 exch def /x1 exch def x1 y1 x2 y1 x2 y2 x1 y2 4 Y } bind def /rarc {rad arcto } bind def /RR { /rad exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def mark newpath { x1 y1 rad add moveto x1 y2 x2 y2 rarc x2 y2 x2 y1 rarc x2 y1 x1 y1 rarc x1 y1 x1 y2 rarc closepath } stopped {x1 y1 x2 y2 R} if cleartomark } bind def /RRR { /rad exch def normalize /y4 exch def /x4 exch def normalize /y3 exch def /x3 exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def newpath normalize moveto mark { x2 y2 x3 y3 rarc x3 y3 x4 y4 rarc x4 y4 x1 y1 rarc x1 y1 x2 y2 rarc closepath } stopped {x1 y1 x2 y2 x3 y3 x4 y4 newpath moveto lineto lineto lineto closepath} if cleartomark } bind def /C { grestore gsave R clip setCurrentScreen } bind def /CP { grestore gsave Y clip setCurrentScreen } bind def /F { FMfonts exch get [FMsetsize 0 0 FMpointsize 0 0] makefont setfont } bind def /Q { /FMpointsize exch def /FMsetsize FMpointsize def F } bind def /QQ { /FMsetsize exch def /FMpointsize exch def F } bind def /T { moveto show } bind def /RF { rotate 0 ne {-1 1 scale} if } bind def /TF { gsave moveto RF show grestore } bind def /P { moveto 0 32 3 2 roll widthshow } bind def /PF { gsave moveto RF 0 32 3 2 roll widthshow grestore } bind def /S { moveto 0 exch ashow } bind def /SF { gsave moveto RF 0 exch ashow grestore } bind def /B { moveto 0 32 4 2 roll 0 exch awidthshow } bind def /BF { gsave moveto RF 0 32 4 2 roll 0 exch awidthshow grestore } bind def /G { gsave newpath normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /Gstrk { savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 0.0 0.0 1.0 5 3 roll arc restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /Gclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GG { gsave newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /GGclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GGstrk { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /A { gsave savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 2 copy 0.0 0.0 1.0 5 3 roll arc round cvi 360 mod exch round cvi 360 mod eq {closepath} if restorematrix PStroke grestore } bind def /Aclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /Astrk { Gstrk } bind def /AA { gsave savematrix newpath 3 index 2 div add exch 4 index 2 div sub exch normalize 3 index 2 div sub exch 4 index 2 div add exch translate rotate scale 0.0 0.0 1.0 5 3 roll arc restorematrix PStroke grestore } bind def /AAclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /AAstrk { GGstrk } bind def /BEGINPRINTCODE { /FMdicttop countdictstack 1 add def /FMoptop count 7 sub def /FMsaveobject save def userdict begin /showpage {} def FMNORMALIZEGRAPHICS 3 index neg 3 index neg translate } bind def /ENDPRINTCODE { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMsaveobject restore } bind def /gn { 0 { 46 mul cf read pop 32 sub dup 46 lt {exit} if 46 sub add } loop add } bind def /cfs { /str sl string def 0 1 sl 1 sub {str exch val put} for str def } bind def /ic [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 {0 hx} {1 hx} {2 hx} {3 hx} {4 hx} {5 hx} {6 hx} {7 hx} {8 hx} {9 hx} {10 hx} {11 hx} {12 hx} {13 hx} {14 hx} {15 hx} {16 hx} {17 hx} {18 hx} {19 hx} {gn hx} {0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14} {15} {16} {17} {18} {19} {gn} {0 wh} {1 wh} {2 wh} {3 wh} {4 wh} {5 wh} {6 wh} {7 wh} {8 wh} {9 wh} {10 wh} {11 wh} {12 wh} {13 wh} {14 wh} {gn wh} {0 bl} {1 bl} {2 bl} {3 bl} {4 bl} {5 bl} {6 bl} {7 bl} {8 bl} {9 bl} {10 bl} {11 bl} {12 bl} {13 bl} {14 bl} {gn bl} {0 fl} {1 fl} {2 fl} {3 fl} {4 fl} {5 fl} {6 fl} {7 fl} {8 fl} {9 fl} {10 fl} {11 fl} {12 fl} {13 fl} {14 fl} {gn fl} ] def /ms { /sl exch def /val 255 def /ws cfs /im cfs /val 0 def /bs cfs /cs cfs } bind def 400 ms /ip { is 0 cf cs readline pop { ic exch get exec add } forall pop } bind def /rip { bis ris copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop ris gis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop gis bis copy pop dup add is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /rip4 { kis cis copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop cis mis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop mis yis copy pop dup dup add is exch cf cs readline pop { ic exch get exec add } forall pop pop yis kis copy pop 3 mul is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /wh { /len exch def /pos exch def ws 0 len getinterval im pos len getinterval copy pop pos len } bind def /bl { /len exch def /pos exch def bs 0 len getinterval im pos len getinterval copy pop pos len } bind def /s1 1 string def /fl { /len exch def /pos exch def /val cf s1 readhexstring pop 0 get def pos 1 pos len add 1 sub {im exch val put} for pos len } bind def /hx { 3 copy getinterval cf exch readhexstring pop pop } bind def /wbytes { dup dup 8 gt { pop 8 idiv mul } { 8 eq {pop} {1 eq {7 add 8 idiv} {3 add 4 idiv} ifelse} ifelse } ifelse } bind def /BEGINBITMAPBWc { 1 {} COMMONBITMAPc } bind def /BEGINBITMAPGRAYc { 8 {} COMMONBITMAPc } bind def /BEGINBITMAP2BITc { 2 {} COMMONBITMAPc } bind def /COMMONBITMAPc { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def cvtProc /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} image bitmapsave restore grestore } bind def 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index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip4 pop cis} {mis} {yis} {kis} true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUECOLOR { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf gis readhexstring pop } { cf bis readhexstring pop } true 3 colorimage bitmapsave restore grestore } bind def 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This ps_prolog file may be freely copied and % distributed in conjunction with documents created using FrameMaker, % FrameMaker+SGML, FrameReader, and FrameViewer as long as this % copyright notice is preserved. /FMDocSave save def % % FrameMaker users specify the proper paper size for each print job in the % "Print" dialog's "Printer Paper Size" "Width" and "Height~ fields. If the % printer that the PS file is sent to does not support the requested paper % size, or if there is no paper tray of the proper size currently installed, % then the job will not be printed. The following flag, if set to true, will % cause the job to print on the default paper in such cases. /FMAllowPaperSizeMismatch false def % % Frame products normally print colors as their true color on a color printer % or as shades of gray, based on luminance, on a black-and white printer. The % following flag, if set to true, forces all non-white colors to print as pure % black. This has no effect on bitmap images. /FMPrintAllColorsAsBlack false def % % Frame products can either set their own line screens or use a printer's % default settings. Three flags below control this separately for no % separations, spot separations and process separations. If a flag % is true, then the default printer settings will not be changed. If it is % false, Frame products will use their own settings from a table based on % the printer's resolution. /FMUseDefaultNoSeparationScreen true def /FMUseDefaultSpotSeparationScreen true def /FMUseDefaultProcessSeparationScreen false def % % For any given PostScript printer resolution, Frame products have two sets of % screen angles and frequencies for printing process separations, which are % recomended by Adobe. The following variable chooses the higher frequencies % when set to true or the lower frequencies when set to false. This is only % effective if the appropriate FMUseDefault...SeparationScreen flag is false. /FMUseHighFrequencyScreens true def % % The following is a set of predefined optimal frequencies and angles for various % common dpi settings. This is taken from "Advances in Color Separation Using % PostScript Software Technology," from Adobe Systems (3/13/89 P.N. LPS 0043) % and corrolated with information which is in various PPD (4.0) files. % % The "dpiranges" figure is the minimum dots per inch device resolution which % can support this setting. The "low" and "high" values are controlled by the % setting of the FMUseHighFrequencyScreens flag above. The "TDot" flags control % the use of the "Yellow Triple Dot" feature whereby the frequency id divided by % three, but the dot function is "trippled" giving a block of 3x3 dots per cell. % % PatFreq is a compromise pattern frequency for ps Level 2 printers which is close % to the ideal WYSIWYG pattern frequency of 9 repetitions/inch but does not beat % (too badly) against the screen frequencies of any separations for that DPI. /dpiranges [ 2540 2400 1693 1270 1200 635 600 0 ] def /CMLowFreqs [ 100.402 94.8683 89.2289 100.402 94.8683 66.9349 63.2456 47.4342 ] def /YLowFreqs [ 95.25 90.0 84.65 95.25 90.0 70.5556 66.6667 50.0 ] def /KLowFreqs [ 89.8026 84.8528 79.8088 89.8026 84.8528 74.8355 70.7107 53.033 ] def /CLowAngles [ 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 ] def /MLowAngles [ 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 ] def /YLowTDot [ true true false true true false false false ] def /CMHighFreqs [ 133.87 126.491 133.843 108.503 102.523 100.402 94.8683 63.2456 ] def /YHighFreqs [ 127.0 120.0 126.975 115.455 109.091 95.25 90.0 60.0 ] def /KHighFreqs [ 119.737 113.137 119.713 128.289 121.218 89.8026 84.8528 63.6395 ] def /CHighAngles [ 71.5651 71.5651 71.5651 70.0169 70.0169 71.5651 71.5651 71.5651 ] def /MHighAngles [ 18.4349 18.4349 18.4349 19.9831 19.9831 18.4349 18.4349 18.4349 ] def /YHighTDot [ false false true false false true true false ] def /PatFreq [ 10.5833 10.0 9.4055 10.5833 10.0 10.5833 10.0 9.375 ] def % % PostScript Level 2 printers contain an "Accurate Screens" feature which can % improve process separation rendering at the expense of compute time. This % flag is ignored by PostScript Level 1 printers. /FMUseAcccurateScreens true def % % The following PostScript procedure defines the spot function that Frame % products will use for process separations. You may un-comment-out one of % the alternative functions below, or use your own. % % Dot function /FMSpotFunction {abs exch abs 2 copy add 1 gt {1 sub dup mul exch 1 sub dup mul add 1 sub } {dup mul exch dup mul add 1 exch sub }ifelse } def % % Line function % /FMSpotFunction { pop } def % % Elipse function % /FMSpotFunction { dup 5 mul 8 div mul exch dup mul exch add % sqrt 1 exch sub } def % % /FMversion (5.5) def /fMLevel1 /languagelevel where {pop languagelevel} {1} ifelse 2 lt def /FMPColor fMLevel1 { false /colorimage where {pop pop true} if } { true } ifelse def /FrameDict 400 dict def systemdict /errordict known not {/errordict 10 dict def errordict /rangecheck {stop} put} if % The readline in PS 23.0 doesn't recognize cr's as nl's on AppleTalk FrameDict /tmprangecheck errordict /rangecheck get put errordict /rangecheck {FrameDict /bug true put} put FrameDict /bug false put mark % Some PS machines read past the CR, so keep the following 3 lines together! currentfile 5 string readline 00 0000000000 cleartomark errordict /rangecheck FrameDict /tmprangecheck get put FrameDict /bug get { /readline { /gstring exch def /gfile exch def /gindex 0 def { gfile read pop dup 10 eq {exit} if dup 13 eq {exit} if gstring exch gindex exch put /gindex gindex 1 add def } loop pop gstring 0 gindex getinterval true } bind def } if /FMshowpage /showpage load def /FMquit /quit load def /FMFAILURE { 2 copy exch = = flush FMshowpage /Helvetica findfont 12 scalefont setfont 72 200 moveto show 72 220 moveto show FMshowpage FMquit } def /FMVERSION { FMversion ne { (Adobe Frame product version does not match ps_prolog! Check installation;) (also check ~/fminit and ./fminit for old versions) FMFAILURE } if } def /fmConcatProcs { /proc2 exch cvlit def/proc1 exch cvlit def/newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }def FrameDict begin [ /ALDsave /FMdicttop /FMoptop /FMpointsize /FMsetsize /FMsaveobject /b /bitmapsave /blut /bpside /bs /bstring /bwidth /c /cf /cs /cynu /depth /edown /fh /fillvals /fw /fx /fy /g /gfile /gindex /grnt /gryt /gstring /height /hh /i /im /indx /is /k /kk /landscape /lb /len /llx /lly /m /magu /manualfeed /n /offbits /onbits /organgle /orgbangle /orgbfreq /orgbproc /orgbxfer /orgfreq /orggangle /orggfreq /orggproc /orggxfer /orghalftone /orgmatrix /orgproc /orgrangle /orgrfreq /orgrproc /orgrxfer /orgxfer /pagesave /paperheight /papersizedict /paperwidth /pos /pwid /r /rad /redt /sl /str /tran /u /urx /ury /val /width /width /ws /ww /x /x1 /x2 /xindex /xpoint /xscale /xx /y /y1 /y2 /yelu /yindex /ypoint /yscale /yy /tintGray ] { 0 def } forall /FmBD {bind def} bind def systemdict /pdfmark known systemdict /currentdistillerparams known and { /fMAcrobat true def /FmPD /pdfmark load def /FmPT /show load def currentdistillerparams /CoreDistVersion get 2000 ge { /FmPD2 /pdfmark load def /FmPA { mark exch /Dest exch 5 3 roll /View [ /XYZ null 6 -2 roll FmDC exch pop null] /DEST FmPD }FmBD } { /FmPD2 /cleartomark load def /FmPA {pop pop pop}FmBD } ifelse } { /fMAcrobat false def /FmPD /cleartomark load def /FmPD2 /cleartomark load def /FmPT /pop load def /FmPA {pop pop pop}FmBD } ifelse /FmDC { transform fMDefaultMatrix defaultmatrix itransform cvi exch cvi exch }FmBD /FmBx { dup 3 index lt {3 1 roll exch} if 1 index 4 index lt {4 -1 roll 3 1 roll exch 4 1 roll} if }FmBD /FMnone 0 def /FMcyan 1 def /FMmagenta 2 def /FMyellow 3 def /FMblack 4 def /FMcustom 5 def /fMNegative false def /FrameSepIs FMnone def /FrameSepBlack 0 def /FrameSepYellow 0 def /FrameSepMagenta 0 def /FrameSepCyan 0 def /FrameSepRed 1 def /FrameSepGreen 1 def /FrameSepBlue 1 def /FrameCurGray 1 def /FrameCurPat null def /FrameCurColors [ 0 0 0 1 0 0 0 1] def /FrameColorEpsilon .001 def /eqepsilon { sub dup 0 lt {neg} if FrameColorEpsilon le } bind def /FrameCmpColorsCMYK { 2 copy 0 get exch 0 get eqepsilon { 2 copy 1 get exch 1 get eqepsilon { 2 copy 2 get exch 2 get eqepsilon { 3 get exch 3 get eqepsilon } {pop pop false} ifelse }{pop pop false} ifelse } {pop pop false} ifelse } bind def /FrameCmpColorsRGB { 2 copy 4 get exch 0 get eqepsilon { 2 copy 5 get exch 1 get eqepsilon { 6 get exch 2 get eqepsilon }{pop pop false} ifelse } {pop pop false} ifelse } bind def /RGBtoCMYK { 1 exch sub 3 1 roll 1 exch sub 3 1 roll 1 exch sub 3 1 roll 3 copy 2 copy le { pop } { exch pop } ifelse 2 copy le { pop } { exch pop } ifelse dup dup dup 6 1 roll 4 1 roll 7 1 roll sub 6 1 roll sub 5 1 roll sub 4 1 roll } bind def /CMYKtoRGB { dup dup 4 -1 roll add 5 1 roll 3 -1 roll add 4 1 roll add 1 exch sub dup 0 lt {pop 0} if 3 1 roll 1 exch sub dup 0 lt {pop 0} if exch 1 exch sub dup 0 lt {pop 0} if exch } bind def /FrameSepInit { 1.0 RealSetgray } bind def /FrameSetSepColor { /FrameSepBlue exch def /FrameSepGreen exch def /FrameSepRed exch def /FrameSepBlack exch def /FrameSepYellow exch def /FrameSepMagenta exch def /FrameSepCyan exch def /FrameSepIs FMcustom def setCurrentScreen } bind def /FrameSetCyan { /FrameSepBlue 1.0 def /FrameSepGreen 1.0 def /FrameSepRed 0.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 1.0 def /FrameSepIs FMcyan def setCurrentScreen } bind def /FrameSetMagenta { /FrameSepBlue 1.0 def /FrameSepGreen 0.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 1.0 def /FrameSepCyan 0.0 def /FrameSepIs FMmagenta def setCurrentScreen } bind def /FrameSetYellow { /FrameSepBlue 0.0 def /FrameSepGreen 1.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 1.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMyellow def setCurrentScreen } bind def /FrameSetBlack { /FrameSepBlue 0.0 def /FrameSepGreen 0.0 def /FrameSepRed 0.0 def /FrameSepBlack 1.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMblack def setCurrentScreen } bind def /FrameNoSep { /FrameSepIs FMnone def setCurrentScreen } bind def /FrameSetSepColors { FrameDict begin [ exch 1 add 1 roll ] /FrameSepColors exch def end } bind def /FrameColorInSepListCMYK { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsCMYK { pop true exit } if } forall dup true ne {pop false} if } bind def /FrameColorInSepListRGB { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsRGB { pop true exit } if } forall dup true ne {pop false} if } bind def /RealSetgray /setgray load def /RealSetrgbcolor /setrgbcolor load def /RealSethsbcolor /sethsbcolor load def end /setgray { FrameDict begin FrameSepIs FMnone eq { RealSetgray } { FrameSepIs FMblack eq { RealSetgray } { FrameSepIs FMcustom eq FrameSepRed 0 eq and FrameSepGreen 0 eq and FrameSepBlue 0 eq and { RealSetgray } { 1 RealSetgray pop } ifelse } ifelse } ifelse end } bind def /setrgbcolor { FrameDict begin FrameSepIs FMnone eq { RealSetrgbcolor } { 3 copy [ 4 1 roll ] FrameColorInSepListRGB { FrameSepBlue eq exch FrameSepGreen eq and exch FrameSepRed eq and { 0 } { 1 } ifelse } { FMPColor { RealSetrgbcolor currentcmykcolor } { RGBtoCMYK } ifelse FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind def /sethsbcolor { FrameDict begin FrameSepIs FMnone eq { RealSethsbcolor } { RealSethsbcolor currentrgbcolor setrgbcolor } ifelse end } bind def FrameDict begin /setcmykcolor where { pop /RealSetcmykcolor /setcmykcolor load def } { /RealSetcmykcolor { 4 1 roll 3 { 3 index add 0 max 1 min 1 exch sub 3 1 roll} repeat RealSetrgbcolor pop } bind def } ifelse userdict /setcmykcolor { FrameDict begin FrameSepIs FMnone eq { RealSetcmykcolor } { 4 copy [ 5 1 roll ] FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and { 0 } { 1 } ifelse } { FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind put fMLevel1 { /patScreenDict 7 dict dup begin <0f1e3c78f0e1c387> [ 45 { pop } {exch pop} .5 2 sqrt] FmBD <0f87c3e1f0783c1e> [ 135 { pop } {exch pop} .5 2 sqrt] FmBD [ 0 { pop } dup .5 2 ] FmBD [ 90 { pop } dup .5 2 ] FmBD <8142241818244281> [ 45 { 2 copy lt {exch} if pop} dup .75 2 sqrt] FmBD <03060c183060c081> [ 45 { pop } {exch pop} .875 2 sqrt] FmBD <8040201008040201> [ 135 { pop } {exch pop} .875 2 sqrt] FmBD end def } { /patProcDict 5 dict dup begin <0f1e3c78f0e1c387> { 3 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <0f87c3e1f0783c1e> { 3 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def <8142241818244281> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -1 -1 moveto 9 9 lineto stroke } bind def <03060c183060c081> { 1 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <8040201008040201> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def end def /patDict 15 dict dup begin /PatternType 1 def /PaintType 2 def /TilingType 3 def /BBox [ 0 0 8 8 ] def /XStep 8 def /YStep 8 def /PaintProc { begin patProcDict bstring known { patProcDict bstring get exec } { 8 8 true [1 0 0 -1 0 8] bstring imagemask } ifelse end } bind def end def } ifelse /tintCMYK { 1 tintGray sub FrameCurColors 0 4 getinterval aload pop 4 index mul 5 1 roll 3 index mul 5 1 roll 2 index mul 5 1 roll mul 4 1 roll }bind def /tintRGB { 1 tintGray sub FrameCurColors 4 3 getinterval aload pop 1 exch sub 3 index mul 1 exch sub 4 1 roll 1 exch sub 2 index mul 1 exch sub 4 1 roll 1 exch sub mul 1 exch sub 3 1 roll }bind def /combineColor { /tintGray 1 1 FrameCurGray sub FrameCurColors 7 get mul sub def FrameSepIs FMnone eq { graymode fMLevel1 or not { [/Pattern [/DeviceCMYK]] setcolorspace tintCMYK FrameCurPat setcolor } { FrameCurColors 3 get 1.0 ge { tintGray RealSetgray } { fMAcrobat not FMPColor graymode and and { tintCMYK RealSetcmykcolor } { tintRGB RealSetrgbcolor } ifelse } ifelse } ifelse } { FrameCurColors 0 4 getinterval aload FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and FrameSepIs FMcustom eq and { tintGray } { 1 } ifelse } { FrameSepIs FMblack eq {tintGray 1.0 exch sub mul 1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop tintGray 1.0 exch sub mul 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop tintGray 1.0 exch sub mul 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop tintGray 1.0 exch sub mul 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse graymode fMLevel1 or not { [/Pattern [/DeviceGray]] setcolorspace FrameCurPat setcolor } { graymode not fMLevel1 and { dup 1 lt {pop FrameCurGray} if } if RealSetgray } ifelse } ifelse } bind def /savematrix { orgmatrix currentmatrix pop } bind def /restorematrix { orgmatrix setmatrix } bind def /fMDefaultMatrix matrix def /fMatrix2 matrix def /dpi 72 0 fMDefaultMatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /freq dpi dup 72 div round dup 0 eq {pop 1} if 8 mul div def /sangle 1 0 fMDefaultMatrix defaultmatrix dtransform exch atan def sangle fMatrix2 rotate fMDefaultMatrix defaultmatrix fMatrix2 concatmatrix dup 0 get /sflipx exch def 3 get /sflipy exch def /screenIndex { 0 1 dpiranges length 1 sub { dup dpiranges exch get 1 sub dpi le {exit} {pop} ifelse } for } bind def /getCyanScreen { FMUseHighFrequencyScreens { CHighAngles CMHighFreqs} {CLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getMagentaScreen { FMUseHighFrequencyScreens { MHighAngles CMHighFreqs } {MLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getYellowScreen { FMUseHighFrequencyScreens { YHighTDot YHighFreqs} { YLowTDot YLowFreqs } ifelse screenIndex dup 3 1 roll get 3 1 roll get { 3 div {2 { 1 add 2 div 3 mul dup floor sub 2 mul 1 sub exch} repeat FMSpotFunction } } {/FMSpotFunction load } ifelse 0.0 exch } bind def /getBlackScreen { FMUseHighFrequencyScreens { KHighFreqs } { KLowFreqs } ifelse screenIndex get 45.0 /FMSpotFunction load } bind def /getSpotScreen { getBlackScreen } bind def /getCompositeScreen { getBlackScreen } bind def /FMSetScreen fMLevel1 { /setscreen load }{ { 8 dict begin /HalftoneType 1 def /SpotFunction exch def /Angle exch def /Frequency exch def /AccurateScreens FMUseAcccurateScreens def currentdict end sethalftone } bind } ifelse def /setDefaultScreen { fMLevel1 { FMPColor { orgrxfer cvx orggxfer cvx orgbxfer cvx orgxfer cvx setcolortransfer } { orgxfer cvx settransfer } ifelse orgfreq organgle orgproc cvx setscreen } { orghalftone sethalftone }ifelse } bind def /setCurrentScreen { FrameSepIs FMnone eq { FMUseDefaultNoSeparationScreen { setDefaultScreen } { getCompositeScreen FMSetScreen } ifelse } { FrameSepIs FMcustom eq { FMUseDefaultSpotSeparationScreen { setDefaultScreen } { getSpotScreen FMSetScreen } ifelse } { FMUseDefaultProcessSeparationScreen { setDefaultScreen } { FrameSepIs FMcyan eq { getCyanScreen FMSetScreen } { FrameSepIs FMmagenta eq { getMagentaScreen FMSetScreen } { FrameSepIs FMyellow eq { getYellowScreen FMSetScreen } { getBlackScreen FMSetScreen } ifelse } ifelse } ifelse } ifelse } ifelse } ifelse } bind def end /FMDOCUMENT { array /FMfonts exch def dup 1 gt {/#copies exch def} {pop} ifelse FrameDict begin 0 ne /manualfeed exch def /paperheight exch def /paperwidth exch def 0 ne /fMNegative exch def 0 ne /edown exch def /yscale exch def /xscale exch def fMLevel1 not { /orghalftone currenthalftone def }if FMPColor { currentcolorscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def cvlit /orgbproc exch def /orgbangle exch def /orgbfreq exch def cvlit /orggproc exch def /orggangle exch def /orggfreq exch def cvlit /orgrproc exch def /orgrangle exch def /orgrfreq exch def currentcolortransfer fMNegative { 1 1 4 { pop { 1 exch sub } fmConcatProcs 4 1 roll } for 4 copy setcolortransfer } if cvlit /orgxfer exch def cvlit /orgbxfer exch def cvlit /orggxfer exch def cvlit /orgrxfer exch def } { currentscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def currenttransfer fMNegative { { 1 exch sub } fmConcatProcs dup settransfer } if cvlit /orgxfer exch def } ifelse end } def /FMENDDOCUMENT { FMDocSave restore } def /FMBEGINPAGE { FrameDict begin /pagesave save def 3.86 setmiterlimit /landscape exch 0 ne def landscape { 90 rotate 0 exch dup /pwid exch def neg translate pop }{ pop /pwid exch def } ifelse edown { [-1 0 0 1 pwid 0] concat } if xscale yscale scale /orgmatrix matrix def gsave } def /FMENDPAGE { grestore pagesave restore end showpage } def /FMFONTDEFINE { FrameDict begin findfont ReEncode 1 index exch definefont FMfonts 3 1 roll put end } def /FMFILLS { FrameDict begin dup array /fillvals exch def dict /patCache exch def end } def /FMFILL { FrameDict begin fillvals 3 1 roll put end } def /FMNORMALIZEGRAPHICS { newpath 1 setlinewidth 0 setlinecap 0 0 0 sethsbcolor 0 setgray } bind def /FMBEGINEPSF { end /FMEPSF save def /showpage {} def FMNORMALIZEGRAPHICS [/fy /fx /fh /fw /ury /urx /lly /llx] {exch def} forall fx fw 2 div add fy fh 2 div add translate rotate fw 2 div neg fh 2 div neg translate fw urx llx sub div fh ury lly sub div scale llx neg lly neg translate /FMdicttop countdictstack 1 add def /FMoptop count def } bind def /FMENDEPSF { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMEPSF restore FrameDict begin } bind def FrameDict begin /setmanualfeed { statusdict /manualfeed true put } bind def /max {2 copy lt {exch} if pop} bind def /min {2 copy gt {exch} if pop} bind def /inch {72 mul} def /pagedimen { paperheight sub abs 16 lt exch paperwidth sub abs 16 lt and {/papername exch def} {pop} ifelse } bind def /setpapername { /papersizedict 14 dict def papersizedict begin /papername /unknown def /Letter 8.5 inch 11.0 inch pagedimen /LetterSmall 7.68 inch 10.16 inch pagedimen /Tabloid 11.0 inch 17.0 inch pagedimen /Ledger 17.0 inch 11.0 inch pagedimen /Legal 8.5 inch 14.0 inch pagedimen /Statement 5.5 inch 8.5 inch pagedimen /Executive 7.5 inch 10.0 inch pagedimen /A3 11.69 inch 16.5 inch pagedimen /A4 8.26 inch 11.69 inch pagedimen /A4Small 7.47 inch 10.85 inch pagedimen /B4 10.125 inch 14.33 inch pagedimen /B5 7.16 inch 10.125 inch pagedimen end } bind def /papersize { papersizedict begin /Letter {lettertray letter} def /LetterSmall {lettertray lettersmall} def /Tabloid {11x17tray 11x17} def /Ledger {ledgertray ledger} def /Legal {legaltray legal} def /Statement {statementtray statement} def /Executive {executivetray executive} def /A3 {a3tray a3} def /A4 {a4tray a4} def /A4Small {a4tray a4small} def /B4 {b4tray b4} def /B5 {b5tray b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end statusdict begin stopped end } bind def /manualpapersize { papersizedict begin /Letter {letter} def /LetterSmall {lettersmall} def /Tabloid {11x17} def /Ledger {ledger} def /Legal {legal} def /Statement {statement} def /Executive {executive} def /A3 {a3} def /A4 {a4} def /A4Small {a4small} def /B4 {b4} def /B5 {b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end stopped } bind def /desperatepapersize { mark statusdict begin /setpageparams where { pop paperwidth paperheight 0 1 {setpageparams} stopped } { true } ifelse { /setpagedevice where { pop 1 dict dup begin /PageSize [ paperwidth paperheight ] def end {setpagedevice} stopped } { true } ifelse } { false } ifelse end {cleartomark true}{cleartomark false}ifelse } bind def /papersizefailure { FMAllowPaperSizeMismatch not { (The requested paper size is not available in any currently-installed tray) (Edit the PS file to "FMAllowPaperSizeMismatch true" to use default tray) FMFAILURE } if } def /DiacriticEncoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /grave /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /.notdef /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /.notdef /AE /Oslash /.notdef /.notdef /.notdef /.notdef /yen /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ordfeminine /ordmasculine /.notdef /ae /oslash /questiondown /exclamdown /logicalnot /.notdef /florin /.notdef /.notdef /guillemotleft /guillemotright /ellipsis /.notdef /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /.notdef /.notdef /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /.notdef /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron ] def /ReEncode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall 0 eq {/Encoding DiacriticEncoding def} if currentdict end } bind def FMPColor { /BEGINBITMAPCOLOR { BITMAPCOLOR} def /BEGINBITMAPCOLORc { BITMAPCOLORc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUECOLOR } def /BEGINBITMAPTRUECOLORc { BITMAPTRUECOLORc } def /BEGINBITMAPCMYK { BITMAPCMYK } def /BEGINBITMAPCMYKc { BITMAPCMYKc } def } { /BEGINBITMAPCOLOR { BITMAPGRAY} def /BEGINBITMAPCOLORc { BITMAPGRAYc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUEGRAY } def /BEGINBITMAPTRUECOLORc { BITMAPTRUEGRAYc } def /BEGINBITMAPCMYK { BITMAPCMYKGRAY } def /BEGINBITMAPCMYKc { BITMAPCMYKGRAYc } def } ifelse /K { FMPrintAllColorsAsBlack { 8 1 roll dup 1 eq 2 index 1 eq and 3 index 1 eq and not {7 {pop} repeat 0 0 0 1 0 0 0} if 8 -1 roll } if FrameCurColors astore pop combineColor } bind def /graymode true def fMLevel1 { /fmGetFlip { fMatrix2 exch get mul 0 lt { -1 } { 1 } ifelse } FmBD } if /setPatternMode { fMLevel1 { 2 index patScreenDict exch known { pop pop patScreenDict exch get aload pop freq mul 5 2 roll fMatrix2 currentmatrix 1 get 0 ne { 3 -1 roll 90 add 3 1 roll sflipx 1 fmGetFlip sflipy 2 fmGetFlip neg mul } { sflipx 0 fmGetFlip sflipy 3 fmGetFlip mul } ifelse 0 lt {exch pop} {pop} ifelse fMNegative { {neg} fmConcatProcs } if bind systemdict /setscreen get exec /FrameCurGray exch def } { /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq sangle landscape {90 add} if {/ypoint exch def /xpoint exch def /xindex xpoint 1 add 2 div bpside mul cvi def /yindex ypoint 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get 1 7 xindex 8 mod sub bitshift and 0 ne fMNegative {not} if {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen offbits offbits onbits add dup 0 ne {div} {pop pop .5} ifelse fMNegative {1.0 exch sub} if /FrameCurGray exch def } ifelse } { pop pop dup patCache exch known { patCache exch get } { dup patDict /bstring 3 -1 roll put patDict 9 PatFreq screenIndex get div dup matrix scale makepattern dup patCache 4 -1 roll 3 -1 roll put } ifelse /FrameCurGray 0 def /FrameCurPat exch def } ifelse /graymode false def combineColor } bind def /setGrayScaleMode { graymode not { /graymode true def fMLevel1 { setCurrentScreen } if } if /FrameCurGray exch def combineColor } bind def /normalize { transform round exch round exch itransform } bind def /dnormalize { dtransform round exch round exch idtransform } bind def /lnormalize { 0 dtransform exch cvi 2 idiv 2 mul 1 add exch idtransform pop } bind def /H { lnormalize setlinewidth } bind def /Z { setlinecap } bind def /PFill { graymode fMLevel1 or not { gsave 1 setgray eofill grestore } if } bind def /PStroke { graymode fMLevel1 or not { gsave 1 setgray stroke grestore } if stroke } bind def /X { fillvals exch get dup type /stringtype eq {8 1 setPatternMode} {setGrayScaleMode} ifelse } bind def /V { PFill gsave eofill grestore } bind def /Vclip { clip } bind def /Vstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /N { PStroke } bind def /Nclip { strokepath clip newpath } bind def /Nstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /M {newpath moveto} bind def /E {lineto} bind def /D {curveto} bind def /O {closepath} bind def /L { /n exch def newpath normalize moveto 2 1 n {pop normalize lineto} for } bind def /Y { L closepath } bind def /R { /y2 exch def /x2 exch def /y1 exch def /x1 exch def x1 y1 x2 y1 x2 y2 x1 y2 4 Y } bind def /rarc {rad arcto } bind def /RR { /rad exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def mark newpath { x1 y1 rad add moveto x1 y2 x2 y2 rarc x2 y2 x2 y1 rarc x2 y1 x1 y1 rarc x1 y1 x1 y2 rarc closepath } stopped {x1 y1 x2 y2 R} if cleartomark } bind def /RRR { /rad exch def normalize /y4 exch def /x4 exch def normalize /y3 exch def /x3 exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def newpath normalize moveto mark { x2 y2 x3 y3 rarc x3 y3 x4 y4 rarc x4 y4 x1 y1 rarc x1 y1 x2 y2 rarc closepath } stopped {x1 y1 x2 y2 x3 y3 x4 y4 newpath moveto lineto lineto lineto closepath} if cleartomark } bind def /C { grestore gsave R clip setCurrentScreen } bind def /CP { grestore gsave Y clip setCurrentScreen } bind def /F { FMfonts exch get [FMsetsize 0 0 FMpointsize 0 0] makefont setfont } bind def /Q { /FMpointsize exch def /FMsetsize FMpointsize def F } bind def /QQ { /FMsetsize exch def /FMpointsize exch def F } bind def /T { moveto show } bind def /RF { rotate 0 ne {-1 1 scale} if } bind def /TF { gsave moveto RF show grestore } bind def /P { moveto 0 32 3 2 roll widthshow } bind def /PF { gsave moveto RF 0 32 3 2 roll widthshow grestore } bind def /S { moveto 0 exch ashow } bind def /SF { gsave moveto RF 0 exch ashow grestore } bind def /B { moveto 0 32 4 2 roll 0 exch awidthshow } bind def /BF { gsave moveto RF 0 32 4 2 roll 0 exch awidthshow grestore } bind def /G { gsave newpath normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /Gstrk { savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 0.0 0.0 1.0 5 3 roll arc restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /Gclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GG { gsave newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /GGclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GGstrk { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /A { gsave savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 2 copy 0.0 0.0 1.0 5 3 roll arc round cvi 360 mod exch round cvi 360 mod eq {closepath} if restorematrix PStroke grestore } bind def /Aclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /Astrk { Gstrk } bind def /AA { gsave savematrix newpath 3 index 2 div add exch 4 index 2 div sub exch normalize 3 index 2 div sub exch 4 index 2 div add exch translate rotate scale 0.0 0.0 1.0 5 3 roll arc restorematrix PStroke grestore } bind def /AAclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /AAstrk { GGstrk } bind def /BEGINPRINTCODE { /FMdicttop countdictstack 1 add def /FMoptop count 7 sub def /FMsaveobject save def userdict begin /showpage {} def FMNORMALIZEGRAPHICS 3 index neg 3 index neg translate } bind def /ENDPRINTCODE { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMsaveobject restore } bind def /gn { 0 { 46 mul cf read pop 32 sub dup 46 lt {exit} if 46 sub add } loop add } bind def /cfs { /str sl string def 0 1 sl 1 sub {str exch val put} for str def } bind def /ic [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 {0 hx} {1 hx} {2 hx} {3 hx} {4 hx} {5 hx} {6 hx} {7 hx} {8 hx} {9 hx} {10 hx} {11 hx} {12 hx} {13 hx} {14 hx} {15 hx} {16 hx} {17 hx} {18 hx} {19 hx} {gn hx} {0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14} {15} {16} {17} {18} {19} {gn} {0 wh} {1 wh} {2 wh} {3 wh} {4 wh} {5 wh} {6 wh} {7 wh} {8 wh} {9 wh} {10 wh} {11 wh} {12 wh} {13 wh} {14 wh} {gn wh} {0 bl} {1 bl} {2 bl} {3 bl} {4 bl} {5 bl} {6 bl} {7 bl} {8 bl} {9 bl} {10 bl} {11 bl} {12 bl} {13 bl} {14 bl} {gn bl} {0 fl} {1 fl} {2 fl} {3 fl} {4 fl} {5 fl} {6 fl} {7 fl} {8 fl} {9 fl} {10 fl} {11 fl} {12 fl} {13 fl} {14 fl} {gn fl} ] def /ms { /sl exch def /val 255 def /ws cfs /im cfs /val 0 def /bs cfs /cs cfs } bind def 400 ms /ip { is 0 cf cs readline pop { ic exch get exec add } forall pop } bind def /rip { bis ris copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop ris gis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop gis bis copy pop dup add is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /rip4 { kis cis copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop cis mis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop mis yis copy pop dup dup add is exch cf cs readline pop { ic exch get exec add } forall pop pop yis kis copy pop 3 mul is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /wh { /len exch def /pos exch def ws 0 len getinterval im pos len getinterval copy pop pos len } bind def /bl { /len exch def /pos exch def bs 0 len getinterval im pos len getinterval copy pop pos len } bind def /s1 1 string def /fl { /len exch def /pos exch def /val cf s1 readhexstring pop 0 get def pos 1 pos len add 1 sub {im exch val put} for pos len } bind def /hx { 3 copy getinterval cf exch readhexstring pop pop } bind def /wbytes { dup dup 8 gt { pop 8 idiv mul } { 8 eq {pop} {1 eq {7 add 8 idiv} {3 add 4 idiv} ifelse} ifelse } ifelse } bind def /BEGINBITMAPBWc { 1 {} COMMONBITMAPc } bind def /BEGINBITMAPGRAYc { 8 {} COMMONBITMAPc } bind def /BEGINBITMAP2BITc { 2 {} COMMONBITMAPc } bind def /COMMONBITMAPc { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def cvtProc /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} image bitmapsave restore grestore } bind def /BEGINBITMAPBW { 1 {} COMMONBITMAP } bind def /BEGINBITMAPGRAY { 8 {} COMMONBITMAP } bind def /BEGINBITMAP2BIT { 2 {} COMMONBITMAP } bind def /COMMONBITMAP { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def cvtProc /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} image bitmapsave restore grestore } bind def /ngrayt 256 array def /nredt 256 array def /nbluet 256 array def /ngreent 256 array def fMLevel1 { /colorsetup { currentcolortransfer /gryt exch def /blut exch def /grnt exch def /redt exch def 0 1 255 { /indx exch def /cynu 1 red indx get 255 div sub def /magu 1 green indx get 255 div sub def /yelu 1 blue indx get 255 div sub def /kk cynu magu min yelu min def /u kk currentundercolorremoval exec def % /u 0 def nredt indx 1 0 cynu u sub max sub redt exec put ngreent indx 1 0 magu u sub max sub grnt exec put nbluet indx 1 0 yelu u sub max sub blut exec put ngrayt indx 1 kk currentblackgeneration exec sub gryt exec put } for {255 mul cvi nredt exch get} {255 mul cvi ngreent exch get} {255 mul cvi nbluet exch get} {255 mul cvi ngrayt exch get} setcolortransfer {pop 0} setundercolorremoval {} setblackgeneration } bind def } { /colorSetup2 { [ /Indexed /DeviceRGB 255 {dup red exch get 255 div exch dup green exch get 255 div exch blue exch get 255 div} ] setcolorspace } bind def } ifelse /fakecolorsetup { /tran 256 string def 0 1 255 {/indx exch def tran indx red indx get 77 mul green indx get 151 mul blue indx get 28 mul add add 256 idiv put} for currenttransfer {255 mul cvi tran exch get 255.0 div} exch fmConcatProcs settransfer } bind def /BITMAPCOLOR { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def fMLevel1 { colorsetup /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} {is} {is} true 3 colorimage } { colorSetup2 /is width depth wbytes string def /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {cf is readhexstring pop} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPCOLORc { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def fMLevel1 { colorsetup /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} {is} {is} true 3 colorimage } { colorSetup2 /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {ip} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPTRUECOLORc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris} {gis} {bis} true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYKc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip4 pop cis} {mis} {yis} {kis} true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUECOLOR { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf gis readhexstring pop } { cf bis readhexstring pop } true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYK { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /mis width string def /yis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf mis readhexstring pop } { cf yis readhexstring pop } { cf kis readhexstring pop } true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUEGRAYc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris gis bis width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAYc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop cis mis yis kis width cgray} image bitmapsave restore grestore } bind def /cgray { /ww exch def /k exch def /y exch def /m exch def /c exch def 0 1 ww 1 sub { /i exch def c i get m i get y i get k i get CMYKtoRGB .144 mul 3 1 roll .587 mul 3 1 roll .299 mul add add c i 3 -1 roll floor cvi put } for c } bind def /gray { /ww exch def /b exch def /g exch def /r exch def 0 1 ww 1 sub { /i exch def r i get .299 mul g i get .587 mul b i get .114 mul add add r i 3 -1 roll floor cvi put } for r } bind def /BITMAPTRUEGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf gis readhexstring pop cf bis readhexstring pop width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /yis width string def /mis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf mis readhexstring pop cf yis readhexstring pop cf kis readhexstring pop width cgray} image bitmapsave restore grestore } bind def /BITMAPGRAY { 8 {fakecolorsetup} COMMONBITMAP } bind def /BITMAPGRAYc { 8 {fakecolorsetup} COMMONBITMAPc } bind def /ENDBITMAP { } bind def end /ALDmatrix matrix def ALDmatrix currentmatrix pop /StartALD { /ALDsave save def savematrix ALDmatrix setmatrix } bind def /InALD { restorematrix } bind def /DoneALD { ALDsave restore } bind def /I { setdash } bind def /J { [] 0 setdash } bind def (5.5) FMVERSION 1 1 0 0 612 792 0 1 6 FMDOCUMENT 0 0 /Times-Italic FMFONTDEFINE 1 0 /Times-Roman FMFONTDEFINE 2 1 /Symbol FMFONTDEFINE 32 FMFILLS 0 0 FMFILL 1 0.1 FMFILL 2 0.3 FMFILL 3 0.5 FMFILL 4 0.7 FMFILL 5 0.9 FMFILL 6 0.97 FMFILL 7 1 FMFILL 8 <0f1e3c78f0e1c387> FMFILL 9 <0f87c3e1f0783c1e> FMFILL 10 FMFILL 11 FMFILL 12 <8142241818244281> FMFILL 13 <03060c183060c081> FMFILL 14 <8040201008040201> FMFILL 16 1 FMFILL 17 0.9 FMFILL 18 0.7 FMFILL 19 0.5 FMFILL 20 0.3 FMFILL 21 0.1 FMFILL 22 0.03 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(2) 316.91 564.09 T (2) 328.47 564.09 T 2 F (\050) 323.62 564.09 T (\051) 335.32 564.09 T 321.54 614.82 321.54 590.82 2 L 7 X V 0 X N 1 18.88 Q ( 1) 47.44 149.59 T FMENDPAGE FMENDDOCUMENT %%EndDocument @endspecial 1417 x @beginspecial 59 @llx 88 @lly 558 @urx 356 @ury 2880 @rwi @setspecial %%BeginDocument: /home/lynx1/majercik/papers/SAT2000/fmcharts/ptree2.ps % % Frame ps_prolog 5.5, for use with Adobe Unix Frame 5.5 products % % This ps_prolog file is Copyright (c) 1986-1996 Adobe Systems, Incoporated. % All rights reserved. This ps_prolog file may be freely copied and % distributed in conjunction with documents created using FrameMaker, % FrameMaker+SGML, FrameReader, and FrameViewer as long as this % copyright notice is preserved. /FMDocSave save def % % FrameMaker users specify the proper paper size for each print job in the % "Print" dialog's "Printer Paper Size" "Width" and "Height~ fields. If the % printer that the PS file is sent to does not support the requested paper % size, or if there is no paper tray of the proper size currently installed, % then the job will not be printed. The following flag, if set to true, will % cause the job to print on the default paper in such cases. /FMAllowPaperSizeMismatch false def % % Frame products normally print colors as their true color on a color printer % or as shades of gray, based on luminance, on a black-and white printer. The % following flag, if set to true, forces all non-white colors to print as pure % black. This has no effect on bitmap images. /FMPrintAllColorsAsBlack false def % % Frame products can either set their own line screens or use a printer's % default settings. Three flags below control this separately for no % separations, spot separations and process separations. If a flag % is true, then the default printer settings will not be changed. If it is % false, Frame products will use their own settings from a table based on % the printer's resolution. /FMUseDefaultNoSeparationScreen true def /FMUseDefaultSpotSeparationScreen true def /FMUseDefaultProcessSeparationScreen false def % % For any given PostScript printer resolution, Frame products have two sets of % screen angles and frequencies for printing process separations, which are % recomended by Adobe. The following variable chooses the higher frequencies % when set to true or the lower frequencies when set to false. This is only % effective if the appropriate FMUseDefault...SeparationScreen flag is false. /FMUseHighFrequencyScreens true def % % The following is a set of predefined optimal frequencies and angles for various % common dpi settings. This is taken from "Advances in Color Separation Using % PostScript Software Technology," from Adobe Systems (3/13/89 P.N. LPS 0043) % and corrolated with information which is in various PPD (4.0) files. % % The "dpiranges" figure is the minimum dots per inch device resolution which % can support this setting. The "low" and "high" values are controlled by the % setting of the FMUseHighFrequencyScreens flag above. The "TDot" flags control % the use of the "Yellow Triple Dot" feature whereby the frequency id divided by % three, but the dot function is "trippled" giving a block of 3x3 dots per cell. % % PatFreq is a compromise pattern frequency for ps Level 2 printers which is close % to the ideal WYSIWYG pattern frequency of 9 repetitions/inch but does not beat % (too badly) against the screen frequencies of any separations for that DPI. /dpiranges [ 2540 2400 1693 1270 1200 635 600 0 ] def /CMLowFreqs [ 100.402 94.8683 89.2289 100.402 94.8683 66.9349 63.2456 47.4342 ] def /YLowFreqs [ 95.25 90.0 84.65 95.25 90.0 70.5556 66.6667 50.0 ] def /KLowFreqs [ 89.8026 84.8528 79.8088 89.8026 84.8528 74.8355 70.7107 53.033 ] def /CLowAngles [ 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 71.5651 ] def /MLowAngles [ 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 18.4349 ] def /YLowTDot [ true true false true true false false false ] def /CMHighFreqs [ 133.87 126.491 133.843 108.503 102.523 100.402 94.8683 63.2456 ] def /YHighFreqs [ 127.0 120.0 126.975 115.455 109.091 95.25 90.0 60.0 ] def /KHighFreqs [ 119.737 113.137 119.713 128.289 121.218 89.8026 84.8528 63.6395 ] def /CHighAngles [ 71.5651 71.5651 71.5651 70.0169 70.0169 71.5651 71.5651 71.5651 ] def /MHighAngles [ 18.4349 18.4349 18.4349 19.9831 19.9831 18.4349 18.4349 18.4349 ] def /YHighTDot [ false false true false false true true false ] def /PatFreq [ 10.5833 10.0 9.4055 10.5833 10.0 10.5833 10.0 9.375 ] def % % PostScript Level 2 printers contain an "Accurate Screens" feature which can % improve process separation rendering at the expense of compute time. This % flag is ignored by PostScript Level 1 printers. /FMUseAcccurateScreens true def % % The following PostScript procedure defines the spot function that Frame % products will use for process separations. You may un-comment-out one of % the alternative functions below, or use your own. % % Dot function /FMSpotFunction {abs exch abs 2 copy add 1 gt {1 sub dup mul exch 1 sub dup mul add 1 sub } {dup mul exch dup mul add 1 exch sub }ifelse } def % % Line function % /FMSpotFunction { pop } def % % Elipse function % /FMSpotFunction { dup 5 mul 8 div mul exch dup mul exch add % sqrt 1 exch sub } def % % /FMversion (5.5) def /fMLevel1 /languagelevel where {pop languagelevel} {1} ifelse 2 lt def /FMPColor fMLevel1 { false /colorimage where {pop pop true} if } { true } ifelse def /FrameDict 400 dict def systemdict /errordict known not {/errordict 10 dict def errordict /rangecheck {stop} put} if % The readline in PS 23.0 doesn't recognize cr's as nl's on AppleTalk FrameDict /tmprangecheck errordict /rangecheck get put errordict /rangecheck {FrameDict /bug true put} put FrameDict /bug false put mark % Some PS machines read past the CR, so keep the following 3 lines together! currentfile 5 string readline 00 0000000000 cleartomark errordict /rangecheck FrameDict /tmprangecheck get put FrameDict /bug get { /readline { /gstring exch def /gfile exch def /gindex 0 def { gfile read pop dup 10 eq {exit} if dup 13 eq {exit} if gstring exch gindex exch put /gindex gindex 1 add def } loop pop gstring 0 gindex getinterval true } bind def } if /FMshowpage /showpage load def /FMquit /quit load def /FMFAILURE { 2 copy exch = = flush FMshowpage /Helvetica findfont 12 scalefont setfont 72 200 moveto show 72 220 moveto show FMshowpage FMquit } def /FMVERSION { FMversion ne { (Adobe Frame product version does not match ps_prolog! Check installation;) (also check ~/fminit and ./fminit for old versions) FMFAILURE } if } def /fmConcatProcs { /proc2 exch cvlit def/proc1 exch cvlit def/newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx }def FrameDict begin [ /ALDsave /FMdicttop /FMoptop /FMpointsize /FMsetsize /FMsaveobject /b /bitmapsave /blut /bpside /bs /bstring /bwidth /c /cf /cs /cynu /depth /edown /fh /fillvals /fw /fx /fy /g /gfile /gindex /grnt /gryt /gstring /height /hh /i /im /indx /is /k /kk /landscape /lb /len /llx /lly /m /magu /manualfeed /n /offbits /onbits /organgle /orgbangle /orgbfreq /orgbproc /orgbxfer /orgfreq /orggangle /orggfreq /orggproc /orggxfer /orghalftone /orgmatrix /orgproc /orgrangle /orgrfreq /orgrproc /orgrxfer /orgxfer /pagesave /paperheight /papersizedict /paperwidth /pos /pwid /r /rad /redt /sl /str /tran /u /urx /ury /val /width /width /ws /ww /x /x1 /x2 /xindex /xpoint /xscale /xx /y /y1 /y2 /yelu /yindex /ypoint /yscale /yy /tintGray ] { 0 def } forall /FmBD {bind def} bind def systemdict /pdfmark known systemdict /currentdistillerparams known and { /fMAcrobat true def /FmPD /pdfmark load def /FmPT /show load def currentdistillerparams /CoreDistVersion get 2000 ge { /FmPD2 /pdfmark load def /FmPA { mark exch /Dest exch 5 3 roll /View [ /XYZ null 6 -2 roll FmDC exch pop null] /DEST FmPD }FmBD } { /FmPD2 /cleartomark load def /FmPA {pop pop pop}FmBD } ifelse } { /fMAcrobat false def /FmPD /cleartomark load def /FmPD2 /cleartomark load def /FmPT /pop load def /FmPA {pop pop pop}FmBD } ifelse /FmDC { transform fMDefaultMatrix defaultmatrix itransform cvi exch cvi exch }FmBD /FmBx { dup 3 index lt {3 1 roll exch} if 1 index 4 index lt {4 -1 roll 3 1 roll exch 4 1 roll} if }FmBD /FMnone 0 def /FMcyan 1 def /FMmagenta 2 def /FMyellow 3 def /FMblack 4 def /FMcustom 5 def /fMNegative false def /FrameSepIs FMnone def /FrameSepBlack 0 def /FrameSepYellow 0 def /FrameSepMagenta 0 def /FrameSepCyan 0 def /FrameSepRed 1 def /FrameSepGreen 1 def /FrameSepBlue 1 def /FrameCurGray 1 def /FrameCurPat null def /FrameCurColors [ 0 0 0 1 0 0 0 1] def /FrameColorEpsilon .001 def /eqepsilon { sub dup 0 lt {neg} if FrameColorEpsilon le } bind def /FrameCmpColorsCMYK { 2 copy 0 get exch 0 get eqepsilon { 2 copy 1 get exch 1 get eqepsilon { 2 copy 2 get exch 2 get eqepsilon { 3 get exch 3 get eqepsilon } {pop pop false} ifelse }{pop pop false} ifelse } {pop pop false} ifelse } bind def /FrameCmpColorsRGB { 2 copy 4 get exch 0 get eqepsilon { 2 copy 5 get exch 1 get eqepsilon { 6 get exch 2 get eqepsilon }{pop pop false} ifelse } {pop pop false} ifelse } bind def /RGBtoCMYK { 1 exch sub 3 1 roll 1 exch sub 3 1 roll 1 exch sub 3 1 roll 3 copy 2 copy le { pop } { exch pop } ifelse 2 copy le { pop } { exch pop } ifelse dup dup dup 6 1 roll 4 1 roll 7 1 roll sub 6 1 roll sub 5 1 roll sub 4 1 roll } bind def /CMYKtoRGB { dup dup 4 -1 roll add 5 1 roll 3 -1 roll add 4 1 roll add 1 exch sub dup 0 lt {pop 0} if 3 1 roll 1 exch sub dup 0 lt {pop 0} if exch 1 exch sub dup 0 lt {pop 0} if exch } bind def /FrameSepInit { 1.0 RealSetgray } bind def /FrameSetSepColor { /FrameSepBlue exch def /FrameSepGreen exch def /FrameSepRed exch def /FrameSepBlack exch def /FrameSepYellow exch def /FrameSepMagenta exch def /FrameSepCyan exch def /FrameSepIs FMcustom def setCurrentScreen } bind def /FrameSetCyan { /FrameSepBlue 1.0 def /FrameSepGreen 1.0 def /FrameSepRed 0.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 1.0 def /FrameSepIs FMcyan def setCurrentScreen } bind def /FrameSetMagenta { /FrameSepBlue 1.0 def /FrameSepGreen 0.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 1.0 def /FrameSepCyan 0.0 def /FrameSepIs FMmagenta def setCurrentScreen } bind def /FrameSetYellow { /FrameSepBlue 0.0 def /FrameSepGreen 1.0 def /FrameSepRed 1.0 def /FrameSepBlack 0.0 def /FrameSepYellow 1.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMyellow def setCurrentScreen } bind def /FrameSetBlack { /FrameSepBlue 0.0 def /FrameSepGreen 0.0 def /FrameSepRed 0.0 def /FrameSepBlack 1.0 def /FrameSepYellow 0.0 def /FrameSepMagenta 0.0 def /FrameSepCyan 0.0 def /FrameSepIs FMblack def setCurrentScreen } bind def /FrameNoSep { /FrameSepIs FMnone def setCurrentScreen } bind def /FrameSetSepColors { FrameDict begin [ exch 1 add 1 roll ] /FrameSepColors exch def end } bind def /FrameColorInSepListCMYK { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsCMYK { pop true exit } if } forall dup true ne {pop false} if } bind def /FrameColorInSepListRGB { FrameSepColors { exch dup 3 -1 roll FrameCmpColorsRGB { pop true exit } if } forall dup true ne {pop false} if } bind def /RealSetgray /setgray load def /RealSetrgbcolor /setrgbcolor load def /RealSethsbcolor /sethsbcolor load def end /setgray { FrameDict begin FrameSepIs FMnone eq { RealSetgray } { FrameSepIs FMblack eq { RealSetgray } { FrameSepIs FMcustom eq FrameSepRed 0 eq and FrameSepGreen 0 eq and FrameSepBlue 0 eq and { RealSetgray } { 1 RealSetgray pop } ifelse } ifelse } ifelse end } bind def /setrgbcolor { FrameDict begin FrameSepIs FMnone eq { RealSetrgbcolor } { 3 copy [ 4 1 roll ] FrameColorInSepListRGB { FrameSepBlue eq exch FrameSepGreen eq and exch FrameSepRed eq and { 0 } { 1 } ifelse } { FMPColor { RealSetrgbcolor currentcmykcolor } { RGBtoCMYK } ifelse FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind def /sethsbcolor { FrameDict begin FrameSepIs FMnone eq { RealSethsbcolor } { RealSethsbcolor currentrgbcolor setrgbcolor } ifelse end } bind def FrameDict begin /setcmykcolor where { pop /RealSetcmykcolor /setcmykcolor load def } { /RealSetcmykcolor { 4 1 roll 3 { 3 index add 0 max 1 min 1 exch sub 3 1 roll} repeat RealSetrgbcolor pop } bind def } ifelse userdict /setcmykcolor { FrameDict begin FrameSepIs FMnone eq { RealSetcmykcolor } { 4 copy [ 5 1 roll ] FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and { 0 } { 1 } ifelse } { FrameSepIs FMblack eq {1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse RealSetgray } ifelse end } bind put fMLevel1 { /patScreenDict 7 dict dup begin <0f1e3c78f0e1c387> [ 45 { pop } {exch pop} .5 2 sqrt] FmBD <0f87c3e1f0783c1e> [ 135 { pop } {exch pop} .5 2 sqrt] FmBD [ 0 { pop } dup .5 2 ] FmBD [ 90 { pop } dup .5 2 ] FmBD <8142241818244281> [ 45 { 2 copy lt {exch} if pop} dup .75 2 sqrt] FmBD <03060c183060c081> [ 45 { pop } {exch pop} .875 2 sqrt] FmBD <8040201008040201> [ 135 { pop } {exch pop} .875 2 sqrt] FmBD end def } { /patProcDict 5 dict dup begin <0f1e3c78f0e1c387> { 3 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <0f87c3e1f0783c1e> { 3 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def <8142241818244281> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -1 -1 moveto 9 9 lineto stroke } bind def <03060c183060c081> { 1 setlinewidth -1 -1 moveto 9 9 lineto stroke 4 -4 moveto 12 4 lineto stroke -4 4 moveto 4 12 lineto stroke} bind def <8040201008040201> { 1 setlinewidth -1 9 moveto 9 -1 lineto stroke -4 4 moveto 4 -4 lineto stroke 4 12 moveto 12 4 lineto stroke} bind def end def /patDict 15 dict dup begin /PatternType 1 def /PaintType 2 def /TilingType 3 def /BBox [ 0 0 8 8 ] def /XStep 8 def /YStep 8 def /PaintProc { begin patProcDict bstring known { patProcDict bstring get exec } { 8 8 true [1 0 0 -1 0 8] bstring imagemask } ifelse end } bind def end def } ifelse /tintCMYK { 1 tintGray sub FrameCurColors 0 4 getinterval aload pop 4 index mul 5 1 roll 3 index mul 5 1 roll 2 index mul 5 1 roll mul 4 1 roll }bind def /tintRGB { 1 tintGray sub FrameCurColors 4 3 getinterval aload pop 1 exch sub 3 index mul 1 exch sub 4 1 roll 1 exch sub 2 index mul 1 exch sub 4 1 roll 1 exch sub mul 1 exch sub 3 1 roll }bind def /combineColor { /tintGray 1 1 FrameCurGray sub FrameCurColors 7 get mul sub def FrameSepIs FMnone eq { graymode fMLevel1 or not { [/Pattern [/DeviceCMYK]] setcolorspace tintCMYK FrameCurPat setcolor } { FrameCurColors 3 get 1.0 ge { tintGray RealSetgray } { fMAcrobat not FMPColor graymode and and { tintCMYK RealSetcmykcolor } { tintRGB RealSetrgbcolor } ifelse } ifelse } ifelse } { FrameCurColors 0 4 getinterval aload FrameColorInSepListCMYK { FrameSepBlack eq exch FrameSepYellow eq and exch FrameSepMagenta eq and exch FrameSepCyan eq and FrameSepIs FMcustom eq and { tintGray } { 1 } ifelse } { FrameSepIs FMblack eq {tintGray 1.0 exch sub mul 1.0 exch sub 4 1 roll pop pop pop} { FrameSepIs FMyellow eq {pop tintGray 1.0 exch sub mul 1.0 exch sub 3 1 roll pop pop} { FrameSepIs FMmagenta eq {pop pop tintGray 1.0 exch sub mul 1.0 exch sub exch pop } { FrameSepIs FMcyan eq {pop pop pop tintGray 1.0 exch sub mul 1.0 exch sub } {pop pop pop pop 1} ifelse } ifelse } ifelse } ifelse } ifelse graymode fMLevel1 or not { [/Pattern [/DeviceGray]] setcolorspace FrameCurPat setcolor } { graymode not fMLevel1 and { dup 1 lt {pop FrameCurGray} if } if RealSetgray } ifelse } ifelse } bind def /savematrix { orgmatrix currentmatrix pop } bind def /restorematrix { orgmatrix setmatrix } bind def /fMDefaultMatrix matrix def /fMatrix2 matrix def /dpi 72 0 fMDefaultMatrix defaultmatrix dtransform dup mul exch dup mul add sqrt def /freq dpi dup 72 div round dup 0 eq {pop 1} if 8 mul div def /sangle 1 0 fMDefaultMatrix defaultmatrix dtransform exch atan def sangle fMatrix2 rotate fMDefaultMatrix defaultmatrix fMatrix2 concatmatrix dup 0 get /sflipx exch def 3 get /sflipy exch def /screenIndex { 0 1 dpiranges length 1 sub { dup dpiranges exch get 1 sub dpi le {exit} {pop} ifelse } for } bind def /getCyanScreen { FMUseHighFrequencyScreens { CHighAngles CMHighFreqs} {CLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getMagentaScreen { FMUseHighFrequencyScreens { MHighAngles CMHighFreqs } {MLowAngles CMLowFreqs} ifelse screenIndex dup 3 1 roll get 3 1 roll get /FMSpotFunction load } bind def /getYellowScreen { FMUseHighFrequencyScreens { YHighTDot YHighFreqs} { YLowTDot YLowFreqs } ifelse screenIndex dup 3 1 roll get 3 1 roll get { 3 div {2 { 1 add 2 div 3 mul dup floor sub 2 mul 1 sub exch} repeat FMSpotFunction } } {/FMSpotFunction load } ifelse 0.0 exch } bind def /getBlackScreen { FMUseHighFrequencyScreens { KHighFreqs } { KLowFreqs } ifelse screenIndex get 45.0 /FMSpotFunction load } bind def /getSpotScreen { getBlackScreen } bind def /getCompositeScreen { getBlackScreen } bind def /FMSetScreen fMLevel1 { /setscreen load }{ { 8 dict begin /HalftoneType 1 def /SpotFunction exch def /Angle exch def /Frequency exch def /AccurateScreens FMUseAcccurateScreens def currentdict end sethalftone } bind } ifelse def /setDefaultScreen { fMLevel1 { FMPColor { orgrxfer cvx orggxfer cvx orgbxfer cvx orgxfer cvx setcolortransfer } { orgxfer cvx settransfer } ifelse orgfreq organgle orgproc cvx setscreen } { orghalftone sethalftone }ifelse } bind def /setCurrentScreen { FrameSepIs FMnone eq { FMUseDefaultNoSeparationScreen { setDefaultScreen } { getCompositeScreen FMSetScreen } ifelse } { FrameSepIs FMcustom eq { FMUseDefaultSpotSeparationScreen { setDefaultScreen } { getSpotScreen FMSetScreen } ifelse } { FMUseDefaultProcessSeparationScreen { setDefaultScreen } { FrameSepIs FMcyan eq { getCyanScreen FMSetScreen } { FrameSepIs FMmagenta eq { getMagentaScreen FMSetScreen } { FrameSepIs FMyellow eq { getYellowScreen FMSetScreen } { getBlackScreen FMSetScreen } ifelse } ifelse } ifelse } ifelse } ifelse } ifelse } bind def end /FMDOCUMENT { array /FMfonts exch def dup 1 gt {/#copies exch def} {pop} ifelse FrameDict begin 0 ne /manualfeed exch def /paperheight exch def /paperwidth exch def 0 ne /fMNegative exch def 0 ne /edown exch def /yscale exch def /xscale exch def fMLevel1 not { /orghalftone currenthalftone def }if FMPColor { currentcolorscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def cvlit /orgbproc exch def /orgbangle exch def /orgbfreq exch def cvlit /orggproc exch def /orggangle exch def /orggfreq exch def cvlit /orgrproc exch def /orgrangle exch def /orgrfreq exch def currentcolortransfer fMNegative { 1 1 4 { pop { 1 exch sub } fmConcatProcs 4 1 roll } for 4 copy setcolortransfer } if cvlit /orgxfer exch def cvlit /orgbxfer exch def cvlit /orggxfer exch def cvlit /orgrxfer exch def } { currentscreen cvlit /orgproc exch def /organgle exch def /orgfreq exch def currenttransfer fMNegative { { 1 exch sub } fmConcatProcs dup settransfer } if cvlit /orgxfer exch def } ifelse end } def /FMENDDOCUMENT { FMDocSave restore } def /FMBEGINPAGE { FrameDict begin /pagesave save def 3.86 setmiterlimit /landscape exch 0 ne def landscape { 90 rotate 0 exch dup /pwid exch def neg translate pop }{ pop /pwid exch def } ifelse edown { [-1 0 0 1 pwid 0] concat } if xscale yscale scale /orgmatrix matrix def gsave } def /FMENDPAGE { grestore pagesave restore end showpage } def /FMFONTDEFINE { FrameDict begin findfont ReEncode 1 index exch definefont FMfonts 3 1 roll put end } def /FMFILLS { FrameDict begin dup array /fillvals exch def dict /patCache exch def end } def /FMFILL { FrameDict begin fillvals 3 1 roll put end } def /FMNORMALIZEGRAPHICS { newpath 1 setlinewidth 0 setlinecap 0 0 0 sethsbcolor 0 setgray } bind def /FMBEGINEPSF { end /FMEPSF save def /showpage {} def FMNORMALIZEGRAPHICS [/fy /fx /fh /fw /ury /urx /lly /llx] {exch def} forall fx fw 2 div add fy fh 2 div add translate rotate fw 2 div neg fh 2 div neg translate fw urx llx sub div fh ury lly sub div scale llx neg lly neg translate /FMdicttop countdictstack 1 add def /FMoptop count def } bind def /FMENDEPSF { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMEPSF restore FrameDict begin } bind def FrameDict begin /setmanualfeed { statusdict /manualfeed true put } bind def /max {2 copy lt {exch} if pop} bind def /min {2 copy gt {exch} if pop} bind def /inch {72 mul} def /pagedimen { paperheight sub abs 16 lt exch paperwidth sub abs 16 lt and {/papername exch def} {pop} ifelse } bind def /setpapername { /papersizedict 14 dict def papersizedict begin /papername /unknown def /Letter 8.5 inch 11.0 inch pagedimen /LetterSmall 7.68 inch 10.16 inch pagedimen /Tabloid 11.0 inch 17.0 inch pagedimen /Ledger 17.0 inch 11.0 inch pagedimen /Legal 8.5 inch 14.0 inch pagedimen /Statement 5.5 inch 8.5 inch pagedimen /Executive 7.5 inch 10.0 inch pagedimen /A3 11.69 inch 16.5 inch pagedimen /A4 8.26 inch 11.69 inch pagedimen /A4Small 7.47 inch 10.85 inch pagedimen /B4 10.125 inch 14.33 inch pagedimen /B5 7.16 inch 10.125 inch pagedimen end } bind def /papersize { papersizedict begin /Letter {lettertray letter} def /LetterSmall {lettertray lettersmall} def /Tabloid {11x17tray 11x17} def /Ledger {ledgertray ledger} def /Legal {legaltray legal} def /Statement {statementtray statement} def /Executive {executivetray executive} def /A3 {a3tray a3} def /A4 {a4tray a4} def /A4Small {a4tray a4small} def /B4 {b4tray b4} def /B5 {b5tray b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end statusdict begin stopped end } bind def /manualpapersize { papersizedict begin /Letter {letter} def /LetterSmall {lettersmall} def /Tabloid {11x17} def /Ledger {ledger} def /Legal {legal} def /Statement {statement} def /Executive {executive} def /A3 {a3} def /A4 {a4} def /A4Small {a4small} def /B4 {b4} def /B5 {b5} def /unknown {unknown} def papersizedict dup papername known {papername} {/unknown} ifelse get end stopped } bind def /desperatepapersize { mark statusdict begin /setpageparams where { pop paperwidth paperheight 0 1 {setpageparams} stopped } { true } ifelse { /setpagedevice where { pop 1 dict dup begin /PageSize [ paperwidth paperheight ] def end {setpagedevice} stopped } { true } ifelse } { false } ifelse end {cleartomark true}{cleartomark false}ifelse } bind def /papersizefailure { FMAllowPaperSizeMismatch not { (The requested paper size is not available in any currently-installed tray) (Edit the PS file to "FMAllowPaperSizeMismatch true" to use default tray) FMFAILURE } if } def /DiacriticEncoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quotesingle /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /grave /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /Adieresis /Aring /Ccedilla /Eacute /Ntilde /Odieresis /Udieresis /aacute /agrave /acircumflex /adieresis /atilde /aring /ccedilla /eacute /egrave /ecircumflex /edieresis /iacute /igrave /icircumflex /idieresis /ntilde /oacute /ograve /ocircumflex /odieresis /otilde /uacute /ugrave /ucircumflex /udieresis /dagger /.notdef /cent /sterling /section /bullet /paragraph /germandbls /registered /copyright /trademark /acute /dieresis /.notdef /AE /Oslash /.notdef /.notdef /.notdef /.notdef /yen /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /ordfeminine /ordmasculine /.notdef /ae /oslash /questiondown /exclamdown /logicalnot /.notdef /florin /.notdef /.notdef /guillemotleft /guillemotright /ellipsis /.notdef /Agrave /Atilde /Otilde /OE /oe /endash /emdash /quotedblleft /quotedblright /quoteleft /quoteright /.notdef /.notdef /ydieresis /Ydieresis /fraction /currency /guilsinglleft /guilsinglright /fi /fl /daggerdbl /periodcentered /quotesinglbase /quotedblbase /perthousand /Acircumflex /Ecircumflex /Aacute /Edieresis /Egrave /Iacute /Icircumflex /Idieresis /Igrave /Oacute /Ocircumflex /.notdef /Ograve /Uacute /Ucircumflex /Ugrave /dotlessi /circumflex /tilde /macron /breve /dotaccent /ring /cedilla /hungarumlaut /ogonek /caron ] def /ReEncode { dup length dict begin { 1 index /FID ne {def} {pop pop} ifelse } forall 0 eq {/Encoding DiacriticEncoding def} if currentdict end } bind def FMPColor { /BEGINBITMAPCOLOR { BITMAPCOLOR} def /BEGINBITMAPCOLORc { BITMAPCOLORc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUECOLOR } def /BEGINBITMAPTRUECOLORc { BITMAPTRUECOLORc } def /BEGINBITMAPCMYK { BITMAPCMYK } def /BEGINBITMAPCMYKc { BITMAPCMYKc } def } { /BEGINBITMAPCOLOR { BITMAPGRAY} def /BEGINBITMAPCOLORc { BITMAPGRAYc} def /BEGINBITMAPTRUECOLOR { BITMAPTRUEGRAY } def /BEGINBITMAPTRUECOLORc { BITMAPTRUEGRAYc } def /BEGINBITMAPCMYK { BITMAPCMYKGRAY } def /BEGINBITMAPCMYKc { BITMAPCMYKGRAYc } def } ifelse /K { FMPrintAllColorsAsBlack { 8 1 roll dup 1 eq 2 index 1 eq and 3 index 1 eq and not {7 {pop} repeat 0 0 0 1 0 0 0} if 8 -1 roll } if FrameCurColors astore pop combineColor } bind def /graymode true def fMLevel1 { /fmGetFlip { fMatrix2 exch get mul 0 lt { -1 } { 1 } ifelse } FmBD } if /setPatternMode { fMLevel1 { 2 index patScreenDict exch known { pop pop patScreenDict exch get aload pop freq mul 5 2 roll fMatrix2 currentmatrix 1 get 0 ne { 3 -1 roll 90 add 3 1 roll sflipx 1 fmGetFlip sflipy 2 fmGetFlip neg mul } { sflipx 0 fmGetFlip sflipy 3 fmGetFlip mul } ifelse 0 lt {exch pop} {pop} ifelse fMNegative { {neg} fmConcatProcs } if bind systemdict /setscreen get exec /FrameCurGray exch def } { /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def freq sangle landscape {90 add} if {/ypoint exch def /xpoint exch def /xindex xpoint 1 add 2 div bpside mul cvi def /yindex ypoint 1 add 2 div bpside mul cvi def bstring yindex bwidth mul xindex 8 idiv add get 1 7 xindex 8 mod sub bitshift and 0 ne fMNegative {not} if {/onbits onbits 1 add def 1} {/offbits offbits 1 add def 0} ifelse } setscreen offbits offbits onbits add dup 0 ne {div} {pop pop .5} ifelse fMNegative {1.0 exch sub} if /FrameCurGray exch def } ifelse } { pop pop dup patCache exch known { patCache exch get } { dup patDict /bstring 3 -1 roll put patDict 9 PatFreq screenIndex get div dup matrix scale makepattern dup patCache 4 -1 roll 3 -1 roll put } ifelse /FrameCurGray 0 def /FrameCurPat exch def } ifelse /graymode false def combineColor } bind def /setGrayScaleMode { graymode not { /graymode true def fMLevel1 { setCurrentScreen } if } if /FrameCurGray exch def combineColor } bind def /normalize { transform round exch round exch itransform } bind def /dnormalize { dtransform round exch round exch idtransform } bind def /lnormalize { 0 dtransform exch cvi 2 idiv 2 mul 1 add exch idtransform pop } bind def /H { lnormalize setlinewidth } bind def /Z { setlinecap } bind def /PFill { graymode fMLevel1 or not { gsave 1 setgray eofill grestore } if } bind def /PStroke { graymode fMLevel1 or not { gsave 1 setgray stroke grestore } if stroke } bind def /X { fillvals exch get dup type /stringtype eq {8 1 setPatternMode} {setGrayScaleMode} ifelse } bind def /V { PFill gsave eofill grestore } bind def /Vclip { clip } bind def /Vstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /N { PStroke } bind def /Nclip { strokepath clip newpath } bind def /Nstrk { currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /M {newpath moveto} bind def /E {lineto} bind def /D {curveto} bind def /O {closepath} bind def /L { /n exch def newpath normalize moveto 2 1 n {pop normalize lineto} for } bind def /Y { L closepath } bind def /R { /y2 exch def /x2 exch def /y1 exch def /x1 exch def x1 y1 x2 y1 x2 y2 x1 y2 4 Y } bind def /rarc {rad arcto } bind def /RR { /rad exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def mark newpath { x1 y1 rad add moveto x1 y2 x2 y2 rarc x2 y2 x2 y1 rarc x2 y1 x1 y1 rarc x1 y1 x1 y2 rarc closepath } stopped {x1 y1 x2 y2 R} if cleartomark } bind def /RRR { /rad exch def normalize /y4 exch def /x4 exch def normalize /y3 exch def /x3 exch def normalize /y2 exch def /x2 exch def normalize /y1 exch def /x1 exch def newpath normalize moveto mark { x2 y2 x3 y3 rarc x3 y3 x4 y4 rarc x4 y4 x1 y1 rarc x1 y1 x2 y2 rarc closepath } stopped {x1 y1 x2 y2 x3 y3 x4 y4 newpath moveto lineto lineto lineto closepath} if cleartomark } bind def /C { grestore gsave R clip setCurrentScreen } bind def /CP { grestore gsave Y clip setCurrentScreen } bind def /F { FMfonts exch get [FMsetsize 0 0 FMpointsize 0 0] makefont setfont } bind def /Q { /FMpointsize exch def /FMsetsize FMpointsize def F } bind def /QQ { /FMsetsize exch def /FMpointsize exch def F } bind def /T { moveto show } bind def /RF { rotate 0 ne {-1 1 scale} if } bind def /TF { gsave moveto RF show grestore } bind def /P { moveto 0 32 3 2 roll widthshow } bind def /PF { gsave moveto RF 0 32 3 2 roll widthshow grestore } bind def /S { moveto 0 exch ashow } bind def /SF { gsave moveto RF 0 exch ashow grestore } bind def /B { moveto 0 32 4 2 roll 0 exch awidthshow } bind def /BF { gsave moveto RF 0 32 4 2 roll 0 exch awidthshow grestore } bind def /G { gsave newpath normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /Gstrk { savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 0.0 0.0 1.0 5 3 roll arc restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /Gclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GG { gsave newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath PFill fill grestore } bind def /GGclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath clip newpath restorematrix } bind def /GGstrk { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath restorematrix currentlinewidth exch setlinewidth PStroke setlinewidth } bind def /A { gsave savematrix newpath 2 index 2 div add exch 3 index 2 div sub exch normalize 2 index 2 div sub exch 3 index 2 div add exch translate scale 2 copy 0.0 0.0 1.0 5 3 roll arc round cvi 360 mod exch round cvi 360 mod eq {closepath} if restorematrix PStroke grestore } bind def /Aclip { newpath savematrix normalize translate 0.0 0.0 moveto dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /Astrk { Gstrk } bind def /AA { gsave savematrix newpath 3 index 2 div add exch 4 index 2 div sub exch normalize 3 index 2 div sub exch 4 index 2 div add exch translate rotate scale 0.0 0.0 1.0 5 3 roll arc restorematrix PStroke grestore } bind def /AAclip { savematrix newpath normalize translate 0.0 0.0 moveto rotate dnormalize scale 0.0 0.0 1.0 5 3 roll arc closepath strokepath clip newpath restorematrix } bind def /AAstrk { GGstrk } bind def /BEGINPRINTCODE { /FMdicttop countdictstack 1 add def /FMoptop count 7 sub def /FMsaveobject save def userdict begin /showpage {} def FMNORMALIZEGRAPHICS 3 index neg 3 index neg translate } bind def /ENDPRINTCODE { count -1 FMoptop {pop pop} for countdictstack -1 FMdicttop {pop end} for FMsaveobject restore } bind def /gn { 0 { 46 mul cf read pop 32 sub dup 46 lt {exit} if 46 sub add } loop add } bind def /cfs { /str sl string def 0 1 sl 1 sub {str exch val put} for str def } bind def /ic [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0223 0 {0 hx} {1 hx} {2 hx} {3 hx} {4 hx} {5 hx} {6 hx} {7 hx} {8 hx} {9 hx} {10 hx} {11 hx} {12 hx} {13 hx} {14 hx} {15 hx} {16 hx} {17 hx} {18 hx} {19 hx} {gn hx} {0} {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14} {15} {16} {17} {18} {19} {gn} {0 wh} {1 wh} {2 wh} {3 wh} {4 wh} {5 wh} {6 wh} {7 wh} {8 wh} {9 wh} {10 wh} {11 wh} {12 wh} {13 wh} {14 wh} {gn wh} {0 bl} {1 bl} {2 bl} {3 bl} {4 bl} {5 bl} {6 bl} {7 bl} {8 bl} {9 bl} {10 bl} {11 bl} {12 bl} {13 bl} {14 bl} {gn bl} {0 fl} {1 fl} {2 fl} {3 fl} {4 fl} {5 fl} {6 fl} {7 fl} {8 fl} {9 fl} {10 fl} {11 fl} {12 fl} {13 fl} {14 fl} {gn fl} ] def /ms { /sl exch def /val 255 def /ws cfs /im cfs /val 0 def /bs cfs /cs cfs } bind def 400 ms /ip { is 0 cf cs readline pop { ic exch get exec add } forall pop } bind def /rip { bis ris copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop ris gis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop gis bis copy pop dup add is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /rip4 { kis cis copy pop is 0 cf cs readline pop { ic exch get exec add } forall pop pop cis mis copy pop dup is exch cf cs readline pop { ic exch get exec add } forall pop pop mis yis copy pop dup dup add is exch cf cs readline pop { ic exch get exec add } forall pop pop yis kis copy pop 3 mul is exch cf cs readline pop { ic exch get exec add } forall pop } bind def /wh { /len exch def /pos exch def ws 0 len getinterval im pos len getinterval copy pop pos len } bind def /bl { /len exch def /pos exch def bs 0 len getinterval im pos len getinterval copy pop pos len } bind def /s1 1 string def /fl { /len exch def /pos exch def /val cf s1 readhexstring pop 0 get def pos 1 pos len add 1 sub {im exch val put} for pos len } bind def /hx { 3 copy getinterval cf exch readhexstring pop pop } bind def /wbytes { dup dup 8 gt { pop 8 idiv mul } { 8 eq {pop} {1 eq {7 add 8 idiv} {3 add 4 idiv} ifelse} ifelse } ifelse } bind def /BEGINBITMAPBWc { 1 {} COMMONBITMAPc } bind def /BEGINBITMAPGRAYc { 8 {} COMMONBITMAPc } bind def /BEGINBITMAP2BITc { 2 {} COMMONBITMAPc } bind def /COMMONBITMAPc { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def cvtProc /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} image bitmapsave restore grestore } bind def /BEGINBITMAPBW { 1 {} COMMONBITMAP } bind def /BEGINBITMAPGRAY { 8 {} COMMONBITMAP } bind def /BEGINBITMAP2BIT { 2 {} COMMONBITMAP } bind def /COMMONBITMAP { /cvtProc exch def /depth exch def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def cvtProc /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} image bitmapsave restore grestore } bind def /ngrayt 256 array def /nredt 256 array def /nbluet 256 array def /ngreent 256 array def fMLevel1 { /colorsetup { currentcolortransfer /gryt exch def /blut exch def /grnt exch def /redt exch def 0 1 255 { /indx exch def /cynu 1 red indx get 255 div sub def /magu 1 green indx get 255 div sub def /yelu 1 blue indx get 255 div sub def /kk cynu magu min yelu min def /u kk currentundercolorremoval exec def % /u 0 def nredt indx 1 0 cynu u sub max sub redt exec put ngreent indx 1 0 magu u sub max sub grnt exec put nbluet indx 1 0 yelu u sub max sub blut exec put ngrayt indx 1 kk currentblackgeneration exec sub gryt exec put } for {255 mul cvi nredt exch get} {255 mul cvi ngreent exch get} {255 mul cvi nbluet exch get} {255 mul cvi ngrayt exch get} setcolortransfer {pop 0} setundercolorremoval {} setblackgeneration } bind def } { /colorSetup2 { [ /Indexed /DeviceRGB 255 {dup red exch get 255 div exch dup green exch get 255 div exch blue exch get 255 div} ] setcolorspace } bind def } ifelse /fakecolorsetup { /tran 256 string def 0 1 255 {/indx exch def tran indx red indx get 77 mul green indx get 151 mul blue indx get 28 mul add add 256 idiv put} for currenttransfer {255 mul cvi tran exch get 255.0 div} exch fmConcatProcs settransfer } bind def /BITMAPCOLOR { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def fMLevel1 { colorsetup /is width depth wbytes string def /cf currentfile def width height depth [width 0 0 height neg 0 height] {cf is readhexstring pop} {is} {is} true 3 colorimage } { colorSetup2 /is width depth wbytes string def /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {cf is readhexstring pop} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPCOLORc { /depth 8 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def fMLevel1 { colorsetup /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height depth [width 0 0 height neg 0 height] {ip} {is} {is} true 3 colorimage } { colorSetup2 /is im 0 lb getinterval def ws 0 lb getinterval is copy pop /cf currentfile def 7 dict dup begin /ImageType 1 def /Width width def /Height height def /ImageMatrix [width 0 0 height neg 0 height] def /DataSource {ip} bind def /BitsPerComponent depth def /Decode [0 255] def end image } ifelse bitmapsave restore grestore } bind def /BITMAPTRUECOLORc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris} {gis} {bis} true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYKc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip4 pop cis} {mis} {yis} {kis} true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUECOLOR { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf gis readhexstring pop } { cf bis readhexstring pop } true 3 colorimage bitmapsave restore grestore } bind def /BITMAPCMYK { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /mis width string def /yis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop } { cf mis readhexstring pop } { cf yis readhexstring pop } { cf kis readhexstring pop } true 4 colorimage bitmapsave restore grestore } bind def /BITMAPTRUEGRAYc { /depth 24 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /ris im 0 width getinterval def /gis im width width getinterval def /bis im width 2 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop ris gis bis width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAYc { /depth 32 def gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /lb width depth wbytes def sl lb lt {lb ms} if /bitmapsave save def /is im 0 lb getinterval def /cis im 0 width getinterval def /mis im width width getinterval def /yis im width 2 mul width getinterval def /kis im width 3 mul width getinterval def ws 0 lb getinterval is copy pop /cf currentfile def width height 8 [width 0 0 height neg 0 height] {width rip pop cis mis yis kis width cgray} image bitmapsave restore grestore } bind def /cgray { /ww exch def /k exch def /y exch def /m exch def /c exch def 0 1 ww 1 sub { /i exch def c i get m i get y i get k i get CMYKtoRGB .144 mul 3 1 roll .587 mul 3 1 roll .299 mul add add c i 3 -1 roll floor cvi put } for c } bind def /gray { /ww exch def /b exch def /g exch def /r exch def 0 1 ww 1 sub { /i exch def r i get .299 mul g i get .587 mul b i get .114 mul add add r i 3 -1 roll floor cvi put } for r } bind def /BITMAPTRUEGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /gis width string def /bis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf gis readhexstring pop cf bis readhexstring pop width gray} image bitmapsave restore grestore } bind def /BITMAPCMYKGRAY { gsave 3 index 2 div add exch 4 index 2 div add exch translate rotate 1 index 2 div neg 1 index 2 div neg translate scale /height exch def /width exch def /bitmapsave save def /is width string def /yis width string def /mis width string def /kis width string def /cf currentfile def width height 8 [width 0 0 height neg 0 height] { cf is readhexstring pop cf mis readhexstring pop cf yis readhexstring pop cf kis readhexstring pop width cgray} image bitmapsave restore grestore } bind def /BITMAPGRAY { 8 {fakecolorsetup} COMMONBITMAP } bind def /BITMAPGRAYc { 8 {fakecolorsetup} COMMONBITMAPc } bind def /ENDBITMAP { } bind def end /ALDmatrix matrix def ALDmatrix currentmatrix pop /StartALD { /ALDsave save def savematrix ALDmatrix setmatrix } bind def /InALD { restorematrix } bind def /DoneALD { ALDsave restore } bind def /I { setdash } bind def /J { [] 0 setdash } bind def (5.5) FMVERSION 1 1 0 0 612 792 0 1 8 FMDOCUMENT 0 0 /Times-Roman FMFONTDEFINE 1 0 /Times-Italic FMFONTDEFINE 2 1 /Symbol FMFONTDEFINE 32 FMFILLS 0 0 FMFILL 1 0.1 FMFILL 2 0.3 FMFILL 3 0.5 FMFILL 4 0.7 FMFILL 5 0.9 FMFILL 6 0.97 FMFILL 7 1 FMFILL 8 <0f1e3c78f0e1c387> FMFILL 9 <0f87c3e1f0783c1e> FMFILL 10 FMFILL 11 FMFILL 12 <8142241818244281> FMFILL 13 <03060c183060c081> FMFILL 14 <8040201008040201> FMFILL 16 1 FMFILL 17 0.9 FMFILL 18 0.7 FMFILL 19 0.5 FMFILL 20 0.3 FMFILL 21 0.1 FMFILL 22 0.03 FMFILL 23 0 FMFILL 24 FMFILL 25 FMFILL 26 <3333333333333333> FMFILL 27 <0000ffff0000ffff> FMFILL 28 <7ebddbe7e7dbbd7e> FMFILL 29 FMFILL 30 <7fbfdfeff7fbfdfe> FMFILL 612 792 0 FMBEGINPAGE 0 FrameSetSepColors FrameNoSep 0 0 0 1 0 0 0 1 K J 3 45 610 791 C 27 45 586 779 C 0 0 0 1 0 0 0 1 K 0 18 Q 0 X (Assignments Responsible for Lea) 79.6 301.83 T (v) 321.72 301.83 T (es in P) 330.45 301.83 T (artial Decision T) 378.18 301.83 T (ree) 498.53 301.83 T 0 24 Q 0 18 Q (Leaf 1:) 91.34 270.94 T (Leaf 2:) 91.34 240.94 T (Leaf 5:) 91.34 210.94 T (Leaf 8:) 91.34 180.94 T (Leaf 13:) 91.34 150.94 T (Leaf 14:) 91.34 120.94 T (Leaf 15:) 91.34 90.94 T 1 24 Q (y) 461.83 268.95 T 0 12 Q (4) 473.01 265.95 T 0 18 Q (1) 500.9 268.95 T 0 14 Q (=) 486.01 268.95 T 1 24 Q (y) 195.83 268.95 T 0 12 Q (1) 207.01 265.95 T 0 18 Q (1) 234.9 268.95 T 0 14 Q (=) 220.01 268.95 T 1 24 Q (y) 284.49 268.95 T 0 12 Q (2) 295.68 265.95 T 0 18 Q (1) 323.57 268.95 T 0 14 Q (=) 308.68 268.95 T 1 24 Q (y) 373.16 268.95 T 0 12 Q (3) 384.35 265.95 T 0 18 Q (1) 412.24 268.95 T 0 14 Q (=) 397.34 268.95 T 1 24 Q (y) 461.83 92.95 T 0 12 Q (4) 473.01 89.95 T 0 18 Q (1) 500.9 92.95 T 0 14 Q (=) 486.01 92.95 T 1 24 Q (y) 195.83 92.95 T 0 12 Q (1) 207.01 89.95 T 0 18 Q (0) 234.9 92.95 T 0 14 Q (=) 220.01 92.95 T 1 24 Q (y) 284.49 92.95 T 0 12 Q (2) 295.68 89.95 T 0 18 Q (0) 323.57 92.95 T 0 14 Q (=) 308.68 92.95 T 1 24 Q (y) 373.16 92.95 T 0 12 Q (3) 384.35 89.95 T 0 18 Q (0) 412.24 92.95 T 0 14 Q (=) 397.34 92.95 T 1 24 Q (y) 461.83 122.28 T 0 12 Q (4) 473.01 119.28 T 0 18 Q (0) 500.9 122.28 T 0 14 Q (=) 486.01 122.28 T 1 24 Q (y) 195.83 122.28 T 0 12 Q (1) 207.01 119.28 T 0 18 Q (0) 234.9 122.28 T 0 14 Q (=) 220.01 122.28 T 1 24 Q (y) 284.49 122.28 T 0 12 Q (2) 295.68 119.28 T 0 18 Q (0) 323.57 122.28 T 0 14 Q (=) 308.68 122.28 T 1 24 Q (y) 373.16 122.28 T 0 12 Q (3) 384.35 119.28 T 0 18 Q (1) 412.24 122.28 T 0 14 Q (=) 397.34 122.28 T 1 24 Q (y) 461.83 151.61 T 0 12 Q (4) 473.01 148.61 T 0 18 Q (1) 500.9 151.61 T 0 14 Q (=) 486.01 151.61 T 1 24 Q (y) 195.83 151.61 T 0 12 Q (1) 207.01 148.61 T 0 18 Q (0) 234.9 151.61 T 0 14 Q (=) 220.01 151.61 T 1 24 Q (y) 284.49 151.61 T 0 12 Q (2) 295.68 148.61 T 0 18 Q (0) 323.57 151.61 T 0 14 Q (=) 308.68 151.61 T 1 24 Q (y) 373.16 151.61 T 0 12 Q (3) 384.35 148.61 T 0 18 Q (1) 412.24 151.61 T 0 14 Q (=) 397.34 151.61 T 1 24 Q (y) 461.83 180.95 T 0 12 Q (4) 473.01 177.95 T 0 18 Q (0) 500.9 180.95 T 0 14 Q (=) 486.01 180.95 T 1 24 Q (y) 195.83 180.95 T 0 12 Q (1) 207.01 177.95 T 0 18 Q (1) 234.9 180.95 T 0 14 Q (=) 220.01 180.95 T 1 24 Q (y) 284.49 180.95 T 0 12 Q (2) 295.68 177.95 T 0 18 Q (0) 323.57 180.95 T 0 14 Q (=) 308.68 180.95 T 1 24 Q (y) 373.16 180.95 T 0 12 Q (3) 384.35 177.95 T 0 18 Q (0) 412.24 180.95 T 0 14 Q (=) 397.34 180.95 T 1 24 Q (y) 461.83 210.28 T 0 12 Q (4) 473.01 207.28 T 0 18 Q (1) 500.9 210.28 T 0 14 Q (=) 486.01 210.28 T 1 24 Q (y) 195.83 210.28 T 0 12 Q (1) 207.01 207.28 T 0 18 Q (1) 234.9 210.28 T 0 14 Q (=) 220.01 210.28 T 1 24 Q (y) 284.49 210.28 T 0 12 Q (2) 295.68 207.28 T 0 18 Q (0) 323.57 210.28 T 0 14 Q (=) 308.68 210.28 T 1 24 Q (y) 373.16 210.28 T 0 12 Q (3) 384.35 207.28 T 0 18 Q (1) 412.24 210.28 T 0 14 Q (=) 397.34 210.28 T 1 24 Q (y) 461.83 239.61 T 0 12 Q (4) 473.01 236.61 T 0 18 Q (0) 500.9 239.61 T 0 14 Q (=) 486.01 239.61 T 1 24 Q (y) 195.83 239.61 T 0 12 Q (1) 207.01 236.61 T 0 18 Q (1) 234.9 239.61 T 0 14 Q (=) 220.01 239.61 T 1 24 Q (y) 284.49 239.61 T 0 12 Q (2) 295.68 236.61 T 0 18 Q (1) 323.57 239.61 T 0 14 Q (=) 308.68 239.61 T 1 24 Q (y) 373.16 239.61 T 0 12 Q (3) 384.35 236.61 T 0 18 Q (1) 412.24 239.61 T 0 14 Q (=) 397.34 239.61 T 0 18 Q (Original F) 58.34 342.83 T (ormula:) 132.57 342.83 T 1 20.33 Q (x) 216.02 342.13 T 0 16 Q (1) 225.88 335.63 T 1 20.33 Q (x) 256.86 342.13 T 0 16 Q (3) 266.71 335.63 T 1 20.33 Q (y) 297.62 342.13 T 0 16 Q (3) 307.42 335.63 T 2 20.33 Q (\332) 238.97 342.13 T (\332) 279.8 342.13 T (\050) 207.26 342.13 T (\051) 316.87 342.13 T 1 F (x) 333.6 342.13 T 0 16 Q (2) 343.46 335.63 T 1 20.33 Q (x) 374.43 342.13 T 0 16 Q (4) 384.29 335.63 T 1 20.33 Q (y) 415.2 342.13 T 0 16 Q (2) 424.99 335.63 T 2 20.33 Q (\332) 356.54 342.13 T (\332) 397.37 342.13 T (\050) 324.83 342.13 T (\051) 434.44 342.13 T 1 F (x) 451.17 342.13 T 0 16 Q (3) 461.03 335.63 T 1 20.33 Q (y) 491.94 342.13 T 0 16 Q (1) 501.74 335.63 T 1 20.33 Q (y) 532.65 342.13 T 0 16 Q (4) 542.45 335.63 T 2 20.33 Q (\332) 474.11 342.13 T (\332) 514.82 342.13 T (\050) 442.41 342.13 T (\051) 551.89 342.13 T 218.06 354.04 223.63 354.04 2 L 0.92 H 2 Z N 299.66 354.04 305.11 354.04 2 L N 335.63 354.04 341.2 354.04 2 L N 376.46 354.04 382.03 354.04 2 L N 534.68 354.04 540.13 354.04 2 L N 3 45 610 791 C 0 0 612 792 C 0 0 0 1 0 0 0 1 K FMENDPAGE FMENDDOCUMENT %%EndDocument @endspecial 246 4355 a Fw(Figur)l(e)h(4.)39 b Fv(Randomized)22 b(lo)r(cal)k(searc)n(h)f(can)f(b)r(e)g(applied)g(to)g(a)h(partial)g(p)r (olicy)f(tree,)h(obtained)246 4446 y(b)n(y)f(sampling,)i(to)g(pro)n (vide)g(an)f(appro)n(ximate)g(answ)n(er)i(for)f(an)g Ft(SSa)-5 b(t)25 b Fv(instance.)i(Bo)n(xes)f(in)g(the)246 4537 y(\014gure)f(represen)n(t)h(decision)g(no)r(des)g(and)f(circles)j (random)c(no)r(des.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e (p.24)p eop %%Page: 25 25 25 24 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(25)246 283 y Fi(sampleevalssat)n Fz(\()p Fl(\036;)15 b(Q;)g(W)e Fz(\))27 b(:=)e Fm(f)342 391 y Fz(if)k Fl(\036)h Fz(is)g(the)g(empt)m(y)h(set,)g(return)f(1)342 499 y(if)f Fl(\036)h Fz(con)m(tains)h(an)f(empt)m(y)h(clause,)f(return) g(0)342 607 y(if)f Fl(Q)p Fz(\()p Fl(x)584 621 y Fn(1)624 607 y Fz(\))c(=)g Fm(9)p Fz(,)30 b(return)534 715 y(max\()p Fi(sampleevalssat)o Fz(\()p Fl(\036)p Fm(d)1421 729 y Fk(x)1461 738 y Fe(1)1496 729 y Fn(=0)1590 715 y Fl(;)15 b(Q;)g(W)e Fz(\))p Fl(;)740 823 y Fi(sampleevalssat)o Fz(\()p Fl(\036)p Fm(d)1423 837 y Fk(x)1463 846 y Fe(1)1498 837 y Fn(=1)1593 823 y Fl(;)i(Q;)g(W)e Fz(\)\))342 931 y(else)30 b(if)f Fl(Q)p Fz(\()p Fl(x)756 945 y Fn(1)796 931 y Fz(\))c(=)1011 868 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1011 868 a Fi(R)1070 868 y currentpoint grestore moveto 1070 868 a 1011 931 a Fz(,)31 b Fm(f)438 1039 y Fz(Let)f Fl(W)686 1053 y Fn(0)756 1039 y Fz(b)s(e)g(the)g(samples)g(in)f Fl(W)43 b Fz(that)31 b(assign)e Fl(x)2128 1053 y Fn(1)2193 1039 y Fz(=)c(0)438 1147 y(Let)30 b Fl(W)686 1161 y Fn(1)756 1147 y Fz(b)s(e)g(the)g(samples)g(in)f Fl(W)43 b Fz(that)31 b(assign)e Fl(x)2128 1161 y Fn(1)2193 1147 y Fz(=)c(1)438 1274 y(return)726 1229 y Fx(j)p Fk(W)813 1238 y Fe(0)847 1229 y Fx(j)p 726 1253 141 4 v 738 1306 a(j)p Fk(W)10 b Fx(j)876 1274 y Fi(sampleevalssat)o Fz(\()p Fl(\036)p Fm(d)1559 1288 y Fk(x)1599 1297 y Fe(1)1634 1288 y Fn(=0)1729 1274 y Fl(;)15 b(Q;)g(W)1967 1288 y Fn(0)2006 1274 y Fz(\)+)729 1372 y Fx(j)p Fk(W)816 1381 y Fe(1)850 1372 y Fx(j)p 729 1396 V 742 1450 a(j)p Fk(W)10 b Fx(j)880 1417 y Fi(sampleevalssat)o Fz(\()p Fl(\036)p Fm(d)1563 1431 y Fk(x)1603 1440 y Fe(1)1638 1431 y Fn(=1)1732 1417 y Fl(;)15 b(Q;)g(W)1970 1431 y Fn(1)2010 1417 y Fz(\))342 1542 y Fm(g)246 1650 y(g)246 1884 y Fw(Figur)l(e)41 b(5.)d Fv(A)h(sto)r(c)n(hastic)i(sampling)f(algorithm)g(for)h Ft(SSa)-5 b(t)39 b Fv(problems)h(c)n(ho)r(oses)h(the)e(b)r(est)246 1974 y(appro)n(ximate)25 b(v)l(alue.)246 2245 y Fz(the)30 b(p)s(olicy)f(is)g(estimated)i(as)f(the)h(prop)s(ortion)d(of)j(the)f Fl(w)j Fz(sampled)c(lea)m(v)m(es)j(that)246 2353 y(are)g(satis\014ed)f (b)m(y)g(the)h(p)s(olicy)-8 b(.)31 b(Let)h Fl(v)j Fz(b)s(e)c(the)h (true)g(v)-5 b(alue)31 b(of)h(the)g(p)s(olicy)e(and)k(^)-48 b Fl(v)246 2461 y Fz(b)s(e)29 b(the)i(estimate)g(found)e(via)h (sampling.)246 2684 y(LEMMA)h(4.)64 b Fj(L)-5 b(et)31 b Fl(\017)25 b(>)g Fz(0)31 b Fj(b)-5 b(e)30 b(a)h(tar)-5 b(get)31 b(appr)-5 b(oximation)34 b(err)-5 b(or.)32 b(The)f(pr)-5 b(ob)g(ability)246 2791 y(that)34 b Fm(j)p Fl(v)23 b Fm(\000)h Fz(^)-49 b Fl(v)t Fm(j)25 b Fl(>)g(\017)32 b Fj(is)h(less)g(than)h Fz(2)p Fl(e)1442 2758 y Fx(\000)p Fn(2)p Fk(\017)1561 2735 y Fe(2)1596 2758 y Fk(w)1652 2791 y Fj(.)362 3014 y Fz(This)28 b(is)i(a)h(direct)e(application)g(of) i(a)f(Cherno\013)g(b)s(ound.)362 3122 y(Th)m(us,)38 b(to)i(b)s(e)e (sure)g(with)g(probabilit)m(y)e(1)26 b Fm(\000)g Fl(\016)42 b Fz(of)d(ha)m(ving)g(an)g(estimate)g(no)246 3230 y(further)28 b(than)h Fl(\017)g Fz(a)m(w)m(a)m(y)i(from)e(the)g(true)g(v)-5 b(alue,)29 b Fl(w)f Fz(=)d Fl(O)s Fz(\()2193 3194 y Fn(1)p 2179 3209 64 4 v 2179 3263 a Fk(\017)2208 3244 y Fe(2)2268 3230 y Fz(log)q(\()2430 3194 y Fn(1)p 2430 3209 36 4 v 2431 3261 a Fk(\016)2476 3230 y Fz(\)\))k(samples)g(are)246 3337 y(su\016cien)m(t.)j(Note)i(that)f(this)e(n)m(um)m(b)s(er)g(do)s (es)h(not)h(dep)s(end)e(on)h Fl(n)p Fz(,)g Fl(m)p Fz(,)h Fl(k)s Fz(,)g(or)f(ho)m(w)246 3445 y(the)e Fr(SSa)-6 b(t)29 b Fz(form)m(ula)h(or)g(the)h(p)s(olicy)d(w)m(as)j(created!)362 3553 y(Lemma)43 b(4)h(directly)e(giv)m(es)i(an)f(appro)m(ximation)f (algorithm)h(for)g Fr(Majsa)-6 b(t)246 3661 y Fz(and)39 b(can)i(b)s(e)f(extended)g(to)h(solv)m(e)g Fr(SSa)-6 b(t)39 b Fz(problems)f(via)i(systematic)h(searc)m(h,)246 3769 y(describ)s(ed)28 b(next.)362 3877 y(Giv)m(en)h(a)g(sample)f Fl(W)42 b Fz(of)29 b(size)g Fl(w)i Fz(of)e(assignmen)m(ts)g(to)h(the)f (randomized)f(v)-5 b(ari-)246 3985 y(ables)30 b(in)f(an)h Fr(SSa)-6 b(t)29 b Fz(form)m(ula,)h(it)g(is)f(straigh)m(tforw)m(ard)i (to)g(recursiv)m(ely)e(compute)246 4093 y(the)39 b(v)-5 b(alue)38 b(of)i(the)f(p)s(olicy)e(with)h(the)h(largest)g(appro)m (ximate)g(v)-5 b(alue.)39 b(Figure)g(5)246 4201 y(sk)m(etc)m(hes)29 b(this)d(pro)s(cedure;)g(as)i(b)s(efore)e Fl(x)1643 4215 y Fn(1)1710 4201 y Fz(refers)h(to)h(the)f(outermost)h(quan)m(ti\014ed) 246 4309 y(v)-5 b(ariable.)362 4417 y(The)27 b(running)d(time)j(of)h Fi(sampleevalssat)e Fz(is)g Fl(w)17 b Fm(\001)d Fz(2)2021 4384 y Fk(E)2081 4417 y Fz(,)27 b(where)g Fl(E)33 b Fz(is)26 b(the)h(n)m(um)m(b)s(er)246 4525 y(of)36 b(existen)m(tial)f(v)-5 b(ariables)34 b(in)h(the)h(form)m(ula.)f(The)h(next)g(theorem)g(sho)m (ws)f(that,)246 4633 y(if)f(the)i(random)f(sample)f Fl(W)48 b Fz(is)35 b(large)h(enough,)f Fi(sampleevalssat)f Fz(giv)m(es)i(a)g (go)s(o)s(d)246 4741 y(appro)m(ximation)c(to)i(the)g(true)f(v)-5 b(alue)33 b(of)h(the)f(giv)m(en)h Fr(SSa)-6 b(t)32 b Fz(form)m(ula)g(with)g(high)246 4848 y(probabilit)m(y)-8 b(.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.25)p eop %%Page: 26 26 26 25 bop 246 100 a Fz(26)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(THEOREM)30 b(5.)70 b Fj(L)-5 b(et)36 b Fz(\()p Fl(\036;)15 b(Q;)g(\022)s Fz(\))37 b Fj(b)-5 b(e)35 b(an)i(instanc)-5 b(e)36 b(of)g Fr(SSa)-6 b(t)35 b Fj(and)h Fl(W)48 b Fj(b)-5 b(e)36 b(a)g(size)246 399 y Fl(w)d Fj(r)-5 b(andom)33 b(sample)f(of)f(the)g(assignments)h(to)f (the)g(r)-5 b(andomize)g(d)34 b(variables)e(in)e Fl(\036)p Fj(.)246 506 y(L)-5 b(et)29 b Fl(P)43 b Fj(b)-5 b(e)29 b(the)h(numb)-5 b(er)30 b(of)f(p)-5 b(ossible)31 b(p)-5 b(olicies)30 b(for)g(the)g Fr(SSa)-6 b(t)28 b Fj(instanc)-5 b(e.)30 b(L)-5 b(et)30 b Fl(v)f Fz(=)246 614 y Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))36 b Fj(b)-5 b(e)35 b(the)g(true)g (value)f(of)h(the)g Fr(SSa)-6 b(t)34 b Fj(formula)i(and)j Fz(^)-48 b Fl(v)38 b Fj(b)-5 b(e)34 b(the)h(estimate)246 722 y(found)41 b(via)h Fi(sampleevalssat)o Fj(.)e(L)-5 b(et)42 b Fl(\017)f(>)f Fz(0)i Fj(b)-5 b(e)41 b(a)g(tar)-5 b(get)43 b(appr)-5 b(oximation)45 b(err)-5 b(or.)246 830 y(The)33 b(pr)-5 b(ob)g(ability)34 b(that)g Fm(j)p Fl(v)24 b Fm(\000)f Fz(^)-48 b Fl(v)s Fm(j)26 b Fl(>)f(\017)32 b Fj(is)h(less)g(than)g Fl(P)13 b Fz(2)p Fl(e)2136 797 y Fx(\000)p Fn(2)p Fk(\017)2255 774 y Fe(2)2291 797 y Fk(w)2347 830 y Fj(.)33 b(Thus,)973 1072 y Fl(w)28 b Fz(=)d Fl(O)1249 944 y Fa(\022)1341 1010 y Fz(1)p 1326 1051 77 4 v 1326 1134 a Fl(\017)1363 1108 y Fn(2)1427 1072 y Fz(log)1560 944 y Fa(\022)1637 1010 y Fz(1)p 1637 1051 46 4 v 1638 1134 a Fl(\016)1692 944 y Fa(\023)1779 1072 y Fz(+)1896 1010 y(1)p 1880 1051 77 4 v 1880 1134 a Fl(\017)1917 1108 y Fn(2)1982 1072 y Fz(log)q(\()p Fl(P)13 b Fz(\))2240 944 y Fa(\023)246 1309 y Fj(samples)44 b(ar)-5 b(e)44 b(su\016cient)f(to)g(b)-5 b(e)43 b(sur)-5 b(e)44 b(with)f(pr)-5 b(ob)g(ability)46 b Fz(1)28 b Fm(\000)f Fl(\016)47 b Fj(of)c(having)h(an)246 1417 y(estimate)33 b(no)g(further)h(than)f Fl(\017)g Fj(away)h(fr)-5 b(om)34 b(the)f(true)f(value.)246 1618 y Fs(Pro)s(of:)63 b Fz(Imagine)28 b(that)h(w)m(e)h(use)e(the)h(sample)f Fl(W)42 b Fz(to)29 b(appro)m(ximately)f(ev)-5 b(aluate)246 1726 y(all)28 b Fl(P)43 b Fz(p)s(olicies.)28 b(By)j(Lemma)f(4,)g(the)g(probabilit)m (y)e(that)i(an)m(y)g(giv)m(en)g(estimate)h(is)246 1834 y(o\013)i(b)m(y)g(more)h(than)f Fl(\017)g Fz(is)f(2)p Fl(e)1203 1801 y Fx(\000)p Fn(2)p Fk(\017)1322 1778 y Fe(2)1357 1801 y Fk(w)1414 1834 y Fz(.)h(Therefore,)g(the)g(probabilit) m(y)e(that)j(at)g(least)246 1942 y(one)28 b(estimate)g(out)g(of)g Fl(P)40 b Fz(is)27 b(o\013)h(b)m(y)g(more)g(than)f Fl(\017)g Fz(is)g(less)g(than)h Fl(P)13 b Fz(2)p Fl(e)2588 1909 y Fx(\000)p Fn(2)p Fk(\017)2707 1886 y Fe(2)2742 1909 y Fk(w)2798 1942 y Fz(.)28 b(If)f(all)246 2050 y Fl(P)45 b Fz(p)s(olicies)31 b(ha)m(v)m(e)j(appro)m(ximate)e(v)-5 b(alues)32 b(within)f Fl(\017)h Fz(of)h(their)f(true)g(v)-5 b(alues,)32 b(then)246 2158 y(the)38 b(maxim)m(um)e(of)i(the)g(appro)m (ximate)g(v)-5 b(alues,)38 b(that)g(is,)i(^)-48 b Fl(v)s Fz(,)38 b(cannot)h(di\013er)d(b)m(y)246 2266 y(more)30 b(than)g Fl(\017)h Fz(from)e(the)i(maxim)m(um)e(of)i(the)f(true)h(v)-5 b(alues,)29 b(that)i(is,)f Fl(v)s Fz(.)p 2958 2266 48 48 v 362 2464 a(The)36 b Fj(sample)41 b(c)-5 b(omplexity)46 b Fz(of)38 b(the)f(algorithm)f(is)g(the)h(n)m(um)m(b)s(er)f(of)h (samples)246 2572 y(needed)47 b(to)i(insure)d(an)i(accurate)i(estimate) f(of)f(the)g(v)-5 b(alue)47 b(of)i(the)f(form)m(ula)246 2680 y(with)43 b(high)g(probabilit)m(y)-8 b(.)42 b(Note)k(that)f(for)f Fr(Sa)-6 b(t)o Fz(,)45 b(the)f(n)m(um)m(b)s(er)f(of)i(p)s(olicies)d(is) 246 2788 y Fl(P)55 b Fz(=)41 b(2)516 2755 y Fk(n)564 2788 y Fz(,)f Fr(Majsa)-6 b(t)n Fz(,)41 b Fl(P)55 b Fz(=)41 b(1,)g(and)f Fr(E-Majsa)-6 b(t)o Fz(,)41 b Fl(P)54 b Fz(=)42 b(2)2314 2755 y Fk(c)2391 2788 y Fm(\024)g Fz(2)2549 2755 y Fk(n)2596 2788 y Fz(.)f(Since)e(the)246 2896 y(sample)d (complexit)m(y)i(for)f Fi(sampleevalssat)f Fz(is)h(logarithmically)e (dep)s(enden)m(t)h(on)246 3004 y Fl(P)13 b Fz(,)27 b(these)g Fr(SSa)-6 b(t)25 b Fz(problems)g(ha)m(v)m(e)j(p)s(olynomial)d(sample)h (complexit)m(y)-8 b(.)27 b(Ho)m(w)m(ev)m(er,)246 3123 y(for)22 b(Alternating)29 b Fr(SSa)-6 b(t)o Fz(,)23 b Fl(P)38 b Fz(=)25 b(\002\(2)1470 3090 y Fn(2)1505 3067 y Ff(n=)p Fe(2)1615 3123 y Fz(\),)d(so)h(Theorem)f(5)h(giv)m(es)g(an)f (exp)s(onen)m(tial)246 3231 y(sample)37 b(b)s(ound.)f(App)s(endix)g(B)i (sho)m(ws)g(that)g(this)f(is)h(a)g(necessary)g(feature)h(of)246 3339 y(the)k(algorithm)e(b)m(y)i(pro)m(viding)e(a)i(family)e(of)i Fr(SSa)-6 b(t)41 b Fz(instances)h(that)h(require)246 3447 y(an)31 b(exp)s(onen)m(tial)f(sized)g(sample)g(for)h Fi(sampleevalssat)f Fz(to)h(giv)m(e)h(accurate)g(results)246 3555 y(with)d(high)g(probabilit)m(y)-8 b(.)246 3769 y(2.3.3.)65 b Fj(R)-5 b(andomize)g(d)36 b(L)-5 b(o)g(c)g(al)35 b(Se)-5 b(ar)g(ch)246 3877 y Fz(The)37 b(basic)g(approac)m(h)h(describ)s(ed)e (in)g(the)i(previous)e(section)i(is)f(to)i(randomly)246 3985 y(sample)33 b(a)i(set)g(of)f(assignmen)m(ts)g(to)h(b)s(e)f(used)g (to)h(ev)-5 b(aluate)35 b(p)s(olicies,)d(then)i(sys-)246 4093 y(tematically)i(searc)m(h)h(the)g(space)g(of)f(p)s(olicies)e(to)k (\014nd)c(one)j(with)e(the)i(b)s(est)f(ap-)246 4201 y(pro)m(ximate)41 b(v)-5 b(alue.)40 b(Although,)g(for)h(man)m(y)f(problems,)g(the)h(n)m (um)m(b)s(er)e(of)i(sam-)246 4309 y(ples)e(required)g(is)h(not)h(to)s (o)h(large,)f(the)g(searc)m(h)g(for)f(the)h(b)s(est)g(p)s(olicy)e(will) f(b)s(e)246 4417 y(exp)s(onen)m(tial)29 b(in)g(the)i(n)m(um)m(b)s(er)e (of)h(existen)m(tial)g(v)-5 b(ariables)29 b(in)g(the)i(form)m(ula.)362 4525 y(Randomized)41 b(lo)s(cal)g(searc)m(h)h(can)g(b)s(e)f(used)g(to)h (substan)m(tially)e(reduce)h(the)246 4633 y(searc)m(h)d(time)f(for)g(a) h(go)s(o)s(d)f(p)s(olicy)-8 b(,)36 b(at)i(the)g(cost)g(of)g(\014nding)d (one)j(that)g(is)e(only)246 4741 y(lo)s(cally)28 b(optimal.)i(As)g (describ)s(ed)e(in)h(Section)h(2.3.2,)j(sto)s(c)m(hastic)e(sampling)d (can)246 4848 y(b)s(e)22 b(used)h(to)h(construct)g(a)f(partial)g(p)s (olicy)e(tree)j(and,)f(giv)m(en)h(a)f(p)s(olicy)-8 b(,)23 b(the)g(partial)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e (p.26)p eop %%Page: 27 27 27 26 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(27)246 291 y(p)s(olicy)29 b(tree)j(can)g(b)s(e)f(ev)-5 b(aluated)31 b(\(its)h(v)-5 b(alue)30 b(is)h(the)g(prop)s(ortion)f(of)h(the)h(sample)246 399 y(corresp)s(onding)k(to)k(satis\014ed)e(lea)m(v)m(es\).)j(The)d (goal)i(is)d(to)j(searc)m(h)g(the)f(space)g(of)246 506 y(p)s(olicies)28 b(to)j(\014nd)e(one)h(with)g(high)f(v)-5 b(alue.)362 614 y(The)24 b Fi(randevalssat)e Fz(algorithm)h(in)g (Figure)h(6)g(tak)m(es)i Fr(SSa)-6 b(t)23 b Fz(form)m(ula)g Fl(\036)h Fz(and)g(re-)246 722 y(turns)e(an)i(appro)m(ximation)f(of)h (its)f(v)-5 b(alue,)24 b(along)g(with)f(the)h(p)s(olicy)e(that)j(appro) m(xi-)246 830 y(mately)d(pro)s(duces)f(this)g(v)-5 b(alue.)22 b(The)f(algorithm's)g(lo)s(cal)h(searc)m(h)h(comp)s(onen)m(t)f(op-)246 938 y(erates)30 b(b)m(y)f(\014rst)f(con)m(v)m(erting)i(the)f(partial)f (p)s(olicy)f(tree)j(to)g(a)f(t)m(w)m(o-lev)m(el)i(Bo)s(olean)246 1046 y(form)m(ula,)38 b(called)g(a)h Fj(tr)-5 b(e)g(eSA)e(T)52 b Fz(form)m(ula)38 b(\(function)g Fi(construct)p 2432 1046 28 4 v 33 w(treesat)p 2721 1046 V 33 w(fo)m(rmula)246 1154 y Fz(in)k(Figure)h(6\).)h(As)f(illustrated)e(in)h(Figure)h(3,)h (the)g(e\013ect)h(of)e(a)h(p)s(olicy)e(on)h(a)246 1262 y(leaf)35 b(can)g(b)s(e)g(summarized)e(b)m(y)i(a)h(Bo)s(olean)f(form)m (ula,)g(called)f(a)i Fj(p)-5 b(ath)38 b(formula)p Fz(,)246 1370 y(for)32 b(that)h(leaf.)g(Eac)m(h)g(random)f(sample)f(can)i(p)s (oten)m(tially)e(pro)s(duce)h(a)h(di\013eren)m(t)246 1478 y(path)44 b(form)m(ula,)f(and)h(the)g(union)f(of)h(all)f(the)h (path)g(form)m(ulae)g(pro)s(duces)f(the)246 1586 y(treeSA)-8 b(T)24 b(form)m(ula.)f(Th)m(us,)g(on)g(one)h(lev)m(el)g(the)g(treeSA)-8 b(T)24 b(form)m(ula)f(is)f(a)i(collection)246 1694 y(of)30 b(clauses,)g(while)f(on)h(another)h(lev)m(el)f(it)g(is)f(a)i (collection)f(of)g(path)h(form)m(ulae.)362 1802 y(A)40 b(satisfying)f(assignmen)m(t)g(to)i(the)f(treeSA)-8 b(T)40 b(form)m(ula)g(corresp)s(onds)e(to)j(a)246 1910 y(setting)25 b(of)g(the)g(decision)e(v)-5 b(ariables)24 b(that)h(satis\014es)f(the)i (original)d Fr(SSa)-6 b(t)23 b Fz(form)m(ula)246 2017 y(along)h(all)f(paths)h(in)f(the)i(partial)e(p)s(olicy)g(tree.)i(In)f (general,)g(it)g(will)e(not)j(b)s(e)e(p)s(ossi-)246 2125 y(ble)i(to)i(\014nd)d(suc)m(h)i(a)h(setting,)f(so)g(the)h(c)m(hallenge) f(here)g(is)f(to)i(\014nd)d(an)i(assignmen)m(t)246 2233 y(to)35 b(the)g(treeSA)-8 b(T)35 b(form)m(ula)f(that)h(maximizes)f(the) h(n)m(um)m(b)s(er)e(of)i(path)g(form)m(ulae)246 2341 y(that)22 b(are)f(satis\014ed.)g(This)f(can)h(b)s(e)g(accomplished)f(b) m(y)h(randomized)g(lo)s(cal)f(searc)m(h.)246 2449 y(In)39 b(the)i(exp)s(erimen)m(ts)e(in)g(this)h(pap)s(er,)f(this)h(is)f(done)h (b)m(y)h(hillclim)m(bing)35 b(on)40 b(an)246 2557 y(ob)5 b(jectiv)m(e)27 b(function)e(that)i(coun)m(ts)g(b)s(oth)e(clauses)h (satis\014ed)g(and)f(path)i(form)m(ulae)246 2665 y(satis\014ed.)f(A)h (satis\014ed)e(path)i(form)m(ula)f(is)g(w)m(eigh)m(ted)h(as)g(hea)m (vily)f(as)h(the)f(n)m(um)m(b)s(er)246 2773 y(of)d(clauses)g(in)g(the)g (original)f Fr(SSa)-6 b(t)22 b Fz(form)m(ula,)h(so)g(that)h(satisfying) e(all)h(the)g(clauses)246 2881 y(in)f(a)i(path)f(form)m(ula)f(con)m (tributes)h(more)h(to)g(the)g(treeSA)-8 b(T)24 b(form)m(ula's)e(v)-5 b(alue)23 b(than)246 2989 y(satisfying)39 b(the)i(same)f(n)m(um)m(b)s (er)g(of)g(clauses)g(scattered)i(o)m(v)m(er)g(more)f(than)f(one)246 3097 y(path)31 b(form)m(ula.)g(The)g(function)g Fi(eval)p 1518 3097 V 32 w(treesat)p 1806 3097 V 33 w(fo)m(rmula)h Fz(calculates)g(the)g(v)-5 b(alue)31 b(of)246 3205 y(the)f(treeSA)-8 b(T)31 b(form)m(ula)f(using)f(this)g(ob)5 b(jectiv)m(e)31 b(function.)362 3313 y(The)f(op)s(eration)g(of)g Fi(randevalssat)f Fz(is)h(similar)d(to)k(that)g(of)g Fr(w)-8 b(alksa)i(t)28 b Fz(\(Kautz)246 3421 y(&)46 b(Selman,)f(1996\).)k(The)d(main)f (di\013erence)h(is)f(in)g(the)i(ob)5 b(jectiv)m(e)47 b(function.)246 3528 y Fr(w)-8 b(alksa)i(t)39 b Fz(searc)m(hes)k(for)e (an)g(assignmen)m(t)h(that)g(maximizes)e(the)i(n)m(um)m(b)s(er)e(of)246 3636 y(satis\014ed)k(clauses.)i(As)f(describ)s(ed)f(ab)s(o)m(v)m(e,)j (the)e(t)m(w)m(o-lev)m(el)i(structure)e(of)h(the)246 3744 y(treeSA)-8 b(T)26 b(form)m(ula)f(requires)f Fi(randevalssat)h Fz(to)h(use)g(an)f(ob)5 b(jectiv)m(e)27 b(function)e(that)246 3852 y(coun)m(ts)37 b(b)s(oth)g(clauses)g(satis\014ed)f(and)h(path)g (form)m(ulae)g(satis\014ed.)f(\(In)h(fact,)h(in)246 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(eci\014ed)e(n)m(um)m(b)s(er)g(of)i(consecutiv)m(e)246 938 y(iterations)c(\(F)-8 b(ailureLimit\))33 b(it)h(restarts)g(with)f (a)i(new)f(random)f(setting)i(of)f(the)246 1046 y(treeSA)-8 b(T)30 b(v)-5 b(ariables.)29 b(A)h(presp)s(eci\014ed)e(n)m(um)m(b)s(er) h(of)h(restarts)g(\(RestartLimit\))g(is)246 1154 y(allo)m(w)m(ed.)362 1262 y(Empirical)e(results)h(with)g Fi(randevalssat)g Fz(are)h(discussed)f(in)g(Section)h(3.4.)246 1490 y(2.3.4.)65 b Fj(R)-5 b(elate)g(d)35 b(Work)246 1598 y Fz(Man)m(y)d(other)g (researc)m(hers)h(ha)m(v)m(e)g(explored)e(appro)m(ximations)g(for)g (probabilit)m(y-)246 1706 y(based)42 b(problems.)f(Roth)i(\(1996\))j (studied)41 b(the)i(complexit)m(y)f(of)h(inference)f(in)246 1814 y(b)s(elief)30 b(net)m(w)m(orks.)j(He)g(sho)m(w)m(ed)f(that)h (computing)e(the)h(probabilit)m(y)e(of)i(a)h(query)246 1922 y(no)s(de)k(\(akin)g(to)i(ev)-5 b(aluating)37 b(a)h Fr(Majsa)-6 b(t)36 b Fz(form)m(ula\))i(is)e(#P-complete.)j(In)e(ad-)246 2030 y(dition,)47 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closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { 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UL LT0 1710 1216 M 179 1 V 179 7 V 179 75 V 180 139 V 179 153 V 179 131 V 179 136 V 179 115 V 179 115 V 179 112 V 180 115 V 179 98 V 179 53 V 179 0 V 179 -39 V 179 -71 V 179 -65 V 179 -71 V 180 -60 V 179 -55 V 179 -48 V 179 -41 V 179 -38 V 179 -38 V 179 -31 V 180 -28 V 179 -28 V 143 -20 V 1710 1216 TriD 1889 1217 TriD 2068 1224 TriD 2247 1299 TriD 2427 1438 TriD 2606 1591 TriD 2785 1722 TriD 2964 1858 TriD 3143 1973 TriD 3322 2088 TriD 3501 2200 TriD 3681 2315 TriD 3860 2413 TriD 4039 2466 TriD 4218 2466 TriD 4397 2427 TriD 4576 2356 TriD 4755 2291 TriD 4934 2220 TriD 5114 2160 TriD 5293 2105 TriD 5472 2057 TriD 5651 2016 TriD 5830 1978 TriD 6009 1940 TriD 6188 1909 TriD 6368 1881 TriD 6547 1853 TriD stroke grestore end showpage %%EndDocument @endspecial 246 1993 a Fw(Figur)l(e)37 b(7.)h Fv(F)-6 b(or)36 b(random)e Fh(n)k Fv(=)g(20)e(v)l(ariable)g(form)n(ulae)f(\()p Fh(k)40 b Fv(=)e(3)d(literals)j(p)r(er)d(clause\),)i(the)246 2084 y(di\016cult)n(y)16 b(of)i(solving)g(random)f 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3926 A 1835 1126 A 2012 1147 A 2188 1324 A 2365 1740 A 2541 2270 A 2718 2694 A 2894 3012 A 3071 3181 A 3247 3197 A 3424 3137 A 3600 3023 A 3777 2901 A 3953 2780 A 4130 2659 A 4306 2543 A 4483 2436 A 4659 2334 A 4836 2255 A 5012 2176 A 5189 2111 A 5365 2050 A 5542 1997 A 5718 1954 A 5895 1911 A 6071 1871 A 6248 1837 A 6424 1806 A 6601 1775 A 6777 1748 A LT2 5814 3626 M (E-MAJSAT \(c=10,t=7\)) Rshow 5994 3626 M 540 0 V 1835 1126 M 177 6 V 176 44 V 177 134 V 176 310 V 177 542 V 176 508 V 177 298 V 176 66 V 177 -56 V 176 -116 V 177 -130 V 176 -124 V 177 -114 V 176 -97 V 177 -84 V 176 -76 V 177 -59 V 176 -60 V 177 -47 V 176 -47 V 177 -42 V 176 -36 V 177 -34 V 176 -35 V 177 -31 V 176 -27 V 177 -28 V 176 -25 V 6174 3626 B 1835 1126 B 2012 1132 B 2188 1176 B 2365 1310 B 2541 1620 B 2718 2162 B 2894 2670 B 3071 2968 B 3247 3034 B 3424 2978 B 3600 2862 B 3777 2732 B 3953 2608 B 4130 2494 B 4306 2397 B 4483 2313 B 4659 2237 B 4836 2178 B 5012 2118 B 5189 2071 B 5365 2024 B 5542 1982 B 5718 1946 B 5895 1912 B 6071 1877 B 6248 1846 B 6424 1819 B 6601 1791 B 6777 1766 B LT3 5814 3326 M (SAT) Rshow 5994 3326 M 540 0 V 1835 1126 M 177 0 V 176 0 V 177 0 V 176 0 V 177 0 V 176 3 V 177 7 V 176 28 V 177 38 V 176 80 V 177 99 V 176 108 V 177 133 V 176 137 V 177 139 V 176 95 V 177 59 V 176 25 V 177 9 V 176 -5 V 177 -24 V 176 -21 V 177 -27 V 176 -37 V 177 -31 V 176 -31 V 177 -33 V 176 -28 V 6174 3326 C 1835 1126 C 2012 1126 C 2188 1126 C 2365 1126 C 2541 1126 C 2718 1126 C 2894 1129 C 3071 1136 C 3247 1164 C 3424 1202 C 3600 1282 C 3777 1381 C 3953 1489 C 4130 1622 C 4306 1759 C 4483 1898 C 4659 1993 C 4836 2052 C 5012 2077 C 5189 2086 C 5365 2081 C 5542 2057 C 5718 2036 C 5895 2009 C 6071 1972 C 6248 1941 C 6424 1910 C 6601 1877 C 6777 1849 C stroke grestore end showpage %%EndDocument @endspecial 246 1989 a Fw(Figur)l(e)k(10.)39 b Fv(The)27 b(p)r(eak)g(di\016cult)n(y)g(of)h(solving)g(di\013eren)n(t)f Ft(SSa)-5 b(t)26 b Fv(instances)i(via)g(DPLL)f(v)l(aries)246 2080 y(with)19 b(the)g(structure)f(of)i(the)f(op)r(erators)h(and)e(the) h(n)n(um)n(b)r(er)e(of)j(clauses.)g(The)g(plot)f(is)g(for)h(random)246 2171 y(form)n(ulae)k(with)f Fh(k)h Fv(=)d(3)j(literals)h(p)r(er)f (clause.)h Ft(Majsa)-5 b(t)23 b Fv(problems)g(app)r(ear)h(to)g(b)r(e)g (the)f(hardest,)246 2262 y(on)i(a)n(v)n(erage.)246 2548 y Fz(ing.)i(Supp)s(orting)e(this)h(observ)-5 b(ation)28 b(is)f(the)h(fact)g(that)h(Alternating)g Fr(SSa)-6 b(t)27 b Fz(and)246 2656 y Fr(E-Majsa)-6 b(t)29 b Fz(\()p Fl(c)d Fz(=)f Fl(n=)p Fz(2\))31 b(b)s(oth)f(consist)g(of)g(half)g(existen)m (tial)g(and)f(half)h(random-)246 2764 y(ized)g(quan)m(ti\014ers,)f(and) h(ha)m(v)m(e)h(v)m(ery)g(similar)d(curv)m(es)i(in)f(Figure)h(10.)362 2872 y(A)37 b(similar)e(observ)-5 b(ation)38 b(concerning)e(problem)g (di\016cult)m(y)g(and)h(quan)m(ti\014er)246 2980 y(t)m(yp)s(e)25 b(w)m(as)h(made)f(b)m(y)g(Gen)m(t)i(&)e(W)-8 b(alsh)25 b(\(1998\);)j(they)d(argue)h(that)g(random)f Fr(QBF)246 3088 y Fz(form)m(ulae)46 b(are)h(trivially)d(easy)k(b)s(ecause)f(of)g (the)g(fact)g(that)h(a)f(single)e(clause)246 3196 y(consisting)34 b(of)h(all)f(\(or)i(nearly)e(all\))h(univ)m(ersal)e(v)-5 b(ariables)34 b(can)i(b)s(e)e(solv)m(ed)h(with)246 3304 y(no)24 b(bac)m(ktrac)m(king)i(at)g(all.)d(Th)m(us,)h(in)g(spite)g(of)g (the)h(fact)h(that)f Fr(QBF)f Fz(is)g(PSP)-8 b(A)m(CE-)246 3412 y(complete,)29 b Fr(QBF)c Fl(<)g Fr(Sa)-6 b(t)28 b Fz(for)h(random)g(CNF)g(form)m(ulae.)g(This)e(e\013ect)k(w)m(as)e (also)246 3520 y(observ)m(ed)i(b)m(y)g(Cadoli,)f(Gio)m(v)-5 b(anardi,)30 b(&)h(Sc)m(haerf)g(\(1997\),)j(who)d(found)e(that)j(in-) 246 3627 y(creasing)20 b(the)h(n)m(um)m(b)s(er)f(of)h(univ)m(ersal)e (quan)m(ti\014ers)h(at)h(the)g(exp)s(ense)g(of)g(existen)m(tial)246 3735 y(quan)m(ti\014ers)29 b(mak)m(es)i(problems)e(easier)h(to)h(solv)m (e.)362 3843 y(Figure)j(11)i(sho)m(ws)f(a)g(similar)d(pattern)j(when)f (the)h(n)m(um)m(b)s(er)f(of)h(literals)e(p)s(er)246 3951 y(clause)i Fl(k)i Fz(=)c(5,)j(although)f(the)h(curv)m(es)f(for)h Fr(Majsa)-6 b(t)33 b Fz(and)i(Alternating)30 b Fr(SSa)-6 b(t)246 4059 y Fz(ha)m(v)m(e)31 b(gro)m(wn)g(closer)f(together.)246 4309 y(3.1.4.)65 b Fj(Thr)-5 b(eshold)246 4417 y Fz(With)39 b Fr(Sa)-6 b(t)o Fz(,)40 b(the)f(p)s(eak)h(hardness)e(of)i(determining) d(whether)i(an)h(instance)f(is)246 4525 y(p)s(ositiv)m(e)31 b(app)s(ears)h(to)i(corresp)s(ond)d(to)i(the)g(p)s(oin)m(t)f(at)h(whic) m(h)f(ab)s(out)g(half)f(of)i(all)246 4633 y(random)26 b(instances)h(are)h(p)s(ositiv)m(e;)f(informal)f(tests)i(supp)s(ort)e (the)h(same)h(conclu-)246 4741 y(sion)39 b(for)h Fr(Majsa)-6 b(t)o Fz(.)40 b(Figure)g(12)i(examines)e(the)h(threshold,)e(sho)m(wing) g(results)246 4848 y(for)45 b Fr(Majsa)-6 b(t)44 b Fz(for)i Fl(k)54 b Fz(=)d(3,)c Fl(n)j Fz(=)h(30,)c(v)-5 b(arying)45 b(the)h(n)m(um)m(b)s(er)f(of)h(clauses)f Fl(m)1780 5847 y Fo(paper.tex;)c(1/09/1999;)h(0:14;)e(p.34)p eop %%Page: 35 35 35 34 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(35)440 1888 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 2880 @rwi @setspecial %%BeginDocument: /u/mlittman/papers/jar00-sat/p10.ps /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -100 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Times-Roman) findfont 300 scalefont setfont 1.000 UL LTb 1710 900 M 63 0 V 4917 0 R -63 0 V -5097 0 R (10) Rshow 1710 1225 M 31 0 V 4949 0 R -31 0 V 1710 1415 M 31 0 V 4949 0 R -31 0 V 1710 1550 M 31 0 V 4949 0 R -31 0 V 1710 1655 M 31 0 V 4949 0 R -31 0 V -4949 85 R 31 0 V 4949 0 R -31 0 V -4949 73 R 31 0 V 4949 0 R -31 0 V -4949 62 R 31 0 V 4949 0 R -31 0 V -4949 56 R 31 0 V 4949 0 R -31 0 V -4949 49 R 63 0 V 4917 0 R -63 0 V -5097 0 R (100) Rshow 1710 2305 M 31 0 V 4949 0 R -31 0 V 1710 2495 M 31 0 V 4949 0 R -31 0 V 1710 2630 M 31 0 V 4949 0 R -31 0 V 1710 2735 M 31 0 V 4949 0 R -31 0 V -4949 85 R 31 0 V 4949 0 R -31 0 V -4949 73 R 31 0 V 4949 0 R -31 0 V -4949 62 R 31 0 V 4949 0 R -31 0 V -4949 56 R 31 0 V 4949 0 R -31 0 V -4949 49 R 63 0 V 4917 0 R -63 0 V -5097 0 R (1000) Rshow 1710 3385 M 31 0 V 4949 0 R -31 0 V 1710 3575 M 31 0 V 4949 0 R -31 0 V 1710 3710 M 31 0 V 4949 0 R -31 0 V 1710 3815 M 31 0 V 4949 0 R -31 0 V -4949 85 R 31 0 V 4949 0 R -31 0 V -4949 73 R 31 0 V 4949 0 R -31 0 V -4949 62 R 31 0 V 4949 0 R -31 0 V -4949 56 R 31 0 V 4949 0 R -31 0 V -4949 49 R 63 0 V 4917 0 R -63 0 V -5097 0 R (10000) Rshow 1710 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (0) Cshow 2421 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (20) Cshow 3133 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (40) Cshow 3844 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (60) Cshow 4556 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (80) Cshow 5267 900 M 0 63 V 0 3177 R 0 -63 V 0 -3477 R (100) 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b(and)e(c)m(hec)m(king)i(those)g(2)2898 473 y Fk(n)p Fx(\000)p Fn(1)246 614 y Fz(assignmen)m(ts.)f(Unit)g (resolution,)g(as)h(describ)s(ed)d(in)i(Section)g(2.1.1,)j(handles)c (this)246 722 y(extreme)31 b(case;)h(other)e(ideas)g(are)h(needed)f(to) h(address)e(this)h(more)g(generally)-8 b(.)362 830 y(Considerations)30 b(suc)m(h)h(as)h(this)e(ha)m(v)m(e)j(prompted)e(a)h(great)h(deal)e(of)g (researc)m(h)246 938 y(in)m(to)37 b(e\016cien)m(t)g(splitting)e (heuristics)g(for)i Fr(Sa)-6 b(t)36 b Fz(\(Ho)s(ok)m(er)i(&)f(Vina)m(y) -8 b(,)37 b(1994;)i(Ba-)246 1046 y(y)m(ardo)20 b(&)g(Sc)m(hrag,)h (1997;)h(Gallo)e(&)g(Urbani,)f(1989;)j(Harc)m(he,)g(Ho)s(ok)m(er,)g(&)d (Thomp-)246 1154 y(son,)37 b(1994;)j(Jeroslo)m(w)d(&)g(W)-8 b(ang,)39 b(1990;)g(Li)e(&)g(An)m(bulagan,)g(1997;)i(Cra)m(wford)246 1262 y(&)27 b(Auton,)g(1996;)j(F)-8 b(reeman,)28 b(1995\).)i(An)d (appropriate)f(splitting)f(heuristic)h(can)246 1370 y(reduce)37 b(the)g(running)e(time)i(of)h(the)f(DPLL)h(algorithm)e(b)m(y)h(sev)m (eral)h(orders)f(of)246 1478 y(magnitude.)21 b(But,)i(do)f(these)h (splitting)d(heuristics)g(impro)m(v)m(e)i(e\016ciency)h(in)e Fr(SSa)-6 b(t)246 1586 y Fz(problems?)362 1694 y(Relativ)m(e)41 b(to)h(its)e(coun)m(terpart)i(in)e Fr(Sa)-6 b(t)40 b Fz(problems,)f(splitting)g(in)h Fr(SSa)-6 b(t)39 b Fz(is)246 1802 y(restricted)20 b(to)i(selection)f(from)g(among)h(a)f(sp)s (eci\014c)f(class)h(of)g(v)-5 b(ariables:)20 b(T)-8 b(o)22 b(ensure)246 1910 y(that)27 b(the)h(v)-5 b(alue)27 b(of)g(the)g(form)m (ula)g(is)f(computed)h(correctly)-8 b(,)28 b(splitting)d(heuristics)246 2017 y(for)j Fr(SSa)-6 b(t)27 b Fz(m)m(ust)i(c)m(ho)s(ose)h(a)f(v)-5 b(ariable)28 b(from)g(the)h(\014rst)f(\(leftmost)h(or)g(outermost\))246 2125 y(blo)s(c)m(k,)h(as)g(describ)s(ed)f(in)g(Section)h(1.2.1.)362 2233 y(The)25 b(reasoning)g(b)s(ehind)e(this)h(restriction)h(is)g(b)s (ecause)g(adjacen)m(t)i(quan)m(ti\014ers)246 2341 y(comm)m(ute)34 b(if)e(they)i(are)f(of)h(the)f(same)h(t)m(yp)s(e,)g(but)e(not)i(if)e (they)i(are)f(of)h(di\013eren)m(t)246 2449 y(t)m(yp)s(es.)25 b(Returning)f(to)i(the)g(example)f(from)g(Section)g(1.2,)i(consider)d (the)i(form)m(ula)874 2555 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 874 2555 a Fi(R)933 2555 y currentpoint grestore moveto 933 2555 a 874 2618 a Fl(x;)1025 2555 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1025 2555 a Fi(R)1084 2555 y currentpoint grestore moveto 1084 2555 a 1025 2618 a Fl(z)t(;)15 b Fm(9)p Fl(y)s(;)g(E)5 b Fz([\()p Fl(x)22 b Fm(_)e Fl(y)s Fz(\)\()p Fl(y)j Fm(_)p 1804 2568 47 4 v 20 w Fl(z)t Fz(\)\()p 1920 2568 52 4 v Fl(x)e Fm(_)p 2074 2568 48 4 v 20 w Fl(y)s Fz(\)])26 b Fm(\025)f Fz(3)p Fl(=)p Fz(5)p Fl(:)246 2787 y Fz(The)e(v)-5 b(alue)24 b(of)g(the)g(form)m(ula)f(is)g (3)p Fl(=)p Fz(4,)i(and)f(this)f(can)h(b)s(e)f(determined)g(b)m(y)g (splitting)246 2895 y(either)31 b(on)h Fl(x)g Fz(or)h Fl(z)j Fz(\014rst)31 b(\(b)s(oth)h(are)h(in)e(the)h(\014rst)g(blo)s(c)m (k\).)g(On)f(the)i(other)f(hand,)246 3003 y(if)40 b Fl(y)45 b Fz(is)c(used)g(as)h(the)g(v)-5 b(ariable)40 b(in)h(the)h(\014rst)f (split,)f(the)i(v)-5 b(alue)41 b(is)g(calculated)246 3111 y(incorrectly)29 b(as)i(1)p Fl(=)p Fz(2.)362 3219 y(A)45 b(h)m(yp)s(othesis)e(is)h(that)h(splitting)d(heuristics)h(w)m (ould)h(ha)m(v)m(e)h(the)g(greatest)246 3327 y(impact)38 b(when)g(there)i(are)f(large)g(blo)s(c)m(ks)f(of)i(in)m(terc)m (hangeable)f(v)-5 b(ariables)38 b(\(the)246 3435 y(extremes)43 b(b)s(eing)e Fr(Sa)-6 b(t)41 b Fz(and)h Fr(Majsa)-6 b(t)n Fz(,)43 b(b)s(oth)e(of)i(whic)m(h)e(consist)h(of)h(a)f(single)246 3543 y(blo)s(c)m(k)24 b(of)h(size)f Fl(n)p Fz(\),)h(and)f(w)m(ould)f (not)i(b)s(e)f(as)h(useful)e(when)h(there)h(w)m(as)g(a)g(great)h(deal) 246 3651 y(of)g(quan)m(ti\014er)e(alternation)h(\(since,)h(barring)e (unit)g(clauses)i(and)f(pure)f(v)-5 b(ariables,)246 3758 y(DPLL)33 b(m)m(ust)h(split)d(on)j(all)e(v)-5 b(ariables)33 b(in)f(the)i(\014rst)f(blo)s(c)m(k)g(b)s(efore)g(splitting)e(on)246 3866 y(a)44 b(v)-5 b(ariable)42 b(from)i(another)g(blo)s(c)m(k\).)g(In) f(the)h(case)h(where)e(blo)s(c)m(ks)g(are)h(single)246 3974 y(v)-5 b(ariables,)28 b(the)h(splitting)e(heuristic)g(can)j(b)s(e) e(exp)s(ected)i(to)g(ha)m(v)m(e)g(no)f(impact)g(on)246 4082 y(p)s(erformance.)362 4190 y(T)-8 b(o)47 b(test)h(this)e(h)m(yp)s (othesis,)f(tests)j(w)m(ere)f(conducted)g(using)f(six)f(splitting)246 4298 y(heuristics)27 b(\(where,)j(in)e(all)g(cases,)j(it)e(is)g (understo)s(o)s(d)e(that)j(the)g(c)m(hoice)h(is)d(made)246 4406 y(from)i(the)g(outermost)h(blo)s(c)m(k)f(of)h(v)-5 b(ariables\):)283 4567 y Fm(\000)73 b Fi(RAND)30 b Fz(c)m(ho)s(oses)h (a)g(v)-5 b(ariable)29 b(randomly)-8 b(,)283 4741 y Fm(\000)73 b Fi(SA)-8 b(TS)p 649 4741 28 4 v 32 w(MOST)28 b Fz(\014nds)f(the)i (literal)f(that)h(satis\014es)f(the)i(most)f(clauses)f(in)g(the)427 4848 y(curren)m(t)j(form)m(ula)e(and)h(c)m(ho)s(oses)h(that)g(v)-5 b(ariable,)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.37)p eop %%Page: 38 38 38 37 bop 246 100 a Fz(38)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)283 291 y Fm(\000)73 b Fi(2S)p 528 291 28 4 v 33 w(JW)21 b Fz(\(2-sided)f(Jeroslo)m(w-W)-8 b(ang,)22 b(describ)s(ed)c(b)m(y)j(Ho)s(ok)m(er)g(&)f(Vina)m(y)g(\(1994\)\))427 399 y(c)m(ho)s(oses)28 b(a)f(v)-5 b(ariable)25 b(whose)h(literals)f (app)s(ear)h(in)f(a)i(large)g(n)m(um)m(b)s(er)e(of)h(short)427 506 y(clauses,)283 687 y Fm(\000)73 b Fi(POS)p 608 687 V 33 w(2S)p 737 687 V 32 w(JW)22 b Fz(\(p)s(ositiv)m(e,)e(2-sided)g (Jeroslo)m(w-W)-8 b(ang,)23 b(describ)s(ed)18 b(b)m(y)j(Ho)s(ok)m(er) 427 795 y(&)33 b(Vina)m(y)g(\(1994\)\))j(is)d Fi(2S)p 1316 795 V 32 w(JW)h Fz(applied)d(to)j(the)g(subset)e(of)i(v)-5 b(ariables)32 b(that)427 903 y(app)s(ear)e(in)f(at)i(least)g(one)g (clause)f(con)m(taining)g(all)f(p)s(ositiv)m(e)g(literals,)283 1083 y Fm(\000)73 b Fi(MOMS)42 b Fz(c)m(ho)s(oses)i(the)f(v)-5 b(ariable)42 b(that)h(has)g(Maxim)m(um)g(Occurrences)f(in)427 1191 y(clauses)31 b(of)f(Minim)m(um)e(Size,)i(and)283 1371 y Fm(\000)73 b Fi(MAX)p 634 1371 V 32 w(UNIT)p Fz(,)26 b(a)g(splitting)d(heuristic)h(used)h(b)m(y)g(Ba)m(y)m(ardo)i(&)f(Sc)m (hrag)g(\(1997\))427 1479 y(in)44 b(some)i(of)f(their)f(w)m(ork)h(on)g (satis\014abilit)m(y)e(algorithms,)h(is)g(similar)e(to)427 1587 y(heuristics)k(used)g(b)m(y)i(the)f(successful)f Fr(POSIT)i Fz(\(F)-8 b(reeman,)48 b(1995\))i(and)427 1695 y Fr(T)-8 b(ablea)n(u)33 b Fz(\(Cra)m(wford)h(&)g(Auton,)g(1996\)) i(solv)m(ers,)e(preferring)e(v)-5 b(ariables)427 1803 y(that)31 b(will)d(lead)i(to)h(a)g(maximal)e(n)m(um)m(b)s(er)g(of)i (unit)e(resolutions.)246 1979 y(With)e(the)g(exception)h(of)g Fi(RAND)p Fz(,)f(these)h(heuristics)e(try)h(to)h(select)h(the)e(v)-5 b(ariable)246 2087 y(that)36 b(will)e(most)j(simplify)32 b(the)37 b(form)m(ula)e(and)h(establish)e(its)i(satis\014abilit)m(y)e (\(or)246 2195 y(lac)m(k)c(thereof)7 b(\))30 b(most)f(quic)m(kly)-8 b(.)29 b Fi(SA)-8 b(TS)p 1552 2195 V 32 w(MOST)28 b Fz(attempts)i(to)g (do)g(this)e(b)m(y)h(c)m(ho)s(os-)246 2303 y(ing)36 b(the)h(literal)f (that)i(will)c(satisfy)-8 b(,)37 b(and)g(th)m(us)g(eliminate,)e(the)j (most)f(clauses.)246 2411 y(The)d(remaining)e(heuristics)h(try)h(to)h (c)m(ho)s(ose)h(a)f(v)-5 b(ariable)33 b(that)i(maximizes)f(the)246 2519 y(n)m(um)m(b)s(er)26 b(of)i(unit)e(clauses)h(lik)m(ely)f(to)i(b)s (e)f(pro)s(duced.)f(The)h(baseline)f(for)i(compar-)246 2627 y(ison)j(is)g(the)h(splitting)e(rule)h(in)g(Figure)g(2,)i(whic)m (h)e(splits)f(on)i(the)g(next)g(v)-5 b(ariable)246 2735 y(in)29 b(the)h(initially)d(sp)s(eci\014ed)i(quan)m(ti\014er)g (ordering.)362 2843 y(T)-8 b(o)23 b(test)h(the)g(p)s(erformance)e(of)h (these)h(heuristics)d(on)i(problems)e(with)h(v)-5 b(arying)246 2951 y(blo)s(c)m(k)28 b(sizes,)g(eigh)m(t)h(sets)g(of)f(100)i(random)d (form)m(ulae)i(w)m(ere)f(generated)i(with)d(the)246 3059 y(follo)m(wing)i(c)m(haracteristics:)732 3237 y(24)i(v)-5 b(ariables,)30 b(24)h(clauses,)f(3)h(literals)e(p)s(er)g(clause)732 3345 y(24)i(v)-5 b(ariables,)30 b(48)h(clauses,)f(3)h(literals)e(p)s (er)g(clause)732 3453 y(24)i(v)-5 b(ariables,)30 b(72)h(clauses,)f(3)h (literals)e(p)s(er)g(clause)732 3561 y(24)i(v)-5 b(ariables,)30 b(96)h(clauses,)f(3)h(literals)e(p)s(er)g(clause)732 3669 y(32)i(v)-5 b(ariables,)30 b(32)h(clauses,)f(3)h(literals)e(p)s (er)g(clause)732 3777 y(32)i(v)-5 b(ariables,)30 b(64)h(clauses,)f(3)h (literals)e(p)s(er)g(clause)732 3885 y(32)i(v)-5 b(ariables,)30 b(96)h(clauses,)f(3)h(literals)e(p)s(er)g(clause)732 3993 y(32)i(v)-5 b(ariables,)30 b(128)h(clauses,)g(3)f(literals)f(p)s (er)h(clause.)246 4201 y(These)20 b(sets)g(w)m(ere)h(generated)g (according)g(to)g Fm(F)1845 4157 y Fk(k)r(;n)1836 4213 y(m)1970 4201 y Fz(\(Section)g(1.4\))g(using)e Fy(makewff)p Fz(.)246 4309 y(In)42 b(order)g(to)h(ensure)f(that)h(the)g(form)m(ulae) f(generated)h(con)m(tained)g(exactly)h Fl(n)246 4417 y Fz(v)-5 b(ariables,)46 b(ho)m(w)m(ev)m(er,)j Fy(makewff)d Fz(w)m(as)i(mo)s(di\014ed)d(to)k(rep)s(eatedly)e(generate)i(a)246 4525 y(form)m(ula)35 b(with)g(the)h(desired)f(c)m(haracteristics)h(un)m (til)f(one)h(w)m(as)g(generated)i(that)246 4633 y(con)m(tained)i(all)f Fl(n)g Fz(v)-5 b(ariables.)39 b(F)-8 b(or)41 b(eac)m(h)g(blo)s(c)m(k)f (size)f(in)g(a)h(sp)s(eci\014ed)f(range)h(of)246 4741 y(blo)s(c)m(k)g(sizes)h(\(1,)i(2,)f(3,)g(4,)g(6,)g(12,)g(and)f(24)h (for)f(problems)e(with)h(24)i(v)-5 b(ariables)246 4848 y(and)29 b(1,)i(2,)g(4,)g(8,)g(16,)h(and)e(32)h(for)f(problems)e(with)h (32)j(v)-5 b(ariables\))29 b(t)m(w)m(o)j(problem)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.38)p eop %%Page: 39 39 39 38 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(39)246 291 y(t)m(yp)s(es)41 b(w)m(ere)h(generated:)h(a)f(t)m(yp)s(e)g Fl(E)k Fz(problem,)41 b(whic)m(h)f(has)h(an)g(existen)m(tially)246 399 y(quan)m(ti\014ed)30 b(outermost)j(blo)s(c)m(k,)e(and)g(an)h Fl(R)g Fz(problem,)f(whic)m(h)f (has)i(a)g(randomly)246 506 y(quan)m(ti\014ed)d(outermost)j(blo)s(c)m (k.)e(The)g(blo)s(c)m(k)h(sizes)f(c)m(hosen)h(w)m(ere)h(all)d(those)j (that)246 614 y(generate)42 b(an)f(equal)f(n)m(um)m(b)s(er)f(of)i (existen)m(tial)f(and)h(randomized)e(quan)m(ti\014ers,)246 722 y(plus)24 b Fr(Sa)-6 b(t)25 b Fz(and)h Fr(E-Majsa)-6 b(t)25 b Fz(\(blo)s(c)m(k)h(size)h(24)g(or)f(32\).)i(In)d(this)g (manner,)h(eac)m(h)i(set)246 830 y(of)33 b(100)i(24-v)-5 b(ariable)33 b(problems)f(generated)i(1400)h(instances,)e(and)g(eac)m (h)h(set)g(of)246 938 y(100)d(32-v)-5 b(ariable)31 b(problems)d (generated)k(1200)g(instances.)362 1046 y(The)43 b Fi(evalssat)g Fz(algorithm)g(w)m(as)i(implemen)m(ted)d(in)h(C)h(and)f(eac)m(h)i (splitting)246 1154 y(heuristic)33 b(w)m(as)j(tested)g(on)g(all)e (10,400)k(problem)c(instances)h(generated)h(as)g(de-)246 1262 y(scrib)s(ed)f(ab)s(o)m(v)m(e.)j(These)f(tests)h(w)m(ere)f (conducted)g(on)g(a)h(Sun)d(Ultra-1)j(running)246 1370 y(SunOS-5.7)22 b(with)g(128)i(MB)g(of)f(RAM.)h(P)m(erformance)g(w)m(as) g(measured)e(b)m(y)h(coun)m(t-)246 1478 y(ing)f(the)h(c)m(hange)g(in)f (the)h(n)m(um)m(b)s(er)e(of)i(recursiv)m(e)f(calls)g(to)i(the)f(cen)m (tral)g(function)e(in)246 1586 y(the)k(algorithm.)f(This)f(function)h (\014rst)g(tests)h(for)g(satisfaction)g(of)g(the)g(form)m(ula;)f(if)246 1694 y(the)i(curren)m(t)g(partial)f(assignmen)m(t)h(is)f(insu\016cien)m (t)f(to)j(establish)d(satisfaction)i(or)246 1802 y(unsatisfaction,)c (the)h(function)f(extends)h(the)h(partial)e(assignmen)m(t.)h(In)f(the)i (most)246 1910 y(general)36 b(case,)h(the)g(function)e(selects)i(an)f (unassigned)e(v)-5 b(ariable,)36 b(\014rst)f(setting)246 2017 y(the)30 b(v)-5 b(ariable)29 b(to)h Fy(True)f Fz(and)g(calling)g (itself)g(recursiv)m(ely)-8 b(,)29 b(and)g(then)h(setting)g(the)246 2125 y(v)-5 b(ariable)33 b(to)i Fy(False)e Fz(and)h(calling)f(itself)h (recursiv)m(ely)-8 b(.)33 b(Whenev)m(er)j(p)s(ossible,)c(as)246 2233 y(in)d(the)h(case)i(of)e(a)h(v)-5 b(ariable)29 b(in)g(a)i(unit)e (clause,)i(one)f(of)h(these)g(recursiv)m(e)f(calls)f(is)246 2341 y(eliminated.)23 b(Measuring)h(the)h(n)m(um)m(b)s(er)f(of)h(calls) f(to)i(this)e(function)g(is)g(equiv)-5 b(alen)m(t)246 2449 y(to)41 b(measuring)f(the)h(size)g(of)g(the)g(DPLL)g(tree)g (generated)h(b)m(y)f(the)g(algorithm)246 2557 y(and)h(is)h(th)m(us)g(a) g(more)h(direct)f(measure)g(of)g(the)h(e\016ciency)f(of)h(the)f (heuristic)246 2665 y(than)j(measuring,)g(for)g(example,)h(CPU)f(time.) h(F)-8 b(or)47 b(p)s(ersp)s(ectiv)m(e,)g(ho)m(w)m(ev)m(er,)246 2773 y(the)40 b(running)d(time,)k(measured)e(in)g(CPU)h(seconds,)g(for) g(a)h(non-)p Fr(Majsa)-6 b(t)38 b Fz(24-)246 2881 y(v)-5 b(ariable,)31 b(48-clause)j(problem)d(w)m(as)i(t)m(ypically)e(less)h (than)h(a)g(second.)g(Running)246 2989 y(times)g(for)g Fr(Majsa)-6 b(t)31 b Fz(problems)h(v)-5 b(aried)32 b(from)h(1)h(to)g(5) g(seconds.)f(A)h(connection)246 3097 y(b)s(et)m(w)m(een)g(these)h(t)m 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(the)g(heuristics)e(tested)246 4068 y(\()p Fi(POS)p 462 4068 28 4 v 32 w(2S)p 590 4068 V 33 w(JW)q Fz(,)k Fi(MAX)p 1023 4068 V 32 w(UNIT)p Fz(,)g Fi(SA)-8 b(TS)p 1554 4068 V 31 w(MOST)p Fz(,)42 b(and)h Fi(MOMS)p Fz(\))e(pro)m(vided)h(some)246 4176 y(impro)m(v)m(emen)m(t)36 b(in)f(p)s(erformance)g(\(reduction)g (in)g(the)h(size)f(of)i(the)f(DPLL)f(tree)246 4284 y(generated\).)28 b(F)-8 b(or)28 b(t)m(yp)s(e)f Fl(E)33 b Fz(problems,)25 b(this)h(impro)m(v)m(emen)m(t)i(w)m(as)f(p)s(ositiv)m(ely)e(cor-)246 4392 y(related)40 b(with)f(blo)s(c)m(k)h(size)h(\(see)g(Figure)f (13\(b\)\).)i(F)-8 b(or)41 b(t)m(yp)s(e)g Fl(R)g Fz(problems)e(the)246 4500 y(impro)m(v)m(emen)m(t)g(w)m(as)h(p)s(ositiv)m(ely)d(correlated)j (with)d(blo)s(c)m(k)i(size)g(up)f(to)i(a)g(blo)s(c)m(k)246 4608 y(size)26 b(of)g(12;)i(all)d(four)g(of)i(these)f(heuristics)f(sho) m(w)h(less)f(impro)m(v)m(emen)m(t)i(for)f(a)h(blo)s(c)m(k)246 4716 y(size)c(of)h(24)g(\()p Fr(Majsa)-6 b(t)22 b Fz(problems,)h(in)f (whic)m(h)g(the)i(form)m(ula)f(cannot)h(b)s(e)f(simpli\014ed)246 4824 y(b)m(y)31 b(puri\014cation\))e(than)i(they)h(do)f(for)g(a)g(blo)s (c)m(k)g(size)g(of)h(12)g(\(see)g(Figure)f(13\(a\)\).)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.39)p eop %%Page: 40 40 40 39 bop 246 100 a Fz(40)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fi(SA)-8 b(TS)p 468 291 28 4 v 31 w(MOST)24 b Fz(and)h Fi(MOMS)f Fz(w)m(ere)h(particularly)e(e\013ectiv)m (e,)k(reducing)d(the)i(n)m(um-)246 399 y(b)s(er)19 b(of)i(recursiv)m(e) e(calls)h(in)f(DPLL)h(b)m(y)h(appro)m(ximately)f(half)f(\(o)m(v)m(er)j (b)s(oth)e(problem)246 506 y(t)m(yp)s(es\))41 b(for)g(a)h(blo)s(c)m(k)f (size)g(of)h(12.)g Fi(RAND)f Fz(p)s(erformed)e(as)j(exp)s(ected)g(on)f (t)m(yp)s(e)246 614 y Fl(R)c Fz(problems,)e(neither)h(impro)m(ving)e (nor)i(degrading)g(p)s(erformance.)g Fi(2S)p 2770 614 V 33 w(JW)q Fz(,)h(a)246 722 y(heuristic)c(whic)m(h)g(migh)m(t)i(ha)m 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b(the)f(heuristics)e(that)i(impro)m(v)m(ed)g(p)s (erformance,)f(the)i(blo)s(c)m(k)e(size)h(did)246 1802 y(not)33 b(ha)m(v)m(e)g(to)h(b)s(e)e(v)m(ery)h(large)f(to)i(pro)s(duce) d(signi\014can)m(t)h(reductions)f(in)g(the)i(size)246 1910 y(of)43 b(the)g(DPLL)g(tree)h(generated.)h(F)-8 b(or)43 b(the)h(four)e(successful)g(heuristics,)f(tree)246 2017 y(size)36 b(w)m(as)h(reduced)f(as)h(so)s(on)f(as)h(blo)s(c)m(ks)f (w)m(ere)h(larger)g(than)f(a)h(single)e(v)-5 b(ariable)246 2125 y(and,)27 b(in)e(some)j(cases,)g(the)g(reduction)e(b)s(ecame)i (signi\014can)m(t)e(ev)m(en)i(for)f(relativ)m(ely)246 2233 y(small)j(blo)s(c)m(k)i(sizes.)g Fi(MAX)p 1169 2233 V 31 w(UNIT)p Fz(,)h Fi(SA)-8 b(TS)p 1689 2233 V 31 w(MOST)p Fz(,)31 b(and)h Fi(MOMS)p Fz(,)e(for)i(example,)246 2341 y(pro)m(vided)37 b(a)m(v)m(erage)k(reductions)c(in)g(tree)i(size)g(of)f (appro)m(ximately)g(23\045,)h(28\045,)246 2449 y(and)29 b(32\045,)j(resp)s(ectiv)m(ely)-8 b(,)30 b(for)g(a)h(blo)s(c)m(k)f (size)g(of)h(six.)362 2557 y(The)39 b(results)f(suggest)i(that)g(a)f (successful)f Fr(Sa)-6 b(t)38 b Fz(heuristic)g(is)g(lik)m(ely)g(to)i(b) s(e)246 2665 y(a)e(successful)g(heuristic)e(in)h Fr(Majsa)-6 b(t)37 b Fz(and)h Fr(SSa)-6 b(t)37 b Fz(problems,)g(but)g(blo)s(c)m(k)h (size)246 2773 y(is)c(a)i(limiting)d(factor.)k(In)e(the)h(case)h(where) e(blo)s(c)m(ks)g(are)h(single)e(v)-5 b(ariables,)35 b(the)246 2881 y(splitting)i(heuristic)i(has)g(no)h(impact)g(on)g(p)s (erformance.)g(It)g(is)f(imp)s(ortan)m(t)g(to)246 2989 y(note,)c(ho)m(w)m(ev)m(er,)h(that)g(the)e(tests)i(w)m(ere)f(conducted) f(on)h(small,)e(random)h(prob-)246 3097 y(lems.)h(F)-8 b(or)36 b(problems)e(with)g(more)i(structure,)f(suc)m(h)g(as)h Fr(SSa)-6 b(t)34 b Fz(form)m(ulae)i(that)246 3205 y(enco)s(de)44 b(planning)e(problems,)h(these)h(results)f(ma)m(y)i(not)g(hold.)e(In)h (addition,)246 3313 y(there)25 b(ma)m(y)g(b)s(e)f(other)h(go)s(o)s(d)g (splitting)d(heuristics)h(that)i(tak)m(e)i(adv)-5 b(an)m(tage)26 b(of)f(this)246 3421 y(structure.)38 b(F)-8 b(or)40 b(example,)f(Ma)5 b(jercik)39 b(&)f(Littman)g(\(1998\))k(rep)s(ort)c(impro)m(v)m(ed)246 3528 y(p)s(erformance)45 b(on)i(suc)m(h)f Fr(SSa)-6 b(t)45 b Fz(problems)f(using)h(a)i Fj(time-or)-5 b(der)g(e)g(d)58 b Fz(splitting)246 3636 y(heuristic)35 b(in)i(whic)m(h)f(v)-5 b(ariables)36 b(with)g(a)i(lo)m(w)m(er)g(time)f(index)g(are)h(c)m (hosen)g(\014rst.)246 3744 y(The)g(success)i(of)f Fi(SA)-8 b(TS)p 1094 3744 V 31 w(MOST)p Fz(,)39 b(whic)m(h)e(is)h (computationally)g(m)m(uc)m(h)i(simpler)246 3852 y(than)k Fi(MAX)p 679 3852 V 32 w(UNIT)g Fz(or)h Fi(MOMS)p Fz(,)e(w)m(as)j (somewhat)f(surprising)c(and)j(w)m(arran)m(ts)246 3960 y(further)29 b(study)-8 b(.)30 b(It)g(is)g(p)s(ossible)e(that)i(p)s (erformance)g(disparities)e(migh)m(t)i(app)s(ear)246 4068 y(in)e(these)i(three)g(splitting)e(heuristics)g(as)i(the)g (problem)e(size)h(increases)h(b)s(ey)m(ond)246 4176 y(that)40 b(of)g(the)g(problems)f(in)f(these)j(tests.)g(Finally)-8 b(,)38 b(it)i(is)f(p)s(ossible)e(that)k(split-)246 4284 y(ting)25 b(heuristics)f(unique)g(to)j(Extended)j Fr(SSa)-6 b(t)25 b Fz(problems)f(could)h(b)s(e)h(dev)m(elop)s(ed.)246 4392 y(Suc)m(h)g(heuristics)f(migh)m(t)i(use)g(the)g(probabilities)c (asso)s(ciated)28 b(with)e(the)h(random)246 4500 y(v)-5 b(ariables)31 b(to)i(select)f(v)-5 b(ariables)31 b(in)g(a)i(w)m(a)m(y)g (that)g(pro)m(vides)e(p)s(oten)m(tially)g(greater)246 4608 y(simpli\014cation)c(of)j(the)h(problem.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.40)p eop %%Page: 41 41 41 40 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(41)571 2168 y @beginspecial 50 @llx 50 @lly 590 @urx 569 @ury 2448 @rwi @setspecial %%BeginDocument: /u/majercik/papers/SAT2000/gnugraphs/heuristics/averages/heur_avs_r.ps /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -100 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 300 scalefont setfont 1.000 UL LTb 1530 900 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-100) Rshow 1530 1813 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-80) Rshow 1530 2727 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-60) Rshow 1530 3640 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-40) Rshow 1530 4554 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-20) Rshow 1530 5468 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0) Rshow 1530 10035 M 63 0 V 8697 0 R -63 0 V -8877 0 R (100) Rshow 1530 9122 M 63 0 V 8697 0 R -63 0 V -8877 0 R (80) Rshow 1530 8208 M 63 0 V 8697 0 R -63 0 V -8877 0 R (60) Rshow 1530 7295 M 63 0 V 8697 0 R -63 0 V -8877 0 R (40) Rshow 1530 6381 M 63 0 V 8697 0 R -63 0 V -8877 0 R (20) Rshow 1530 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (0) Cshow 2231 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (2) Cshow 2932 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (4) Cshow 3632 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (6) Cshow 4333 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (8) Cshow 5034 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (10) Cshow 5735 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (12) Cshow 6436 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (14) Cshow 7136 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (16) Cshow 7837 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (18) Cshow 8538 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (20) Cshow 9239 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (22) Cshow 9940 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (24) Cshow 1.000 UL LTa 1530 5468 M 8760 0 V 1.000 UL LTb 1530 900 M 8760 0 V 0 9135 V -8760 0 V 0 -9135 V 300 5467 M currentpoint gsave translate 90 rotate 0 0 M (AVERAGE PERCENT INCREASE IN DPLL TREE SIZE) Cshow grestore 5910 150 M (BLOCK SIZE) Cshow 2.000 UP 1.000 UL LT0 8904 9822 M (RANDOM \(RAND\)) Rshow 9084 9822 M 846 0 V 10 -4451 R 5735 5624 L 3632 5203 L -700 353 V 2581 5342 L -350 109 V -351 17 V 9940 5371 Pls 5735 5624 Pls 3632 5203 Pls 2932 5556 Pls 2581 5342 Pls 2231 5451 Pls 1880 5468 Pls 9507 9822 Pls 2.000 UP 1.000 UL LT1 8904 9522 M (2-SIDED JEROSLOW-WANG \(2S_JW\)) Rshow 9084 9522 M 846 0 V 10 -4215 R 5735 5465 L 3632 5112 L -700 542 V 2581 5244 L -350 180 V -351 44 V 9940 5307 Crs 5735 5465 Crs 3632 5112 Crs 2932 5654 Crs 2581 5244 Crs 2231 5424 Crs 1880 5468 Crs 9507 9522 Crs 2.000 UP 1.000 UL LT2 8904 9222 M (POS. 2-SIDED JEROSLOW-WANG \(POS_2S_JW\)) Rshow 9084 9222 M 846 0 V 10 -4400 R 5735 4311 L 3632 4784 L -700 316 V -351 46 V -350 225 V 9940 4822 Star 5735 4311 Star 3632 4784 Star 2932 5100 Star 2581 5146 Star 2231 5371 Star 9507 9222 Star 2.000 UP 1.000 UL LT3 8904 8922 M (MAXIMIZE UNIT RESOLUTIONS \(MAX_UNIT\)) Rshow 9084 8922 M 846 0 V 10 -4583 R 5735 3433 L 3632 4158 L -700 648 V -351 124 V -350 254 V -351 284 V 9940 4339 Box 5735 3433 Box 3632 4158 Box 2932 4806 Box 2581 4930 Box 2231 5184 Box 1880 5468 Box 9507 8922 Box 2.000 UP 1.000 UL LT4 8904 8622 M (SATISFIES MOST CLAUSES \(SATS_MOST\)) Rshow 9084 8622 M 846 0 V 10 -3867 R 5735 2983 L 3632 4120 L -700 521 V -351 207 V -350 364 V -351 256 V 9940 4755 BoxF 5735 2983 BoxF 3632 4120 BoxF 2932 4641 BoxF 2581 4848 BoxF 2231 5212 BoxF 1880 5468 BoxF 9507 8622 BoxF 2.000 UP 1.000 UL LT5 8904 8322 M (MAX. OCCURRS. IN MIN-SIZED CLAUSES \(MOMS\)) Rshow 9084 8322 M 846 0 V 10 -4174 R 5735 2722 L 3632 3972 L -700 506 V -351 227 V -350 407 V -351 356 V 9940 4148 Circle 5735 2722 Circle 3632 3972 Circle 2932 4478 Circle 2581 4705 Circle 2231 5112 Circle 1880 5468 Circle 9507 8322 Circle stroke grestore end showpage %%EndDocument @endspecial 1269 2298 a(\(a\))31 b(\014rst)f(blo)s(c)m(k)g Fl(R)571 4302 y @beginspecial 50 @llx 50 @lly 590 @urx 569 @ury 2448 @rwi @setspecial %%BeginDocument: /u/majercik/papers/SAT2000/gnugraphs/heuristics/averages/heur_avs_e.ps /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -100 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 300 scalefont setfont 1.000 UL LTb 1530 900 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-100) Rshow 1530 1813 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-80) Rshow 1530 2727 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-60) Rshow 1530 3640 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-40) Rshow 1530 4554 M 63 0 V 8697 0 R -63 0 V -8877 0 R (-20) Rshow 1530 5468 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0) Rshow 1530 10035 M 63 0 V 8697 0 R -63 0 V -8877 0 R (100) Rshow 1530 9122 M 63 0 V 8697 0 R -63 0 V -8877 0 R (80) Rshow 1530 8208 M 63 0 V 8697 0 R -63 0 V -8877 0 R (60) Rshow 1530 7295 M 63 0 V 8697 0 R -63 0 V -8877 0 R (40) Rshow 1530 6381 M 63 0 V 8697 0 R -63 0 V -8877 0 R (20) Rshow 1530 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (0) Cshow 2231 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (2) Cshow 2932 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (4) Cshow 3632 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (6) Cshow 4333 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (8) Cshow 5034 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (10) Cshow 5735 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (12) Cshow 6436 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (14) Cshow 7136 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (16) Cshow 7837 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (18) Cshow 8538 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (20) Cshow 9239 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (22) Cshow 9940 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (24) Cshow 1.000 UL LTa 1530 5468 M 8760 0 V 1.000 UL LTb 1530 900 M 8760 0 V 0 9135 V -8760 0 V 0 -9135 V 300 5467 M currentpoint gsave translate 90 rotate 0 0 M (AVERAGE PERCENT INCREASE IN DPLL TREE SIZE) Cshow grestore 5910 150 M (BLOCK SIZE) Cshow 2.000 UP 1.000 UL LT0 8904 9822 M (RANDOM \(RAND\)) Rshow 9084 9822 M 846 0 V 10 -2351 R 5735 5941 L 3632 5385 L -700 -10 V -351 169 V 2231 5410 L -351 58 V 9940 7471 Pls 5735 5941 Pls 3632 5385 Pls 2932 5375 Pls 2581 5544 Pls 2231 5410 Pls 1880 5468 Pls 9507 9822 Pls 2.000 UP 1.000 UL LT1 8904 9522 M (2-SIDED JEROSLOW-WANG \(2S_JW\)) Rshow 9084 9522 M 846 0 V 10 -1450 R 5735 5692 L 3632 5463 L 2932 5346 L -351 299 V 2231 5314 L -351 154 V 9940 8072 Crs 5735 5692 Crs 3632 5463 Crs 2932 5346 Crs 2581 5645 Crs 2231 5314 Crs 1880 5468 Crs 9507 9522 Crs 2.000 UP 1.000 UL LT2 8904 9222 M (POS. 2-SIDED JEROSLOW-WANG \(POS_2S_JW\)) Rshow 9084 9222 M 846 0 V 10 -4750 R 5735 4704 L 3632 4868 L -700 131 V -351 273 V -350 62 V -351 134 V 9940 4472 Star 5735 4704 Star 3632 4868 Star 2932 4999 Star 2581 5272 Star 2231 5334 Star 1880 5468 Star 9507 9222 Star 2.000 UP 1.000 UL LT3 8904 8922 M (MAXIMIZE UNIT RESOLUTIONS \(MAX_UNIT\)) Rshow 9084 8922 M 846 0 V 10 -4901 R 5735 4596 L -2103 58 V -700 166 V -351 196 V -350 165 V -351 287 V 9940 4021 Box 5735 4596 Box 3632 4654 Box 2932 4820 Box 2581 5016 Box 2231 5181 Box 1880 5468 Box 9507 8922 Box 2.000 UP 1.000 UL LT4 8904 8622 M (SATISFIES MOST CLAUSES \(SATS_MOST\)) Rshow 9084 8622 M 846 0 V 10 -5826 R 5735 3843 L 3632 4217 L -700 622 V -351 60 V -350 210 V -351 359 V 9940 2796 BoxF 5735 3843 BoxF 3632 4217 BoxF 2932 4839 BoxF 2581 4899 BoxF 2231 5109 BoxF 1880 5468 BoxF 9507 8622 BoxF 2.000 UP 1.000 UL LT5 8904 8322 M (MAX. OCCURRS. 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BoxF 9507 9522 BoxF 2.000 UP 1.000 UL LT3 8904 9222 M (OTHER SSAT) Rshow 9084 9222 M 846 0 V 3061 7201 M 3381 4752 L 3720 3105 L 447 -951 V 532 -297 V 642 -541 V 766 -231 V 7022 965 L 8121 921 L 9439 906 L 3061 5923 M 320 -196 V 3720 4284 L 4167 2868 L 532 -795 V 642 -624 V 766 -322 V 7022 975 L 8121 928 L 9439 907 L 3061 7638 M 3381 5696 L 3720 4493 L 4167 2637 L 532 -564 V 642 -678 V 766 -281 V 7022 983 L 8121 929 L 9439 906 L 3061 7623 M 3381 4280 L 339 310 V 4167 2966 L 532 -774 V 642 -706 V 766 -315 V 915 -169 V 8121 938 L 9439 908 L 3061 5947 M 320 -305 V 3720 4419 L 4167 2504 L 532 -507 V 642 -558 V 766 -314 V 7022 987 L 8121 928 L 9439 906 L 3061 5691 M 320 -393 V 3720 3893 L 4167 2724 L 532 -963 V 642 -334 V 766 -302 V 7022 988 L 8121 927 L 9439 906 L 3061 7740 M 3381 5524 L 3720 4250 L 4167 2613 L 532 -512 V 642 -729 V 766 -289 V 7022 968 L 8121 922 L 9439 907 L 3061 8131 M 3381 5515 L 3720 4142 L 4167 2683 L 532 -697 V 642 -464 V 766 -363 V 915 -153 V 8121 928 L 9439 906 L 3061 7201 Box 3381 4752 Box 3720 3105 Box 4167 2154 Box 4699 1857 Box 5341 1316 Box 6107 1085 Box 7022 965 Box 8121 921 Box 9439 906 Box 3061 5923 Box 3381 5727 Box 3720 4284 Box 4167 2868 Box 4699 2073 Box 5341 1449 Box 6107 1127 Box 7022 975 Box 8121 928 Box 9439 907 Box 3061 7638 Box 3381 5696 Box 3720 4493 Box 4167 2637 Box 4699 2073 Box 5341 1395 Box 6107 1114 Box 7022 983 Box 8121 929 Box 9439 906 Box 3061 7623 Box 3381 4280 Box 3720 4590 Box 4167 2966 Box 4699 2192 Box 5341 1486 Box 6107 1171 Box 7022 1002 Box 8121 938 Box 9439 908 Box 3061 5947 Box 3381 5642 Box 3720 4419 Box 4167 2504 Box 4699 1997 Box 5341 1439 Box 6107 1125 Box 7022 987 Box 8121 928 Box 9439 906 Box 3061 5691 Box 3381 5298 Box 3720 3893 Box 4167 2724 Box 4699 1761 Box 5341 1427 Box 6107 1125 Box 7022 988 Box 8121 927 Box 9439 906 Box 3061 7740 Box 3381 5524 Box 3720 4250 Box 4167 2613 Box 4699 2101 Box 5341 1372 Box 6107 1083 Box 7022 968 Box 8121 922 Box 9439 907 Box 3061 8131 Box 3381 5515 Box 3720 4142 Box 4167 2683 Box 4699 1986 Box 5341 1522 Box 6107 1159 Box 7022 1006 Box 8121 928 Box 9439 906 Box 9507 9222 Box stroke grestore end showpage %%EndDocument @endspecial 246 2616 a Fw(Figur)l(e)38 b(14.)h Fv(The)e(partial)h(p)r (olicy)f(tree)g(pro)r(duced)g(b)n(y)f(random)g(sampling)h(can)g(b)r(e)g (solv)n(ed)246 2707 y(exactly)22 b(using)h(a)h(DPLL-st)n(yle)e(approac) n(h.)h(Appro)n(ximation)f(error)h(decreases)h(as)g(sample)e(size)246 2798 y(increases.)246 3122 y Fz(times,)33 b(measured)h(in)f(CPU)g (seconds,)i(for)e(12-v)-5 b(ariable,)35 b(24-clause)g(problems,)246 3230 y(with)29 b(4086)j(assignmen)m(t)e(samples,)g(v)-5 b(aried)29 b(from)h(1)h(to)g(5)g(seconds.)362 3337 y(Figure)23 b(14)h(sho)m(ws)e(results)g(for)h(the)h Fi(sampleevalssat)e Fz(algorithm)g(on)h(problems)246 3445 y(with)i(12)j(v)-5 b(ariables,)26 b(24)i(clauses,)f(and)f(3)h(literals)f(p)s(er)g(clause;) h(the)g(graph)f(sho)m(ws)246 3553 y(mean)32 b(squared)f(error)h(in)f (the)h(v)-5 b(alue)32 b(estimate)h(as)f(a)h(function)e(of)h(the)g(n)m (um)m(b)s(er)246 3661 y(of)38 b(sampled)e(assignmen)m(ts)h(for)h(all)f (problem)f(t)m(yp)s(es.)i(Since)e(the)i(algorithm)f(is)246 3769 y(p)s(erforming)f(p)s(erfect)i(optimization)f(on)i(the)f(partial)f (tree,)j(an)m(y)e(error)g(is)g(due)246 3877 y(en)m(tirely)33 b(to)h(the)h(incompleteness)d(of)i(the)h(partial)d(tree)j(due)e(to)i (sampling.)d(As)246 3985 y(exp)s(ected,)39 b(the)f(mean)h(and)f(v)-5 b(ariance)38 b(of)g(the)h(squared)f(error)g(approac)m(h)g(zero)246 4093 y(as)i(the)g(n)m(um)m(b)s(er)f(of)h(assignmen)m(t)f(samples)g (increases,)h(and)g(the)g(partial)e(tree)246 4201 y(constructed)30 b(approac)m(hes)h(the)g(full)d(p)s(olicy)h(tree.)362 4309 y(Figure)i(15)i(sho)m(ws)f(the)g(same)h(graphs)e(for)h(the)g (results)f(of)h(the)h Fi(randevalssat)246 4417 y Fz(algorithm.)20 b(\(Note)j(that)f(the)g(results)e(for)h(all)f(27)i(sets)g(of)f (problems)f(w)m(ere)i(similar;)246 4525 y(w)m(e)33 b(sho)m(w)g(the)g (results)f(only)g(for)g(the)h(set)h(of)f(problems)e(with)h(12)h(v)-5 b(ariables,)32 b(24)246 4633 y(clauses,)j(and)f(3)h(literal)f(p)s(er)f (clause.\))j(Here,)g(the)f(mean)g(and)f(v)-5 b(ariance)35 b(of)g(the)246 4741 y(squared)28 b(error)g(decrease)i(as)f(the)g(n)m (um)m(b)s(er)e(of)i(assignmen)m(t)f(samples)g(increases.)246 4848 y(But,)k(ev)m(en)g(in)e(the)i(limiting)c(case,)33 b(where)e(the)h(n)m(um)m(b)s(er)e(of)i(assignmen)m(ts)f(sam-)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.43)p eop %%Page: 44 44 44 43 bop 246 100 a Fz(44)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)440 2514 y @beginspecial 50 @llx 50 @lly 590 @urx 569 @ury 2880 @rwi @setspecial %%BeginDocument: /u/majercik/papers/SAT2000/gnugraphs/randalg/treesat_mult_3.12.24.ps /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -100 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 300 scalefont setfont 1.000 UL LTb 1530 900 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0) Rshow 1530 2205 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.01) Rshow 1530 3510 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.02) Rshow 1530 4815 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.03) Rshow 1530 6120 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.04) Rshow 1530 7425 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.05) Rshow 1530 8730 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.06) Rshow 1530 10035 M 63 0 V 8697 0 R -63 0 V -8877 0 R (0.07) Rshow 1530 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (1) Cshow 2189 900 M 0 31 V 0 9104 R 0 -31 V 2575 900 M 0 31 V 0 9104 R 0 -31 V 2849 900 M 0 31 V 0 9104 R 0 -31 V 3061 900 M 0 31 V 0 9104 R 0 -31 V 3234 900 M 0 31 V 0 9104 R 0 -31 V 3381 900 M 0 31 V 0 9104 R 0 -31 V 3508 900 M 0 31 V 0 9104 R 0 -31 V 3620 900 M 0 31 V 0 9104 R 0 -31 V 3720 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (10) Cshow 4379 900 M 0 31 V 0 9104 R 0 -31 V 4765 900 M 0 31 V 0 9104 R 0 -31 V 5039 900 M 0 31 V 0 9104 R 0 -31 V 5251 900 M 0 31 V 0 9104 R 0 -31 V 5424 900 M 0 31 V 0 9104 R 0 -31 V 5571 900 M 0 31 V 0 9104 R 0 -31 V 5698 900 M 0 31 V 0 9104 R 0 -31 V 5810 900 M 0 31 V 0 9104 R 0 -31 V 5910 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (100) Cshow 6569 900 M 0 31 V 0 9104 R 0 -31 V 6955 900 M 0 31 V 0 9104 R 0 -31 V 7229 900 M 0 31 V 0 9104 R 0 -31 V 7441 900 M 0 31 V 0 9104 R 0 -31 V 7614 900 M 0 31 V 0 9104 R 0 -31 V 7761 900 M 0 31 V 0 9104 R 0 -31 V 7888 900 M 0 31 V 0 9104 R 0 -31 V 8000 900 M 0 31 V 0 9104 R 0 -31 V 8100 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (1000) Cshow 8759 900 M 0 31 V 0 9104 R 0 -31 V 9145 900 M 0 31 V 0 9104 R 0 -31 V 9419 900 M 0 31 V 0 9104 R 0 -31 V 9631 900 M 0 31 V 0 9104 R 0 -31 V 9804 900 M 0 31 V 0 9104 R 0 -31 V 9951 900 M 0 31 V 0 9104 R 0 -31 V 10078 900 M 0 31 V 0 9104 R 0 -31 V 10190 900 M 0 31 V 0 9104 R 0 -31 V 10290 900 M 0 63 V 0 9072 R 0 -63 V 0 -9372 R (10000) Cshow 1.000 UL LTb 1530 900 M 8760 0 V 0 9135 V -8760 0 V 0 -9135 V 300 5467 M currentpoint gsave translate 90 rotate 0 0 M (MEAN SQUARED ERROR IN FORMULA VALUE ESTIMATE) Cshow grestore 5910 150 M (NUMBER OF ASSIGNMENTS SAMPLED) Cshow 2.000 UP 1.000 UL LT2 8904 9822 M (SAT) Rshow 9084 9822 M 846 0 V 3061 900 M 320 0 V 339 0 V 447 0 V 532 0 V 642 0 V 766 0 V 915 0 V 1099 0 V 1318 0 V 3061 900 Star 3381 900 Star 3720 900 Star 4167 900 Star 4699 900 Star 5341 900 Star 6107 900 Star 7022 900 Star 8121 900 Star 9439 900 Star 9507 9822 Star 2.000 UP 1.000 UL LT4 8904 9522 M (MAJSAT) Rshow 9084 9522 M 846 0 V 3061 1733 M 320 -355 V 339 -4 V 447 -52 V 532 -213 V 642 -95 V 766 -77 V 915 -23 V 8121 904 L 1318 -3 V 3061 1733 BoxF 3381 1378 BoxF 3720 1374 BoxF 4167 1322 BoxF 4699 1109 BoxF 5341 1014 BoxF 6107 937 BoxF 7022 914 BoxF 8121 904 BoxF 9439 901 BoxF 9507 9522 BoxF 2.000 UP 1.000 UL LT3 8904 9222 M (OTHER SSAT) Rshow 9084 9222 M 846 0 V 3061 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b(,)26 b(or)e(p)s(olicy)-8 b(,)22 b(that)i(sp)s(eci\014es) 246 3528 y(ho)m(w)e(eac)m(h)h(existen)m(tial)e(v)-5 b(ariable)20 b(should)g(b)s(e)i(set)g(giv)m(en)g(the)g(settings)g(of)g(the)g(v)-5 b(ari-)246 3636 y(ables)31 b(that)i(precede)f(it)g(in)f(the)h(quan)m (ti\014er)f(ordering,)g(but)h(this)f(only)g(happ)s(ens)246 3744 y(for)36 b(those)g(situations)f(represen)m(ted)h(b)m(y)h(paths)e (in)g(the)i(random)e(sample.)h(The)246 3852 y(algorithm)44 b(implicitly)d(assumes)k(that)h(p)s(erformance)e(on)h(missing)e(branc)m (hes)246 3960 y(is)38 b(the)h(same)h(as)f(the)g(a)m(v)m(erage)j(p)s (erformance)c(o)m(v)m(er)j(all)d(paths)g(in)g(the)h(partial)246 4068 y(tree,)27 b(but)e(do)s(esn't)h(actually)g(\014nd)f(a)h(plan)f (that)i(ac)m(hiev)m(es)g(this)e(p)s(erformance.)h(A)246 4176 y(topic)h(for)f(future)g(w)m(ork)h(is)f(extending)h(the)g(sto)s(c) m(hastic)h(sampling)c(algorithm)i(to)246 4284 y(return)21 b(successful)g(plans,)g(p)s(erhaps)f(in)h(an)h(iterativ)m(e)h(approac)m 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Fr(SSa)-6 b(t)34 b Fz(problems)g(exactly)j(and)e (e\016cien)m(tly)-8 b(.)36 b(F)-8 b(uture)36 b(w)m(ork)246 722 y(will)26 b(b)s(e)i(to)h(implemen)m(t)f(and)g(test)h(the)g (algorithm,)f(comparing)g(it)g(to)i(the)f(split-)246 830 y(ting)42 b(heuristics)e(examined)i(in)f(Section)h(3.3.)i(Although) e(they)h(are)f(not)h(w)m(ell)246 938 y(understo)s(o)s(d)27 b(theoretically)-8 b(,)29 b(the)h(threshold)d(pruning)g(rules)h (describ)s(ed)f(in)h(Sec-)246 1046 y(tion)20 b(2.1.2)j(could)e(b)s(e)f (helpful)f(in)h(a)h(practical)g(implemen)m(tation)f(of)h(the)h(univ)m (ersal)246 1154 y(searc)m(h)31 b(algorithm.)362 1262 y(There)45 b(has)h(b)s(een)g(a)g(recen)m(t)h(\015urry)d(of)i(activit)m (y)h(in)e(analyzing)g(common)246 1370 y(Da)m(vis-Putnam)26 b(algorithms,)g(and)f(pro)m(ving)h(that)h(these)g(algorithms)e(are)i (guar-)246 1478 y(an)m(teed)i(to)g(solv)m(e)f(an)m(y)h Fl(k)s Fz(-SA)-8 b(T)28 b(instance)g(in)f(sub)s(exp)s(onen)m(tial-time) f(\(Monien)i(&)246 1586 y(Sp)s(ec)m(k)m(enmey)m(er,)c(1985;)h(P)m (aturi,)e(Pudlak,)f(&)h(Zane,)g(1997;)i(P)m(aturi)e Fj(et)j(al.)p Fz(,)e(1998\).)246 1694 y(\(F)-8 b(or)36 b(example,)f(a)h(v)m(ery)f (simple)e(randomized)h(DPLL-t)m(yp)s(e)i(algorithm)e(solv)m(es)246 1802 y(3-CNF)28 b Fr(Sa)-6 b(t)26 b Fz(in)g(w)m(orst-case)i(time)f(2) 1514 1769 y Fn(\(2)p Fk(=)p Fn(3\))p Fk(n)1722 1802 y Fz(.\))h(It)f(w)m(ould)f(b)s(e)h(in)m(teresting)f(to)i(sho)m(w)246 1910 y(that)j(these)f(analyses)h(carry)f(o)m(v)m(er)i(to)f(the)f Fr(SSa)-6 b(t)29 b Fz(setting.)362 2017 y(In)h(addition)f(to)i (Resolution-based)f(metho)s(ds,)g(there)h(are)g(sev)m(eral)g(alterna-) 246 2125 y(tiv)m(e)26 b(metho)s(ds)f(for)g(solving)f(satis\014abilit)m (y)g(instances.)h(In)g(particular,)f(algebraic)246 2233 y(approac)m(hes)35 b(based)f(on)g(discrete)g(v)m(ersions)g(of)g(the)h (Grobner)f(basis)f(algorithm)246 2341 y(are)43 b(lo)s(oking)e (promising)f(for)j(solving)e(satis\014abilit)m(y)f(problems)h(\(Clegg,) j(Ed-)246 2449 y(monds,)24 b(&)g(Impagliazzo,)h(1996\).)i(F)-8 b(uture)25 b(w)m(ork)g(will)d(examine)j(to)g(what)g(exten)m(t)246 2557 y(these)30 b(metho)s(ds)g(carry)h(o)m(v)m(er)g(to)g(the)g Fr(SSa)-6 b(t)29 b Fz(domain.)362 2665 y(A)d(n)m(um)m(b)s(er)g(of)g (recen)m(t)i(studies)d(ha)m(v)m(e)i(sho)m(wn)f(that)h(the)g (satis\014abilit)m(y)d(frame-)246 2773 y(w)m(ork)31 b(is)f(useful)f (for)i(examining)f(a)i(v)-5 b(ariet)m(y)31 b(of)g(searc)m(h)h (problems.)e(An)h(exciting)246 2881 y(elemen)m(t)i(of)g(the)g(presen)m (t)g(w)m(ork)g(is)f(that)i(it)e(illustrates)f(ho)m(w)i(the)g (satis\014abilit)m(y)246 2989 y(approac)m(h)f(applies)e(outside)i(of)g Fr(Sa)-6 b(t)o Fz('s)32 b(complexit)m(y)g(class,)g(greatly)h(expanding) 246 3097 y(the)d(realm)g(of)h(problems)d(that)j(can)g(b)s(e)f(attac)m (k)m(ed.)246 3330 y Fr(A)m(ckno)n(wledgments)246 3536 y Fz(Thanks)i(to)i(P)m(ank)-5 b(a)5 b(j)34 b(Agarw)m(al,)g(Julien)d (Basc)m(h,)j(Ian)f(Gen)m(t,)i(Judy)d(Goldsmith,)246 3644 y(Moises)40 b(Goldszmidt,)f(Henry)g(Kautz,)i(Don)f(Lo)m(v)m(eland,)h (Matt)g(P)m(eters,)g(John)246 3752 y(Reif,)19 b(Bart)i(Selman,)f(and)f (T)-8 b(ob)m(y)21 b(W)-8 b(alsh)20 b(for)g(feedbac)m(k)h(and)f (suggestions.)h(Avrim)246 3859 y(Blum)32 b(suggested)i(the)g(use)f(of)g (sampling)f(and)h(Cherno\013)f(b)s(ounds,)g(and)g(Noam)246 3967 y(Shazeer)20 b(pro)m(vided)f(the)i(insigh)m(t)d(b)s(ehind)g(the)i (coun)m(ter)h(example)f(in)f(App)s(endix)f(B.)246 4075 y(W)-8 b(e)35 b(w)m(ould)e(also)i(lik)m(e)e(to)i(thank)f(Stev)m(e)i(Co) s(ok,)e(Mik)m(e)h(Mollo)m(y)-8 b(,)35 b(and)f(esp)s(ecially)246 4183 y(Dimitris)22 b(Ac)m(hlioptas)i(for)h(illuminating)20 b(con)m(v)m(ersations)26 b(on)e(the)h(0{1)h(threshold)246 4291 y(prop)s(erties)i(of)j Fr(Sa)-6 b(t)29 b Fz(and)h Fr(Majsa)-6 b(t)29 b Fz(form)m(ulae.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.47)p eop %%Page: 48 48 48 47 bop 246 100 a Fz(48)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)1417 291 y Fs(App)s(endix)1057 498 y(A.)73 b(Correctness)36 b(of)f Fi(evalssat)246 705 y Fz(This)24 b(section)j(pro)m(v)m(es)h (that)f(the)g Fi(evalssat)e Fz(algorithm)h(describ)s(ed)e(in)i(Section) g(2.1)246 812 y(pro)s(duces)j(the)h(correct)i(v)-5 b(alue)30 b(of)g(the)h(giv)m(en)f Fr(SSa)-6 b(t)29 b Fz(instance)h(\()p Fl(\036;)15 b(Q;)g(\022)s Fz(\).)246 989 y(LEMMA)31 b(6.)65 b Fj(F)-7 b(or)32 b(al)5 b(l)32 b Fl(n)p Fj(,)e(for)i(al)5 b(l)31 b(formulae)i Fl(\036)e Fj(with)h(at)f(most)h Fl(n)f Fj(variables,)h(for)246 1097 y(al)5 b(l)38 b(thr)-5 b(esholds)41 b Fl(\022)851 1112 y Fk(l)877 1097 y Fj(,)d Fl(\022)986 1112 y Fk(h)1031 1097 y Fj(,)f(such)i(that)g Fl(\022)1543 1112 y Fk(l)1604 1097 y Fm(\024)c Fl(\022)1753 1112 y Fk(h)1798 1097 y Fj(,)i Fi(evalssat)o Fz(\()p Fl(\036;)15 b(Q;)g(\022)2441 1112 y Fk(l)2468 1097 y Fl(;)g(\022)2551 1112 y Fk(h)2596 1097 y Fz(\))39 b Fj(r)-5 b(eturns)39 b(a)246 1205 y(value)34 b Fl(v)j Fj(such)d(that:)h(\(i\))f(if)f Fl(\022)1254 1220 y Fk(l)1307 1205 y Fm(\024)27 b Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))29 b Fm(\024)e Fl(\022)1923 1220 y Fk(h)1967 1205 y Fj(,)34 b(then)g(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))28 b(=)g Fl(v)s Fj(;)34 b(\(ii\))f(if)246 1313 y(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b Fm(\024)e Fl(\022)759 1328 y Fk(l)784 1313 y Fj(,)33 b(then)g Fl(v)28 b Fm(\024)d Fl(\022)1258 1328 y Fk(l)1284 1313 y Fj(;)32 b(and)i(\(iii\))e(if)h(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b Fm(\025)e Fl(\022)2313 1328 y Fk(h)2357 1313 y Fj(,)32 b(then)i Fl(v)28 b Fm(\025)d Fl(\022)2831 1328 y Fk(h)2876 1313 y Fj(.)246 1489 y Fs(Pro)s(of:)70 b Fz(The)32 b(correctness)h(of)g(the)f(algorithm)g(will)e(b)s(e)h(pro)m (v)m(ed)i(b)m(y)f(induction)246 1597 y(on)38 b(the)i(n)m(um)m(b)s(er)d (of)i(v)-5 b(ariables)38 b(of)h(the)g(form)m(ula.)g(If)f(there)h(are)g (no)g(v)-5 b(ariables,)246 1705 y(then)29 b Fl(\036)h Fz(either)f(consists)g(of)g(no)h(clauses)f(\()p Fj(i.e.)p Fz(,)h(it)f(con)m(tains)h(the)f(empt)m(y)h(set\),)h(or)246 1813 y(it)24 b(con)m(tains)g(an)g(empt)m(y)h(clause.)g(If)f Fl(\036)g Fz(consists)g(of)g(no)h(clauses,)f(then)g Fl(\036)h Fz(ev)-5 b(aluates)246 1921 y(to)29 b(1)g(b)m(y)g(de\014nition.)d(Lik)m (ewise,)i(if)g Fl(\036)g Fz(con)m(tains)h(an)g(empt)m(y)g(clause,)f (this)g(implies)246 2029 y(that)j Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))27 b(=)e(0,)31 b(and)e(th)m(us)h(the)h(lemma)f(holds)f(in)g (this)g(case.)362 2137 y(No)m(w,)41 b(assume)f(that)h Fl(\036)f Fz(has)g Fl(n)i(>)f Fz(0)g(v)-5 b(ariables,)39 b(and)h(as)g(usual,)f(the)i(v)-5 b(ari-)246 2245 y(ables)32 b(are)i(named)f Fl(x)979 2259 y Fn(1)1018 2245 y Fl(;)15 b(:)g(:)g(:)i(;)e(x)1272 2259 y Fk(n)1352 2245 y Fz(where)33 b Fl(x)1670 2259 y Fn(1)1742 2245 y Fz(is)f(the)i(outermost)g(quan)m (ti\014ed)e(v)-5 b(ari-)246 2352 y(able.)42 b(As)g(ab)s(o)m(v)m(e,)h (the)g(lemma)e(holds)g(if)g Fl(\036)i Fz(consists)e(of)h(no)h(clauses)e (or)i(con-)246 2460 y(tains)31 b(an)h(empt)m(y)h(clause.)f(In)f(all)g (other)h(cases,)i(a)e(single)f(v)-5 b(ariable)31 b(is)g(remo)m(v)m(ed.) 246 2568 y(The)38 b(\014rst)f(case)j(is)d(where)h(a)h(v)-5 b(ariable)37 b Fl(x)1692 2582 y Fk(i)1758 2568 y Fz(is)h(remo)m(v)m(ed) h(b)s(ecause)f(it)g(o)s(ccurs)g(in)246 2676 y(a)48 b(singleton)g (clause)g(in)f Fl(\036)i Fz(with)e(sign)g Fl(b)p Fz(,)i(and)f Fl(Q)p Fz(\()p Fl(x)2152 2690 y Fk(i)2180 2676 y Fz(\))56 b(=)f Fm(9)p Fz(.)48 b(In)g(this)f(case,)246 2784 y Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))44 b(=)f Fj(val)q Fz(\()p Fl(\036)p Fm(d)992 2799 y Fk(x)1032 2809 y Ff(i)1059 2799 y Fn(=)p Fk(b)1148 2784 y Fl(;)15 b(Q)p Fz(\))41 b(b)s(ecause)h(when)e Fl(x)1984 2798 y Fk(i)2053 2784 y Fz(is)g(set)h(to)h(1)28 b Fm(\000)f Fl(b)p Fz(,)41 b Fl(\036)g Fz(ev)-5 b(alu-)246 2892 y(ates)41 b(to)h(0.)f(Then,)f(b)m (y)h(the)f(inductiv)m(e)g(h)m(yp)s(othesis,)f(the)i(algorithm)f (outputs)246 3000 y(a)e(correct)h(v)-5 b(alue)38 b(on)g Fl(\036)p Fm(d)1111 3015 y Fk(x)1151 3025 y Ff(i)1178 3015 y Fn(=)p Fk(b)1267 3000 y Fz(.)g(Similarly)-8 b(,)35 b(if)i Fl(x)1885 3014 y Fk(i)1952 3000 y Fz(is)g(a)h(unit)f(v)-5 b(ariable)37 b(in)g Fl(\036)h Fz(and)246 3108 y Fl(Q)p Fz(\()p Fl(x)405 3122 y Fk(i)433 3108 y Fz(\))49 b(=)g Fm(8)p Fz(,)44 b(then)h Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))51 b(=)d(0)d(and,)g(th)m(us,)f(the)h(algorithm)f(outputs)g (a)246 3216 y(correct)h(v)-5 b(alue.)43 b(Finally)-8 b(,)43 b(if)g Fl(x)1326 3230 y Fk(i)1397 3216 y Fz(is)g(a)i(unit)d(v)-5 b(ariable)43 b(and)g Fl(Q)p Fz(\()p Fl(x)2501 3230 y Fk(i)2529 3216 y Fz(\))48 b(=)2789 3153 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 2789 3153 a Fi(R)2848 3153 y currentpoint grestore moveto 2848 3153 a 2789 3216 a Fz(,)c(then)246 3324 y Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))45 b(=)e(\()p Fj(val)p Fz(\()p Fl(\036)p Fm(d)1027 3339 y Fk(x)1067 3349 y Ff(i)1094 3339 y Fn(=)p Fk(b)1184 3324 y Fl(;)15 b(Q)p Fz(\))28 b(+)f Fj(val)p Fz(\()p Fl(\036)p Fm(d)1697 3339 y Fk(x)1737 3349 y Ff(i)1764 3339 y Fn(=1)p Fx(\000)p Fk(b)1944 3324 y Fl(;)15 b(Q)p Fz(\)\))p Fl(=)p Fz(2)45 b(=)e Fj(val)p Fz(\()p Fl(\036)p Fm(d)2615 3339 y Fk(x)2655 3349 y Ff(i)2682 3339 y Fn(=)p Fk(b)2771 3324 y Fl(;)15 b(Q)p Fz(\))p Fl(=)p Fz(2.)246 3432 y(In)37 b(this)f(case,)j(when)e (calling)f Fi(evalssat)g Fz(recursiv)m(ely)g(on)i Fl(\036)p Fm(d)2335 3447 y Fk(x)2375 3457 y Ff(i)2402 3447 y Fn(=)p Fk(b)2491 3432 y Fz(,)g(the)f(lo)m(w)h(and)246 3540 y(high)i(b)s(ounds) f Fl(\022)829 3555 y Fk(l)896 3540 y Fz(and)i Fl(\022)1127 3555 y Fk(h)1213 3540 y Fz(m)m(ust)h(b)s(e)f(shifted)f(b)m(y)h(a)h (factor)h(of)e(t)m(w)m(o.)i(That)f(is,)246 3648 y(when)e Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))44 b(=)f Fl(x)p Fz(,)e Fj(val)p Fz(\()p Fl(\036)p Fm(d)1357 3663 y Fk(x)1397 3673 y Ff(i)1424 3663 y Fn(=)p Fk(b)1513 3648 y Fl(;)15 b(Q)p Fz(\))42 b(is)e(2)p Fl(x)p Fz(;)h(th)m(us)g(if)f Fl(\022)2315 3663 y Fk(l)2383 3648 y Fm(\024)j Fl(x)f Fm(\024)h Fl(\022)2748 3663 y Fk(h)2792 3648 y Fz(,)e(then)246 3756 y(b)m(y)32 b(induction)d Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)1202 3771 y Fk(x)1242 3781 y Ff(i)1269 3771 y Fn(=)p Fk(b)1358 3756 y Fl(;)15 b(Q;)g Fz(2)p Fl(\022)1598 3771 y Fk(l)1625 3756 y Fl(;)g Fz(2)p Fl(\022)1753 3771 y Fk(h)1798 3756 y Fz(\))33 b(will)c(return)i(the)h(v)-5 b(alue)31 b(2)p Fl(x)i Fz(since)246 3863 y(2)p Fl(\022)334 3878 y Fk(l)405 3863 y Fm(\024)46 b Fz(2)p Fl(x)g Fm(\024)f Fz(2)p Fl(\022)869 3878 y Fk(h)914 3863 y Fz(.)d(Similarly)-8 b(,)40 b(the)j(algorithm)e(is)h(correct)i(for)e Fl(x)j Fm(\024)h Fl(\022)2821 3878 y Fk(l)2889 3863 y Fz(and)246 3971 y Fl(x)25 b Fm(\025)g Fl(\022)462 3986 y Fk(h)506 3971 y Fz(.)362 4079 y(The)20 b(second)g(case)i(is)d(where)h Fl(x)1391 4093 y Fk(i)1440 4079 y Fz(is)f(a)i(pure)e(v)-5 b(ariable)20 b(with)f(sign)g Fl(b)i Fz(and)f Fl(Q)p Fz(\()p Fl(x)2876 4093 y Fk(i)2904 4079 y Fz(\))26 b(=)246 4187 y Fm(8)p Fz(.)48 b(In)h(this)e(case,)k Fj(val)p Fz(\()p Fl(\036;)15 b(Q)p Fz(\))58 b(=)e(min)n(\()p Fj(val)q Fz(\()p Fl(\036)p Fm(d)1894 4202 y Fk(x)1934 4212 y Ff(i)1961 4202 y Fn(=)p Fk(b)2050 4187 y Fl(;)15 b(Q)p Fz(\))p Fl(;)g Fj(val)r Fz(\()p Fl(\036)p Fm(d)2479 4202 y Fk(x)2519 4212 y Ff(i)2546 4202 y Fn(=1)p Fx(\000)p Fk(b)2725 4187 y Fl(;)g(Q)p Fz(\)\))58 b(=)246 4295 y Fj(val)p Fz(\()p Fl(\036)p Fm(d)486 4310 y Fk(x)526 4320 y Ff(i)553 4310 y Fn(=1)p Fx(\000)p Fk(b)733 4295 y Fl(;)15 b(Q)p Fz(\),)42 b(so)g(b)m(y)g(induction)e(the)i(algorithm)f(is)g(correct.)i(Similarly) -8 b(,)246 4403 y(when)29 b Fl(x)535 4417 y Fk(i)593 4403 y Fz(is)h(a)h(pure)e(v)-5 b(ariable)29 b(and)h Fl(Q)p Fz(\()p Fl(x)1645 4417 y Fk(i)1673 4403 y Fz(\))c(=)f Fm(9)p Fz(,)30 b(then)330 4572 y Fj(val)q Fz(\()p Fl(\036;)15 b(Q)p Fz(\))26 b(=)f(max)q(\()p Fj(val)p Fz(\()p Fl(\036)p Fm(d)1245 4587 y Fk(x)1285 4597 y Ff(i)1312 4587 y Fn(=)p Fk(b)1402 4572 y Fl(;)15 b(Q)p Fz(\))p Fl(;)g Fj(val)q Fz(\()p Fl(\036)p Fm(d)1830 4587 y Fk(x)1870 4597 y Ff(i)1897 4587 y Fn(=1)p Fx(\000)p Fk(b)2077 4572 y Fl(;)g(Q)p Fz(\)\))26 b(=)f Fj(val)p Fz(\()p Fl(\036)p Fm(d)2621 4587 y Fk(x)2661 4597 y Ff(i)2688 4587 y Fn(=)p Fk(b)2778 4572 y Fl(;)15 b(Q)p Fz(\))p Fl(:)362 4741 y Fz(The)41 b(third)f(case)j(is)e(when)g Fl(x)1407 4755 y Fn(1)1488 4741 y Fz(is)g(remo)m(v)m(ed)h(and)g Fl(Q)p Fz(\()p Fl(x)2311 4755 y Fn(1)2350 4741 y Fz(\))j(=)f Fm(9)p Fz(.)d(Let)i Fl(v)2881 4755 y Fn(0)2965 4741 y Fz(=)246 4848 y Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)669 4862 y Fk(x)709 4871 y Fe(1)744 4862 y Fn(=0)838 4848 y Fl(;)15 b(Q;)g(\022)1033 4863 y Fk(l)1060 4848 y Fl(;)g(\022)1143 4863 y Fk(h)1187 4848 y Fz(\).)36 b(If)f Fj(val)q Fz(\()p Fl(\036)p Fm(d)1620 4862 y Fk(x)1660 4871 y Fe(1)1695 4862 y Fn(=0)1789 4848 y Fl(;)15 b(Q)p Fz(\))34 b Fm(\025)f Fl(\022)2117 4863 y Fk(h)2162 4848 y Fz(,)i(then)g Fl(v)2478 4862 y Fn(0)2551 4848 y Fm(\025)e Fl(\022)2698 4863 y Fk(h)2778 4848 y Fz(b)m(y)i(the)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e (p.48)p eop %%Page: 49 49 49 48 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(49)246 291 y(induction)29 b(h)m(yp)s(othesis,)i(so)h Fl(v)1283 305 y Fn(0)1354 291 y Fz(is)f(a)i(correct)g(v)-5 b(alue.)31 b(Otherwise,)g Fl(v)2592 305 y Fn(0)2660 291 y Fl(<)c(\022)2801 306 y Fk(h)2846 291 y Fz(.)32 b(Let)246 399 y Fl(v)290 413 y Fn(1)354 399 y Fz(=)25 b Fi(evalssat)o Fz(\()p Fl(\036)p Fm(d)873 413 y Fk(x)913 422 y Fe(1)949 413 y Fn(=1)1043 399 y Fl(;)15 b(Q;)g Fz(max)q(\()p Fl(\022)1443 414 y Fk(l)1469 399 y Fl(;)g(v)1553 413 y Fn(0)1593 399 y Fz(\))p Fl(;)g(\022)1711 414 y Fk(h)1756 399 y Fz(\).)30 b(There)e(are)i(sev)m (eral)f(simple)e(cases)246 506 y(to)e(c)m(hec)m(k:)h(If)e Fj(val)q Fz(\()p Fl(\036)p Fm(d)941 520 y Fk(x)981 529 y Fe(1)1016 520 y Fn(=0)1110 506 y Fl(;)15 b(Q)p Fz(\))26 b Fm(\025)f Fl(\022)1422 521 y Fk(l)1448 506 y Fz(,)f(then)h Fl(v)1743 520 y Fn(0)1807 506 y Fz(will)c(equal)j Fj(val)q Fz(\()p Fl(\036)p Fm(d)2447 520 y Fk(x)2487 529 y Fe(1)2522 520 y Fn(=0)2617 506 y Fl(;)15 b(Q)p Fz(\))25 b(b)m(y)f(the)246 614 y(inductiv)m(e)31 b(h)m(yp)s(othesis,)h(so)h(max\()p Fl(\022)1479 629 y Fk(l)1505 614 y Fl(;)15 b(v)1589 628 y Fn(0)1629 614 y Fz(\))30 b(=)f Fl(v)1838 628 y Fn(0)1877 614 y Fz(,)k(and)g(th)m(us)f(it)h(is)e(easy)j(to)g(c)m(hec)m(k)246 722 y(that)g(max\()p Fl(v)694 736 y Fn(0)734 722 y Fl(;)15 b(v)818 736 y Fn(1)858 722 y Fz(\))33 b(is)f(a)i(correct)h(answ)m(er.)e (Similarly)-8 b(,)30 b(it)j(can)h(b)s(e)e(c)m(hec)m(k)m(ed)k(that)246 830 y(the)30 b(answ)m(er)g(returned)g(is)f(correct)j(if)d Fj(val)p Fz(\()p Fl(\036)p Fm(d)1791 844 y Fk(x)1831 853 y Fe(1)1867 844 y Fn(=0)1961 830 y Fl(;)15 b(Q)p Fz(\))26 b Fl(<)f(\022)2273 845 y Fk(l)2299 830 y Fz(.)362 938 y(The)j(fourth)g(case,)j(when)c Fl(x)1325 952 y Fn(1)1394 938 y Fz(is)h(remo)m(v)m(ed)h(and)g Fl(Q)p Fz(\()p Fl(x)2178 952 y Fn(1)2217 938 y Fz(\))d(=)f Fm(8)p Fz(,)k(is)e(v)m(ery)j(similar) 246 1046 y(to)h(the)f(ab)s(o)m(v)m(e)i(case.)362 1154 y(The)g(\014fth)g(and)h(\014nal)e(case)j(is)e(when)g Fl(x)1713 1168 y Fn(1)1785 1154 y Fz(is)g(remo)m(v)m(ed)i(and)e Fl(Q)p Fz(\()p Fl(x)2581 1168 y Fn(1)2621 1154 y Fz(\))e(=)2845 1091 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 2845 1091 a Fi(R)2903 1091 y currentpoint grestore moveto 2903 1091 a 2845 1154 a Fz(.)j(Let)246 1262 y Fl(v)293 1229 y Fx(0)290 1286 y Fn(0)366 1262 y Fz(=)j Fj(val)p Fz(\()p Fl(\022)s Fm(d)705 1276 y Fk(x)745 1285 y Fe(1)779 1276 y Fn(=0)874 1262 y Fl(;)15 b(Q)p Fz(\))37 b(and)g(let)g Fl(v)1427 1229 y Fx(0)1424 1286 y Fn(1)1499 1262 y Fz(=)g Fj(val)p Fz(\()p Fl(\022)s Fm(d)1839 1276 y Fk(x)1879 1285 y Fe(1)1913 1276 y Fn(=1)2008 1262 y Fl(;)15 b(Q)p Fz(\).)37 b(If)g Fl(v)2362 1229 y Fx(0)2359 1286 y Fn(0)2435 1262 y Fl(<)f Fz(2)p Fl(\022)2630 1277 y Fk(l)2681 1262 y Fm(\000)24 b Fz(1)37 b(then)246 1370 y(\()p Fl(v)328 1337 y Fx(0)325 1394 y Fn(0)390 1370 y Fz(+)25 b Fl(v)533 1337 y Fx(0)530 1394 y Fn(1)570 1370 y Fz(\))p Fl(=)p Fz(2)40 b Fl(<)e Fz(\(2)p Fl(\022)967 1385 y Fk(l)1019 1370 y Fm(\000)25 b Fz(1)h(+)f(1\))p Fl(=)p Fz(2)40 b(=)f Fl(\022)1645 1385 y Fk(l)1670 1370 y Fz(.)g(If)f Fl(v)1880 1337 y Fx(0)1877 1394 y Fn(0)1955 1370 y Fl(>)g Fz(2)p Fl(\022)2152 1385 y Fk(h)2197 1370 y Fz(,)h(then)f(\()p Fl(v)2558 1337 y Fx(0)2555 1394 y Fn(0)2620 1370 y Fz(+)25 b Fl(v)2763 1337 y Fx(0)2760 1394 y Fn(1)2800 1370 y Fz(\))p Fl(=)p Fz(4)40 b Fl(>)246 1478 y Fz(\(2)p Fl(\022)369 1493 y Fk(h)414 1478 y Fz(\))p Fl(=)p Fz(2)29 b(=)f Fl(\022)710 1493 y Fk(h)755 1478 y Fz(.)k(So,)g(no)g(detailed)f(answ)m(er)h(is)f (needed)h(if)f Fl(v)2271 1445 y Fx(0)2268 1502 y Fn(0)2336 1478 y Fl(<)d Fz(2)p Fl(\022)2523 1493 y Fk(l)2570 1478 y Fm(\000)21 b Fz(1)33 b(or)f Fl(v)2900 1445 y Fx(0)2897 1502 y Fn(0)2965 1478 y Fl(>)246 1586 y Fz(2)p Fl(\022)334 1601 y Fk(h)379 1586 y Fz(.)f(Th)m(us,)g(it)f(su\016ces)h(to)i (calculate)e Fl(v)1629 1600 y Fn(0)1696 1586 y Fz(=)26 b Fi(evalssat)o Fz(\()p Fl(\022)s Fm(d)2208 1600 y Fk(x)2248 1609 y Fe(1)2282 1600 y Fn(=0)2377 1586 y Fl(;)15 b(Q;)g Fz(2)p Fl(\022)2617 1601 y Fk(l)2664 1586 y Fm(\000)21 b Fz(1)p Fl(;)15 b Fz(2)p Fl(\022)2929 1601 y Fk(h)2975 1586 y Fz(\).)246 1694 y(If)41 b(\()p Fl(v)427 1708 y Fn(0)494 1694 y Fz(+)27 b(1\))p Fl(=)p Fz(2)46 b Fl(<)d(\022)965 1709 y Fk(l)991 1694 y Fz(,)f(then)f(\()p Fl(v)1358 1661 y Fx(0)1355 1718 y Fn(0)1422 1694 y Fz(+)27 b Fl(v)1567 1661 y Fx(0)1564 1718 y Fn(1)1604 1694 y Fz(\))p Fl(=)p Fz(2)45 b Fl(<)e(\022)1931 1709 y Fk(l)1957 1694 y Fz(,)f(so)f(return)g Fl(v)2480 1708 y Fn(0)2519 1694 y Fl(=)p Fz(2)h(\(an)m(ything)246 1802 y(less)37 b(than)h Fl(\022)684 1817 y Fk(l)710 1802 y Fz(\).)h(If)f Fl(v)952 1816 y Fn(0)991 1802 y Fl(=)p Fz(2)i Fm(\025)e Fl(\022)1273 1817 y Fk(h)1318 1802 y Fz(,)g(then)g(\()p Fl(v)1678 1769 y Fx(0)1675 1826 y Fn(0)1741 1802 y Fz(+)25 b Fl(v)1884 1769 y Fx(0)1881 1826 y Fn(1)1920 1802 y Fz(\))p Fl(=)p Fz(2)40 b Fm(\025)e Fl(v)2238 1816 y Fn(0)2278 1802 y Fl(=)p Fz(2)h Fm(\025)g Fl(\022)2560 1817 y Fk(h)2604 1802 y Fz(,)g(so)f(return)246 1910 y Fl(v)290 1924 y Fn(0)329 1910 y Fl(=)p Fz(2.)26 b(Otherwise,)d(calculate)i Fl(v)1336 1924 y Fn(1)1400 1910 y Fz(=)g Fi(evalssat)o Fz(\()p Fl(\022)s Fm(d)1911 1924 y Fk(x)1951 1933 y Fe(1)1985 1924 y Fn(=1)2080 1910 y Fl(;)15 b Fz(2)p Fl(\022)2208 1925 y Fk(l)2242 1910 y Fm(\000)8 b Fl(v)2365 1924 y Fn(0)2405 1910 y Fl(;)15 b Fz(2)p Fl(\022)2533 1925 y Fk(h)2586 1910 y Fm(\000)8 b Fl(v)2709 1924 y Fn(0)2748 1910 y Fz(\).)25 b(No)m(w,)246 2017 y(if)39 b Fl(v)383 2031 y Fn(1)466 2017 y Fl(<)j Fz(2)p Fl(\022)667 2032 y Fk(l)720 2017 y Fm(\000)27 b Fl(v)862 2031 y Fn(0)902 2017 y Fz(,)41 b(then)f(\()p Fl(v)1267 1984 y Fx(0)1264 2042 y Fn(0)1332 2017 y Fz(+)26 b Fl(v)1476 1984 y Fx(0)1473 2042 y Fn(1)1513 2017 y Fz(\))p Fl(=)p Fz(2)44 b Fl(<)e(\022)1838 2032 y Fk(l)1864 2017 y Fz(,)f(and)f(if)g Fl(v)2255 2031 y Fn(1)2337 2017 y Fl(>)j Fz(2)p Fl(\022)2539 2032 y Fk(h)2611 2017 y Fm(\000)27 b Fl(v)2753 2031 y Fn(0)2792 2017 y Fz(,)41 b(then)246 2125 y(\()p Fl(v)328 2092 y Fx(0)325 2150 y Fn(0)385 2125 y Fz(+)20 b Fl(v)523 2092 y Fx(0)520 2150 y Fn(1)559 2125 y Fz(\))p Fl(=)p Fz(2)27 b Fl(>)e(\022)850 2140 y Fk(h)894 2125 y Fz(.)p 2958 2125 48 48 v 726 2553 a Fs(B.)73 b(A)35 b(Di\016cult)h(Instance)e(for)h Fi(sampleevalssat)246 2777 y Fz(This)26 b(section)j(sho)m(ws)g(that)g Fi(sampleevalssat)e Fz(can)i(require)e(exp)s(onen)m(tially)g(man)m(y)246 2884 y(samples)g(to)i(accurately)g(appro)m(ximate)f(some)h Fr(SSa)-6 b(t)27 b Fz(problems.)g(Consider)f(the)246 2992 y Fl(n)p Fz(-v)-5 b(ariable)29 b Fr(SSa)-6 b(t)29 b Fz(form)m(ula)579 3125 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 579 3125 a Fi(R)638 3125 y currentpoint grestore moveto 638 3125 a 579 3188 a Fl(y)624 3202 y Fn(1)663 3188 y Fl(;)762 3125 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 762 3125 a Fi(R)821 3125 y currentpoint grestore moveto 821 3125 a 762 3188 a Fl(y)807 3202 y Fn(2)846 3188 y Fl(;)15 b(:)g(:)g(:)i(;)1107 3125 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1107 3125 a Fi(R)1166 3125 y currentpoint grestore moveto 1166 3125 a 1107 3188 a Fl(y)1152 3202 y Fk(n)p Fx(\000)p Fn(2)1289 3188 y Fl(;)e Fm(9)p Fl(x)1432 3202 y Fk(n)p Fx(\000)p Fn(1)1569 3188 y Fl(;)1668 3125 y gsave currentpoint currentpoint translate 180 neg rotate neg exch neg exch translate 1668 3125 a Fi(R)1727 3125 y currentpoint grestore moveto 1727 3125 a 1668 3188 a Fl(y)1713 3202 y Fk(n)1760 3188 y Fz(\(\()p Fl(x)1882 3202 y Fk(n)p Fx(\000)p Fn(1)2040 3188 y Fm(_)p 2120 3138 92 4 v 19 w Fl(y)2165 3202 y Fk(n)2212 3188 y Fz(\)\()p 2282 3138 190 4 v Fl(x)2334 3202 y Fk(n)p Fx(\000)p Fn(1)2492 3188 y Fm(_)20 b Fl(y)2618 3202 y Fk(n)2664 3188 y Fz(\)\))p Fl(:)246 3384 y Fz(The)31 b(v)-5 b(alue)31 b(of)g(this)g(form)m(ula)f(is)h(1)p Fl(=)p Fz(2,)i(since)e(the)g(form)m(ula)g(is)f Fy(True)h Fz(if)f(and)h(only)246 3492 y(if)e Fl(x)381 3506 y Fk(n)p Fx(\000)p Fn(1)543 3492 y Fz(=)c Fl(y)684 3506 y Fk(n)731 3492 y Fz(,)30 b(whic)m(h)f(happ)s(ens)f(half)h(of)h(the)h(time)e (regardless)h(of)g(the)g(v)-5 b(alue)30 b(of)246 3600 y Fl(x)298 3614 y Fk(n)p Fx(\000)p Fn(1)435 3600 y Fz(.)362 3717 y(The)d(n)m(um)m(b)s(er)f(of)h(p)s(olicies)f(for)h(this)f Fr(SSa)-6 b(t)26 b Fz(form)m(ula)g(is)h Fl(P)38 b Fz(=)25 b(2)2501 3684 y Fn(2)2536 3661 y Ff(n)p Fq(\000)p Fe(2)2689 3717 y Fz(since)i(the)246 3825 y(setting)g(for)g Fl(x)729 3839 y Fk(n)p Fx(\000)p Fn(1)894 3825 y Fz(can)h(b)s(e)e(conditioned)g (on)i(the)f(preceding)g Fl(n)14 b Fm(\000)g Fz(2)27 b(randomized)246 3933 y(v)-5 b(ariables.)362 4041 y(Let)41 b Fl(W)52 b Fz(b)s(e)40 b(a)g(size)g Fl(w)j Fz(sample)c(of)h(the)g(assignmen)m(ts)g (to)h(the)f(randomized)246 4149 y(v)-5 b(ariables)29 b(in)g(the)h(form)m(ula.)g(The)g(estimated)h(v)-5 b(alue)30 b(of)g(the)h(form)m(ula)e(is)861 4258 y Fa(X)842 4455 y Fk(\013)p Fx(2)p Fk(W)1026 4345 y Fz(matc)m(h)q(\()p Fl(\013;)15 b(W)m(;)g(n)p Fz(\))p Fl(=)p Fz(matc)m(h)s(\()p Fl(\013;)g(W)m(;)g(n)22 b Fm(\000)e Fz(2\))p Fl(;)246 4633 y Fz(where)32 b(matc)m(h)q(\()p Fl(\013;)15 b(W)m(;)g(i)p Fz(\))36 b(is)c(the)h(n)m(um)m(b)s(er)f(of)i(samples)e(in)g Fl(W)45 b Fz(that)34 b(matc)m(h)g Fl(\013)f Fz(in)246 4741 y(the)e(setting)h(of)g(the)g(\014rst)f Fl(i)h Fz(v)-5 b(ariables.)30 b(The)h(justi\014cation)f(for)i(this)e(expression)246 4848 y(is)36 b(that)h(the)h(existen)m(tial)e(v)-5 b(ariable)36 b(immediately)g(ab)s(o)m(v)m(e)i(an)m(y)f(giv)m(en)g(sampled)1780 5847 y Fo(paper.tex;)k(1/09/1999;)h(0:14;)e(p.49)p eop %%Page: 50 50 50 49 bop 246 100 a Fz(50)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fz(leaf)42 b(will)d(b)s(e)j(set)g(to)h(maximize)f (the)g(n)m(um)m(b)s(er)f(of)h(sampled)f(lea)m(v)m(es)i(b)s(elo)m(w)f (it)246 399 y(that)37 b(are)h(made)f Fy(True)n Fz(.)h(That)e(is,)h(it)f (will)e(b)s(e)j(set)g(to)h(matc)m(h)g(the)f(ma)5 b(jorit)m(y)37 b(of)246 506 y(the)29 b(assignmen)m(ts)g(to)h Fl(y)1059 520 y Fk(n)1135 506 y Fz(immediately)d(b)s(elo)m(w)i(it.)g(F)-8 b(or)30 b(example,)f(if)f(a)i(decision)246 614 y(no)s(de)f(has)h(only)g (one)h(leaf)f(b)s(eneath,)g(the)h(v)-5 b(alue)29 b(of)i(that)g(leaf)f (will)e(b)s(e)i(1.)362 722 y(F)-8 b(or)31 b(an)m(y)g(sample,)f Fl(\013)p Fz(,)h(with)e(probabilit)m(y)f(at)j(least)f(1)21 b Fm(\000)f Fl(w)r(=)p Fz(2)2449 689 y Fk(n)p Fx(\000)p Fn(1)2587 722 y Fz(,)909 909 y(matc)m(h)q(\()p Fl(\013;)15 b(W)m(;)g(n)p Fz(\))27 b(=)e(matc)m(h)q(\()p Fl(\013;)15 b(W)m(;)g(n)21 b Fm(\000)f Fz(2\))246 1095 y(b)s(ecause)42 b(the)g(second)g(set)g(is)f(only)g(larger)h(than)f(the)h(\014rst)f(set) i(if)e(there)h(is)f(a)246 1203 y(sample)31 b(that)j(matc)m(hes)g Fl(\013)f Fz(in)e(its)h(assignmen)m(t)h(to)g Fl(y)2062 1217 y Fn(1)2101 1203 y Fl(;)15 b(:)g(:)g(:)i(;)e(y)2348 1217 y Fk(n)p Fx(\000)p Fn(2)2517 1203 y Fz(but)32 b(di\013ers)f(in)246 1311 y(its)39 b(assignmen)m(t)h(to)h Fl(y)1026 1325 y Fk(n)1072 1311 y Fz(.)g(Th)m(us,)e(the)h(exp)s(ected)g(v)-5 b(alue)40 b(of)g(the)g Fi(sampleevalssat)246 1419 y Fz(estimate)31 b Fl(E)5 b 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y(Beame,)18 b(P)-6 b(.,)19 b(and)f(Pitassi,)i(T.)j(1996.)h(Simpli\014ed)17 b(and)h(impro)n(v)n(ed)f(resolution)j(lo)n(w)n(er)f(b)r(ounds.)j(In)362 3394 y Fw(37th)28 b(A)n(nnual)g(Symp)l(osium)g(on)g(F)-6 b(oundations)29 b(of)e(Computer)i(Scienc)l(e)p Fv(,)e(274{282.)37 b(IEEE.)246 3485 y(Beame,)22 b(P)-6 b(.;)22 b(Karp,)g(R.;)g(Pitassi,)i (T.;)f(and)e(Saks,)h(M.)29 b(1998.)h(On)21 b(the)g(complexit)n(y)g(of)h (unsatis\014-)362 3576 y(abilit)n(y)h(of)h(random)e Fh(k)r Fv(-CNF)h(form)n(ulas.)30 b(In)22 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)28 b(of)c(the)i(Thirtieth)g(A)n(nnual)g(A)n(CM)362 3667 y(Symp)l(osium)i(on)f(The)l(ory)i(of)e(Computing)p Fv(,)g(561{571.)246 3758 y(Ben-Sasson,)c(E.,)g(and)f(Wigderson,)h(A.)29 b(1999.)h(Short)22 b(pro)r(ofs)i(are)f(narro)n(w:)g(Resolution)g(made)362 3848 y(simple.)34 b(T)-6 b(o)26 b(app)r(ear)g(in)g(STOC)g(1999.)246 3939 y(Birn)n(baum,)j(E.,)i(and)f(Lozinskii,)i(E.)f(L.)48 b(1999.)i(The)30 b(go)r(o)r(d)i(old)e(Da)n(vis-Putnam)f(pro)r(cedure) 362 4030 y(helps)d(coun)n(ting)f(mo)r(dels.)35 b Fw(Journal)28 b(of)f(A)n(rti\014cial)h(Intel)t(ligenc)l(e)g(R)l(ese)l(ar)l(ch)f Fv(10:457{477.)246 4121 y(Cadoli,)21 b(M.;)g(Gio)n(v)l(anardi,)g(A.;)f (and)g(Sc)n(haerf,)g(M.)26 b(1997.)g(Exp)r(erimen)n(tal)20 b(analysis)h(of)g(the)e(com-)362 4212 y(putational)29 b(cost)g(of)h(ev)l(aluating)f(quan)n(ti\014ed)e(Bo)r(olean)k(form)n (ulae.)44 b(In)28 b Fw(Fifth)i(Confer)l(enc)l(e)362 4303 y(of)k(the)h(Italian)f(Asso)l(ciation)i(for)e(A)n(rti\014cial)h(Intel)t (ligenc)l(e)g(\(AI*IA'97\))p Fv(,)e(n)n(um)n(b)r(er)e(1321,)362 4394 y(207{218.)37 b(Springer-V)-6 b(erlag.)246 4485 y(Cadoli,)28 b(M.;)f(Gio)n(v)l(anardi,)g(A.;)g(and)f(Sc)n(haerf,)h(M.) 37 b(1998.)h(An)26 b(algorithm)g(to)h(ev)l(aluate)f(quan-)362 4576 y(ti\014ed)f(Bo)r(olean)i(form)n(ulae.)34 b(In)24 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)e(the)h(Fifte)l(enth)g (National)f(Confer)l(enc)l(e)i(on)362 4667 y(A)n(rti\014cial)e(Intel)t (ligenc)l(e)h(\(AAAI-98\))p Fv(,)f(262{267.)38 b(The)26 b(AAAI)e(Press/The)j(MIT)f(Press.)246 4758 y(Ch)n(v\023)-38 b(atal,)24 b(V.,)g(and)f(Reed,)g(B.)31 b(1992.)h(Mic)n(k)24 b(gets)g(some)f(\(the)g(o)r(dds)h(are)g(on)f(his)h(side\).)31 b(In)22 b Fw(33nd)362 4848 y(A)n(nnual)28 b(Symp)l(osium)f(on)h(F)-6 b(oundations)29 b(of)f(Computer)g(Scienc)l(e)p Fv(,)f(620{627.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.50)p eop %%Page: 51 51 51 50 bop 1144 100 a Fn(Sto)r(c)n(hastic)25 b(Bo)r(olean)g (Satis\014abilit)n(y)807 b Fz(51)246 291 y Fv(Clegg,)34 b(M.;)f(Edmonds,)f(J.;)h(and)f(Impagliazzo,)i(R.)54 b(1996.)i(Using)32 b(the)g(Gr\177)-38 b(obner)33 b(basis)g(al-)362 381 y(gorithm)e(to)g (\014nd)f(pro)r(ofs)i(of)g(unsatis\014abilit)n(y)-6 b(.)51 b(In)31 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)f(the)g(Twenty-Eighth)362 472 y(A)n(nnual)28 b(A)n(CM)f(Symp)l(osium)h(on)f(The)l(ory)i(of)e (Computing)p Fv(,)g(174{183.)246 563 y(Condon,)20 b(A.;)g(F)-6 b(eigen)n(baum,)20 b(J.;)h(Lund,)e(C.;)i(and)e(Shor,)h(P)-6 b(.)25 b(1997.)i(Random)19 b(debators)h(and)g(the)362 654 y(hardness)32 b(of)g(appro)n(ximating)f(sto)r(c)n(hastic)i (functions.)53 b Fw(SIAM)32 b(Journal)i(on)g(Computing)362 745 y Fv(26\(2\):369{400.)246 836 y(Condon,)50 b(A.)106 b(1992.)i(The)50 b(complexit)n(y)f(of)h(sto)r(c)n(hastic)h(games.)107 b Fw(Information)50 b(and)362 927 y(Computation)27 b Fv(96\(2\):203{224.)246 1018 y(Cra)n(wford,)c(J.)f(M.,)g(and)e(Auton,)h (L.)g(D.)27 b(1996.)h(Exp)r(erimen)n(tal)21 b(results)h(in)f(the)f (crosso)n(v)n(er)i(p)r(oin)n(t)362 1109 y(in)j(random)g(3SA)-6 b(T.)34 b Fw(A)n(rti\014cial)28 b(Intel)t(ligenc)l(e)e Fv(81\(1-2\):31|57.)246 1200 y(Da)n(vis,)f(M.;)h(Logemann,)f(G.;)h(and) f(Lo)n(v)n(eland,)g(D.)33 b(1962.)i(A)24 b(mac)n(hine)g(program)h(for)h (theorem)362 1291 y(pro)n(ving.)34 b Fw(Communic)l(ations)28 b(of)f(the)i(A)n(CM)c Fv(5:394{397.)246 1381 y(Dec)n(h)n(ter,)40 b(R.)78 b(1996.)i(Buc)n(k)n(et)39 b(elimination:)j(A)d(unifying)i (framew)n(ork)g(for)g(probabilistic)362 1472 y(inference.)64 b(In)34 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)39 b(of)d(the)h(12th)g(Confer)l (enc)l(e)h(on)f(Unc)l(ertainty)h(in)e(A)n(rti\014cial)362 1563 y(Intel)t(ligenc)l(e)28 b(\(UAI-96\))p Fv(,)f(211{219.)37 b(Morgan)27 b(Kaufmann)d(Publishers.)246 1654 y(F)-6 b(reeman,)38 b(J.)i(W.)73 b(1995.)j Fw(Impr)l(ovements)41 b(to)f(pr)l(op)l(ositional)h(satis\014ability)g(se)l(ar)l(ch)g(algo-) 362 1745 y(rithms)p Fv(.)e(Ph.D.)28 b(Dissertation,)g(Departmen)n(t)e (of)i(Computer)e(and)h(Information)g(Science,)362 1836 y(Univ)n(ersit)n(y)e(of)h(P)n(ennsylv)l(ania.)246 1927 y(F)-6 b(riedgut,)23 b(E.)32 b(1997.)g(Necessary)24 b(and)f(su\016cien) n(t)h(conditions)g(for)g(sharp)g(thresholds)g(of)g(graph)362 2018 y(prop)r(erties,)j(and)e(the)g Fh(k)r Fv(-SA)-6 b(T)25 b(problem.)33 b(Preprin)n(t.)246 2109 y(F)-6 b(rieze,)20 b(A.,)g(and)f(Suen,)g(S.)24 b(1996.)h(Analysis)20 b(of)h(t)n(w)n(o)e (simple)h(heuristics)g(on)f(a)h(random)f(instance)362 2200 y(of)26 b(k-SA)-6 b(T.)33 b Fw(Journal)c(of)e(A)n(lgorithms)f Fv(20\(2\):312{355.)246 2291 y(Gallo,)46 b(G.,)g(and)e(Urbani,)h(G.)92 b(1989.)h(Algorithms)44 b(for)i(testing)f(the)f(satis\014abilit)n(y)i (of)362 2381 y(prop)r(ositional)27 b(form)n(ulae.)35 b Fw(Journal)28 b(of)g(L)l(o)l(gic)g(Pr)l(o)l(gr)l(amming)e Fv(7:45{61.)246 2472 y(Gen)n(t,)i(I.,)h(and)f(W)-6 b(alsh,)29 b(T.)43 b(1998.)i(Bey)n(ond)28 b(NP:)h(the)f(QSA)-6 b(T)27 b(phase)i(transition.)43 b(T)-6 b(ec)n(hnical)362 2563 y(Rep)r(ort)25 b(APES-05-1998.)246 2654 y(Go)r(erdt,)g(A.)33 b(1996.)i(A)24 b(threshold)h(for)h(unsatis\014abilit)n(y)-6 b(.)33 b Fw(Journal)27 b(of)g(Computer)h(and)f(System)362 2745 y(Scienc)l(es)g Fv(53:469{486.)246 2836 y(Gu,)37 b(J.;)i(Purdom,)e(P)-6 b(.)38 b(W.;)g(F)-6 b(ranco,)38 b(J.;)h(and)e(W)-6 b(ah,)37 b(B.)i(J.)70 b(1997.)i(Algorithms)37 b(for)i(the)362 2927 y(satis\014abilit)n(y)26 b(\(SA)-6 b(T\))24 b(problem:)h(A)f(surv)n(ey)-6 b(.)33 b(In)24 b Fw(Satis\014ability)k(\(SA)-6 b(T\))28 b(Pr)l(oblem)p Fv(,)d(19{151.)362 3018 y(DIMA)n(CS,)g(American)g(Mathematical)i(So)r (ciet)n(y)-6 b(.)246 3109 y(Harc)n(he,)35 b(F.;)h(Ho)r(ok)n(er,)f(J.)h (N.;)g(and)e(Thompson,)h(G.)63 b(1994.)h(A)35 b(computational)g(study)f (of)362 3200 y(satis\014abilit)n(y)g(algorithms)g(for)g(prop)r (ositional)h(logic.)58 b Fw(ORSA)34 b(Journal)h(on)g(Computing)362 3291 y Fv(6:423|435.)246 3381 y(Ho)r(ok)n(er,)f(J.)h(N.,)f(and)f(Vina)n (y)-6 b(,)34 b(V.)58 b(1994.)j(Branc)n(hing)34 b(rules)h(for)g (satis\014abilit)n(y)-6 b(.)59 b(T)-6 b(ec)n(hnical)362 3472 y(Rep)r(ort)39 b(GSIA)g(W)-6 b(orking)39 b(P)n(ap)r(er)h(1994-09,) i(Carnegie)f(Mellon)f(Univ)n(ersit)n(y)-6 b(.)75 b(Revised)362 3563 y(Jan)n(uary)26 b(1995.)246 3654 y(Jeroslo)n(w,)40 b(R.,)d(and)g(W)-6 b(ang,)37 b(J.)69 b(1990.)i(Solving)37 b(prop)r(ositional)i(satis\014abilit)n(y)f(problems.)362 3745 y Fw(A)n(nnals)28 b(of)f(Mathematics)i(and)f(AI)d Fv(1:167|187.)246 3836 y(Kamath,)38 b(A.;)h(Mot)n(w)n(ani,)h(R.;)f(P)n (alem,)g(K.;)g(and)f(Spirakis,)h(P)-6 b(.)73 b(1995.)i(T)-6 b(ail)40 b(b)r(ounds)e(for)362 3927 y(o)r(ccupancy)31 b(and)g(the)g(satis\014abilit)n(y)i(threshold)e(conjecture.)53 b Fw(R)l(andom)33 b(Structur)l(es)j(and)362 4018 y(A)n(lgorithms)26 b Fv(7\(1\):59{80.)246 4109 y(Kautz,)18 b(H.,)g(and)g(Selman,)g(B.)k (1996.)i(Pushing)18 b(the)g(en)n(v)n(elop)r(e:)g(Planning,)h(prop)r (ositional)h(logic,)362 4200 y(and)26 b(sto)r(c)n(hastic)i(searc)n(h.) 38 b(In)26 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(of)d(the)i(Thirte)l (enth)g(National)f(Confer)l(enc)l(e)h(on)362 4291 y(A)n(rti\014cial)d (Intel)t(ligenc)l(e)p Fv(,)g(1194{1201.)38 b(AAAI)24 b(Press/The)j(MIT)f(Press.)246 4381 y(Kearns,)32 b(M.;)h(Mansour,)g (Y.;)e(and)h(Ng,)g(A.)f(Y.)53 b(1999.)h(A)31 b(sparse)h(sampling)g (algorithm)g(for)362 4472 y(near-optimal)25 b(planning)h(large)h(mark)n (o)n(v)d(decision)j(pro)r(cesses.)36 b(Submitted.)246 4563 y(Kirousis,)c(L.)g(M.;)g(Kranakis,)g(E.;)g(Krizanc,)g(D.;)g(and)f (Stamatiou,)g(Y.)g(C.)52 b(1998.)h(Appro)n(x-)362 4654 y(imating)31 b(the)f(unsatis\014abilit)n(y)h(threshold)g(of)h(random)e (form)n(ulas.)50 b Fw(R)l(andom)33 b(Structur)l(es)362 4745 y(and)28 b(A)n(lgorithms)e Fv(12\(3\):253{269.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.51)p eop %%Page: 52 52 52 51 bop 246 100 a Fz(52)828 b Fn(Littman,)23 b(Ma)t(jercik,)f(and)j (Pitassi)246 291 y Fv(Li,)i(C.)g(M.,)h(and)e(An)n(bulagan.)37 b(1997.)i(Heuristics)27 b(based)g(on)f(unit)g(propagation)i(for)f (satis\014a-)362 381 y(bilit)n(y)h(problems.)43 b(In)28 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)33 b(of)d(the)h(Fifte)l(enth)g (International)h(Joint)f(Confer)l(enc)l(e)362 472 y(on)d(A)n (rti\014cial)f(Intel)t(ligenc)l(e)p Fv(,)g(366|371.)246 563 y(Littman,)39 b(M.)i(L.;)f(Goldsmith,)g(J.;)h(and)f(Mundhenk,)f(M.) 77 b(1998.)i(The)40 b(computational)362 654 y(complexit)n(y)28 b(of)i(probabilistic)h(plan)e(existence)h(and)f(ev)l(aluation.)46 b Fw(Journal)31 b(of)g(A)n(rti\014cial)362 745 y(Intel)t(ligenc)l(e)d (R)l(ese)l(ar)l(ch)g Fv(9:1{36.)246 836 y(Littman,)34 b(M.)h(L.)61 b(1997.)h(Probabilistic)36 b(prop)r(ositional)h(planning:) e(Represen)n(tations)f(and)362 927 y(complexit)n(y)-6 b(.)42 b(In)29 b Fw(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)d(the)h(F)-6 b(ourte)l(enth)33 b(National)e(Confer)l(enc)l(e)h(on)f(A)n(rti\014cial) 362 1018 y(Intel)t(ligenc)l(e)p Fv(,)26 b(748{754.)37 b(AAAI)25 b(Press/The)i(MIT)f(Press.)246 1109 y(Lusena,)c(C.;)h (Goldsmith,)f(J.;)h(and)f(Mundhenk,)e(M.)29 b(1998.)h(Nonappro)n (ximabilit)n(y)21 b(results)h(for)362 1200 y(Mark)n(o)n(v)e(decision)g (pro)r(cesses.)26 b(T)-6 b(ec)n(hnical)21 b(Rep)r(ort)e(UK)g(CS)h(Dept) f(TR)h(275-98,)h(Univ)n(ersit)n(y)362 1291 y(of)26 b(Ken)n(tuc)n(ky)-6 b(.)246 1381 y(Ma)t(jercik,)26 b(S.)f(M.,)h(and)e(Littman,)g(M.)h(L.)33 b(1998.)i(MAXPLAN:)24 b(A)g(new)h(approac)n(h)g(to)f(proba-)362 1472 y(bilistic)f(planning.)29 b(In)21 b(Simmons,)g(R.;)h(V)-6 b(eloso,)23 b(M.;)g(and)e(Smith,)g(S.,)h(eds.,)h Fw(Pr)l(o)l(c)l(e)l(e) l(dings)j(of)362 1563 y(the)32 b(F)-6 b(ourth)33 b(International)g 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b(1998.)h(P)n(ersonal)27 b(comm)n(unication.)246 3291 y(Roth,)i(D.)47 b(1996.)h(On)30 b(the)f(hardness)h(of)h(appro)n (ximate)e(reasoning.)48 b Fw(A)n(rti\014cial)32 b(Intel)t(ligenc)l(e) 362 3381 y Fv(82\(1{2\):273{302.)246 3472 y(Selman,)22 b(B.;)i(Kautz,)f(H.;)g(and)g(Cohen,)g(B.)30 b(1996.)i(Lo)r(cal)24 b(searc)n(h)g(strategies)g(for)g(satis\014abilit)n(y)362 3563 y(testing.)35 b(In)25 b(Johnson,)h(D.)g(S.,)g(and)f(T)-6 b(ric)n(k,)26 b(M.)h(A.,)e(eds.,)i Fw(Cliques,)g(Coloring,)g(and)h (Satis-)362 3654 y(\014ability)p Fv(.)f(American)f(Mathematical)h(So)r (ciet)n(y)-6 b(.)37 b(521{531.)k(DIMA)n(CS)26 b(Series)h(in)g(Discrete) 362 3745 y(Mathematics)f(and)f(Theoretical)j(Computer)d(Science,)h(v.)f (26.)246 3836 y(Selman,)34 b(B.;)i(Mitc)n(hell,)g(D.;)f(and)f(Lev)n (esque,)g(H.)61 b(1996.)i(Generating)36 b(hard)e(satis\014abilit)n(y) 362 3927 y(problems.)g Fw(A)n(rti\014cial)27 b(Intel)t(ligenc)l(e)f Fv(81:17{29.)246 4018 y(Shac)n(h)n(ter,)45 b(R.)h(D.)94 b(1986.)i(Ev)l(aluating)47 b(in\015uence)e(diagrams.)95 b Fw(Op)l(er)l(ations)48 b(R)l(ese)l(ar)l(ch)362 4109 y Fv(34\(6\):871{882.)246 4200 y(Umans,)25 b(C.)34 b(1999.)i(P)n (ersonal)27 b(comm)n(unication.)1780 5847 y Fo(paper.tex;)41 b(1/09/1999;)h(0:14;)e(p.52)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF