%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: survey.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips survey.dvi -o %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX Output 1998.05.22:0052 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 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b(formalized)f(b)m(y)i(Co)s(ok)f(and)g(Rec)m(kho)m(w)i([)p Fr(36)p Fw(],)f(there)g(exists)f(a)h(prop)s(ositional)456 3757 y(pro)s(of)f(system)i(giving)f(rise)f(to)j(short)e(\(p)s (olynomial-size\))f(pro)s(ofs)g(of)i(all)f(tautologies)456 3864 y(if)25 b(and)g(only)h(if)f Fx(NP)i Fw(equals)e Fx(co-NP)r Fw(.)39 b(Co)s(ok)26 b(and)g(Rec)m(kho)m(w)h(w)m(ere)g(the)g (\014rst)e(to)i(prop)s(ose)456 3972 y(a)k(program)h(of)f(researc)m(h)i (aimed)d(at)j(attac)m(king)g(the)e Fx(NP)h Fw(v)m(ersus)f Fx(co-NP)j Fw(problem)29 b(b)m(y)456 4080 y(systematically)c(studying)g (and)g(pro)m(ving)g(strong)i(lo)m(w)m(er)f(b)s(ounds)e(for)i(standard)f (pro)s(of)456 4188 y(systems)32 b(of)g(increasing)f(complexit)m(y)-8 b(.)46 b(This)30 b(program)i(has)g(sev)m(eral)g(imp)s(ortan)m(t)f(side) 456 4296 y(e\013ects.)605 4404 y(First,)e(standard)e(pro)s(of)h (systems)h(are)g(in)m(teresting)f(in)f(their)h(o)m(wn)h(righ)m(t.)40 b(Almost)456 4512 y(all)30 b(theorem-pro)m(ving)i(systems)g(implemen)m (t)e(a)i(deterministic)d(or)j(randomized)f(pro-)456 4620 y(cedure)40 b(that)i(is)e(based)g(on)h(a)g(standard)f(prop)s(ositional) e(pro)s(of)j(system,)i(and)e(th)m(us)456 4728 y(upp)s(er)32 b(and)j(lo)m(w)m(er)g(b)s(ounds)d(on)j(these)g(systems)g(shed)f(ligh)m (t)g(on)h(the)g(inheren)m(t)e(com-)456 4836 y(plexit)m(y)j(of)h(an)m(y) h(theorem-pro)m(ving)f(system)h(up)s(on)d(whic)m(h)h(it)h(is)f(based.) 61 b(The)37 b(most)p 456 4942 499 4 v 605 5001 a Fq(1)640 5033 y Fp(Computer)27 b(Science)i(and)g(Engineering,)h(Univ)n(ersit)n (y)e(of)h(W)-6 b(ashington,)30 b(Seattle,)g(W)-9 b(A)29 b(98195-)456 5124 y(2350,)e Fo(beame@cs.washington.edu)605 5184 y Fq(2)640 5216 y Fp(Computer)c(Science,)h(Univ)n(ersit)n(y)f(of)i (Arizona,)g(T)-6 b(ucson,)24 b(AZ)g(85721,)i Fo(toni@cs.arizona.edu) 1932 5315 y Fs(1)p eop %%Page: 2 2 2 1 bop 456 425 a Fw(striking)37 b(example)i(is)f(Resolution)h(on)g (whic)m(h)f(almost)h(all)f(prop)s(ositional)f(theorem)456 533 y(pro)m(v)m(ers)30 b(\(and)g(ev)m(en)h(\014rst-order)f(theorem)h (pro)m(v)m(ers\))g(are)g(based.)605 641 y(Secondly)-8 b(,)27 b(and)g(of)g(equal)g(or)g(greater)h(imp)s(ortance,)f(lo)m(w)m (er)g(b)s(ounds)e(on)i(standard)456 749 y(pro)s(of)34 b(systems)i(additionally)d(pro)m(v)m(e)j(that)h(a)f(certain)f(class)g (of)h(algorithms)f(for)g(the)456 857 y(satis\014abilit)m(y)28 b(problem)h(will)e(fail)i(to)i(run)e(in)g(p)s(olynomial-time.)605 965 y(This)k(program)h(has)h(led)e(to)j(man)m(y)e(b)s(eautiful)f (results)g(as)i(w)m(ell)e(as)i(to)h(new)e(con-)456 1073 y(nections)24 b(with)g(circuit)f(complexit)m(y)i(within)d(the)j(last)g (t)m(w)m(en)m(t)m(y)i(y)m(ears.)40 b(In)24 b(this)f(article,)456 1181 y(w)m(e)31 b(will)d(try)i(to)i(highligh)m(t)c(some)j(of)g(the)g (main)e(disco)m(v)m(eries,)i(with)e(emphasis)h(on)g(the)456 1289 y(in)m(terpla)m(y)c(b)s(et)m(w)m(een)i(logic,)g(\(circuit\))f (complexit)m(y)g(theory)-8 b(,)29 b(and)e(com)m(binatorics)g(that)456 1397 y(has)39 b(arisen.)67 b(W)-8 b(e)40 b(omit)g(all)e(pro)s(ofs;)43 b(see)d([)p Fr(91)q Fw(])g(for)f(a)h(quite)f(readable)g(surv)m(ey)g (that)456 1504 y(includes)28 b(detailed)h(pro)s(ofs)h(of)g(man)m(y)h (of)f(the)h(earlier)e(results.)605 1612 y(In)46 b(section)g(2,)51 b(w)m(e)c(de\014ne)f(v)-5 b(arious)45 b(pro)s(of)h(systems)h(that)g (will)c(b)s(e)j(discussed)456 1720 y(throughout)41 b(this)g(article.)74 b(In)42 b(section)g(3,)j(w)m(e)d(review)f(some)h(of)g(the)g(main)f(lo)m (w)m(er)456 1828 y(b)s(ounds)27 b(that)k(ha)m(v)m(e)g(b)s(een)e(pro)m (v)m(en)h(for)g(standard)f(pro)s(of)g(systems,)h(emphasizing)e(the)456 1936 y(com)m(binatorial)38 b(tec)m(hniques)h(and)f(connections)i(to)f (circuit)f(complexit)m(y)h(that)h(ha)m(v)m(e)456 2044 y(b)s(een)33 b(sho)m(wn.)51 b(Finally)32 b(in)g(section)j(4,)g(w)m(e)g (list)d(some)j(of)f(the)g(main)f(op)s(en)g(questions)456 2152 y(and)c(promising)f(directions)h(in)g(the)i(area.)1246 2408 y Fr(2.)52 b(Prop)s(ositional)36 b(pro)s(of)g(systems)605 2570 y Fw(What)24 b(exactly)g(is)e(a)h(prop)s(ositional)e(pro)s(of)7 b(?)37 b(Co)s(ok)23 b(and)f(Rec)m(kho)m(w)j(w)m(ere)e(p)s(ossibly)456 2678 y(the)40 b(\014rst)g(to)h(mak)m(e)g(this)e(and)h(related)g (questions)f(precise.)70 b(They)40 b(sa)m(w)h(that)f(it)g(is)456 2786 y(useful)28 b(to)j(separate)g(the)f(idea)g(of)g(pro)m(viding)e(a)j (pro)s(of)e(from)h(that)h(of)f(b)s(eing)f(e\016cien)m(t.)456 2894 y(Since)23 b(there)i(are)g(only)f(\014nitely)f(man)m(y)i(truth)f (assignmen)m(ts)g(to)i(c)m(hec)m(k,)i(wh)m(y)c(not)h(allo)m(w)456 3001 y(the)38 b(statemen)m(t)h(itself)e(as)h(a)h(pro)s(of)7 b(?)62 b(What)38 b(extra)h(v)-5 b(alue)37 b(is)g(there)h(in)f(a)h (\014lled)e(out)456 3109 y(truth)h(table)h(or)g(a)g(deriv)-5 b(ation)37 b(using)g(some)h(axiom/inference)g(sc)m(heme?)64 b(The)38 b(k)m(ey)456 3217 y(observ)-5 b(ation)32 b(is)f(that)i(a)g (pro)s(of)f(is)f(easy)i(to)g(c)m(hec)m(k,)i(unlik)m(e)30 b(the)j(statemen)m(t)h(itself.)46 b(Of)456 3325 y(course)26 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(ofs.)1724 4174 y Fs(3)1802 4207 y Fw(Prop)s(ert)m(y)f(\(1\))i(ensures) d(that)i(the)g(system)f Fm(S)33 b Fw(is)456 4315 y(logically)22 b(b)s(oth)h(sound)g(and)h(complete.)39 b(The)23 b Fn(c)-5 b(omplexity)26 b Fm(comp)2685 4337 y Fk(S)2759 4315 y Fw(of)f(a)f(prop)s(ositional)456 4423 y(pro)s(of)h(system)i Fm(S)k Fw(is)25 b(then)h(de\014ned)f(to)j(b)s(e)d(the)i(smallest)e(b)s (ounding)f(function)h Fm(b)g Fw(:)h Fj(N)37 b Fl(!)456 4531 y Fj(N)48 b Fw(on)36 b(the)h(lengths)e(of)i(the)f(pro)s(ofs)f(in)g Fm(S)41 b Fw(as)c(a)f(function)f(of)h(the)h(tautologies)g(b)s(eing)p 456 4668 499 4 v 605 4727 a Fq(3)640 4759 y Fp(Co)r(ok)29 b(and)g(Rec)n(kho)n(w's)g(de\014nition)f(is)i(formally)f(di\013eren)n (t)f(although)h(essen)n(tially)h(equiv)l(alen)n(t)456 4850 y(to)e(this)h(one.)43 b(They)29 b(de\014ne)e(a)i(pro)r(of)h (system)d(as)j(a)e(p)r(olynomial-time)g(computable)g Fi(onto)h Fp(function)456 4942 y Fh(f)g Fp(:)22 b(\006)620 4910 y Fg(\003)678 4942 y Ff(!)p Fp(T)-6 b(A)n(UT)22 b(whic)n(h)g(can)h(b)r(e)f(though)n(t)g(of)i(as)f(mapping)e(eac)n(h)i (string)g(view)n(ed)g(as)g(p)r(oten)n(tial)g(pro)r(of)456 5033 y(on)n(to)28 b(the)h(tautology)g(it)g(pro)n(v)n(es.)43 b(The)29 b(analogous)i(function)d Fh(f)37 b Fp(in)29 b(our)g(case)g(w)n(ould)g(map)f(\()p Fh(F)r(;)13 b(p)p Fp(\))28 b(to)456 5124 y Fh(F)34 b Fp(if)24 b Fh(S)t Fp(\()p Fh(F)r(;)14 b(p)p Fp(\))23 b(w)n(ere)h(true)f(and)h(w)n(ould)g (map)f(it)g(to)h(a)g(trivial)h(tautology)-6 b(,)25 b(\()p Fh(x)12 b Ff(_)i(:)p Fh(x)p Fp(\),)23 b(otherwise.)36 b(In)23 b(the)456 5216 y(con)n(v)n(erse)i(direction)h(one)g(w)n(ould)g (de\014ne)f Fh(S)t Fp(\()p Fh(F)r(;)14 b(p)p Fp(\))25 b(to)g(b)r(e)h(true)f(i\013)h Fh(f)8 b Fp(\()p Fh(p)p Fp(\))21 b(=)g Fh(F)11 b Fp(.)p eop %%Page: 3 3 3 2 bop 456 425 a Fw(pro)m(v)m(ed,)31 b(i.e.)40 b(for)30 b(all)g Fm(F)13 b Fw(,)1194 590 y Fm(F)39 b Fl(2)25 b Fw(T)-8 b(A)m(UT)25 b Fl(,)h(9)p Fm(p:)p Fl(j)p Fm(p)p Fl(j)f(\024)g Fm(b)p Fw(\()p Fl(j)p Fm(F)13 b Fl(j)p Fw(\))p Fm(:)31 b(S)5 b Fw(\()p Fm(F)s(;)15 b(p)p Fw(\))p Fm(:)456 754 y Fw(E\016cien)m(t)28 b(pro)s(of)f(systems)h(corresp)s (ond)f(to)i(those)g(of)f(p)s(olynomial)e(complexit)m(y;)j(these)456 862 y(are)h(called)g Fn(p-b)-5 b(ounde)g(d)p Fw(.)605 970 y(Giv)m(en)31 b(these)g(de\014nitions,)d(man)m(y)j(natural)e (questions)h(arise:)41 b(Ho)m(w)31 b(e\016cien)m(t)g(are)456 1078 y(existing)39 b(pro)s(of)h(systems?)71 b(Ho)m(w)41 b(can)g(one)g(compare)g(the)f(relativ)m(e)h(e\016ciencies)f(of)456 1186 y(pro)s(of)f(systems?)70 b(Can)40 b(one)h(classify)e(pro)s(of)h (systems)g(using)f(reduction)g(as)i(w)m(e)f(do)456 1294 y(languages?)45 b(Is)32 b(there)g(a)h(pro)s(of)e(system)i(of)f(optimal) f(complexit)m(y)h(\(up)f(to)i(a)f(p)s(olyno-)456 1402 y(mial\)?)605 1510 y(The)f(k)m(ey)g(to)s(ol)g(for)g(comparing)g(pro)s (of)f(systems)h(is)f(p-sim)m(ulation.)41 b(A)31 b(pro)s(of)f(sys-)456 1618 y(tem)21 b Fm(T)33 b Fn(p-simulates)22 b Fw(a)f(pro)s(of)f(system) h Fm(S)k Fw(i\013)20 b(there)h(is)f(a)h(p)s(olynomial-time)d (computable)456 1726 y(function)29 b Fm(f)39 b Fw(mapping)29 b(pro)s(ofs)g(in)g Fm(S)36 b Fw(in)m(to)30 b(pro)s(ofs)f(in)g Fm(T)13 b Fw(,)31 b(that)g(is)e(for)h(all)f Fm(F)39 b Fl(2)25 b Fw(T)-8 b(A)m(UT,)456 1834 y Fm(S)5 b Fw(\()p Fm(F)s(;)15 b(p)p Fw(\))31 b Fl(,)g Fm(T)13 b Fw(\()p Fm(F)s(;)i(f)10 b Fw(\()p Fm(p)p Fw(\)\).)52 b(W)-8 b(e)35 b(use)f(non-standard)e(notation)i(and)g(write)f Fm(S)j Fl(\024)3199 1848 y Fk(p)3269 1834 y Fm(T)47 b Fw(in)456 1966 y(this)38 b(case.)69 b(Clearly)-8 b(,)41 b(it)e(implies)e(that)j Fm(comp)2075 1988 y Fk(T)2171 1966 y Fl(\024)g Fm(comp)2491 1918 y Fk(O)r Fs(\(1\))2491 1995 y Fk(S)2641 1966 y Fw(.)68 b(\(One)39 b(sa)m(ys)i(that)f Fm(T)456 2074 y Fn(we)-5 b(akly)32 b(p-simulates)e Fm(S)k Fw(i\013)28 b(w)m(e)i(ha)m(v)m(e)g (this)e(latter)h(condition)f(but)g(w)m(e)h(do)g(not)g(kno)m(w)g(if)456 2182 y(suc)m(h)h(a)i(reducing)d(function)h Fm(f)40 b Fw(exists.\))j(One)30 b(sa)m(ys)i(that)g Fm(S)k Fw(and)30 b Fm(T)44 b Fw(are)31 b Fn(p-e)-5 b(quivalent)456 2290 y Fw(i\013)38 b(eac)m(h)h(p-sim)m(ulates)f(the)h(other.)66 b(Ob)m(viously)-8 b(,)39 b(t)m(w)m(o)h(p-equiv)-5 b(alen)m(t)38 b(pro)s(of)g(systems)456 2398 y(either)29 b(are)i(b)s(oth)f(p-b)s (ounded)e(or)i(neither)g(is.)605 2597 y Fr(2.1.)53 b(F)-9 b(rege)23 b(and)h(extended-F)-9 b(rege)23 b(pro)s(ofs.)46 b Fw(Co)s(ok)20 b(and)g(Rec)m(kho)m(w)h(did)e(more)456 2705 y(than)k(merely)g(formalize)f(the)i(in)m(tuitiv)m(e)e(general)i (notions)f(of)g(the)h(e\016ciency)f(of)h(prop)s(o-)456 2813 y(sitional)k(pro)s(ofs.)40 b(They)29 b(also)h(iden)m(ti\014ed)f(t) m(w)m(o)i(ma)5 b(jor)30 b(classes)g(of)h(p-equiv)-5 b(alen)m(t)29 b(pro)s(of)456 2921 y(systems)40 b(whic)m(h)g(they)h(called)f(F)-8 b(rege)42 b(and)e(extended-F)-8 b(rege)43 b(systems)e(in)e(honor)h(of) 456 3029 y(Gottlob)j(F)-8 b(rege)43 b(who)f(made)h(some)f(of)h(the)f (\014rst)g(attempts)h(to)g(formalize)f(mathe-)456 3137 y(matics)30 b(based)g(on)h(logic)f(and)g(set)h(theory)g([)p Fr(41)q(,)k(42)p Fw(])c(\(and)g(whose)f(w)m(ork)h(is)e(no)m(w)i(b)s (est)456 3245 y(kno)m(wn)23 b(as)i(the)f(unfortunate)g(victim)f(of)i (Russell's)d(famous)i(parado)m(x)g(concerning)g(the)456 3353 y(set)31 b(of)f(all)f(sets)i(that)g(are)g(not)f(mem)m(b)s(ers)g (of)h(themselv)m(es\).)605 3461 y(A)24 b(F)-8 b(rege)25 b(system)e Fl(F)33 b Fw(is)22 b(de\014ned)g(in)g(terms)i(of)f(a)h (\014nite,)g(implicationally)c(complete)456 3569 y(\(enough)g(to)i (deriv)m(e)e(ev)m(ery)i(true)e(statemen)m(t\))j(set)f Fl(A)2221 3583 y Fe(F)2302 3569 y Fw(of)f(axioms)f(and)h(inference)f (rules.)456 3686 y(The)j(general)h(form)f(of)h(an)g(inference)f(rule)g (is)f(written)h(as)2430 3645 y Fk(A)2483 3654 y Fq(1)2518 3645 y Fk(;:::)10 b(;A)2681 3657 y Fd(k)p 2430 3665 289 4 v 2546 3718 a Fk(B)2752 3686 y Fw(where)24 b Fm(A)3077 3700 y Fs(1)3116 3686 y Fm(;)15 b(:)g(:)g(:)32 b(;)15 b(A)3401 3701 y Fk(k)456 3794 y Fw(and)30 b Fm(B)36 b Fw(are)c(prop)s(ositional)d(form)m(ulas;)i(the)g(rule)g(is)f(an)h (axiom)g(if)g Fm(k)f Fw(=)d(0.)44 b(A)31 b(form)m(ula)456 3902 y Fm(H)44 b Fw(follo)m(ws)37 b(from)g(form)m(ulas)g Fm(G)1553 3916 y Fs(1)1592 3902 y Fm(;)15 b(:)g(:)g(:)32 b(;)15 b(G)1881 3917 y Fk(k)1961 3902 y Fw(using)36 b(this)h(inference) g(rule)f(if)h(there)h(is)f(a)456 4010 y(consisten)m(t)e(set)g(of)g (substitutions)e Fm(\033)38 b Fw(of)d(form)m(ulas)f(for)h(the)g(v)-5 b(ariables)33 b(app)s(earing)h(in)456 4118 y(the)c(rule)f(suc)m(h)h (that)h Fm(G)1268 4132 y Fk(i)1321 4118 y Fw(=)25 b Fm(A)1485 4085 y Fk(\033)1485 4144 y(i)1562 4118 y Fw(for)31 b Fm(i)25 b Fw(=)g(1)p Fm(;)15 b(:)g(:)g(:)32 b(;)15 b(k)34 b Fw(and)c Fm(H)i Fw(=)25 b Fm(B)2652 4085 y Fk(\033)2699 4118 y Fw(.)605 4226 y(F)-8 b(or)41 b(a)h(F)-8 b(rege)42 b(system)f Fl(F)9 b Fw(,)43 b(a)e(t)m(ypical)g(set)g(of)g(axioms)f Fl(A)2623 4240 y Fe(F)2724 4226 y Fw(migh)m(t)h(include)d(the)456 4334 y(axiom)29 b(of)h(the)f(excluded)g(middle)p 1666 4313 200 4 v 1666 4365 a Fk(A)p Fe(_:)p Fk(A)1905 4334 y Fw(or)h(iden)m(tit)m(y)p 2361 4313 177 4 v 2361 4365 a Fk(A)p Fe(!)p Fk(A)2576 4334 y Fw(as)g(w)m(ell)e(as)i(the)g(cut)g (rule)466 4419 y Fk(A)p Fe(_)p Fk(C)q(;)p Fe(:)p Fk(C)5 b Fe(_)p Fk(B)p 466 4439 377 4 v 575 4491 a(A)p Fe(_)p Fk(B)881 4460 y Fw(or)30 b Fn(mo)-5 b(dus)34 b(p)-5 b(onens)1582 4419 y Fk(A;A)p Fe(!)p Fk(B)p 1582 4439 253 4 v 1680 4491 a(B)1845 4460 y Fw(.)40 b(A)30 b(pro)s(of)f(of)h(a)g(tautology)h Fm(F)43 b Fw(in)29 b Fl(F)39 b Fw(consists)456 4568 y(of)c(a)h (\014nite)e(sequence)i Fm(F)1322 4582 y Fs(1)1362 4568 y Fm(;)15 b(:)g(:)g(:)32 b(;)15 b(F)1637 4582 y Fk(r)1711 4568 y Fw(of)36 b(form)m(ulas,)g(called)e Fn(lines)p Fw(,)k(suc)m(h)d(that)h Fm(F)47 b Fw(=)33 b Fm(F)3406 4582 y Fk(r)456 4676 y Fw(and)25 b(eac)m(h)j Fm(F)887 4690 y Fk(j)950 4676 y Fw(either)e(is)f(an)h(an)g(instance)g(of)g(an)h (axiom)f(in)f Fl(A)2548 4690 y Fe(F)2635 4676 y Fw(or)h(follo)m(ws)f (from)h(some)456 4784 y(previous)f(lines)g Fm(F)1076 4798 y Fk(i)1100 4807 y Fq(1)1154 4784 y Fm(:)15 b(:)g(:)32 b(;)15 b(F)1389 4798 y Fk(i)1413 4810 y Fd(k)1482 4784 y Fw(for)27 b Fm(i)1649 4798 y Fs(1)1689 4784 y Fm(;)15 b(:)g(:)g(:)32 b(;)15 b(i)1937 4799 y Fk(k)2005 4784 y Fm(<)25 b(j)33 b Fw(using)25 b(some)i(inference)f(rule)f(of)i Fl(A)3358 4798 y Fe(F)3419 4784 y Fw(.)456 4892 y(An)38 b(equiv)-5 b(alen)m(t)38 b(w)m(a)m(y)i(of)f(using)f(a)h(F)-8 b(rege)41 b(system)e(w)m(orks)f(bac)m(kw)m(ards)i(from)e Fl(:)p Fm(F)52 b Fw(to)456 5000 y(deriv)m(e)30 b(a)h(con)m(tradiction)f (suc)m(h)h(as)g Fm(p)20 b Fl(^)g(:)p Fm(p)p Fw(.)40 b(The)30 b(size)h(of)g(a)g(F)-8 b(rege)32 b(pro)s(of)e(is)f(t)m(ypically)456 5108 y(de\014ned)c(to)i(b)s(e)f(the)h(total)h(n)m(um)m(b)s(er)d(of)i (sym)m(b)s(ols)e(o)s(ccurring)g(in)g(the)i(pro)s(of.)39 b(The)26 b(pro)s(of)456 5216 y(can)43 b(also)f(b)s(e)g(tree-lik)m(e)h (or)g(dag-lik)m(e:)65 b(in)42 b(the)h(tree-lik)m(e)g(case,)k(eac)m(h)d (in)m(termediate)p eop %%Page: 4 4 4 3 bop 456 425 a Fw(form)m(ula)28 b(can)i(b)s(e)e(at)i(used)f(at)h (most)f(once)h(in)e(subsequen)m(t)g(deriv)-5 b(ations;)29 b(in)f(the)h(more)456 533 y(general)43 b(dag-lik)m(e)h(case,)k(an)43 b(in)m(termediate)g(form)m(ula)g(can)g(b)s(e)g(used)g(un)m(b)s (oundedly)456 641 y(man)m(y)35 b(times.)56 b(Kra)5 b(j)-10 b(\023)-35 b(\020)m(\024)-43 b(cek)36 b([)p Fr(59)q Fw(])f(has)h(sho)m (wn)e(that)j(for)e(F)-8 b(rege)37 b(systems,)g(there)f(is)e(not)456 749 y(m)m(uc)m(h)c(loss)g(in)f(e\016ciency)h(in)f(going)i(from)f(a)g (dag-lik)m(e)h(pro)s(of)e(to)i(a)g(tree-lik)m(e)g(pro)s(of.)605 857 y(V)-8 b(arious)34 b(F)-8 b(rege)36 b(systems)e(\(whic)m(h)f(ha)m (v)m(e)i(also)f(b)s(een)g(called)f(Hilb)s(ert)f(systems)i(or)456 965 y(Hilb)s(ert-st)m(yle)23 b(deduction)h(systems\))h(app)s(ear)g (frequen)m(tly)f(in)f(logic)i(textb)s(o)s(oks.)39 b(Ho)m(w-)456 1073 y(ev)m(er,)29 b(it)f(is)g(di\016cult)e(to)j(\014nd)e(t)m(w)m(o)i (logic)f(textb)s(o)s(oks)h(that)g(de\014ne)f(precisely)e(the)j(same)456 1181 y(suc)m(h)24 b(system.)40 b(Co)s(ok)25 b(and)f(Rec)m(kho)m(w)j (sho)m(w)m(ed)e(that)h(these)f(distinctions)e(do)i(not)h(mat-)456 1289 y(ter;)36 b(namely)-8 b(,)34 b(all)f(F)-8 b(rege)35 b(systems)f(are)g(p-equiv)-5 b(alen)m(t.)50 b(F)-8 b(urthermore,)35 b(they)f(sho)m(w)m(ed)456 1397 y(that)d(F)-8 b(rege)31 b(systems)g(w)m(ere)g(also)f(p-equiv)-5 b(alen)m(t)29 b(to)j(another)e(class)g(of)h(pro)s(of)e(systems)456 1504 y(app)s(earing)j(frequen)m(tly)h(in)g(logic)g(textb)s(o)s(oks)h (called)f(sequen)m(t)i(calculus)d(or)i(Gen)m(tzen)456 1612 y(systems.)39 b(These)28 b(systems)f(manipulate)f(pairs)g(of)i (sequences)g(\(or)g(sets\))g(of)g(form)m(ulas,)456 1720 y(written)34 b(as)i(\000)e Fl(!)h Fw(\001,)i(where)e(\000)g(and)h(\001) f(are)h(sequences)g(of)g(form)m(ulas)f(with)f(the)i(in-)456 1828 y(tended)22 b(in)m(terpretation)h(b)s(eing)f(that)h(the)h (conjunction)e(of)h(the)g(form)m(ulas)f(in)g(\000)h(implies)456 1936 y(the)28 b(disjunction)e(of)j(the)g(form)m(ulas)e(in)h(\001.)39 b(Therefore)29 b(to)g(pro)m(v)m(e)h(a)e(form)m(ula)g Fm(F)42 b Fw(in)27 b(the)456 2044 y(sequen)m(t)j(calculus,)g(one)g(pro) m(v)m(es)h(the)g(corresp)s(onding)d(sequen)m(t)j Fl(!)25 b Fm(F)13 b Fw(.)605 2152 y(In)20 b(an)m(y)h(sequen)m(t)g(calculus)e (system,)k(there)e(is)f(an)g(underlying)e(basis)h(set)j(of)e(connec-) 456 2260 y(tiv)m(es,)30 b Fm(B)5 b Fw(.)40 b Fm(B)34 b Fw(can)29 b(consist)g(of)h(un)m(b)s(ounded)d(fan-in)h(or)i(b)s (ounded)d(fan-in)h(connectiv)m(es,)456 2368 y(and)d(the)h(only)f (requiremen)m(t)g(is)g(that)h Fm(B)k Fw(b)s(e)c(a)g(complete)g(basis.) 38 b(\(T)m(ypically)-8 b(,)26 b Fm(B)k Fw(is)25 b(the)456 2476 y(standard)i(basis)f(consisting)h(of)h Fl(^)p Fw(,)g Fl(_)g Fw(and)f Fl(:)p Fw(.\))40 b(The)27 b(only)g(initial)e(sequen)m (t)k(is)e Fm(A)e Fl(!)g Fm(A)456 2584 y Fw(for)30 b(an)m(y)g(form)m (ula)g Fm(A)h Fw(de\014ned)e(o)m(v)m(er)i Fm(B)5 b Fw(.)605 2692 y(Additionally)19 b(there)j(are)h(three)f(t)m(yp)s(es)g(of)g (rules)f(for)h(deriving)e(new)h(sequen)m(ts)i(from)456 2800 y(previous)34 b(ones:)52 b(\(i\))36 b(structural)f(rules;)i (\(ii\))e(logical)h(rules;)h(and)f(\(iii\))e(the)i(cut)g(rule.)456 2908 y(Structural)30 b(rules)h(do)h(not)h(actually)f(manipulate)f(the)h (underlying)d(form)m(ulas)j(of)g(the)456 3015 y(sequen)m(t,)d(but)e (instead)h(they)g(allo)m(w)f(one)i(to)g(op)s(erate)f(on)g(the)h (sequences)f(of)g(form)m(ulas)456 3123 y(as)f(sets)h(rather)f(than)g (sequences.)40 b(A)28 b(t)m(ypical)f(structural)f(rule)g(is)g(con)m (traction)j(whic)m(h)456 3231 y(allo)m(ws)37 b(us)h(to)g(deriv)m(e)g (\000)p Fm(;)15 b(A)39 b Fl(!)f Fw(\001)g(from)g(\000)p Fm(;)15 b(A;)g(A)39 b Fl(!)f Fw(\001.)64 b(The)37 b(logical)h(rules)f (allo)m(w)456 3339 y(us)28 b(to)i(build)c(larger)i(form)m(ulas)g(from)h (previous)e(ones,)j(according)f(to)h(the)f(truth-table)456 3447 y(de\014nition)19 b(of)k(eac)m(h)g(of)f(the)h(connectiv)m(es)g(in) e Fm(B)5 b Fw(.)37 b(More)23 b(precisely)-8 b(,)23 b(for)f(eac)m(h)h (connectiv)m(e)456 3555 y(in)31 b Fm(B)5 b Fw(,)32 b(there)h(are)g(t)m (w)m(o)g(logical)f(rules,)g(one)h(for)f(in)m(tro)s(ducing)e(the)i (connectiv)m(e)i(on)e(the)456 3663 y(left)g(and)g(one)h(for)f(in)m(tro) s(ducing)f(the)i(connectiv)m(e)g(on)g(the)g(righ)m(t.)47 b(F)-8 b(or)33 b(example,)g(if)f Fl(^)456 3771 y Fw(is)j(in)f Fm(B)5 b Fw(,)37 b(then)f(the)g Fl(^)p Fw(-left)f(rule)g(w)m(ould)g (allo)m(w)g(us)g(to)i(deriv)m(e)e(\000)p Fm(;)15 b(A)24 b Fl(^)g Fm(B)39 b Fl(!)34 b Fw(\001)i(from)456 3879 y(\000)p Fm(;)15 b(A;)g(B)35 b Fl(!)c Fw(\001,)j(and)f(the)h Fl(^)p Fw(-righ)m(t)f(rule)f(w)m(ould)g(allo)m(w)h(us)g(to)h(deriv)m(e) g(\000)c Fl(!)g Fm(A)23 b Fl(^)f Fm(B)5 b(;)15 b Fw(\001)456 3987 y(from)30 b(\000)25 b Fl(!)g Fm(A;)15 b Fw(\001)30 b(and)g(\000)25 b Fl(!)g Fm(B)5 b(;)15 b Fw(\001.)605 4095 y(Of)22 b(particular)f(imp)s(ortance)h(for)h(sequen)m(t)g(calculi) e(is)h(the)h(cut)g(rule:)36 b(F)-8 b(rom)23 b(\000)p Fm(;)15 b(A)25 b Fl(!)456 4203 y Fw(\001)42 b(and)f(\000)k Fl(!)g Fm(A;)15 b Fw(\001,)46 b(deriv)m(e)c(\000)i Fl(!)h Fw(\001.)76 b(Gen)m(tzen)44 b(sho)m(w)m(ed)f(that)f(the)h(cut)f(rule)f (is)456 4311 y(unnecessary)g(but)h(it)f(ma)m(y)i(ha)m(v)m(e)g(a)g(h)m (uge)f(impact)g(on)g(pro)s(of)g(length.)75 b(If)42 b(the)g(cut)456 4419 y(rule)25 b(is)g(remo)m(v)m(ed)j(then)e(one)g(obtains)g(a)h(m)m (uc)m(h)g(w)m(eak)m(er)g(system)g(than)f(F)-8 b(rege)28 b(systems.)456 4526 y(This)34 b(system)j(is)e(kno)m(wn)h(as)g(analytic) g(tableaux)g(or)h(cut-free)g(LK.)f(While)f(analytic)456 4634 y(tableaux)g(are)h(often)f(more)h(e\016cien)m(t)g(than)f(truth)f (tables,)j(somewhat)f(surprisingly)456 4742 y(they)20 b(cannot)i(ev)m(en)f(p-sim)m(ulate)f(truth)g(tables)g(b)s(ecause)h (their)e(w)m(orst-case)k(complexit)m(y)456 4850 y(is)29 b(\012\()p Fm(n)p Fw(!\))i(rather)f(than)g Fm(O)s Fw(\()p Fm(n)p Fw(2)1486 4817 y Fk(n)1533 4850 y Fw(\))h(for)f(truth)f(tables)i (\(see)g([)p Fr(91)q Fw(]\).)605 4958 y(The)39 b(other)g(ma)5 b(jor)40 b(class)f(of)g(pro)s(of)f(systems)i(iden)m(ti\014ed)d(b)m(y)i (Co)s(ok)g(and)g(Rec)m(k-)456 5066 y(ho)m(w)25 b(includes)f(systems)h (that)i(p)s(ermit)d(one)i(to)g(extend)g(F)-8 b(rege)27 b(pro)s(ofs)e(b)m(y)h(in)m(tro)s(ducing)456 5174 y(new)g(prop)s (ositional)f(v)-5 b(ariables)25 b(to)j(stand)f(for)g(arbitrary)f(form)m (ulas)g(app)s(earing)f(in)h(the)p eop %%Page: 5 5 5 4 bop 456 425 a Fw(pro)s(of.)50 b(All)33 b(suc)m(h)g(systems,)i(whic) m(h)e(are)h(called)f(extended-F)-8 b(rege,)37 b(are)d(p-equiv)-5 b(alen)m(t)456 533 y(to)32 b(eac)m(h)h(other.)45 b(These)32 b(systems)f(app)s(ear)h(to)g(b)s(e)f(m)m(uc)m(h)h(more)g(succinct)f (than)h(F)-8 b(rege)456 641 y(pro)s(ofs)21 b(and)h(can)h(con)m(v)m (enien)m(tly)f(express)g(man)m(y)h(mathem)m(tical)f(argumen)m(ts)h (quite)f(nat-)456 749 y(urally)-8 b(.)38 b(It)29 b(has)g(b)s(een)f(sho) m(wn)g(b)m(y)h(Do)m(wd)g(\(unpublished\))c(and)j(also)h(b)m(y)g(Kra)5 b(j)-10 b(\023)-35 b(\020)m(\024)-43 b(cek)29 b(and)456 857 y(Pudl\023)-45 b(ak)20 b([)p Fr(60)q Fw(])j(that)f(extended-F)-8 b(rege)25 b(pro)s(ofs)c(are)i(also)f(p-equiv)-5 b(alen)m(t)21 b(to)i(substitution-)456 965 y(F)-8 b(rege)36 b(\(sF\))h(pro)s(ofs.)54 b(\(In)35 b(sF,)h(one)f(is)g(allo)m(w)m(ed)g(to)h(use)f(eac)m(h)h(line) e(of)i(a)f(F)-8 b(rege)37 b(pro)s(of)456 1073 y(immediately)c(as)i(if)f (it)g(w)m(ere)i(an)e(axiom;)k(that)d(is,)g(new)g(lines)e(follo)m(w)h (from)g(existing)456 1181 y(ones)c(b)m(y)g(substituting)e(arbitrary)i (form)m(ulas)f(for)h(their)g(prop)s(ositional)d(v)-5 b(ariables.\))605 1373 y Fr(2.2.)53 b(CNF)46 b(refutations)h(and)g (Resolution.)f Fw(Using)40 b(the)i(construction)e(of)456 1481 y(the)22 b(standard)g(reduction)f(from)h(SA)-8 b(T)22 b(to)h(3-SA)-8 b(T,)23 b(one)f(can)h(tak)m(e)h(an)e(arbitrary)f(prop)s (o-)456 1589 y(sitional)k(form)m(ula)h Fm(F)40 b Fw(and)27 b(con)m(v)m(ert)i(it)d(to)i(a)g(CNF)f(or)g(3-CNF)h(form)m(ula)e(in)g (suc)m(h)h(a)g(w)m(a)m(y)456 1697 y(that)32 b(it)f(has)g(only)g(p)s (olynomially)d(larger)j(size)g(and)g(is)f(unsatis\014able)g(i\013)g (the)i(original)456 1804 y(form)m(ula)37 b(w)m(as)i(a)g(tautology)-8 b(.)67 b(T)-8 b(o)39 b(do)f(this)f(one)i(adds)f(new)g(v)-5 b(ariables)37 b Fm(x)3012 1818 y Fk(A)3107 1804 y Fw(to)j(stand)456 1912 y(for)33 b(eac)m(h)h(of)g(its)f(subform)m(ulas)e Fm(A)j Fw(and)f(clauses)g(to)h(sp)s(ecify)e(that)i(the)g(v)-5 b(alue)33 b(at)h(eac)m(h)456 2020 y(connectiv)m(e)e(is)f(computed)h (correctly)g(as)g(w)m(ell)e(as)i(one)g(clause)f(of)h(the)g(form)f Fl(:)p Fm(x)3233 2034 y Fk(F)3292 2020 y Fw(.)44 b(In)456 2128 y(this)36 b(w)m(a)m(y)-8 b(,)40 b(one)e(can)f(consider)f(an)m(y)i (sound)d(and)i(complete)h(system)f(that)h(pro)s(duces)456 2236 y(refutations)29 b(for)i(CNF)f(form)m(ulas)g(as)g(a)h(general)f (prop)s(ositional)e(pro)s(of)i(system.)605 2344 y(In)e(the)h(1960's)i (sev)m(eral)e(suc)m(h)g(refutation)g(systems)g(w)m(ere)g(dev)m(elop)s (ed.)40 b(The)28 b(most)456 2452 y(p)s(o)m(w)m(erful)d(of)j(these)g (systems)f(is)f(Resolution)g([)p Fr(83)q Fw(],)j(whic)m(h)d(in)f(its)i (prop)s(ositional)e(form)456 2560 y(is)j(a)h(v)m(ery)h(sp)s(ecialized)d (form)i(of)g(a)h(F)-8 b(rege)31 b(pro)s(of)d(system)h(that)h(can)g (only)e(manipulate)456 2668 y(clauses)i(and)f(has)i(only)e(one)i (inference)e(rule,)h(the)g(resolution)f(rule)1661 2809 y Fm(A)20 b Fl(_)g Fm(x;)45 b(B)25 b Fl(_)20 b(:)p Fm(x)p 1661 2849 579 4 v 1829 2933 a(A)g Fl(_)g Fm(B)456 3057 y Fw(called)g Fn(r)-5 b(esolution)26 b(on)f(variable)g Fm(x)c Fw(and)g(is)g(a)g(sp)s(ecial)f(form)h(of)h(cut.)38 b(The)21 b(con)m(tradictory)456 3165 y(form)m(ula)31 b(to)i(b)s(e)f(deriv)m(ed)f(is)g(simply)f(an)i(empt)m(y)h(clause)f (\(whic)m(h)f(can)i(b)s(e)e(seen)i(as)f(the)456 3273 y(result)d(of)h(resolving)f(on)i(clauses)f Fm(p)g Fw(and)g Fl(:)p Fm(p)p Fw(\).)605 3381 y(Resolution)21 b(w)m(as)h(pre-dated)g(b) m(y)f(t)m(w)m(o)i(systems)f(kno)m(wn)f(as)h(Da)m(vis-Putnam)g(pro)s (ce-)456 3489 y(dures)i(whic)m(h)h(are)i(still)d(the)i(most)g(widely)e (used)h(in)g(prop)s(ositional)e(theorem)k(pro)m(ving.)456 3597 y(The)i(general)h(idea)f(of)h(these)g(pro)s(cedures)e(is)h(to)h (con)m(v)m(ert)i(a)e(problem)e(on)h Fm(n)h Fw(v)-5 b(ariables)456 3705 y(to)26 b(problems)d(on)i Fm(n)10 b Fl(\000)g Fw(1)25 b(v)-5 b(ariables)24 b(b)m(y)h(eliminating)e(all)h(references)i(to)g (some)f(v)-5 b(ariable.)456 3812 y(The)37 b(former)g([)p Fr(39)p Fw(])h(whic)m(h)e(w)m(e)i(call)f(DP)h(do)s(es)f(this)g(b)m(y)g (applying)e(all)i(p)s(ossible)e(uses)456 3920 y(of)i(the)g(resolution)f (rule)f(on)i(a)h(giv)m(en)f(v)-5 b(ariable)36 b(to)h(eliminate)f(it.)60 b(The)37 b(latter)g([)p Fr(38)q Fw(],)456 4028 y(whic)m(h)30 b(w)m(e)i(call)f(DLL)g(and)g(is)g(the)h(form)f(used)g(to)s(da)m(y)-8 b(,)33 b(branc)m(hes)e(based)g(on)h(the)g(p)s(os-)456 4136 y(sible)h(truth)h(assignmen)m(ts)h(to)g(a)h(giv)m(en)f(v)-5 b(ariable;)36 b(although)f(at)g(\014rst)g(this)e(do)s(es)i(not)456 4244 y(lo)s(ok)26 b(lik)m(e)g(Resolution,)h(it)g(is)f(an)h(easy)g (argumen)m(t)h(to)f(sho)m(w)g(that)h(this)e(second)h(form)f(is)456 4352 y(equiv)-5 b(alen)m(t)27 b(to)h(the)g(sp)s(ecial)f(class)g(of)h (tree-lik)m(e)g(Resolution)f(pro)s(ofs.)39 b(As)27 b(a)h(pro)s(of)f (sys-)456 4460 y(tem,)34 b(Resolution)d(is)h(strictly)g(stronger)h (than)f(DP)h([)p Fr(49)q Fw(])g(whic)m(h)f(is)f(strictly)h(stronger)456 4568 y(than)j(DLL)i([)p Fr(91)p Fw(].)58 b(The)36 b(reasons)g(for)g (DLL's)g(p)s(opularit)m(y)e(are)j(related)f(to)h(its)e(pro)s(of)456 4676 y(searc)m(h)c(prop)s(erties)d(whic)m(h)h(w)m(e)i(discuss)e(b)s (elo)m(w.)605 4784 y(A)36 b(more)g(general)h(but)e(still)f(restricted)i (form)f(of)i(Resolution)d(is)i(called)f Fn(r)-5 b(e)g(gular)456 4892 y Fw(Resolution,)33 b(whic)m(h)f(w)m(as)i(in)m(tro)s(duced)e(and)h (analyzed)g(b)m(y)g(Tseitin)f([)p Fr(88)q Fw(].)50 b(A)33 b(regular)456 5000 y(Resolution)g(refutation)i(is)f(a)h(Resolution)f (refutation)g(whose)h(underlying)d(directed)456 5108 y(acyclic)h(graph)h(has)f(the)h(prop)s(ert)m(y)f(that)h(along)g(eac)m (h)h(path)e(from)h(the)g(ro)s(ot)g(\(empt)m(y)456 5216 y(clause\))26 b(to)i(a)e(leaf)h(\(initial)d(clause\),)j(eac)m(h)h(v)-5 b(ariable)25 b(is)h(resolv)m(ed)g(up)s(on)f(at)j(most)e(once.)p eop %%Page: 6 6 6 5 bop 456 425 a Fw(It)38 b(is)f(not)i(to)s(o)f(hard)g(to)h(see)f (that)h(an)m(y)g(minimal)c(tree-lik)m(e)k(Resolution)e(refutation)456 533 y(is)h(regular;)k(also,)g(the)d(DP)g(algorithm)f(trivially)e(pro)s (duces)i(a)h(regular)f(Resolution)456 641 y(pro)s(of.)h(F)-8 b(or)31 b(a)f(p)s(erio)s(d)e(of)h(15)i(y)m(ears)g(after)f(Tseitin's)e (analysis,)h(although)g(there)h(w)m(ere)456 749 y(impro)m(v)m(emen)m (ts)25 b(in)f(the)i(b)s(ounds)d(deriv)m(ed)i(for)g(regular)f (resolution)g([)p Fr(48)q Fw(],)j(understand-)456 857 y(ing)i(general)i(resolution)d(seemed)j(out)g(of)f(reac)m(h.)605 1112 y Fr(2.3.)53 b(Circuit-complexit)m(y-based)c(Pro)s(of)g(Systems.)c Fw(One)d(of)h(the)g(most)456 1219 y(p)s(o)m(w)m(erful)21 b(insigh)m(ts)f(that)j(has)g(dev)m(elop)s(ed)e(in)g(the)i(study)f(of)g (prop)s(ositional)e(pro)s(of)h(com-)456 1327 y(plexit)m(y)e(is)h(that)h (there)g(is)e(a)i(parallel)e(b)s(et)m(w)m(een)i(circuit-based)e (complexit)m(y)h(classes)h(and)456 1435 y(prop)s(ositional)26 b(pro)s(of)i(systems.)40 b(This)27 b(insigh)m(t,)h(noticed)h(b)m(y)g (sev)m(eral)g(researc)m(hers)g(in-)456 1543 y(dep)s(enden)m(tly)-8 b(,)24 b(but)g(\014rst)f(used)h(in)f(a)i(signi\014can)m(t)e(w)m(a)m(y)i (b)m(y)g(Ajtai)f([)p Fr(4)q(,)k(1)p Fw(])c(and)g(Buss)g([)p Fr(29)q Fw(],)456 1651 y(has)40 b(greatly)h(broadened)f(the)g(range)h (of)g(pro)s(of)f(systems)g(that)h(are)g(t)m(ypically)f(con-)456 1759 y(sidered)h(and)i(has)g(led)g(to)h(new)f(tec)m(hniques)g(for)g (analyzing)g(pro)s(of)f(systems.)80 b(W)-8 b(e)456 1867 y(\014rst)41 b(brie\015y)f(outline)h(the)i(general)f(form)f(of)i(this)e (corresp)s(ondence)g(and)h(then)g(w)m(e)456 1975 y(re-examine)30 b(and)g(re\014ne)f(some)i(of)g(the)f(pro)s(of)g(systems)g(ab)s(o)m(v)m (e)i(in)d(this)g(ligh)m(t.)605 2083 y(T)m(ypically)-8 b(,)26 b(circuit-based)g(complexit)m(y)g(classes)h(are)g(de\014ned)e(b) m(y)i(giving)e(a)i(struc-)456 2191 y(tural)h(c)m(haracterization)i(of)g (a)f(class)g(of)h(circuits)d(and)i(then)g(placing)f(some)h(b)s(ound)e (on)456 2299 y(the)32 b(size)g(of)g(the)h(circuits)d(in)m(v)m(olv)m (ed.)46 b(F)-8 b(or)33 b(man)m(y)f(circuit-based)f(complexit)m(y)h (classes)456 2407 y Fx(C)38 b Fw(this)g(size)g(b)s(ound)f(is)h(p)s (olynomial.)62 b(F)-8 b(or)40 b(an)m(y)f(suc)m(h)f(class)h Fx(C)f Fw(w)m(e)h(can)g(consider)f(a)456 2515 y(F)-8 b(rege-st)m(yle)39 b(pro)s(of)d(system)h(whose)g(lines)e(are)i (circuits)e(with)h(the)h(same)g(structural)456 2623 y(c)m (haracterization)30 b(as)g(the)g(circuits)e(de\014ning)f Fx(C)j Fw(but)f(whic)m(h)f(do)h(not)h(necessarily)e(sat-)456 2730 y(isfy)35 b(the)h(size)h(b)s(ound.)56 b(The)36 b(set)h(of)f (circuits)f(of)i(this)e(t)m(yp)s(e)i(m)m(ust)f(b)s(e)g(closed)g(under) 456 2838 y(substitution)j(in)m(to)i(an)m(y)h(form)m(ula)e(app)s(earing) g(in)g(an)h(axiom)g(or)h(inference)e(rule)g(of)456 2946 y(the)32 b(system.)46 b(Although)31 b(the)h(notation)h(is)e(not)h (precise)g(in)e(general,)j(w)m(e)g(call)e(suc)m(h)h(a)456 3054 y(pro)s(of)h(system)h Fx(C)p Fw(-F)-8 b(rege.)55 b(\(Since)33 b(w)m(e)i(can)f(alw)m(a)m(ys)h(assume)f(that)h(our)e(goal) i(form)m(ula)456 3162 y(is)e(in)f(CNF,)i(there)h(is)e(no)g(problem)g (represen)m(ting)g(it)g(in)g Fx(C)p Fw(-F)-8 b(rege)36 b(for)d(virtually)f(an)m(y)456 3270 y(non-trivial)c Fx(C)p Fw(.\))605 3378 y(F)-8 b(or)26 b(example,)g(the)f(complexit)m(y)g (class)f(corresp)s(onding)f(to)j(the)f(set)g(of)h(all)d(p)s(olyno-)456 3486 y(mial)28 b(size)i(prop)s(ositional)d(form)m(ulas)h(is)h Fx(NC)1959 3449 y Fs(1)2028 3486 y Fw(so)h Fx(NC)2261 3449 y Fs(1)2301 3486 y Fw(-F)-8 b(rege)31 b(w)m(ould)e(just)g(b)s(e)g (another)456 3594 y(name)22 b(for)g(F)-8 b(rege.)40 b(It)22 b(is)g(also)g(easy)h(to)g(observ)m(e)g(that)g(the)f(extension)g(rule)g (of)g(extended-)456 3702 y(F)-8 b(rege)36 b(pro)s(ofs)d(builds)e (circuits)i(in)h(terms)g(of)h(the)f(original)f(prop)s(ositional)f(v)-5 b(ariables)456 3810 y(in)30 b(whic)m(h)h(the)h(new)g(v)-5 b(ariables)30 b(giv)m(e)j(the)f(v)-5 b(alues)31 b(computed)h(b)m(y)g (sub-circuits.)43 b(Th)m(us)456 3918 y(extended-F)-8 b(rege)29 b(could)d(also)g(b)s(e)h(called)f Fx(P/p)s(oly)p Fw(-F)-8 b(rege.)42 b(Resolution)25 b(is)h(a)i(F)-8 b(rege)28 b(sys-)456 4026 y(tem)h(that)h(manipulates)d(simple)g(clauses)h(\(or,)i (alternativ)m(ely)-8 b(,)30 b(terms)f(if)e(one)j(views)e(it)456 4134 y(dually)h(as)k(a)f(pro)s(of)f(system)h(rather)g(than)f(a)i (refutation)e(system\))i(but)e(there)h(isn't)f(a)456 4241 y(con)m(v)m(enien)m(t)37 b(name)f(for)g(this)g(complexit)m(y)g (class)g(of)g(depth)g(1)g(form)m(ulas.)58 b(Note)38 b(that)456 4349 y(with)c(this)h(in)m(tuition)f(it)h(is)g(clear)h(that)g (extended-Resolution,)h(the)f(natural)f(gener-)456 4457 y(alization)g(of)h(Resolution)f(that)i(p)s(ermits)d(the)i(in)m(tro)s (duction)e(of)i(new)g(prop)s(ositional)456 4565 y(v)-5 b(ariables,)26 b(is)g(p-equiv)-5 b(alen)m(t)26 b(to)h(extended-F)-8 b(rege)29 b(since)d(it)h(clearly)f(can)h(generate)h(an)m(y)456 4673 y(circuit)h(in)g Fx(P/p)s(oly)h Fw(with)f(a)i(p)s(olynomial)d(n)m (um)m(b)s(er)h(of)h(extensions.)605 4781 y(A)k(v)m(ery)g(natural)f(new) h(pro)s(of)f(system)h(arising)e(from)i(this)f(corresp)s(ondence)g(is)g (a)456 4889 y(generalization)d(of)i(Resolution)e(to)i(arbitrary)e (constan)m(t-depth,)i(un)m(b)s(ounded)d(fan-in)456 4997 y(form)m(ulas/circuits)18 b(\(for)i(constan)m(t)i(depth)d(there)i(is)e (no)h(di\013erence)g(b)s(et)m(w)m(een)h(p)s(olynomial-)456 5108 y(size)42 b(form)m(ulas)g(and)g(circuits\).)76 b(This)41 b(new)h(system,)k Fx(A)m(C)2514 5071 y Fs(0)2554 5108 y Fw(-F)-8 b(rege,)47 b(also)c(kno)m(wn)f(as)456 5216 y(constan)m(t-)f(or)f(b)s(ounded-depth)d(F)-8 b(rege,)43 b(\014rst)c(arose)i(from)e(the)h(study)f(of)g(b)s(ounded)p eop %%Page: 7 7 7 6 bop 456 425 a Fw(\014rst-order)36 b(arithmetic)h(\(the)g(w)m(ord)g (`b)s(ounded')f(in)g(that)i(case)g(deriv)m(es)f(from)f(p)s(oly-)456 536 y(nomial)e(complexit)m(y)h(b)s(ounds)e(rather)i(than)g(from)g(the)g (depth\).)55 b Fx(A)m(C)2883 499 y Fs(0)2922 536 y Fw(-F)-8 b(rege)37 b(pro)s(ofs)456 644 y(deriv)m(e)31 b(from)h(translations)e (of)j(pro)s(ofs)e(in)f(certain)i(systems)g(of)g(b)s(ounded)e (\014rst-order)456 752 y(arithmetic)38 b([)p Fr(69)q(,)46 b(28)p Fw(])40 b(whic)m(h)e(are)i(restrictions/extensions)f(of)g(P)m (eano)i(arithmetic)456 860 y(that)g(mo)s(del)f(feasible)g(inference.)71 b(Although)40 b(these)h(motiv)-5 b(ations)41 b(are)g(imp)s(ortan)m(t) 456 968 y(in)e(the)i(study)f(of)i(constructiv)m(e)f(logic,)j(space)d (considerations)f(do)h(not)g(p)s(ermit)e(us)456 1076 y(to)d(go)h(in)m(to)f(detail)g(ab)s(out)f(them;)k(w)m(e)e(refer)f(the)g (in)m(terested)g(reader)g(to)h([)p Fr(59)q Fw(])f(where)456 1183 y(man)m(y)c(of)h(these)f(connections)h(are)f(describ)s(ed)f(in)g (detail.)45 b(T)m(ypically)-8 b(,)32 b(lo)m(w)m(er)h(b)s(ounds)456 1291 y(on)28 b(the)h(size)f(of)h Fx(A)m(C)1123 1255 y Fs(0)1162 1291 y Fw(-F)-8 b(rege)31 b(pro)s(ofs)c(can)i(sho)m(w)f(that) i(related)e(\014rst-order)g(tautologies)456 1399 y(are)42 b(unpro)m(v)-5 b(able)40 b(in)g(a)i(giv)m(en)g(system)f(of)h(b)s (ounded)e(arithmetic.)73 b(These)42 b(transla-)456 1507 y(tions)28 b(are)h(analogous)g(to)h(those)f(of)g(F)-8 b(urst,)30 b(Saxe,)g(and)e(Sipser)f([)p Fr(46)q Fw(],)i(and)f(Sipser)f ([)p Fr(86)q Fw(])456 1615 y(whic)m(h)f(con)m(v)m(ert)i(oracle)g (computations)f(in)f(the)h(p)s(olynomial)e(hierarc)m(h)m(y)i(to)h (constan)m(t-)456 1723 y(depth)35 b(un)m(b)s(ounded)e(fan-in)i (circuits.)56 b(In)36 b(addition)e(to)j Fx(A)m(C)2548 1686 y Fs(0)2587 1723 y Fw(-F)-8 b(rege,)40 b(pro)s(of)35 b(systems)456 1831 y(for)29 b Fx(A)m(C)710 1794 y Fs(0)750 1831 y Fw([)p Fm(p)p Fw(]-F)-8 b(rege)32 b(and)d Fx(TC)1416 1794 y Fs(0)1456 1831 y Fw(-F)-8 b(rege)31 b(and)f(their)f(sub)s (classes)f(ha)m(v)m(e)j(also)f(b)s(een)f(studied)456 1939 y(extensiv)m(ely)-8 b(.)605 2047 y(The)35 b(corresp)s(ondence)h(b) s(et)m(w)m(een)g(circuit)f(classes)g(and)h(pro)s(of)f(systems)h(has)f (not)456 2155 y(only)k(b)s(een)g(fruitful)e(in)h(dev)m(eloping)h(ideas) h(for)g(new)f(pro)s(of)g(systems.)70 b(It)40 b(has)f(also)456 2263 y(b)s(een)23 b(the)h(a)m(v)m(en)m(ue)h(for)e(applying)f(circuit)g (lo)m(w)m(er)i(b)s(ound)e(tec)m(hniques)h(to)i(prop)s(ositional)456 2371 y(pro)s(ofs.)37 b(Some)21 b(of)h(the)g(ma)5 b(jor)21 b(progress)h(of)g(the)f(last)h(decade)g(building)c(on)j(the)h(original) 456 2479 y(insigh)m(t)39 b(due)i(to)g(Ajtai)g([)p Fr(4)q(,)47 b(1)p Fw(],)d(has)d(b)s(een)f(in)g(ac)m(hieving)h(lo)m(w)m(er)g(b)s (ounds)e(for)h Fx(A)m(C)3375 2442 y Fs(0)3414 2479 y Fw(-)456 2587 y(F)-8 b(rege)34 b(pro)s(of)f(systems)g(and)f(their)g (extensions.)49 b(In)32 b(general,)i(the)g(in)m(tuition)d(for)h(this) 456 2694 y(approac)m(h)g(is)e(that)i(an)m(y)g(tautology)h(that)f(needs) g(to)g(use)f(in)f(its)h(pro)s(of)g(some)h(concept)456 2802 y(that)24 b(is)e(not)h(represen)m(table)g(in)f(complexit)m(y)h (class)h Fx(C)f Fw(will)e(not)i(b)s(e)g(e\016cien)m(tly)g(pro)m(v)-5 b(able)456 2910 y(in)29 b Fx(C)p Fw(-F)-8 b(rege.)922 3107 y Fr(3.)53 b(The)34 b(State)g(of)h(the)g(Art)f(in)h(Pro)s(of)h (Complexit)m(y)605 3269 y Fw(W)-8 b(e)32 b(giv)m(e)g(a)f(quic)m(k)g (tour)g(of)g(the)g(state)h(of)g(the)f(art)g(in)f(prop)s(ositional)e (pro)s(of)i(com-)456 3377 y(plexit)m(y)-8 b(.)39 b(As)27 b(is)g(alw)m(a)m(ys)h(inevitable)e(in)g(a)i(short)f(surv)m(ey)g(suc)m (h)h(as)f(ours,)h(space)g(consid-)456 3485 y(erations)h(do)g(not)h(p)s (ermit)d(us)i(to)h(do)f(justice)g(to)h(the)f(full)e(range)j(of)g (results)e(a)m(v)-5 b(ailable.)456 3593 y(Although)25 b(w)m(e)i(will)d(not)i(alw)m(a)m(ys)h(emphasize)f(the)h(connections)f (v)m(ery)h(strongly)-8 b(,)27 b(man)m(y)456 3701 y(of)j(these)h (results)f(ha)m(v)m(e)i(b)s(een)d(inspired)f(or)i(deriv)m(ed)g(from)g (metho)s(ds)g(in)f(circuit)g(com-)456 3809 y(plexit)m(y)-8 b(,)22 b(and)f(con)m(v)m(ersely)h(lo)m(w)m(er)f(b)s(ounds)e(for)i (particular)e(pro)s(of)h(systems)h(imply)e(lo)m(w)m(er)456 3916 y(b)s(ounds)25 b(for)j(restricted)g(families)e(of)i(algorithms)f (for)h(solving)f(SA)-8 b(T.)28 b(W)-8 b(e)29 b(exp)s(ect)f(this)456 4024 y(cross-fertilization)h(to)i(con)m(tin)m(ue.)605 4132 y(In)h(k)m(eeping)i(with)d(the)j(general)f(program)g(of)g(pro)m (ving)g(that)g(pro)s(of)g(systems)g(are)456 4240 y(not)g(e\016cien)m (t,)j(m)m(uc)m(h)d(w)m(ork)h(has)g(b)s(een)e(dev)m(oted)j(to)f(pro)m (ving)f(lo)m(w)m(er)h(b)s(ounds)e(on)h(the)456 4348 y(sizes)f(of)h(pro) s(ofs)f(of)g(sp)s(eci\014c)g(tautologies.)48 b(One)32 b(can)h(broadly)f(c)m(haracterize)i(sev)m(eral)456 4456 y(classes)41 b(of)h(form)m(ulas)f(for)h(whic)m(h)e(these)j(lo)m(w)m(er) f(b)s(ounds)d(ha)m(v)m(e)k(b)s(een)e(sho)m(wn.)75 b(The)456 4564 y(\014rst)27 b(t)m(w)m(o)j(classes)e(consist)g(of)g(the)g(prop)s (ositional)e(translations)h(of)h(com)m(binatorial)f(or)456 4672 y(coun)m(ting)35 b(principles.)54 b(The)36 b(\014rst)f(of)h(these) g(in)m(v)m(olv)m(es)g(highly)e(symmetric)h(coun)m(ting)456 4780 y(principles.)63 b(A)39 b(canonical)g(example)g(here)g(is)f(the)h (translation)g(of)g(the)g(pigeonhole)456 4888 y(principle,)23 b(whic)m(h)i(prohibits)e(1-1)k(functions)e(from)g Fm(m)h Fw(to)h Fm(n)e Fw(for)h Fm(m)f(>)g(n)p Fw(,)h(as)h(an)e(unsat-)456 4996 y(is\014able)k(CNF)h(form)m(ula)g Fl(:)p Fm(P)13 b(H)7 b(P)1603 4963 y Fk(m)1590 5018 y(n)1700 4996 y Fw(with)30 b Fm(mn)g Fw(atoms)h Fm(p)2387 5010 y Fk(ij)2478 4996 y Fw(denoting)f(whether)g(or)h(not)456 5104 y Fm(i)d Fw(is)e(mapp)s(ed)h(to)h Fm(j)5 b Fw(.)41 b(Related)28 b(principles)c(include)i(the)i(coun)m(ting)f(principles)d Fm(C)7 b(ount)3398 5071 y Fk(n)3398 5126 y(p)456 5216 y Fw(for)36 b Fm(n)f Fl(6\021)g Fw(0)41 b(\(mo)s(d)30 b Fm(p)p Fw(\))36 b(whic)m(h)g(express)g(the)g(prop)s(ert)m(y)g(that)h Fm(n)f Fw(cannot)h(b)s(e)f(p)s(erfectly)p eop %%Page: 8 8 8 7 bop 456 425 a Fw(partitioned)32 b(in)m(to)j(sets)f(of)h(size)f Fm(p)g Fw(and)g(on)m(to-)q Fm(P)13 b(H)7 b(P)2273 392 y Fk(m)2260 448 y(n)2374 425 y Fw(whic)m(h)33 b(only)g(prohibits)f (bijec-)456 533 y(tions)i(rather)h(than)f(all)g(1-1)i(functions.)53 b(The)35 b(second)g(class)f(consists)h(of)g(m)m(uc)m(h)g(less)456 641 y(symmetric)21 b(coun)m(ting)g(principles,)f(suc)m(h)i(as)g(the)g (`o)s(dd-c)m(harged)g(graph')f(principles)e(for)456 749 y(b)s(ounded-degree)24 b(graphs,)h(one)h(for)f(eac)m(h)h(graph,)g(whic) m(h)e(express)h(the)g(prop)s(ert)m(y)g(that)456 857 y(the)31 b(sum)g(of)g(the)h(degrees)g(in)e(an)m(y)i(of)g(its)f(sub-graphs)f(is)g (ev)m(en;)j(these)f(are)g(of)g(partic-)456 965 y(ular)26 b(in)m(terest)j(when)e(the)h(underlying)d(graph)j(is)f(a)i(b)s (ounded-degree)e(expander)g([)p Fr(90)q Fw(].)456 1073 y(The)d(third)g(class)h(of)g(form)m(ulas)g(are)g(`min)m(term-maxterm')h (form)m(ulas)e(asso)s(ciated)i(with)456 1181 y(an)m(y)35 b(monotone)g(function,)g(whic)m(h)f(express)g(the)h(fact)h(that)g(an)m (y)f(min)m(term)f(and)g(an)m(y)456 1289 y(maxterm)k(of)h(suc)m(h)f(a)g (function)f(m)m(ust)i(o)m(v)m(erlap;)k(these)38 b(are)h(of)g (particular)d(in)m(terest)456 1397 y(when)e(the)j(function)d(is)h(kno)m (wn)g(to)i(require)e(exp)s(onen)m(tial-size)g(monotone)i(circuits.)456 1504 y(A)d(related)g(family)f(of)h(form)m(ulas)f(is)h(obtained)f(b)m(y) h(taking)g(a)h(language)g Fm(L)f Fw(that)g(is)g(in)456 1612 y Fx(NP)23 b Fl(\\)g Fx(coNP)r Fw(,)36 b(and)f(writing)e(the)i (form)m(ula)f(expressing)g(the)i(fact)g(that)f(an)m(y)h(instance)456 1720 y Fm(x)i Fw(cannot)h(ha)m(v)m(e)g(b)s(oth)f(a)h(`y)m(es')g (witness)f(and)f(a)i(`no')g(witness.)63 b(Finally)-8 b(,)39 b(there)g(are)456 1828 y Fm(k)s Fw(-CNF)31 b(form)m(ulas)e (randomly)g(c)m(hosen)i(from)f(an)g(appropriate)f(distribution.)605 1936 y(W)-8 b(e)29 b(summarize)d(kno)m(wn)h(b)s(ounds)e(for)i(these)h (form)m(ulas)e(b)m(y)h(considering)f(the)h(v)-5 b(ar-)456 2044 y(ious)38 b(pro)s(of)h(systems)g(one)h(b)m(y)g(one.)68 b(F)-8 b(our)40 b(basic)f(metho)s(ds)f(can)i(b)s(e)f(iden)m(ti\014ed)f (for)456 2152 y(pro)m(ving)f(these)i(lo)m(w)m(er)g(b)s(ounds:)55 b(\(1\))39 b(the)g(b)s(ottlenec)m(k)g(coun)m(ting)f(metho)s(d;)43 b(\(2\))c(the)456 2260 y(metho)s(d)25 b(of)i(restrictions;)g(\(3\))g (the)f(in)m(terp)s(olation)f(metho)s(d;)i(\(4\))h(algebraic)e(metho)s (ds.)456 2368 y(The)37 b(\014rst)h(three)h(metho)s(ds)e(are)i(quite)f (sp)s(eci\014c)f(whereas)h(the)h(last)f(metho)s(d)g(is)f(the)456 2476 y(y)m(oungest,)31 b(probably)e(the)h(most)h(p)s(o)m(w)m(erful,)e (and)h(already)g(has)g(man)m(y)h(facets.)605 2584 y(Some)38 b(of)g(the)g(state)h(of)f(our)g(kno)m(wledge)g(of)g(pro)s(of)f (complexit)m(y)g(lo)m(w)m(er)i(b)s(ounds)456 2692 y(can)30 b(b)s(e)g(summarized)f(b)m(y)h(the)h(follo)m(wing)d(c)m(hain)836 2923 y(Resolution)c Fm(<)1353 2937 y Fk(p)1418 2923 y Fx(A)m(C)1534 2886 y Fs(0)1574 2923 y Fw(-F)-8 b(rege)27 b Fm(<)1914 2937 y Fk(p)1979 2923 y Fx(A)m(C)2095 2886 y Fs(0)2134 2923 y Fw([)p Fm(p)p Fw(]-F)-8 b(rege)28 b Fl(\024)2571 2937 y Fk(p)2635 2923 y Fx(TC)2755 2886 y Fs(0)2795 2923 y Fw(-F)-8 b(rege)r Fm(:)456 3155 y Fw(The)32 b(separations)h(matc)m(h)h(all)e(the)h(ma)5 b(jor)33 b(circuit)e(complexit)m(y)i(separations)g(kno)m(wn)456 3263 y(with)c(the)h(notable)g(exception)h(of)f(the)h(last)f(one.)41 b(Ho)m(w)m(ev)m(er,)33 b(the)d(prosp)s(ect)g(for)g(pro)s(of)456 3371 y(complexit)m(y)23 b(seems)i(b)s(etter)f(than)g(that)h(for)e (circuit)g(complexit)m(y:)38 b(The)23 b(results)g(of)h([)p Fr(80)q Fw(])456 3479 y(indicate)j(that)i(to)h(mak)m(e)f(further)e (progress)i(in)e(circuit)g(lo)m(w)m(er)i(b)s(ounds)d(will)g(lik)m(ely)h (re-)456 3587 y(quire)33 b(v)m(ery)i(new,)g(nonconstructiv)m(e)f(tec)m (hniques.)53 b(Ho)m(w)m(ev)m(er)36 b(suc)m(h)e(barriers)f(do)h(not)456 3695 y(curren)m(tly)c(exist)h(in)f(pro)s(of)g(complexit)m(y:)42 b(that)32 b(is,)f(pro)m(ving)f(sup)s(erp)s(olynomial)d(lo)m(w)m(er)456 3803 y(b)s(ounds)f(for)j(F)-8 b(rege)31 b(systems)e(migh)m(t)g(b)s(e)f (no)h(harder)f(than)h(what)g(w)m(e)g(curren)m(tly)f(kno)m(w)456 3910 y(ho)m(w)34 b(to)g(do.)52 b(The)33 b(tec)m(hniques)g(used)h(to)g (obtain)g(some)g(of)g(the)g(most)h(recen)m(t)g(results)456 4018 y(in)c(pro)s(of)g(complexit)m(y)i(use)f(insigh)m(ts)f(from)h (areas)h(of)f(mathematics)h(apparen)m(tly)f(un-)456 4126 y(related)e(to)h(those)g(applied)d(to)j(sho)m(w)f(circuit)g(complexit)m (y)g(b)s(ounds.)605 4460 y Fr(3.1.)53 b(Resolution.)46 b Fw(Resolution)41 b(is)h(the)g(most)g(w)m(ell-studied)e(mo)s(del.)75 b(Exp)s(o-)456 4568 y(nen)m(tial)28 b(lo)m(w)m(er)h(b)s(ounds)e(are)i (no)m(w)g(kno)m(wn)g(for)g(all)f(of)h(the)g(ma)5 b(jor)29 b(classes)g(of)g(form)m(ulas)456 4676 y(listed)e(ab)s(o)m(v)m(e.)41 b(The)28 b(\014rst)g(sup)s(erp)s(olynomial)d(lo)m(w)m(er)k(b)s(ound)d (for)j(Resolution)e(w)m(as)i(ob-)456 4784 y(tained)f(b)m(y)g(Tseitin)f (in)g(the)i(1960's)i(for)d(the)h(o)s(dd-c)m(harged)f(graph)g (tautologies)i(in)d(the)456 4892 y(sp)s(ecial)h(case)i(of)g(regular)f (Resolution)f([)p Fr(88)q Fw(].)41 b(In)m(terestingly)-8 b(,)29 b(obtaining)g(an)g(impro)m(v)m(e-)456 5000 y(men)m(t)j(of)g (this)f(b)s(ound)f(to)i(an)g(exp)s(onen)m(tial)f(one)h(b)m(y)g(Galil)e ([)p Fr(48)q Fw(])i(w)m(as)g(a)g(driving)e(force)456 5108 y(b)s(ehind)23 b(some)k(of)g(the)g(early)f(w)m(ork)h(in)f(the)h (dev)m(elopmen)m(t)g(of)f(the)h(theory)g(of)g(expander)456 5216 y(graphs)i([)p Fr(47)q Fw(].)p eop %%Page: 9 9 9 8 bop 605 425 a Fw(There)28 b(w)m(as)g(a)h(15+)f(y)m(ear)h(gap)g(b)s (efore)e(the)i(\014rst)e(sup)s(erp)s(olynomial)d(lo)m(w)m(er)k(b)s (ound)456 533 y(for)d(pro)s(ofs)f(in)g(general)i(Resolution)e(w)m(as)i (obtained)f(b)m(y)g(Hak)m(en)i([)p Fr(51)p Fw(])f(who)f(sho)m(w)m(ed)h (ex-)456 641 y(p)s(onen)m(tial)c(lo)m(w)m(er)i(b)s(ounds)d(for)i(the)h (pigeonhole)f(principle.)35 b(Subsequen)m(tly)-8 b(,)24 b(exp)s(onen-)456 749 y(tial)30 b(b)s(ounds)f(ha)m(v)m(e)k(also)e(b)s (een)f(sho)m(wn)h(for)g(the)h(o)s(dd-c)m(harged)f(graph)g(form)m(ulas)f ([)p Fr(90)q Fw(],)456 857 y(random)25 b Fm(k)s Fw(-CNF)h(form)m(ulas)f (with)f(v)-5 b(arious)25 b(clause/v)-5 b(ariable)25 b(ratios)h([)p Fr(32)q(,)j(45)q(,)g(10)q(,)h(9)p Fw(],)456 965 y(and)20 b(min)m(term-maxterm)i(form)m(ulas)e([)p Fr(74)q Fw(].)38 b(The)21 b(pro)s(ofs)f(of)i(all)e(of)i(the)g(strongest)g(forms)456 1073 y(of)32 b(these)g(b)s(ounds)d(for)j(Resolution,)f(other)h(than)g (those)g(for)g(the)g(min)m(term-maxterm)456 1181 y(form)m(ulas,)d(in)m (v)m(olv)m(e)i(a)g(tec)m(hnique)f(kno)m(wn)g(as)g Fn(b)-5 b(ottlene)g(ck)34 b(c)-5 b(ounting)31 b Fw(due)e(to)i(Hak)m(en.)605 1289 y(In)23 b(this)g(metho)s(d,)i(one)g(views)e(the)h(pro)s(of)g(as)g (a)g(directed)g(acyclic)g(graph)f(of)i(clauses)456 1397 y(and)38 b(views)g(the)h(truth)f(assignmen)m(ts)g(as)h(\015o)m(wing)f (from)h(the)g(ro)s(ot)g(of)g(the)g(directed)456 1504 y(acyclic)27 b(graph)f(to)i(a)f(leaf,)h(where)e(an)h(assignmen)m(t)g (\015o)m(ws)f(through)h(a)g(clause)g Fm(C)33 b Fw(if)26 b(and)456 1612 y(only)32 b(if:)46 b(\(i\))33 b(it)g(\015o)m(ws)h (through)e(the)i(paren)m(t)g(clause)f(of)h Fm(C)39 b Fw(and)33 b(\(ii\))g(the)h(assignmen)m(t)456 1720 y(falsi\014es)19 b Fm(C)7 b Fw(.)37 b(Eac)m(h)22 b(assignmen)m(t)f(can)h(b)s(e)e(seen)h (to)h(\015o)m(w)g(through)e(a)i(unique)d(path)i(in)f(an)m(y)456 1828 y(Resolution)28 b(refutation.)39 b(The)29 b(idea)f(is)h(to)g(sho)m (w)g(that)h(for)f(the)g(form)m(ula)f(in)g(question,)456 1936 y(there)41 b(m)m(ust)g(exist)g(a)g(large)g(set)h(of)f(truth)g (assignmen)m(ts)f(with)g(the)h(prop)s(ert)m(y)g(that)456 2044 y(eac)m(h)c(m)m(ust)f(pass)f(through)g(a)i(large)f(clause.)57 b(Since)35 b(a)i(large)f(clause)f(cannot)i(falsify)456 2152 y(to)s(o)26 b(man)m(y)h(assignmen)m(ts,)f(this)f(implies)f(that)i (there)h(m)m(ust)e(exist)h(man)m(y)g(large)h(clauses)456 2260 y(and)i(hence)i(the)f(pro)s(of)g(m)m(ust)g(b)s(e)g(large.)605 2368 y(An)44 b(essen)m(tial)g(lemma)g(in)f(an)m(y)i(b)s(ottlenec)m(k)g (coun)m(ting)f(argumen)m(t)h(is)e(to)i(sho)m(w)456 2476 y(that)34 b(an)m(y)h(Resolution)d(refutation)i(of)g Fm(F)47 b Fw(m)m(ust)34 b(in)m(v)m(olv)m(e)g(a)h(large)f(clause.)51 b(Recen)m(tly)-8 b(,)456 2584 y(it)33 b(has)g(b)s(een)h(sho)m(wn)f([)p Fr(17)p Fw(],)j(using)c(ideas)h(from)g([)p Fr(33)q Fw(],)i(that)g(for)e (a)h(suitable)f(c)m(hoice)h(of)456 2692 y(parameters)g(this)g(lemma)f (is)h(also)g(su\016cien)m(t,)h(namely)f(an)m(y)g(Resolution)g (refutation)456 2800 y(of)c(small)f(size)h(can)h(b)s(e)f(con)m(v)m (erted)i(in)m(to)e(a)h(refutation)f(with)f(only)g(small)g(clauses.)605 2908 y(Another)38 b(metho)s(d)g(used)f(to)j(obtain)d(exp)s(onen)m(tial) h(Resolution)f(lo)m(w)m(er)h(b)s(ounds,)456 3015 y(used)c(for)h (example)g(for)g(the)g(min)m(term-maxterm)g(form)m(ulas,)h(is)e(the)h (metho)s(d)g(of)g(in-)456 3123 y(terp)s(olation,)29 b(whic)m(h)g(will)f (b)s(e)i(discussed)e(in)h(section)i(3.3.)605 3231 y(In)i(addition)f(to) i(lo)m(w)m(er)g(b)s(ounds)d(for)i(general)h(resolution,)f(there)h(is)e (also)i(practi-)456 3339 y(cal)k(in)m(terest)g(in)f(understanding)f (the)i(b)s(eha)m(vior)f(of)i(the)f(sp)s(ecial)f(cases)i(of)f(DP)h(and) 456 3447 y(DLL)g(algorithms.)68 b(\(See)40 b([)p Fr(67)q(,)46 b(34)p Fw(].\))70 b(Random)39 b Fm(k)s Fw(-CNF)h(form)m(ulas)f(ha)m(v)m (e)i(b)s(een)e(of)456 3555 y(particular)24 b(in)m(terest)h(in)g(this)f (regard)i(and)f(there)h(is)e(a)i(v)-5 b(ariet)m(y)26 b(of)g(results)f(giving)f(more)456 3663 y(precise)f(b)s(ounds)f(on)i (their)f(prop)s(erties)g(b)s(oth)h(as)g(pro)s(of)f(systems)i(and)e(as)i (satis\014abilit)m(y)456 3771 y(algorithms)k(at)i(v)-5 b(arious)29 b(clause/v)-5 b(ariable)30 b(ratios)g([)p Fr(32)q(,)35 b(45,)g(10)q(,)g(9)p Fw(].)605 4023 y Fr(3.2.)53 b Fx(A)m(C)936 3986 y Fs(0)976 4023 y Fr(-F)-9 b(rege)23 b(systems)g(and)g(their)g(extensions.)46 b Fw(While)19 b(Hak)m(en's)i(b)s(ound)456 4131 y(for)30 b(resolution)f(w)m(as)h(a)h (ma)5 b(jor)30 b(breakthrough,)g(it)g(is)f(the)i(pap)s(er)e(b)m(y)h (Ajtai)g([)p Fr(4)q Fw(])g(giving)456 4241 y(sup)s(er-p)s(olynomial)19 b(lo)m(w)m(er)24 b(b)s(ounds)d(for)j(pro)s(ofs)e(of)i(the)g(pigeonhole) e(principle)e(in)i Fx(A)m(C)3375 4205 y Fs(0)3414 4241 y Fw(-)456 4349 y(F)-8 b(rege)39 b(systems)e(that)h(has)f(formed)g(the) h(basis)e(of)h(m)m(uc)m(h)h(of)f(the)h(researc)m(h)g(in)e(pro)s(of)456 4457 y(complexit)m(y)i(o)m(v)m(er)j(the)e(last)g(decade.)67 b(As)39 b(men)m(tioned)g(ab)s(o)m(v)m(e,)k(this)38 b(ga)m(v)m(e)j(the)e (\014rst)456 4565 y(connection)31 b(b)s(et)m(w)m(een)h(the)f(tec)m (hniques)g(of)g(circuit)f(complexit)m(y)h(and)f(those)i(of)f(pro)s(of) 456 4673 y(complexit)m(y)-8 b(.)56 b(Ajtai's)36 b(result)e(has)h(b)s (een)g(impro)m(v)m(ed)g(to)h(exp)s(onen)m(tial)f(lo)m(w)m(er)h(b)s (ounds)456 4781 y(and)g(these)h(apply)f(to)h(all)f(the)h(symmetric)g (coun)m(ting)f(principles)e(in)h(the)i(\014rst)g(class)456 4889 y(ab)s(o)m(v)m(e,)26 b(e.g.)e([)p Fr(14)q(,)j(73,)g(62)q Fw(].)38 b(Despite)24 b(this)e(success,)j(no)e(lo)m(w)m(er)h(b)s(ounds) d(are)j(kno)m(wn)f(for)456 5000 y Fx(A)m(C)572 4963 y Fs(0)611 5000 y Fw(-F)-8 b(rege)35 b(pro)s(ofs)e(of)g(form)m(ulas)g(in) f(the)h(other)h(classes)f(ab)s(o)m(v)m(e)i(although)d(there)i(are)456 5108 y(certain)h(other)h(tautologies)g(for)f(whic)m(h)g(w)m(e)h(kno)m (w)f(a)h(sup)s(erp)s(olynomial)c(separation)456 5216 y(as)e(a)h(function)e(of)i(the)f(depth)g([)p Fr(58)q Fw(].)p eop %%Page: 10 10 10 9 bop 605 425 a Fw(The)37 b Fn(r)-5 b(estriction)40 b(metho)-5 b(d)p Fw(,)41 b(b)m(y)36 b(whic)m(h)g(the)h(lo)m(w)m(er)h(b) s(ounds)c(ab)s(o)m(v)m(e)k(w)m(ere)g(sho)m(wn)456 533 y(comes)24 b(from)g(Ajtai's)g(pap)s(er)f([)p Fr(4)p Fw(].)39 b(The)24 b(essence)g(of)h(this)e(idea)g(is)g(to)i(apply)d(restrictions) 456 641 y(to)31 b(try)f(to)i(simplify)27 b(eac)m(h)k(of)g(the)g(form)m (ulas)e(in)g(the)i(pro)s(of)f(y)m(et)i(lea)m(v)m(e)g(the)e(input)f (tau-)456 749 y(tology)i(still)e(highly)g(non-trivial.)39 b(Th)m(us)30 b(the)h(basic)f(metho)s(d)g(is)g(v)m(ery)h(similar)e(to)i (the)456 857 y(random-restriction)f(metho)s(d,)i(used)f(to)i(sho)m(w)f (that)h Fx(A)m(C)2448 820 y Fs(0)2519 857 y Fw(cannot)g(compute)f (parit)m(y)-8 b(.)456 965 y(Ho)m(w)m(ev)m(er,)38 b(it)d(is)f (necessarily)f(more)i(complex:)50 b(Since)34 b(the)h(circuits)f(app)s (earing)f(in)h(a)456 1073 y(sound)g(pro)s(of)g(alw)m(a)m(ys)i(compute)f (the)g(constan)m(t)i(function)d(1,)j(the)e(usual)f(simpli\014ca-)456 1181 y(tion)25 b(induced)g(b)m(y)h(restrictions)f(applied)f(to)j (circuits)e(m)m(ust)h(b)s(e)f(replaced)h(b)m(y)g(one)h(that)456 1289 y(includes)34 b(a)j(form)f(of)h(appro)m(ximation)e(as)i(w)m(ell.) 2134 1256 y Fs(4)2232 1289 y Fw(Using)e(this)h(metho)s(d,)i(one)f(sho)m (ws)456 1397 y(that)32 b(if)e(the)i(pro)s(of)f(is)g(to)s(o)h(short,)g (then)f(there)h(exists)f(a)h(restriction)e(suc)m(h)i(that)g(after)456 1504 y(applying)20 b(the)j(restriction)f(to)h(the)g(short)f(pro)s(of,)i (what)e(results)g(is)f(a)i(v)m(ery)g(trivial)e(pro)s(of)456 1612 y(of)31 b(a)g(form)m(ula)f(of)h(the)g(same)g(basic)f(form,)h(but)f (on)h(a)g(reduced)f(n)m(um)m(b)s(er)f(of)i(v)-5 b(ariables.)456 1720 y(Then)35 b(a)i(con)m(tradiction)f(can)h(b)s(e)f(reac)m(hed)h(b)m (y)f(sho)m(wing)g(that)h(suc)m(h)f(a)h(trivial)d(pro)s(of)456 1828 y(cannot)d(exist.)605 1936 y(Once)k(it)f(is)f(kno)m(wn)i(that)g (the)g(pigeonhole)e(principle)f(is)h(not)i(pro)m(v)-5 b(able)34 b(in)f Fx(A)m(C)3375 1899 y Fs(0)3414 1936 y Fw(-)456 2044 y(F)-8 b(rege)30 b(one)f(can)g(immediately)e(obtain)h (a)h(stronger)g(system)g(b)m(y)f(adding)f Fm(P)13 b(H)7 b(P)3197 2011 y Fk(n)p Fs(+1)3184 2067 y Fk(n)3363 2044 y Fw(as)456 2152 y(an)28 b(axiom)f(sc)m(hema)i(for)f(arbitrary)f Fm(n)p Fw(;)i(i.e.,)f(one)h(is)e(p)s(ermitted)g(to)h(deriv)m(e)g(lines) e(in)h(the)456 2260 y(pro)s(of)f(b)m(y)i(substituting)d(arbitrary)h (form)m(ulas)h(for)g(the)g(v)-5 b(ariables)27 b(of)g(some)h Fm(P)13 b(H)7 b(P)3282 2227 y Fk(n)p Fs(+1)3269 2282 y Fk(n)3419 2260 y Fw(.)456 2368 y(Ajtai)37 b([)p Fr(5)q Fw(])h(sho)m(w)m(ed)g(that)g(ev)m(en)g(in)f(this)f(stronger)i(system)g Fm(C)7 b(ount)2800 2335 y Fk(n)2800 2392 y Fs(2)2884 2368 y Fw(do)s(es)37 b(not)h(ha)m(v)m(e)456 2476 y(p)s(olynomial-size) 21 b(pro)s(ofs.)37 b(There)23 b(is)f(no)m(w)i(some)f(quite)g(in)m (teresting)g(structure)g(kno)m(wn)456 2584 y(ab)s(out)31 b(the)g(relativ)m(e)h(pro)s(of)e(strength)i(of)f(these)h(augmen)m(ted)g Fx(A)m(C)2716 2547 y Fs(0)2755 2584 y Fw(-F)-8 b(rege)33 b(systems)f(in)456 2692 y(whic)m(h)d(some)h(axiom)h(sc)m(heme)g(is)e (added)h(to)h(the)g(basic)e(system.)605 2800 y(In)g(the)h(system)f(ab)s (o)m(v)m(e)i(in)d(whic)m(h)g Fm(P)13 b(H)7 b(P)2020 2767 y Fk(n)p Fs(+1)2007 2822 y Fk(n)2187 2800 y Fw(w)m(as)30 b(added)f(as)h(an)f(axiom)g(sc)m(hema,)456 2908 y(all)43 b(of)i(the)g Fm(C)7 b(ount)1141 2875 y Fk(n)1141 2930 y(p)1231 2908 y Fw(principles)41 b(in)i(fact)j(are)f(no)m(w)f(kno)m(wn) g(to)i(require)d(exp)s(onen-)456 3019 y(tial)34 b(size)h(pro)s(ofs)f([) p Fr(15,)40 b(82)q Fw(];)e(con)m(v)m(ersely)-8 b(,)37 b(giv)m(en)e(an)m(y)g Fm(C)7 b(ount)2600 2986 y Fk(n)2600 3042 y(p)2681 3019 y Fw(axiom)35 b(sc)m(hema,)i(an)m(y)456 3165 y(b)s(ounded-depth)28 b(F)-8 b(rege)33 b(pro)s(ofs)e(of)h Fm(P)13 b(H)7 b(P)1935 3132 y Fk(n)p Fs(+1)1922 3188 y Fk(n)2103 3165 y Fw(or)32 b(on)m(to-)q Fm(P)13 b(H)7 b(P)2645 3121 y Fk(n)p Fs(+)p Fk(p)2779 3098 y Fd(\017)j Fq(log)i Fd(n)2632 3177 y Fk(n)2983 3165 y Fw(requires)30 b(ex-)456 3273 y(p)s(onen)m(tial)e(size)j(but)e(on)m(to-)r Fm(P)13 b(H)7 b(P)1623 3240 y Fk(n)p Fs(+1)1610 3296 y Fk(n)1790 3273 y Fw(is)29 b(trivial)f([)p Fr(11)q Fw(].)41 b(Th)m(us)29 b Fm(P)13 b(H)7 b(P)2827 3240 y Fk(n)p Fs(+1)2814 3296 y Fk(n)2995 3273 y Fw(is)29 b(exp)s(onen-)456 3381 y(tially)c(stronger)i(than)f(on)m(to-)r Fm(P)13 b(H)7 b(P)1676 3348 y Fk(n)p Fs(+1)1663 3404 y Fk(n)1813 3381 y Fw(.)39 b(Quite)26 b(precise)h(conditions)e(are)i(no)m(w)g(kno)m(wn) 456 3489 y(under)c(whic)m(h)g(exp)s(onen)m(tial)h(separations)g(exist)h (b)s(et)m(w)m(een)g(the)g(v)-5 b(arious)24 b Fm(C)7 b(ount)3182 3456 y Fk(n)3182 3512 y(p)3252 3489 y Fw(prin-)456 3601 y(ciples)29 b(in)g(this)g(con)m(text)j([)p Fr(23)q(,)j(13,)g(11)q Fw(].)605 3709 y(The)44 b(pro)s(ofs)h(for)f(these)i(results)d(b)s(egin) h(with)f(the)j(same)f(restriction)f(metho)s(d)456 3817 y(strategy)28 b(describ)s(ed)c(ab)s(o)m(v)m(e.)41 b(Ho)m(w)m(ev)m(er,) 29 b(with)c(the)i(additional)d(axiom)j(sc)m(hema)g(there)456 3925 y(is)45 b(the)i(further)f(requiremen)m(t)g(that)h(eac)m(h)h(line)d (in)g(the)i(pro)s(of)f(where)h(the)f(axiom)456 4033 y(sc)m(hema)31 b(is)e(applied)f(b)s(e)i(simpli\014ed)d(just)i(as)i(the)g(rest)f(of)h (the)f(pro)s(of)g(is.)40 b(The)29 b(need)i(to)456 4141 y(pro)m(v)m(e)j(this)f(latter)h(requiremen)m(t)f(motiv)-5 b(ated)34 b(the)g(in)m(tro)s(duction)d(of)j(Nullstellensatz)456 4249 y(pro)s(ofs)27 b([)p Fr(13)q Fw(])h(\(see)h(b)s(elo)m(w\))f(and)g (the)g(exp)s(onen)m(tial)g(separations)g(are)g(all)f(deriv)m(ed)h(from) 456 4357 y(lo)m(w)m(er)i(b)s(ounds)e(on)j(the)f(degrees)h(of)g(suc)m(h) f(pro)s(ofs.)605 4464 y(The)42 b(metho)s(d)g(in)m(v)m(olving)e (Nullstellensatz)h(degree)i(lo)m(w)m(er)f(b)s(ounds)e(is)i(not)g(the) 456 4572 y(only)30 b(one)i(that)g(has)f(b)s(een)f(used)h(for)g (obtaining)f(suc)m(h)h(separations.)43 b(In)31 b(pro)m(ving)f(the)456 4680 y(\014rst)41 b(sup)s(er-p)s(olynomial)d(separations)k(b)s(et)m(w)m (een)h(the)g(v)-5 b(arious)41 b Fm(C)7 b(ount)2981 4647 y Fk(n)2981 4703 y(p)3027 4680 y Fw(,)45 b(Ajtai)e([)p Fr(2)p Fw(])456 4788 y(used)26 b(certain)i(prop)s(erties)e(of)i (concisely)f(represen)m(ted)h(symmetric)e(systems)i(of)g(linear)p 456 4942 499 4 v 605 5001 a Fq(4)640 5033 y Fp([)p Fc(4)p Fp(])f(used)f(the)g(language)h(of)g(forcing)h(to)e(describ)r(e)h(this)g (appro)n(ximation;)f(the)g(cleanest)h(w)n(a)n(y)g(of)456 5124 y(expressing)j(this)g(appro)n(ximation)g(is)g(in)g(terms)f(of)i (so-called)g Fh(k)r Fp(-ev)l(aluations)g(whic)n(h)f(are)g(describ)r(ed) 456 5216 y(in)25 b([)p Fc(13)q(,)k(91,)g(11)q Fp(])d(and)f(are)h(a)g (mo)r(di\014cation)g(of)g(the)g(de\014nitions)f(in)h([)p Fc(62)q Fp(].)p eop %%Page: 11 11 11 10 bop 456 425 a Fw(equations)25 b([)p Fr(3)p Fw(])h(whic)m(h)e(he)h (pro)m(v)m(ed)h(using)e(structural)g(results)g(in)g(the)h(theory)h(of)g (repre-)456 533 y(sen)m(tations)k(of)h(the)g(symmetric)f(group)g([)p Fr(54)p Fw(])h(o)m(v)m(er)h Fm(GF)13 b Fw(\()p Fm(p)p Fw(\))30 b(to)h(sho)m(w)g(that)g(the)f(axiom)456 641 y(sc)m(hemas)g(are)g(appropriately)d(simpli\014ed.)37 b(This)28 b(tec)m(hnique)h(has)g(recen)m(tly)h(b)s(een)f(em-)456 749 y(plo)m(y)m(ed)38 b(to)i(giv)m(e)f(sup)s(er-p)s(olynomial)c(lo)m(w) m(er)j(b)s(ounds)f(for)h(v)m(ery)h(p)s(o)m(w)m(erful)f(algebraic)456 857 y(pro)s(of)29 b(systems)i(\(see)g(sections)f(3.4)i(and)e(3.5\).)605 1029 y Fr(3.3.)53 b(Cutting)36 b(Planes)h(and)f(In)m(terp)s(olation.)45 b Fw(The)31 b(metho)s(d)h(of)g(using)e(cut-)456 1137 y(ting)j(planes)f(for)i(inference)f(in)f(the)i(study)f(of)h(p)s(olytop) s(es)e(in)h(in)m(teger)h(programming)456 1245 y(w)m(as)c(\014rst)f (describ)s(ed)f(b)m(y)i(Gomory)h([)p Fr(50)q Fw(],)g(mo)s(di\014ed)c (and)j(sho)m(wn)g(to)g(b)s(e)g(complete)g(b)m(y)456 1353 y(Ch)m(v\023)-45 b(atal)32 b([)p Fr(30)q Fw(],)h(and)e(\014rst)h (analyzed)g(for)g(its)f(e\016ciency)i(as)f(a)g(pro)s(of)g(system)g(in)f ([)p Fr(37)q Fw(].)456 1460 y(It)h(is)f(one)h(of)g(t)m(w)m(o)h(classes) f(of)g(pro)s(of)f(systems)h(dev)m(elop)s(ed)f(b)m(y)h(represen)m(ting)f (unsatis-)456 1568 y(\014ably)f(problems)g(as)h(in)m(teger)h(or)g(0-1)h (programming)d(problems,)g(the)i(other)g(b)s(eing)e(a)456 1676 y(collection)j(of)i(systems)f(due)f(to)i(Lo)m(v)-5 b(asz)35 b(and)f(Sc)m(hrijv)m(er)f([)p Fr(63)q Fw(])h(whic)m(h)f(are)h (describ)s(ed)456 1784 y(in)29 b(detail)g(in)g(the)i(op)s(en)f (problems)e(section.)605 1892 y(Cutting)37 b(Planes)g(pro)s(ofs)g (manipulate)g(in)m(teger)h(linear)e(inequalities:)54 b(One)37 b(can)456 2000 y(add)31 b(inequalities)f(or)i(m)m(ultiply)e (them)i(b)m(y)h(p)s(ositiv)m(e)e(constan)m(ts)i(but)f(the)h(truly)d(p)s (o)m(w-)456 2108 y(erful)e(rule)i(is)f(the)i(rounded)d(division)g (rule:)620 2259 y Fm(ca)707 2273 y Fs(1)747 2259 y Fm(x)799 2273 y Fs(1)859 2259 y Fw(+)20 b Fm(ca)1037 2273 y Fs(2)1076 2259 y Fm(x)1128 2273 y Fs(2)1188 2259 y Fw(+)g Fm(:)15 b(:)g(:)h(ca)1487 2274 y Fk(k)1530 2259 y Fm(x)1582 2274 y Fk(k)1650 2259 y Fl(\025)25 b Fm(b)83 b Fl(\))g Fm(a)2090 2273 y Fs(1)2130 2259 y Fm(x)2182 2273 y Fs(1)2241 2259 y Fw(+)20 b Fm(a)2380 2273 y Fs(2)2420 2259 y Fm(x)2472 2273 y Fs(2)2531 2259 y Fw(+)g Fm(:)15 b(:)g(:)h(a)2791 2274 y Fk(k)2834 2259 y Fm(x)2886 2274 y Fk(k)2954 2259 y Fl(\025)25 b(d)p Fm(b=c)p Fl(e)p Fm(:)456 2409 y Fw(A)32 b(refutation)g(of)g(a)h(set)f(of)h(in)m(teger)f(linear)f(inequalities)f (is)h(a)i(sequence)f(of)g(inequali-)456 2517 y(ties,)f(where)f(eac)m(h) i(inequalit)m(y)d(is)h(either)g(one)h(of)g(the)g(original)e(ones)i(or)g (follo)m(ws)f(from)456 2625 y(previous)25 b(inequalities)f(b)m(y)i (applying)f(one)i(of)g(the)g(ab)s(o)m(v)m(e)g(rules,)g(and)f(where)g (the)h(\014nal)456 2733 y(inequalit)m(y)h(is)g(1)e Fl(\025)f Fw(0.)41 b(T)-8 b(o)30 b(refute)g(a)g(CNF)f(form)m(ula,)h(one)g (\014rst)f(con)m(v)m(erts)i(eac)m(h)g(clause)456 2841 y(in)m(to)h(an)f(equiv)-5 b(alen)m(t)32 b(in)m(teger)g(linear)f (inequalit)m(y)-8 b(.)44 b(Cutting)31 b(planes)g(pro)s(ofs)g(can)h (sim-)456 2949 y(ulate)k(Resolution)f(e\016cien)m(tly)h(and)f(easily)h (pro)m(v)m(e)h(all)e(of)h(the)h(symmetric)e(coun)m(ting)456 3057 y(tautologies)c(men)m(tioned)f(ab)s(o)m(v)m(e.)605 3165 y(Exp)s(onen)m(tial)f(lo)m(w)m(er)h(b)s(ounds)f(ha)m(v)m(e)i(b)s (een)f(sho)m(wn)f(for)h(cutting)h(planes)e(pro)s(ofs)g(of)456 3273 y(the)f(min)m(term-maxterm)f(form)m(ulas)g(using)f(a)i(metho)s(d)g (called)f Fn(interp)-5 b(olation)p Fw(.)42 b(In)27 b(this)456 3381 y(metho)s(d)i(one)i(b)s(egins)d(with)h(an)h(unsatis\014able)e (form)m(ula)h(of)h(the)h(form)e Fm(F)39 b Fw(=)25 b Fm(A)p Fw(\()p Fm(x;)15 b(z)t Fw(\))21 b Fl(^)456 3489 y Fm(B)5 b Fw(\()p Fm(y)s(;)15 b(z)t Fw(\))35 b(where)f(w)m(e)i(view)e Fm(x)g Fw(and)g Fm(y)k Fw(eac)m(h)e(as)f(a)g(v)m(ector)h(of)f(`priv)-5 b(ate')35 b(v)-5 b(ariables,)35 b(and)456 3597 y Fm(z)i Fw(as)c(a)g(v)m(ector)i(of)e(`shared')f(v)-5 b(ariables.)47 b(After)34 b(an)m(y)f(assignmen)m(t)f Fm(\034)43 b Fw(to)34 b(the)f(common)456 3705 y Fm(z)j Fw(v)-5 b(ariables)31 b(is)h(made,)h(in)e(the)h(remaining)f(form)m(ula)g Fm(A)p Fw(\()p Fm(x;)15 b(\034)10 b Fw(\))23 b Fl(^)e Fm(B)5 b Fw(\()p Fm(y)s(;)15 b(\034)10 b Fw(\),)34 b(it)e(m)m(ust)g(b)s(e)456 3812 y(the)g(case)h(that)f(either)f Fm(A)h Fw(is)f(unsatis\014able)f (or)i Fm(B)k Fw(is)31 b(unsatis\014able.)43 b(The)32 b(asso)s(ciated)456 3920 y(in)m(terp)s(olation)c(problem)g(for)h Fm(F)43 b Fw(tak)m(es)31 b(as)f(input)d(an)j(assignmen)m(t)f Fm(\034)40 b Fw(to)30 b(the)g(common)456 4028 y(v)-5 b(ariables,)30 b(and)g(outputs)g Fm(A)i Fw(only)e(when)g Fm(A)p Fw(\()p Fm(x;)15 b(\034)10 b Fw(\))32 b(is)e(unsatis\014able,)f (and)h(outputs)h Fm(B)456 4136 y Fw(only)25 b(when)h Fm(B)5 b Fw(\()p Fm(y)s(;)15 b(\034)10 b Fw(\))27 b(is)f (unsatis\014able.)37 b(Of)26 b(course,)i(sometimes)e(b)s(oth)g Fm(A)h Fw(and)f Fm(B)31 b Fw(ma)m(y)456 4244 y(b)s(e)37 b(acceptable)i(answ)m(ers.)62 b(\(This)37 b(problem)f(is)h(called)g (the)h(in)m(terp)s(olation)e(problem)456 4352 y(since)26 b(it)h(is)f(equiv)-5 b(alen)m(t)26 b(to)i(Craig)e(In)m(terp)s(olation.) 39 b(In)26 b(the)h(t)m(ypical)g(form)m(ulation,)g Fm(G)f Fw(is)456 4460 y(a)35 b(tautological)g(form)m(ula)f(in)f(the)i(form)f Fm(A)1931 4427 y Fe(0)1955 4460 y Fw(\()p Fm(x;)15 b(z)t Fw(\))33 b Fl(!)g Fm(B)2394 4427 y Fe(0)2417 4460 y Fw(\()p Fm(y)s(;)15 b(z)t Fw(\).)54 b(In)34 b(our)h(form)m(ulation)456 4568 y Fm(F)43 b Fw(is)29 b Fl(:)p Fm(G)p Fw(,)h Fm(A)h Fw(is)e Fm(A)1094 4535 y Fe(0)1117 4568 y Fw(,)i(and)f Fm(B)35 b Fw(is)29 b Fl(:)p Fm(B)1680 4535 y Fe(0)1702 4568 y Fw(.\))605 4676 y(F)-8 b(or)29 b(arbitrary)e Fm(F)13 b Fw(,)29 b(it)f(ma)m(y)g(b)s(e)g(v)m(ery)h(di\016cult)d(to)j(solv)m(e) f(the)h(in)m(terp)s(olation)d(prob-)456 4784 y(lem)37 b(asso)s(ciated)h(with)f Fm(F)13 b Fw(:)55 b(if)37 b Fm(L)h Fw(is)f(a)h(decision)e(problem)g(in)h Fx(NP)26 b 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641 y(p)s(olynomial-time)g (solution.)49 b(This)31 b(p)s(ossibilit)m(y)f(w)m(as)k(\014rst)e (suggested)i(b)m(y)g(Kra)5 b(j)-10 b(\023)-35 b(\020)m(\024)-43 b(cek.)456 749 y(If)33 b(this)f(situation)g(exists)h(for)g(pro)s(of)g (system)h Fm(S)5 b Fw(,)34 b(then)f(w)m(e)h(sa)m(y)g(that)g Fm(S)39 b Fw(has)33 b(the)g Fn(fe)-5 b(a-)456 857 y(sible)42 b(interp)-5 b(olation)45 b(pr)-5 b(op)g(erty)p Fw(.)75 b(There)41 b(is)f(also)h(a)g(monotone)h(v)m(ersion)e(of)h(feasible)456 965 y(in)m(terp)s(olation.)51 b(Namely)-8 b(,)36 b Fm(F)45 b Fw(=)32 b Fm(A)p Fw(\()p Fm(x;)15 b(z)t Fw(\))24 b Fl(^)f Fm(B)5 b Fw(\()p Fm(y)s(;)15 b(z)t Fw(\))35 b(is)f(monotone)h (if)e Fm(z)39 b Fw(o)s(ccurs)34 b(only)456 1073 y(p)s(ositiv)m(ely)d (in)h Fm(A)p Fw(,)i(and)e(in)g(this)g(case)i(the)g(in)m(terp)s(olation) d(problem)h(is)g(monotone.)49 b Fm(S)456 1181 y Fw(has)39 b(the)h(monotone)h(feasible)d(in)m(terp)s(olation)h(prop)s(ert)m(y)g (if)g(whenev)m(er)g Fm(F)53 b Fw(is)39 b(mono-)456 1289 y(tone)27 b(and)f(has)h(a)g(short)g Fm(S)5 b Fw(-pro)s(of,)27 b(then)g(the)g(asso)s(ciated)g(in)m(terp)s(olation)e(problem)h(has)456 1397 y(p)s(olynomial-size)i(\(uniform\))g(monotone)k(circuits.)605 1504 y(Razb)s(oro)m(v)g([)p Fr(79)q Fw(])g(and)f(indep)s(enden)m(tly)e (Bonet,)k(Pitassi)d(and)h(Raz)h([)p Fr(20)q Fw(])g(w)m(ere)g(the)456 1612 y(\014rst)k(to)i(use)e(the)i(ab)s(o)m(v)m(e)g(idea)e(to)i(obtain)f (exp)s(onen)m(tial)f(lo)m(w)m(er)h(b)s(ounds)e(for)h(certain)456 1720 y(pro)s(of)43 b(systems.)83 b(In)44 b([)p Fr(79)q Fw(],)k(a)d(form)m(ula)f(\(formalizing)f(that)i(SA)-8 b(T)44 b(do)s(es)g(not)h(ha)m(v)m(e)456 1828 y(p)s(olynomial-size)19 b(circuits\))h(is)g(constructed)i(with)e(the)i(prop)s(ert)m(y)e(that)i (the)g(asso)s(ciated)456 1936 y(in)m(terp)s(olan)m(t)36 b(problem)g(has)h(no)h(p)s(olynomial-time)d(circuits,)j(under)e (cryptographic)456 2044 y(assumptions.)62 b([)p Fr(20)q Fw(])38 b(constructs)g(a)h(monotone)g(form)m(ula)e(with)g(the)h(prop)s (ert)m(y)g(that)456 2152 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b(in)m(terp)s(olation)e(can)j(th)m(us)f(giv)m(e)g(v)m(ery) h(go)s(o)s(d)f(lo)m(w)m(er)g(b)s(ounds)e(for)i(man)m(y)456 3123 y(pro)s(of)29 b(systems)h(\(sometimes)h(only)e(under)g (cryptographic)h(assumptions\).)39 b(In)29 b(addi-)456 3231 y(tion)h(to)i(the)g(unconditional)c(lo)m(w)m(er)k(b)s(ounds)d(men) m(tioned)i(ab)s(o)m(v)m(e,)i(conditional)c(lo)m(w)m(er)456 3339 y(b)s(ounds)19 b(ha)m(v)m(e)j(b)s(een)f(sho)m(wn)f(for)h(all)f(of) i(the)f(follo)m(wing)f(systems:)36 b(Resolution,)22 b(Cutting)456 3447 y(Planes,)31 b(Nullstellensatz)e([)p Fr(33)q Fw(],)j(P)m (olynomial)e(Calculus)g([)p Fr(75)p Fw(],)j(as)e(w)m(ell)g(as)g(an)m(y) h(pro)s(of)456 3555 y(system)f(where)g(the)g(underlying)d(form)m(ulas)i (in)g(the)h(pro)s(of)g(ha)m(v)m(e)h(small)e(probabilistic)456 3663 y(comm)m(unication)40 b(complexit)m(y)g([)p Fr(20)q Fw(].)72 b(Unfortunately)39 b(there)i(are)g(strong)g(negativ)m(e)456 3771 y(results,)g(sho)m(wing)f(that)g(the)h(in)m(terp)s(olation)d (metho)s(d)i Fn(c)-5 b(annot)41 b Fw(b)s(e)f(applied)e(to)j(giv)m(e)456 3879 y(lo)m(w)m(er)36 b(b)s(ounds)e(for)i(the)g(follo)m(wing)e(pro)s (of)i(systems,)h(again)f(under)f(v)-5 b(arious)35 b(crypto-)456 3987 y(graphic)27 b(assumptions:)38 b(Extended)28 b(F)-8 b(rege)30 b([)p Fr(61)q Fw(],)f(F)-8 b(rege,)30 b Fx(TC)2572 3950 y Fs(0)2612 3987 y Fw(-F)-8 b(rege)30 b([)p Fr(21)q Fw(],)f(and)f(ev)m(en)456 4098 y Fx(A)m(C)572 4061 y Fs(0)611 4098 y Fw(-F)-8 b(rege)32 b([)p Fr(64)q(,)j(19)q Fw(].)605 4205 y(A)c(disadv)-5 b(an)m(tage)31 b(of)g(the)f(in)m(terp)s (olation)f(metho)s(d)h(for)g(obtaining)g(lo)m(w)m(er)h(b)s(ounds)456 4313 y(is)26 b(that)h(it)g(applies)e(only)h(to)h(form)m(ulas)f(of)i(a)f (v)m(ery)g(sp)s(ecial)f(form.)39 b(Th)m(us,)27 b(for)f(example,)456 4421 y(nothing)g(is)h(kno)m(wn)g(ab)s(out)g(the)h(length)f(of)h(the)f (shortest)h(Cutting)f(Planes)g(pro)s(ofs)g(for)456 4529 y(either)i(the)i(o)s(dd-c)m(harged)f(graph)g(form)m(ulas)g(or)g(for)g (random)g(form)m(ulas.)605 4784 y Fr(3.4.)53 b(Algebraic)33 b(pro)s(of)g(systems.)45 b Fw(The)28 b(Nullstellensatz)e([)p Fr(13)q Fw(])i(and)g(P)m(olyno-)456 4892 y(mial)38 b(Calculus)g([)p Fr(33)q Fw(])i(pro)s(of)f(systems)h(are)g(based)g(on)f(a)i(sp)s(ecial)d (case)j(of)f(Hilb)s(ert's)456 5000 y(famous)28 b(Nullstellensatz)f (whic)m(h)g(relates)i(the)g(question)f(of)h(the)f(non-existence)h(of)g (si-)456 5108 y(m)m(ultaneous)24 b(zeros)h(of)g(a)g(family)e(of)i(m)m (ultiv)-5 b(ariable)22 b(p)s(olynomials)g(in)i(certain)g(\014elds)g(to) 456 5216 y(the)f(question)f(of)h(the)g(existence)g(of)g(co)s(e\016cien) m(t)h(p)s(olynomials)c(witnessing)h(that)i(1)h(is)e(in)p eop %%Page: 13 13 13 12 bop 456 425 a Fw(the)29 b(ideal)e(generated)j(b)m(y)f(these)g(p)s (olynomials.)38 b(\(In)28 b(fact)i(they)f(use)f(a)i(generalization)456 533 y(of)g(this)f(sp)s(ecial)g(case)j(to)f(rings)e(as)i(w)m(ell)e(as)h (\014elds.\))605 641 y(T)-8 b(o)30 b(use)f(this)f(relation)h(one)g (\014rst)g(expresses)g(an)g(unsatis\014able)e(Bo)s(olean)j(form)m(ula) 456 749 y(as)41 b(a)g(system)h(of)f(constan)m(t-degree)i(p)s(olynomial) c(equations)i(in)e(some)j(p)s(olynomial)456 857 y(ring.)59 b(F)-8 b(or)38 b(the)g(prop)s(ositional)c(v)m(ersions)j(of)g(the)g (symmetric)g(coun)m(ting)g(principles,)456 965 y(these)i(translations)f (are)h(quite)g(natural,)i(for)d(example)h Fm(P)13 b(H)7 b(P)2693 932 y Fk(m)2680 987 y(n)2799 965 y Fw(when)38 b(translated)456 1073 y(has)31 b(p)s(olynomials)1125 1005 y Fb(P)1221 1100 y Fk(j)t Fe(\024)p Fk(n)1370 1073 y Fm(x)1422 1087 y Fk(ij)1503 1073 y Fl(\000)21 b Fw(1)27 b(=)f(0)32 b(for)f(eac)m(h)h Fm(i)27 b Fl(\024)f Fm(m)p Fw(,)32 b(as)f(w)m(ell)f(as)i Fm(x)2940 1087 y Fk(ij)3000 1073 y Fm(x)3052 1088 y Fk(i)3076 1069 y Fg(0)3099 1088 y Fk(j)3162 1073 y Fw(=)26 b(0)32 b(for)456 1191 y(eac)m(h)37 b Fm(i)f Fl(6)p Fw(=)e Fm(i)869 1158 y Fe(0)928 1191 y Fl(\024)h Fm(m)h Fw(and)g Fm(j)k Fl(\024)35 b Fm(n)p Fw(.)58 b(More)37 b(generally)-8 b(,)38 b(there)e(are)h(natural)e(lo)m (w-degree)456 1299 y(translations)41 b(of)h(arbitrary)f(CNF)h(form)m (ulas)f(that)i(are)f(similar)e(to)j(those)f(used)g(in)456 1407 y(probabilistically)28 b(c)m(hec)m(k)-5 b(able)33 b(pro)s(ofs)f(\(PCP\))h([)p Fr(8,)38 b(40)p Fw(].)48 b(T)-8 b(o)33 b(use)f(these)h(mec)m(hanisms)456 1515 y(to)44 b(detect)i(0-1)f(solutions)d(only)-8 b(,)48 b(w)m(e)c(add)g (the)g(equations)g Fm(x)2641 1482 y Fs(2)2709 1515 y Fl(\000)29 b Fm(x)48 b Fw(=)g(0)c(for)g(eac)m(h)456 1628 y(v)-5 b(ariable)29 b Fm(x)p Fw(.)42 b(Hilb)s(ert's)29 b(Nullstellensatz)g(implies)f(that)j(suc)m(h)g(a)g(system)g Fl(f)3050 1605 y Fm(~)3036 1628 y(Q)p Fw(\()o Fm(~)-44 b(x)q Fw(\))26 b(=)f(0)p Fl(g)456 1736 y Fw(do)s(es)30 b(not)g(ha)m(v)m(e)i(a)e(solution)f(if)g(and)h(only)g(if)f(there)h (exists)g(a)h(family)e(of)h(m)m(ulti-v)-5 b(ariate)456 1850 y(p)s(olynomials)987 1827 y Fm(~)974 1850 y(P)56 b Fw(suc)m(h)42 b(that)1515 1782 y Fb(P)1611 1877 y Fk(i)1654 1850 y Fm(P)1712 1864 y Fk(i)1741 1850 y Fw(\()o Fm(~)-44 b(x)p Fw(\))p Fm(Q)1935 1864 y Fk(i)1963 1850 y Fw(\()o Fm(~)g(x)q Fw(\))46 b Fl(\021)g Fw(1.)79 b(It)43 b(is)f(easy)h(to)h (see)g(that)f(the)456 1961 y Fm(x)508 1928 y Fs(2)567 1961 y Fl(\000)20 b Fm(x)25 b Fw(=)g(0)31 b(equations)f(guaran)m(tee)i (that)f(degree)g(at)g(most)g Fm(n)f Fw(is)f(su\016cien)m(t.)605 2069 y(The)g(complexit)m(y)h(in)f(the)h(Nullstellensatz)e(pro)s(of)h (system)h(is)f(simply)e(the)j(size)g(of)456 2177 y(the)e(dense)g (represen)m(tation)h(of)f(the)h(co)s(e\016cien)m(t)g(p)s(olynomials)d (and)i(th)m(us)g(of)g(the)h(form)456 2286 y Fm(n)511 2253 y Fk(O)r Fs(\()p Fk(d)p Fs(\))700 2286 y Fw(where)38 b Fm(d)h Fw(is)f(the)h(largest)h(degree)f(required.)65 b(In)38 b(the)h(P)m(olynomial)f(Calculus)456 2394 y(pro)s(of)31 b(system)h(one)g(do)s(es)g(not)h(need)e(to)i(explicitly)d(write)h(do)m (wn)h(these)g(p)s(olynomials)456 2502 y(all)41 b(at)h(once)h(but)f (rather)g(one)g(can)g(giv)m(e)h(a)f(deriv)-5 b(ation)41 b(that)i(demonstrates)f(their)456 2610 y(existence)d(in)m(v)m(olving)f 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Fm(n)p Fw(\))33 b([)p Fr(26)p Fw(].)49 b(It)33 b(is)f(ev)m(en)i(p)s (ossible)d(that)i(certain)456 3581 y(principles)41 b(ma)m(y)46 b(b)s(e)e(pro)m(v)m(ed)h(using)f(degree)i(2)f(using)f(P)m(olynomial)f (Calculus)g(but)456 3689 y(require)29 b(degree)i(\012\()1146 3623 y Fl(p)p 1222 3623 55 4 v 66 x Fm(n)o Fw(\))g(Nullstellensatz)d (pro)s(ofs)i([)p Fr(33)q Fw(].)605 3797 y(Although)i(it)g(is)g(trivial) e(to)k(pro)m(v)m(e)f Fm(C)7 b(ount)2085 3764 y Fk(n)2085 3819 y(r)2164 3797 y Fw(in)31 b(constan)m(t)j(degree)f(in)f Fj(Z)3100 3811 y Fk(r)3133 3797 y Fw(,)i(degree)456 3910 y(lo)m(w)m(er)g(b)s(ounds)f(of)h Fm(n)1180 3877 y Fs(\012\(1\))1359 3910 y Fw(ha)m(v)m(e)i(b)s(een)e(sho)m(wn)f(in)h Fj(Z)2247 3924 y Fk(r)2315 3910 y Fw(for)g Fm(P)13 b(H)7 b(P)2683 3877 y Fk(m)2670 3933 y(n)2784 3910 y Fw([)p Fr(12)p Fw(],)36 b Fm(C)7 b(ount)3255 3877 y Fk(n)3255 3933 y(s)3336 3910 y Fw(for)456 4035 y(most)27 b Fm(s)e Fl(6)p Fw(=)g Fm(r)30 b Fw([)p Fr(23)q Fw(])e(and)f(on)m(to-)q Fm(P)13 b(H)7 b(P)1696 4002 y Fk(n)p Fs(+)p Fk(r)1828 3979 y Fd(!)r Fq(\(1\))1683 4058 y Fk(n)1982 4035 y Fw([)p Fr(11)q Fw(].)40 b(So)27 b(far,)h(all)f(of)g(the)h(Nullstellensatz)456 4143 y(b)s(ounds)33 b(men)m(tioned)i(w)m(ere)h(sho)m(wn)e(using)g(the)i (notion)f(of)g(a)h(dual)e Fn(design)i Fw([)p Fr(12)q(,)k(25)q Fw(].)456 4251 y(This)28 b(is)h(t)m(ypically)g(a)i(com)m(binatorial)e (construction)h(that)h(guaran)m(tees)h(that)f(one)f(can)456 4359 y(deriv)m(e)37 b(a)g(solution)f(to)j(a)e(set)h(of)g(dual)e (equations)h(that)h(express)f(the)h(co)s(e\016cien)m(t)g(of)456 4467 y(the)j(constan)m(t)h(term)e(in)1344 4399 y Fb(P)1440 4494 y Fk(i)1483 4467 y Fm(P)1541 4481 y Fk(i)1597 4467 y Fl(\001)27 b Fm(Q)1721 4481 y Fk(i)1790 4467 y Fw(as)41 b(a)g(linear)e(com)m(bination)h(of)h(higher)f(degree)456 4575 y(co)s(e\016cien)m(ts)46 b(in)e(the)h(indeterminate)f(co)s (e\016cien)m(ts)i(of)f(the)h Fm(P)2642 4589 y Fk(i)2670 4575 y Fw(.)86 b(Since)3033 4507 y Fb(P)3129 4602 y Fk(i)3172 4575 y Fm(P)3230 4589 y Fk(i)3289 4575 y Fl(\001)30 b Fm(Q)3416 4589 y Fk(i)456 4683 y Fw(is)37 b(supp)s(osed)f(to)j (represen)m(t)f(the)g(p)s(olynomial)d(1)k(this)e(w)m(ould)g(b)s(e)g (imp)s(ossible.)60 b([)p Fr(23)q Fw(])456 4791 y(in)m(tro)s(duced)26 b(a)i(nice)f(tec)m(hnique)h(\(used)f(in)g([)p Fr(11)p Fw(]\))i(that)f(mak)m(es)h(suc)m(h)e(designs)g(easier)h(to)456 4899 y(construct.)605 5007 y(Using)d(a)g(more)h(general)f(class)g(of)h (designs)e(the)i(degree)g(b)s(ounds)d(for)i Fm(P)13 b(H)7 b(P)3173 4974 y Fk(m)3160 5029 y(n)3265 5007 y Fw(w)m(ere)456 5115 y(w)m(ere)25 b(impro)m(v)m(ed)g(to)h(degree)g Fm(n=)p Fw(2)10 b(+)g(1)25 b(b)m(y)g(Razb)s(oro)m(v)h([)p Fr(76)q Fw(].)39 b(Remark)-5 b(ably)d(,)27 b(Razb)s(oro)m(v's)p eop %%Page: 14 14 14 13 bop 456 425 a Fw(result)39 b(also)i(applies)e(to)j(the)f(P)m (olynomial)e(Calculus)g(pro)s(of)h(system)h(and)f(w)m(as)h(not)456 533 y(only)28 b(the)i(\014rst)f(non-trivial)e(lo)m(w)m(er)j(b)s(ound)e (sho)m(wn)h(for)g(that)h(system)g(but)f(also)g(is)g(one)456 641 y(of)39 b(strongest)h(kno)m(wn)f(for)g(the)h(Nullstellensatz)e (system.)68 b(It)39 b(in)m(v)m(olv)m(es)h(an)f(explicit)456 749 y(computation)h(of)g(the)g(Gro)s(ebner)f(basis)g(of)i(the)f(ideal)f (generated)i(b)m(y)f(the)g Fm(P)13 b(H)7 b(P)3378 716 y Fk(m)3365 771 y(n)456 857 y Fw(equations.)37 b(Razb)s(oro)m(v)21 b(also)g(extends)f(this)g(lo)m(w)m(er)g(b)s(ound)f(to)i(obtain)f (stronger,)j(nearly)456 965 y(linear,)35 b(degree)i(lo)m(w)m(er)f(b)s (ounds)e(in)g(P)m(olynomial)h(Calculus)e(for)j(related)g(tautologies) 456 1073 y(as)30 b(a)h(function)e(of)i(the)f(n)m(um)m(b)s(er)f(of)i(v) -5 b(ariables.)605 1181 y(Recen)m(tly)d(,)29 b(Kra)5 b(jicek)26 b([)p Fr(57)p Fw(])h(has)f(sho)m(wn)g(non-constan)m(t)h (degree)h(lo)m(w)m(er)e(b)s(ounds)f(for)456 1289 y Fm(C)7 b(ount)712 1256 y Fk(n)712 1311 y(s)789 1289 y Fw(in)31 b(the)h(P)m(olynomial)f(Calculus)e(pro)s(of)i(system)i(o)m(v)m(er)g Fj(Z)2707 1303 y Fk(r)2772 1289 y Fw(b)m(y)f(extending)f(the)456 1397 y(results)k(of)h(Ajtai)h(regarding)e(symmetric)h(linear)f (equations)h(and)g(the)g(structure)g(of)456 1504 y(represen)m(tations) 30 b(of)g(the)h(symmetric)f(group)g([)p Fr(3)p Fw(])h(men)m(tioned)f (earlier.)605 1730 y Fr(3.5.)53 b Fx(A)m(C)936 1693 y Fs(0)976 1730 y Fw([)p Fm(r)s Fw(])p Fr(-F)-9 b(rege)28 b(systems.)45 b Fw(W)-8 b(e)26 b(ha)m(v)m(e)g(already)f(seen)g(ho)m(w)g (one)g(can)g(extend)456 1841 y Fx(A)m(C)572 1804 y Fs(0)611 1841 y Fw(-F)-8 b(rege)31 b(pro)s(ofs)c(b)m(y)h(adding)f(axioms)i(for)f (coun)m(ting)g(mo)s(dulo)f Fm(r)s Fw(.)39 b(A)29 b(far)f(more)h(gen-) 456 1949 y(eral,)43 b(and)d(in)g(some)h(sense,)j(more)d(natural)f(w)m (a)m(y)i(to)g(add)e(the)h(p)s(o)m(w)m(er)g(of)g(mo)s(dular)456 2057 y(coun)m(ting)27 b(to)h(a)g(pro)s(of)e(system)i(is)e(to)i(include) d(it)i(fundamen)m(tally)f(in)g(the)h(structure)g(of)456 2165 y(the)34 b(ob)5 b(jects)36 b(ab)s(out)e(whic)m(h)f(one)i(reasons;) i(that)e(is,)g(to)g(in)m(tro)s(duce)f(mo)s(dular)e(coun)m(t-)456 2273 y(ing)39 b(connectiv)m(es)h(in)m(to)g(the)g(lines)e(of)i(the)g (pro)s(ofs)f(themselv)m(es)i(and)e(new)g(inference)456 2380 y(rules)30 b(for)i(manipulating)d(these)k(form)m(ulas.)45 b Fx(A)m(C)2134 2344 y Fs(0)2173 2380 y Fw([)p Fm(r)s Fw(]-F)-8 b(rege)34 b(is)e(precisely)e(suc)m(h)i(a)g(sys-)456 2488 y(tem.)44 b(Ev)m(en)32 b(the)f(P)m(olynomial)f(Calculus)g(pro)s (of)g(system)i(mo)s(dulo)e Fm(r)k Fw(ma)m(y)e(b)s(e)f(view)m(ed)456 2596 y(as)38 b(a)g(subsystem)f(of)h(an)g Fx(A)m(C)1465 2559 y Fs(0)1504 2596 y Fw([)p Fm(r)s Fw(]-F)-8 b(rege)40 b(pro)s(of)d(system)h(in)f(whic)m(h)f(all)h(the)h(mo)s(dular)456 2704 y(connectiv)m(es)31 b(are)g(at)g(the)f(top)h([)p Fr(71)q Fw(].)605 2815 y(A)m(t)26 b(presen)m(t)e(there)h(are)g(no)f(lo) m(w)m(er)h(b)s(ounds)d(kno)m(wn)i(for)h Fx(A)m(C)2608 2778 y Fs(0)2647 2815 y Fw([)p Fm(r)s Fw(]-F)-8 b(rege,)28 b(ev)m(en)d(when)456 2923 y Fm(r)35 b Fw(is)c(a)i(prime.)46 b(One)32 b(program)g(for)g(obtaining)g(suc)m(h)g(b)s(ounds)e(w)m(as)j (laid)e(out)i(in)e([)p Fr(23)q Fw(].)456 3033 y(where)24 b(it)g(is)f(sho)m(wn)h(ho)m(w)h(to)g(con)m(v)m(ert)h(an)f Fx(A)m(C)1994 2997 y Fs(0)2034 3033 y Fw([)p Fm(p)p Fw(]-F)-8 b(rege)26 b(pro)s(of)e(of)h Fm(F)37 b Fw(to)26 b(a)f(P)m(olynomial)456 3141 y(Calculus)i(pro)s(of)j(of)g(a)g(system)g(that)h(in)m(v)m(olv)m (es)f(the)g(p)s(olynomials)d(for)i Fm(F)43 b Fw(plus)28 b(certain)456 3249 y(lo)m(w)j(degree)i(extension)e(p)s(olynomials)e (\(whic)m(h)i(mimic)f(the)i(lo)m(w)f(degree)i(appro)m(xima-)456 3360 y(tions)c(used)h(b)m(y)g([)p Fr(77)q(,)35 b(87)p Fw(])c(for)f Fx(A)m(C)1621 3323 y Fs(0)1661 3360 y Fw([)p Fm(p)p Fw(])g(lo)m(w)m(er)h(b)s(ounds\).)605 3586 y Fr(3.6.)53 b(F)-9 b(rege)42 b(systems)f(and)h Fx(TC)1835 3549 y Fs(0)1874 3586 y Fr(-F)-9 b(rege)42 b(systems.)j Fw(Just)36 b(as)h Fx(A)m(C)3067 3549 y Fs(0)3106 3586 y Fw([)p Fm(r)s Fw(]-F)-8 b(rege)456 3696 y(pro)s(ofs)19 b(include)g(coun)m(ting)i(mo)s (dulo)e Fm(r)k Fw(as)e(a)g(\014rst-class)f(concept,)k(so)d Fx(TC)2897 3659 y Fs(0)2937 3696 y Fw(-F)-8 b(rege)22 b(pro)s(ofs)456 3804 y(include)31 b(coun)m(ting)j(up)f(to)i(a)f (threshold)e(as)j(a)f(\014rst-class)f(concept.)53 b(Lo)s(osely)33 b(sp)s(eak-)456 3912 y(ing,)c Fx(TC)752 3875 y Fs(0)792 3912 y Fw(-F)-8 b(rege)32 b(pro)s(ofs)e(are)g(pro)s(ofs)g(where)g(the)g (underlying)e(class)i(of)g(form)m(ulas)g(are)456 4020 y(small-w)m(eigh)m(t,)36 b(constan)m(t-depth)h(threshold)d(form)m (ulas.)55 b(Th)m(us,)36 b(for)g(example,)h(Cut-)456 4128 y(ting)e(Planes)g(pro)s(ofs)f(can)i(b)s(e)f(view)m(ed)g(as)h(a)g(sp)s (ecial)e(case)j(of)f(restricted)f Fx(TC)3161 4091 y Fs(0)3200 4128 y Fw(-F)-8 b(rege)456 4236 y(pro)s(ofs.)605 4344 y(W)g(e)22 b(only)f(ha)m(v)m(e)h(partial)e(results)g(on)h Fx(TC)1944 4307 y Fs(0)1984 4344 y Fw(-F)-8 b(rege)23 b(pro)s(ofs.)36 b(Maciel)22 b(and)e(Pitassi)g([)p Fr(65)q Fw(])456 4452 y(ha)m(v)m(e)27 b(sho)m(wn)e(pro)s(of-theoretic)h (analogues)g(of)g(circuit)f(complexit)m(y)g(constructions)h([)p Fr(7)p Fw(])456 4563 y(to)f(relate)g Fx(TC)928 4526 y Fs(0)968 4563 y Fw(-)g(and)f Fx(A)m(C)1310 4526 y Fs(0)1350 4563 y Fw([)p Fm(p)p Fw(]-F)-8 b(rege)27 b(pro)s(ofs.)38 b(In)24 b(particular)f(they)i(sho)m(w)g(ho)m(w)g(one)g(can)456 4673 y(con)m(v)m(ert)g(p)s(olynomial-size)d Fx(A)m(C)1522 4636 y Fs(0)1562 4673 y Fw([2]-F)-8 b(rege)27 b(pro)s(ofs)c(in)m(to)h (restricted)g(quasip)s(olynomial-)456 4784 y(size)30 b(depth)f(3)i Fx(TC)1081 4747 y Fs(0)1121 4784 y Fw(-F)-8 b(rege)32 b(pro)s(ofs.)605 4892 y(The)43 b(p)s(oten)m(tial)g(list)g(of) h(candidate)f(hard)g(problems)f(for)h Fx(TC)2813 4855 y Fs(0)2853 4892 y Fw(-F)-8 b(rege)45 b(is)e(quite)456 5000 y(short,)33 b(in)e(part)h(b)s(ecause)h(there)f(are)h(e\016cien)m (t)g Fx(TC)2209 4963 y Fs(0)2249 5000 y Fw(-F)-8 b(rege)34 b(pro)s(ofs)e(for)g(so)h(man)m(y)f(for-)456 5108 y(m)m(ulas)e(for)i (whic)m(h)e(lo)m(w)m(er)i(b)s(ounds)e(are)i(kno)m(wn)f(in)f(other)i (systems)f(and)h(b)s(ecause)f(the)456 5216 y(basic)26 b(tec)m(hniques)h(for)g(dealing)f(with)f(these)j(form)m(ulas)e (fundamen)m(tally)g(break)h(do)m(wn.)p eop %%Page: 15 15 15 14 bop 456 425 a Fw(There)34 b(are)h(p)s(olynomial-size)d Fx(TC)1645 388 y Fs(0)1684 425 y Fw(-F)-8 b(rege)37 b(pro)s(ofs)d(of)g (all)g(of)h(the)f(symmetric)g(coun)m(t-)456 533 y(ing)h(principles)e([) p Fr(29)q Fw(])k(as)f(w)m(ell)g(as)h(the)f(o)s(dd-c)m(harged)h(graph)f (principles)d([)p Fr(90)q Fw(].)59 b(And,)456 644 y(as)37 b(men)m(tioned)g(ab)s(o)m(v)m(e,)j(the)d(in)m(terp)s(olation)e(metho)s (d)i(cannot)h(apply)d(to)j Fx(TC)3161 607 y Fs(0)3200 644 y Fw(-F)-8 b(rege)456 752 y(assuming)28 b(that)i(factoring)g(Blum)f (in)m(tegers)h(is)e(hard)h([)p Fr(21)q Fw(].)41 b(Since)28 b Fx(TC)2888 715 y Fs(0)2928 752 y Fw(-F)-8 b(rege)31 b(pro)s(ofs)456 860 y(are)j(sp)s(ecial)f(cases)h(of)h(F)-8 b(rege)35 b(pro)s(ofs,)f(the)h(same)f(problems)e(apply)h(to)i(F)-8 b(rege)35 b(pro)s(ofs)456 968 y(as)f(w)m(ell.)51 b(Some)35 b(candidates)e(for)h(hard)g(tautologies)h(for)f(these)g(systems)h(ha)m (v)m(e)g(b)s(een)456 1076 y(suggested)c(in)e([)p Fr(18)q Fw(].)605 1255 y Fr(3.7.)53 b(Optimal)28 b(pro)s(of)i(systems.)45 b Fw(Researc)m(h)26 b(in)f(pro)s(of)g(complexit)m(y)g(w)m(as)h(orig-) 456 1363 y(inally)34 b(motiv)-5 b(ated)37 b(in)e(part)h(as)h(a)g(w)m(a) m(y)g(of)g(pro)m(ving)f Fx(NP)g Fl(6)p Fw(=)f Fx(co-NP)q Fw(,)k(b)m(y)d(pro)m(ving)g(su-)456 1471 y(p)s(erp)s(olynomial)g(lo)m (w)m(er)41 b(b)s(ounds)d(for)j(increasingly)d(p)s(o)m(w)m(erful)h(pro)s (of)h(systems.)71 b(An)456 1579 y(imp)s(ortan)m(t)27 b(question)g(is)g(whether)h(suc)m(h)g(a)g(c)m(hain)g(of)g(results)f (will)e(ev)m(er)k(actually)f(lead)456 1687 y(us)38 b(to)i(a)g(pro)s(of) f(of)g Fx(NP)i Fl(6)p Fw(=)f Fx(co-NP)r Fw(.)68 b(In)38 b(other)i(w)m(ords,)h(is)e(there)g(an)h(optimal)e(pro)s(of)456 1795 y(system?)45 b(This)30 b(question)h(is)g(quite)h(imp)s(ortan)m(t)f (and)g(is)g(still)f(op)s(en.)45 b(Ho)m(w)m(ev)m(er,)35 b(some)456 1903 y(partial)25 b(results)g(ha)m(v)m(e)j(b)s(een)d (obtained)h([)p Fr(66)q(,)k(60)q(,)g(16)q Fw(])c(relating)g(this)f (existence)i(to)g(the)456 2011 y(equiv)-5 b(alence)29 b(of)i(certain)f(complexit)m(y)g(classes.)605 2191 y Fr(3.8.)53 b(Pro)s(of)43 b(Searc)m(h.)j Fw(While)36 b(lengths)h(of)g (pro)s(ofs)f(are)h(imp)s(ortan)m(t,)i(it)d(is)g(also)456 2299 y(imp)s(ortan)m(t)42 b(to)i(b)s(e)f(able)g(to)h Fn(\014nd)g Fw(pro)s(ofs)e(quic)m(kly)-8 b(.)79 b(Clearly)-8 b(,)46 b(if)c(w)m(e)i(kno)m(w)f(that)h(a)456 2407 y(pro)s(of)27 b(system)h Fm(S)34 b Fw(is)27 b(not)h(p)s(olynomially-b)s(ounded,)c (then)k(no)g(e\016cien)m(t)g(deterministic)456 2515 y(pro)s(cedure)h (can)h(exist)g(that)g(will)e(pro)s(duce)h(short)g(pro)s(ofs)h(of)g(all) f(tautologies.)41 b(But)30 b(is)456 2623 y(it)f(p)s(ossible)f(to)j (\014nd)d(short)i(pro)s(ofs)f(of)i(all)e(tautologies)i(that)f(ha)m(v)m (e)i(short)d(pro)s(ofs?)40 b(T)-8 b(o)456 2730 y(this)29 b(end,)i([)p Fr(21)q Fw(])f(de\014nes)g(a)i(pro)s(of)e(system)g Fm(S)36 b Fw(to)c(b)s(e)e Fn(automatizable)j Fw(if)d(there)h(exists)f (a)456 2838 y(deterministic)g(algorithm)i(that)i(tak)m(es)g(as)f(input) e(a)j(tautology)g Fm(F)13 b Fw(,)34 b(and)e(outputs)g(an)456 2946 y Fm(S)5 b Fw(-pro)s(of)30 b(of)g Fm(F)43 b Fw(in)30 b(time)g(p)s(olynomial)d(in)i(the)i(size)f(of)h(the)f(shortest)h Fm(S)5 b Fw(-pro)s(of)30 b(of)g Fm(F)13 b Fw(.)605 3054 y(Automatizabilit)m(y)25 b(is)g(v)m(ery)i(imp)s(ortan)m(t)e(for)h (automated)h(theorem)g(pro)m(ving,)f(and)456 3162 y(is)f(v)m(ery)i (similar)e(to)i(an)g(older)f(concept,)i Fm(k)s Fw(-pro)m(v)-5 b(abilit)m(y)d(.)39 b(The)26 b Fm(k)s Fw(-pro)m(v)-5 b(abilit)m(y)26 b(problem)456 3270 y(for)k(a)g(pro)s(of)f(system)i Fm(S)k Fw(is)29 b(as)h(follo)m(ws.)40 b(The)30 b(input)e(is)h(a)i(form) m(ula)e Fm(F)13 b Fw(,)31 b(and)e(a)i(n)m(um)m(b)s(er)456 3378 y Fm(k)s Fw(,)26 b(and)e(the)h(input)d(should)h(b)s(e)h(accepted)i (if)d(and)h(only)g(if)g Fm(F)37 b Fw(has)25 b(an)f Fm(S)5 b Fw(-pro)s(of)24 b(of)h(size)g(at)456 3486 y(most)33 b Fm(k)s Fw(.)49 b(Clearly)32 b(a)i(pro)s(of)e(system)h Fm(S)39 b 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(that)g(\014nding)d(go)s(o)s(d)j(estimates)g(of)g(the)g(pro)s(of)456 4349 y(length)31 b(is)g(hard,)h(ev)m(en)h(for)f(v)m(ery)g(simple)e(pro) s(of)h(systems)h(suc)m(h)g(as)g(Resolution.)45 b(But)456 4457 y(it)30 b(ma)m(y)i(still)d(b)s(e)h(that)h(most)h(pro)s(of)e (systems)h(are)g(automatizable.)43 b(Ho)m(w)m(ev)m(er,)34 b(under)456 4565 y(stronger)j(assumptions,)g(one)g(can)h(sho)m(w)e (that)i(man)m(y)f(pro)s(of)f(systems)h(are)h(not)f(au-)456 4673 y(tomatizable.)63 b(These)37 b(results)f(are)i(sho)m(wn)f(b)m(y)h (exploiting)e(a)i(connection)g(b)s(et)m(w)m(een)456 4781 y(in)m(terp)s(olation)30 b(and)h(automatizabilit)m(y)-8 b(.)45 b(In)31 b(particular,)g(it)h(can)g(b)s(e)f(sho)m(wn)h(that)g(if) f(a)456 4889 y(pro)s(of)24 b(system)i Fm(S)31 b Fw(do)s(es)25 b(not)h(ha)m(v)m(e)h(feasible)d(in)m(terp)s(olation,)h(then)h(this)e (implies)f(that)j Fm(S)456 4997 y Fw(is)h(not)i(automatizable.)41 b(Th)m(us,)28 b(feasible)g(in)m(terp)s(olation)f(giv)m(es)i(us)f(a)i (formal)e(tradeo\013)456 5105 y(b)s(et)m(w)m(een)37 b(the)f(complexit)m (y/strength)h(of)g Fm(S)k Fw(and)36 b(the)h(abilit)m(y)e(to)i Fn(\014nd)g Fw(short)f(pro)s(ofs)456 5216 y(quic)m(kly)-8 b(.)42 b(Using)30 b(this)g(connection,)i(it)f(has)g(b)s(een)f(sho)m(wn) g(that)i Fx(A)m(C)2774 5179 y Fs(0)2814 5216 y Fw(-F)-8 b(rege)33 b(pro)s(ofs)d(as)p eop %%Page: 16 16 16 15 bop 456 425 a Fw(w)m(ell)32 b(as)i(an)m(y)h(pro)s(of)e(system)h (that)g(can)g(p-sim)m(ulate)f Fx(A)m(C)2412 388 y Fs(0)2451 425 y Fw(-F)-8 b(rege,)37 b(is)c(not)h(automatiz-)456 533 y(able,)c(under)f(cryptographic)g(assumptions.)39 b(\(See)31 b([)p Fr(61)q(,)k(21)q(,)g(64,)g(19)p Fw(].\))605 641 y(Are)28 b(there)g(an)m(y)f(pro)s(of)g(systems)h(that)g(are)g (automatizable?)40 b(Both)28 b(the)g(Nullstel-)456 749 y(lensatz)i(and)g(P)m(olynomial)f(Calculus)f(pro)s(of)i(systems)g(as)h (w)m(ell)e(as)i(DLL)f(are)h(actually)456 857 y(searc)m(h)e(pro)s (cedures)f(as)h(w)m(ell)e(as)i(nondeterministic)d(algorithms.)39 b(But,)30 b(unlik)m(e)d(DLL,)456 965 y(the)36 b(Nullstellensatz)f(and)h (P)m(olynomial)g(Calculus)e(algorithms)i(are)h(guaran)m(teed)h(to)456 1073 y(\014nd)22 b(short)i(pro)s(ofs)f(if)f(they)i(exist)g([)p Fr(13)q(,)j(33)q Fw(];)f(that)f(is,)f(they)h(are)f(automatizable.)39 b(\(An)m(y)456 1181 y(pro)s(of)33 b(of)h(degree)h Fm(d)f Fw(in)f Fm(n)g Fw(v)-5 b(ariables)33 b(ma)m(y)i(b)s(e)e(found)g(using)f (linear)h(algebra)h(in)f(time)456 1289 y Fm(n)511 1256 y Fk(O)r Fs(\()p Fk(d)p Fs(\))661 1289 y Fw(.\))47 b(Nullstellensatz)31 b(pro)s(ofs)g(ma)m(y)j(b)s(e)d(exp)s(onen)m(tially)g(smaller)g(than)h (b)s(ounded-)456 1397 y(depth)c(F)-8 b(rege)31 b(pro)s(ofs,)e(but)g (they)g(also)g(ma)m(y)h(b)s(e)f(exp)s(onen)m(tially)f(larger)h(than)g (Resolu-)456 1505 y(tion)21 b(pro)s(ofs)h([)p Fr(33)q Fw(].)38 b(P)m(olynomial)21 b(Calculus)f(pro)s(ofs)h(are)i(at)g(w)m (orst)f(quasi-p)s(olynomially)456 1613 y(larger)27 b(than)g(the)h(b)s (est)f(pro)s(ofs)g(under)f(an)m(y)i(DLL)g(algorithm)e(and)h(from)h (this)e(one)i(can)456 1721 y(deriv)m(e)39 b(a)i(metho)s(d)f(of)g(searc) m(hing)g(for)g(short)g(DLL)g(pro)s(ofs)f(that)i(is)e(guaran)m(teed)j (to)456 1829 y(succeed)34 b([)p Fr(33)q Fw(].)52 b(In)34 b([)p Fr(10)q Fw(],)h(this)e(algorithm)g(is)h(con)m(v)m(erted)h(to)g(a) g(direct)e(searc)m(h)i(pro)s(ce-)456 1937 y(dure)29 b(for)i(suc)m(h)g (pro)s(ofs.)41 b(It)31 b(app)s(ears)f(fruitful)e(to)k(in)m(v)m (estigate)g(P)m(olynomial)e(Calculus)456 2045 y(pro)s(ofs)36 b(as)h(a)g(theorem-pro)m(ving)g(to)s(ol)g(to)g(see)h(if)e(they)h(can)g (b)s(e)f(re\014ned)g(to)i(comp)s(ete)456 2153 y(with)29 b(DLL)h(algorithms.)40 b(Preliminary)27 b(results)i(of)i(this)e(form)h (app)s(ear)g(in)f([)p Fr(33)p Fw(].)1528 2355 y Fr(4.)53 b(Op)s(en)34 b(Problems)456 2517 y Fa(Find)k(hard)g(t)-6 b(a)n(utologies)36 b(f)n(or)i Fx(TC)1868 2481 y Fs(0)1907 2517 y Fa(-Frege)h(and)e(Frege.)52 b Fw(No)35 b(examples)f(of)456 2625 y(tautologies)42 b(are)h(kno)m(wn)e(for)h(whic)m(h)e(F)-8 b(rege)44 b(pro)s(ofs)d(ev)m(en)h(require)f(a)h(sup)s(er-linear)456 2733 y(n)m(um)m(b)s(er)35 b(of)i(distinct)e(subform)m(ulas.)57 b(One)36 b(di\016cult)f(problem)f(that)k(is)d(faced)i(when)456 2841 y(trying)e(to)h(pro)m(v)m(e)h(lo)m(w)m(er)f(b)s(ounds)e(for)i(F)-8 b(rege)37 b(or)f(Extended)f(F)-8 b(rege)38 b(systems)e(is)f(that)456 2949 y(there)f(is)g(a)h(surprising)c(lac)m(k)k(of)f(hard)g(candidate)g (tautologies.)54 b(Most)36 b(of)e(the)h(lo)m(w)m(er)456 3057 y(b)s(ounds)42 b(pro)m(v)m(en)i(th)m(us)g(far)h(ha)m(v)m(e)g(b)s (een)f(for)g(v)-5 b(arious)43 b(coun)m(ting)h(principles,)h(all)e(of) 456 3165 y(whic)m(h)28 b(ha)m(v)m(e)j(p)s(olynomial-size)c Fx(TC)1684 3128 y Fs(0)1724 3165 y Fw(-F)-8 b(rege)31 b(or)f(F)-8 b(rege)31 b(pro)s(ofs.)39 b(\(Ev)m(en)30 b(the)g(min)m(term-)456 3273 y(maxterm)f(form)m(ulas)e(can)i(b)s(e)g (view)m(ed)f(as)h(an)f(application)g(of)g(a)i(coun)m(ting)e (principle.\))605 3381 y(Some)40 b(candidate)g(hard)g(examples)g(ha)m (v)m(e)h(b)s(een)f(suggested)h(in)e([)p Fr(18)p Fw(])i(including)456 3489 y(random)32 b Fm(k)s Fw(-CNF)i(form)m(ulas.)49 b(Another)33 b(family)f(of)h(examples)g(that)h(is)f(of)g(particular)456 3597 y(in)m(terest)43 b(for)g(complexit)m(y)g(theory)-8 b(,)47 b(are)d(at)g(least)f(as)g(hard)g(to)g(pro)m(v)m(e)h(as)g(man)m (y)f(of)456 3705 y(the)33 b(classes)g(of)g(form)m(ulas)g(discussed)e (in)h(this)g(pap)s(er,)h(but)f(are)i(only)e(b)s(eliev)m(ed)g(to)i(b)s (e)456 3812 y(tautological)28 b(are)g(particular)e(form)m(ulas)h(giv)m (en)g(b)m(y)h(Razb)s(oro)m(v)g([)p Fr(79)q Fw(])g(stating)g(that)g Fx(NP)456 3920 y Fw(is)h(not)i(con)m(tained)f(in)f Fx(P/p)s(oly)p Fw(.)605 4028 y(There)22 b(is)g(an)h(ev)m(en)h(larger)e(gap)h(in)f (tautologies)h(that)h(seem)f(to)h(separate)f(extended)456 4136 y(F)-8 b(rege)32 b(from)e(F)-8 b(rege)32 b(systems.)41 b(In)29 b(fact,)j(w)m(e)f(kno)m(w)g(of)f(no)h(con)m(vincing)e(com)m (binatorial)456 4244 y(tautologies)k(that)g(migh)m(t)f(ha)m(v)m(e)h(p)s (olynomial-size)d(extended)i(F)-8 b(rege)34 b(pro)s(ofs,)e(but)g(re-) 456 4352 y(quire)39 b(exp)s(onen)m(tial-size)h(F)-8 b(rege)42 b(pro)s(ofs.)70 b(In)40 b([)p Fr(18)q Fw(],)j(sev)m(eral)e(tautologies) g(based)f(on)456 4460 y(linear)34 b(algebra)h(are)h(suggested)g(to)h (giv)m(e)f(a)f(quasip)s(olynomial)d(separation)k(b)s(et)m(w)m(een)456 4568 y(extended)31 b(F)-8 b(rege)34 b(and)d(F)-8 b(rege)33 b(systems.)45 b(A)32 b(v)m(ery)g(simple)e(suc)m(h)i(example,)g (suggested)456 4676 y(b)m(y)27 b(Co)s(ok)h(and)f(Rac)m(k)m(o\013,)j(is) d(the)g(prop)s(ositional)e(form)i(of)h(the)g(Bo)s(olean)g(matrix)e(pro) s(d-)456 4784 y(uct)k(implication)e Fm(AB)i Fw(=)25 b Fm(I)32 b Fl(\))25 b Fm(B)5 b(A)25 b Fw(=)g Fm(I)7 b Fw(.)456 4892 y Fa(Ho)n(w)48 b(hard)h(are)f(random)g Fm(k)s Fa(-CNF)i(f)n(ormulas?)80 b Fw(The)44 b(only)g(lo)m(w)m(er)h(b)s (ounds)456 5000 y(kno)m(wn)23 b(for)h(unsatis\014abilit)m(y)d(pro)s (ofs)i(of)h(random)f(form)m(ulas)g(are)h(for)g(forms)g(of)g(Resolu-)456 5108 y(tion.)37 b(What)23 b(ab)s(out)f(other)g(pro)s(of)g(systems)g (suc)m(h)g(as:)37 b(b)s(ounded-depth)20 b(F)-8 b(rege)23 b(pro)s(ofs?)456 5216 y(cutting)34 b(planes)g(pro)s(ofs?)53 b(Nullstellensatz)33 b(or)i(p)s(olynomial)d(calculus)h(pro)s(ofs?)53 b(The)p eop %%Page: 17 17 17 16 bop 456 425 a Fw(absence)28 b(of)f(random)g(unsatis\014able)e (form)m(ulas)i(in)f(the)i(list)e(of)i(lo)m(w)m(er)g(b)s(ounds)d(for)i (sys-)456 533 y(tems)39 b(other)h(than)f(Resolution)g(is)f(quite)h (notew)m(orth)m(y)-8 b(,)44 b(esp)s(ecially)38 b(giv)m(en)h(the)h(lac)m (k)456 641 y(of)45 b(go)s(o)s(d)h(upp)s(er)d(b)s(ounds)h(for)h(pro)s (ofs)g(of)h(these)g(form)m(ulas)f(in)f(an)m(y)i(system,)k(ev)m(en)456 749 y(extended-F)-8 b(rege.)605 857 y(Man)m(y)25 b Fx(NP)p Fw(-complete)g(graph)f(problems)e(are)j(easy)g(on)f(the)g(a)m(v)m (erage)j(for)d(the)g(natu-)456 965 y(ral)g(random)h(graph)g(probabilit) m(y)d(distributions.)36 b(Random)25 b Fm(k)s Fw(-CNF)g(form)m(ulas)g (under)456 1073 y(the)j(analogous)h(probabilit)m(y)d(distributions)e (seem)29 b(surprisingly)24 b(hard)j(in)g(the)i(region)456 1181 y(of)39 b(probabilities)d(for)k(whic)m(h)e(the)i(form)m(ulas)e (are)i(lik)m(ely)e(unsatis\014able)g([)p Fr(32)q(,)45 b(9)p Fw(].)69 b(Is)456 1289 y Fm(k)s Fw(-UNSA)-8 b(T)31 b(hard)e(on)h(the)h(a)m(v)m(erage)i(in)c(this)g(sense?)605 1397 y(F)-8 b(or)32 b(example,)g(the)g(b)s(est)f(upp)s(er)f(b)s(ound)g (for)h(an)m(y)h(searc)m(h)g(algorithm)f(for)g(unsat-)456 1511 y(is\014abilit)m(y)25 b(pro)s(ofs)j(of)h(random)f Fm(m)p Fw(-clause)g Fm(n)p Fw(-v)-5 b(ariable)27 b(3-CNF)j(form)m(ulas) e(is)f(2)3154 1478 y Fk(O)r Fs(\()p Fk(n)3280 1455 y Fq(2)3315 1478 y Fk(=m)p Fs(\))456 1619 y Fw(with)46 b(probabilit)m(y)f(1)33 b Fl(\000)e Fm(o)p Fw(\(1\))49 b(in)e Fm(n)g Fw(and)g(this)g(is)g(tigh)m(t)h(for)g(a)g(class)g(of)g (DLL)f(al-)456 1727 y(gorithms)40 b([)p Fr(9)q Fw(].)74 b(F)-8 b(or)43 b(Resolution)d(the)i(lo)m(w)m(er)f(b)s(ound)f(for)h (this)g(problem)f(is)g(nearly)456 1846 y(2)501 1813 y Fs(\012\()p Fk(n)622 1790 y Fq(5)657 1813 y Fk(=m)754 1790 y Fq(4)789 1813 y Fs(\))851 1846 y Fw(with)29 b(probabilit)m(y)f (1)21 b Fl(\000)f Fm(O)s Fw(\(1\))31 b(in)e Fm(n)h Fw([)p Fr(9)p Fw(].)41 b(Can)30 b(these)h(b)s(e)f(impro)m(v)m(ed?)456 1957 y Fa(Superpol)-6 b(ynomial)24 b(lo)n(wer)i(bounds)g(f)n(or)h Fx(A)m(C)2229 1920 y Fs(0)2269 1957 y Fw([)p Fm(r)s Fw(])g Fa(Frege?.)37 b Fw(Studying)22 b Fx(A)m(C)3281 1920 y Fs(0)3320 1957 y Fw([)p Fm(r)s Fw(]-)456 2065 y(F)-8 b(rege)45 b(is)e(a)i(natural)e(next)h(step)h(in)d(pro)m(ving)i(lo)m(w)m (er)g(b)s(ounds)e(for)i(pro)s(of)f(systems,)456 2173 y(in)c(particular)g(when)g Fm(r)k Fw(is)d(a)h(prime.)70 b(W)-8 b(e)41 b(already)f(men)m(tioned)g(the)h(program)g(for)456 2281 y(obtaining)30 b(suc)m(h)i(lo)m(w)m(er)g(b)s(ounds)e(in)g([)p Fr(23)q Fw(].)46 b(F)-8 b(or)33 b(this)d(system)j(w)m(e)f(do)g(ha)m(v)m (e)h(a)f(natural)456 2389 y(candidate)43 b(for)h(a)h(hard)e(tautology) -8 b(,)49 b(namely)43 b Fm(C)7 b(ount)2378 2356 y Fk(n)2378 2411 y(p)2468 2389 y Fw(for)44 b(prime)e Fm(p)48 b Fl(6)p Fw(=)g Fm(r)s Fw(.)81 b(Suc)m(h)456 2501 y(a)41 b(lo)m(w)m(er)h(b)s (ound)d(w)m(ould)i(further)f(the)h(circuit/pro)s(of)f(system)i(corresp) s(ondence)e(b)m(y)456 2609 y(extending)i(pro)s(of)g(complexit)m(y)g(lo) m(w)m(er)h(b)s(ounds)e(to)i(the)g(natural)f(analogue)h(of)g(the)456 2717 y(Razb)s(oro)m(v-Smolensky)30 b(circuit)f(lo)m(w)m(er)h(b)s(ounds) f([)p Fr(77,)35 b(87)q Fw(].)456 2825 y Fa(Pol)-6 b(ynomial)36 b(Calculus)f(in)j(a)f(theorem)e(pr)n(o)n(ver?)52 b(Better)35 b(Resolution)456 2932 y(pr)n(oof)j(sear)n(ch?)53 b Fw(Designing)34 b(e\016cien)m(t)i(theorem)g(pro)m(v)m(ers)f(for)g(the)h(prop)s (ositional)456 3040 y(calculus)f(is)h(an)h(imp)s(ortan)m(t)f(practical) h(question.)59 b(T)-8 b(o)38 b(date,)h(DLL)e(algorithms)f(are)456 3148 y(the)26 b(c)m(hampion)f(theorem)h(pro)m(v)m(ers)g(although)f (they)h(are)g(theoretically)f(quite)g(w)m(eak)i(as)456 3256 y(pro)s(of)22 b(systems.)38 b(A)23 b(recen)m(t)h(c)m(hallenger)e (seems)h(promising:)35 b(A)23 b(v)-5 b(arian)m(t)22 b(of)h(the)g(Gro)s (eb-)456 3364 y(ner)33 b(basis)h(algorithm)f(has)h(b)s(een)g(used)f(to) i(\014nd)e(P)m(olynomial)g(Calculus)f(pro)s(ofs)h([)p Fr(33)q Fw(])456 3472 y(and)h(build)e(a)k(fairly)e(e\016cien)m(t)h (theorem)h(pro)m(v)m(er.)56 b(Can)34 b(this)h(b)s(e)f(tuned)h(to)h (comp)s(ete)456 3580 y(with)29 b(DLL)h(algorithms?)605 3688 y(Clegg)41 b Fn(et)h(al.)f Fw([)p Fr(33)q Fw(])g(also)g(giv)m(e)g (sim)m(ulations)e(of)i(DLL)g(and)f(Resolution)f(b)m(y)i(the)456 3796 y(P)m(olynomial)20 b(Calculus.)36 b(One)21 b(metho)s(d)g(of)h (impro)m(ving)e(the)j(comp)s(etitiv)m(eness)e(of)h(P)m(oly-)456 3904 y(nomial)i(Calculus)f(as)j(a)g(theorem)g(pro)m(v)m(er)g(w)m(ould)e (b)s(e)h(to)i(impro)m(v)m(e)e(these)h(sim)m(ulations.)456 4012 y(This)c(is)i(also)g(related)h(to)g(the)g(question)f(of)h(impro)m (ving)d(the)j(more)g(direct)f(metho)s(ds)g(for)456 4120 y(pro)s(of)34 b(searc)m(h)i(for)f(Resolution)g(and)g(DLL)g([)p Fr(10)q(,)40 b(17)q Fw(])c(that)g(w)m(ere)f(inspired)e(b)m(y)i(these) 456 4228 y(P)m(olynomial)29 b(Calculus)f(sim)m(ulations.)456 4336 y Fa(New)36 b(pr)n(oof)f(systems)g(fr)n(om)h Fx(NP)p Fa(-complete)f(pr)n(oblems.)45 b Fw(An)32 b(app)s(ealing)f(di-)456 4443 y(rection)g(in)f(pro)s(of)h(complexit)m(y)h(is)e(the)i(p)s (ossibilit)m(y)c(of)k(using)e(natural)g(domains)h(that)456 4551 y(con)m(tain)39 b(NP-hard)e(problems)g(to)j(seek)f(out)g(new)f (and)g(in)m(teresting)g(pro)s(of)f(systems)456 4659 y(whic)m(h)c (reason)h(ab)s(out)g(ob)5 b(jects)35 b(from)f(radically)e(di\013eren)m (t)i(domains)f(from)h(Bo)s(olean)456 4767 y(form)m(ulas.)57 b(Cutting)35 b(planes)g(come)i(from)f(in)m(teger)g(programming,)h (Nullstellensatz)456 4875 y(and)f(P)m(olynomial)f(Calculus)g(systems)h (come)i(from)e(systems)h(of)g(p)s(olynomial)d(equa-)456 4983 y(tions.)59 b(Pitassi)36 b(and)g(Urquhart)g([)p Fr(72)q Fw(])h(considered)f(the)h(Ha)5 b(jos)38 b(calculus)d(for)i (non-3-)456 5091 y(colorabilit)m(y)h(whic)m(h)h(they)i(found,)g (surprisingly)-8 b(,)39 b(to)i(b)s(e)f(equiv)-5 b(alen)m(t)39 b(to)i(extended-)456 5199 y(F)-8 b(rege)32 b(pro)s(ofs.)39 b(It)31 b(is)e(lik)m(ely)g(that)i(there)g(is)e(more)i(to)g(b)s(e)f (mined)e(in)h(this)h(searc)m(h.)p eop %%Page: 18 18 18 17 bop 456 425 a Fa(Weak)42 b(pigeonhole)h(principle)g(and)h(the)f (limits)h(of)f(Resolution.)67 b Fw(It)39 b(is)456 533 y(kno)m(wn)e(that)i(for)f Fm(n)g(<)f(m)h(<)g(n)1586 500 y Fs(2)1625 533 y Fm(=)15 b Fw(log)i Fm(n)p Fw(,)40 b Fm(P)13 b(H)7 b(P)2163 500 y Fk(m)2150 556 y(n)2268 533 y Fw(requires)36 b(sup)s(erp)s(olynomial-size)456 641 y(Resolution)20 b(pro)s(ofs)g([)p Fr(51)q(,)25 b(27)p Fw(].)38 b(Originally)18 b(it)j(w)m(as)h(conjectured)g(that)g(for)f(an) m(y)h Fm(m)j(>)g(n)p Fw(,)456 749 y(an)m(y)41 b(Resolution)e (refutation)h(of)h Fm(P)13 b(H)7 b(P)1870 716 y Fk(m)1857 771 y(n)1977 749 y Fw(w)m(ould)39 b(require)g(size)i(exp)s(onen)m(tial) e(in)h Fm(n)p Fw(.)456 857 y(Ho)m(w)m(ev)m(er,)45 b(this)40 b(conjecture)h(w)m(as)h(sho)m(wn)e(to)h(b)s(e)f(false)h(for)f(large)h (enough)g Fm(m)f Fw([)p Fr(24)q Fw(].)456 965 y(Moreo)m(v)m(er,)47 b(it)41 b(app)s(ears)g(that)h(the)g(standard)f(metho)s(d,)j(the)e(b)s (ottlenec)m(k)g(coun)m(ting)456 1073 y(tec)m(hnique,)30 b(cannot)h(b)s(e)f(applied)f(to)i(obtain)f(lo)m(w)m(er)h(b)s(ounds)d (for)i Fm(m)c(>)f(n)2975 1040 y Fs(2)3014 1073 y Fw(.)41 b(An)30 b(in)m(ter-)456 1181 y(esting)g(op)s(en)h(problem)e(is)i(to)g (pro)m(v)m(e)h(lo)m(w)m(er)g(b)s(ounds)d(for)h Fm(P)13 b(H)7 b(P)2664 1148 y Fk(m)2651 1203 y(n)2762 1181 y Fw(for)31 b Fm(m)26 b(>)g(n)3160 1148 y Fs(2)3199 1181 y Fw(.)43 b(This)456 1289 y(w)m(ould)38 b(lik)m(ely)h(giv)m(e)h(rise)f (to)h(a)g(new)g(lo)m(w)m(er)g(b)s(ound)e(metho)s(d)h(for)g(Resolution.) 68 b(Ad-)456 1397 y(ditionally)30 b(the)j(complexit)m(y)g(of)h(the)f(w) m(eak)h(pigeonhole)e(principle)e(in)i(v)-5 b(arious)32 b(pro)s(of)456 1504 y(systems)22 b(is)e(in)m(teresting)i(in)e(its)i(o)m (wn)g(righ)m(t:)36 b(it)21 b(is)g(kno)m(wn)h([)p Fr(70)p Fw(])g(that)h(one)f(can)g(pro)m(v)m(e)h(the)456 1612 y(existence)35 b(of)g(in\014nitely)d(man)m(y)j(primes)e(in)h(systems)g (as)i(w)m(eak)f(as)h(p)s(olynomial-size,)456 1720 y(b)s(ounded-depth)29 b(F)-8 b(rege,)34 b(assuming)c(the)i(w)m(eak-pigeonhole)g(principle)c (as)k(an)f(axiom)456 1828 y(sc)m(hema.)40 b(More)29 b(generally)-8 b(,)29 b(the)f(w)m(eak)h(pigeonhole)e(principle)e(can)j(b)s(e)f(used)h (to)g(carry)456 1936 y(out)44 b(most)h(com)m(binatorial)e(coun)m(ting)h (argumen)m(ts,)k(and)c(is)f(closely)h(connected)h(to)456 2044 y(appro)m(ximate)30 b(coun)m(ting.)605 2152 y(P)m(artial)h (results)f(for)h(the)g(w)m(eak)h(pigeonhole)e(principle)e(ha)m(v)m(e)k (recen)m(tly)g(b)s(een)e(ob-)456 2260 y(tained)38 b([)p Fr(81)q Fw(].)67 b(There)39 b(is)f(a)i(v)m(ery)f(tigh)m(t)h(connection) f(b)s(et)m(w)m(een)h(regular)e(Resolution)456 2368 y(refutations)30 b(and)h(read-once)i(branc)m(hing)d(programs)h(whic)m(h)f(generalizes)h (the)h(equiv-)456 2476 y(alence)41 b(b)s(et)m(w)m(een)g(tree)g (resolution)e(and)h(DLL)h(men)m(tioned)f(earlier.)71 b(Let)41 b Fm(F)53 b Fw(b)s(e)40 b(an)456 2584 y(unsatis\014able)33 b(form)m(ula)i(in)f(conjunctiv)m(e)i(normal)f(form.)56 b(The)35 b(searc)m(h)i(problem)d(as-)456 2692 y(so)s(ciated)29 b(with)e Fm(F)42 b Fw(tak)m(es)30 b(as)g(input)c(a)k(truth)e(assignmen) m(t)h Fm(\034)38 b Fw(to)30 b(the)f(underlying)d(v)-5 b(ari-)456 2800 y(ables)27 b(of)h Fm(F)13 b Fw(,)28 b(and)f(outputs)g (a)h(clause)g(in)e Fm(F)41 b Fw(that)28 b(is)f(set)h(to)g(false)g(b)m (y)f Fm(\034)10 b Fw(.)40 b(Kra)5 b(j)-10 b(\023)-35 b(\020)m(\024)-43 b(cek)28 b([)p Fr(59)q Fw(])456 2908 y(has)37 b(sho)m(wn)g(that)h(for)f(an)m(y)h Fm(F)13 b Fw(,)40 b(the)d(minimal)e(size)i(Resolution)g(refutation)g(of)h Fm(F)50 b Fw(is)456 3015 y(essen)m(tially)35 b(equiv)-5 b(alen)m(t)35 b(to)i(the)g(minimal)c(size)j(read-once)h(branc)m(hing)e (program)h(to)456 3123 y(solv)m(e)c(the)g(related)g(searc)m(h)g (problem.)44 b(This)30 b(idea)h(w)m(as)h(exploited)f(in)f([)p Fr(81)q Fw(])i(to)h(obtain)456 3231 y(some)g(restricted)g(lo)m(w)m(er)h (b)s(ounds)d(for)i(Resolution)g(pro)s(ofs)f(of)i(w)m(eak)g(v)m(ersions) f(of)g(the)456 3339 y(pigeonhole)c(principle.)605 3447 y(Another,)h(ev)m(en)g(older,)f(principle)d(for)j(whic)m(h)f(no)h(sup)s (erp)s(olynomial)c(Resolution)456 3555 y(lo)m(w)m(er)32 b(b)s(ounds)e(are)j(kno)m(wn)f(is)g(the)g(domino-tiling)e(principle.)43 b(More)33 b(details)f(ab)s(out)456 3663 y(this)d(old)g(problem)g(can)i (b)s(e)f(found)f(in)g([)p Fr(89)p Fw(].)456 3771 y Fa(Lo)n(v)-8 b(asz-Schrijver)42 b(pr)n(oof)j(systems.)71 b Fw(A)41 b(v)-5 b(ariet)m(y)42 b(of)f(inference)f(systems)i(for)456 3879 y(01-programming)29 b(are)g(describ)s(ed)e(b)m(y)i(Lo)m(v)-5 b(asz)31 b(and)d(Sc)m(hrijv)m(er)g([)p Fr(63)q Fw(])h(in)f(a)i(pap)s (er)e(that)456 3987 y(is)f(primarily)e(concerned)k(with)e(their)h (implications)d(for)k(linear)e(programming.)38 b(Lik)m(e)456 4095 y(cutting)i(planes)f(these)i(pro)s(of)f(systems)g(represen)m(t)h (statemen)m(ts)h(using)d(systems)h(of)456 4203 y(linear)34 b(inequalities,)i(but)g(unlik)m(e)e(cutting)i(planes,)h(they)g(replace) f(rounding)e(b)m(y)i(an)456 4311 y(abilit)m(y)24 b(to)j(tak)m(e)h (linear)c(com)m(binations)i(of)g(deriv)m(ed)f(degree)i(2)g(terms)f(in)f (in)m(termediate)456 4419 y(steps)36 b(obtained)f(b)m(y)h(m)m (ultiplying)d(certain)j(inequalities)e(and)i(including)c(the)37 b(equa-)456 4526 y(tions)28 b Fm(x)729 4493 y Fs(2)794 4526 y Fw(=)d Fm(x)k Fw(pro)m(vided)f(all)g(degree)i(2)f(terms)g (cancel.)41 b(\(There)29 b(are)g(sev)m(eral)h(v)m(ersions)456 4634 y(dep)s(ending)25 b(on)k(whether)e(one)i(can)g(\(1\))g(add)f(the)g (squares)g(of)h(arbitrary)e(linear)f(terms)456 4742 y(to)41 b(the)g(appropriate)f(side)g(of)h(the)g(inequalities)d(or)j(\(2\))h(m)m (ultiply)c(inequalities)g(b)m(y)456 4850 y Fm(x)f Fw(or)h(1)26 b Fl(\000)f Fm(x)38 b Fw(or,)i(more)e(generally)-8 b(,)40 b(\(3\))f(m)m(ultiply)c(b)m(y)j(an)m(y)g(linear)f(term)h(previously)456 4958 y(sho)m(wn)26 b(to)i(b)s(e)f(p)s(ositiv)m(e.\))39 b(Lo)m(v)-5 b(asz)28 b(and)f(Sc)m(hrijv)m(er)f(pro)m(v)m(e)i(a)f(n)m (um)m(b)s(er)f(of)i(prop)s(erties)d(of)456 5066 y(their)30 b(systems)h(that)h(allo)m(w)f(one)g(to)h(precisely)e(determine)g(the)i (depth)e(of)i(the)f(pro)s(ofs)456 5174 y(in)m(v)m(olv)m(ed)f(but)f (they)i(do)f(not)h(consider)e(the)i(issue)e(of)h(pro)s(of)g(size)g (directly)-8 b(.)p eop %%Page: 19 19 19 18 bop 605 425 a Fw(It)29 b(turns)f(out)h(that,)h(despite)e(the)h (absence)h(of)f(the)g(rounded)f(division)d(of)30 b(cutting)456 533 y(planes,)22 b(one)h(can)f(easily)f(sim)m(ulate)g(Resolution)g(and) g(pro)m(v)m(e)i(the)f(pigeonhole)f(principle)456 641 y(in)h(p)s(olynomial)e(size)k(in)e(their)g(w)m(eak)m(est)k(system.)38 b(\(T)-8 b(o)25 b(see)f(an)f(example)g(of)h(the)f(system)456 749 y(in)31 b(action)j(consider)e(that)h(if)f(1)23 b Fl(\000)e Fm(a)h Fl(\000)g Fm(b)30 b Fl(\025)f Fw(0,)35 b(1)22 b Fl(\000)g Fm(a)g Fl(\000)f Fm(c)30 b Fl(\025)g Fw(0,)k(1)23 b Fl(\000)e Fm(b)h Fl(\000)g Fm(c)30 b Fl(\025)f Fw(0)34 b(holds)456 857 y(for)f(0)e Fl(\024)f Fm(a)h Fl(\024)f Fw(1,)35 b(0)c Fl(\024)g Fm(b)f Fl(\024)h Fw(1,)k(0)c Fl(\024)f Fm(c)h Fl(\024)g Fw(1)j(then)f(at)h(most)g(one)g(of)g Fm(a;)15 b(b;)g(c)34 b Fw(is)f(1)h(and)f(so)456 965 y Fm(a)10 b Fw(+)g Fm(b)g Fw(+)g Fm(c)27 b Fl(\024)d Fw(1.)40 b(This)24 b(is)g(obtained)h(b)m(y)g(computing)g(0)h Fl(\024)f Fm(a)p Fw(\(1)10 b Fl(\000)g Fm(a)g Fl(\000)g Fm(b)p Fw(\))28 b(=)d Fm(a)10 b Fl(\000)g Fm(a)3130 932 y Fs(2)3180 965 y Fl(\000)g Fm(ab)26 b Fw(=)456 1073 y Fl(\000)p Fm(ab)p Fw(,)35 b(0)c Fl(\024)g Fm(a)p Fw(\(1)24 b Fl(\000)e Fm(a)h Fl(\000)f Fm(c)p Fw(\))32 b(=)g Fm(a)22 b Fl(\000)h Fm(a)1682 1040 y Fs(2)1744 1073 y Fl(\000)f Fm(ac)32 b Fw(=)f Fl(\000)p Fm(ac)p Fw(,)k(and)f(0)d Fl(\024)h Fw(\(1)23 b Fl(\000)f Fm(a)p Fw(\)\(1)i Fl(\000)f Fm(b)f Fl(\000)h Fm(c)p Fw(\))32 b(=)456 1181 y(1)13 b Fl(\000)g Fm(a)g Fl(\000)g Fm(b)g Fl(\000)g Fm(c)g Fw(+)g Fm(ab)g Fw(+)g Fm(ac)p Fw(.)39 b(Adding)25 b(these)j(inequalities)c(leads)i(to) i(the)f(desired)e(result.\))605 1289 y(Ho)m(w)m(ev)m(er)42 b(other)f(problems)d(are)i(less)g(clear.)70 b(In)39 b(particular,)i(Lo) m(v)-5 b(asz)42 b(\(priv)-5 b(ate)456 1397 y(comm)m(unication\))33 b(suggested)h(the)f(problem)f(of)h(deriving)e(the)i(maxim)m(um)f(indep) s(en-)456 1504 y(den)m(t)22 b(set)g(size)f(of)h(graphs)f(that)i(can)f (b)s(e)f(expressed)g(as)h(the)g(line)e(graphs)h(of)h(o)s(dd)e(cliques.) 456 1612 y(One)25 b(can)g(easily)g(sho)m(w)g(that)h(this)f(problem)e (is)i(completely)g(equiv)-5 b(alen)m(t)25 b(to)h(the)g(parit)m(y)456 1720 y(principle)37 b(and)j(requires)g(linear)f(depth)h(in)g(the)h(n)m (um)m(b)s(er)f(of)h(v)m(ertices,)j(but)c(this)g(is)456 1828 y(far)32 b(from)g(either)g(an)h(algorithm)e(or)i(a)g(lo)m(w)m(er)g (b)s(ound.)45 b(This)31 b(question)h(is)f(in)m(timately)456 1936 y(tied)k(to)i(the)g(question)f(of)g(whether)g(these)h(systems)f (can)h(sim)m(ulate)e(cutting)h(planes)456 2044 y(e\016cien)m(tly)-8 b(.)48 b(Assuming)32 b(that)h(they)g(cannot,)i(it)d(ma)m(y)i(b)s(e)e (ev)m(en)i(more)f(in)m(teresting)g(to)456 2152 y(consider)d(what)h(the) h(com)m(bination)f(of)h(these)g(new)f(systems)g(with)g(cutting)g (planes)f(is)456 2260 y(capable)g(of.)456 2368 y Fa(Odd-char)n(ged)35 b(graphs)h(hard)f(f)n(or)h Fm(AC)2050 2335 y Fs(0)2090 2368 y Fa(-Frege?)46 b Fw(Urquhart)32 b(has)h(suggested)456 2476 y(that)28 b(a)g(study)f(of)h(Tseitin's)e(o)s(dd-c)m(harged)i (graph)f(tautologies)i([)p Fr(88)q Fw(])f(for)f(appropriate)456 2584 y(graphs)22 b([)p Fr(90)q Fw(])i(in)e(b)s(ounded-depth)f(F)-8 b(rege)25 b(systems)f(migh)m(t)f(also)g(lead)h(to)g(lo)m(w)m(er)g(b)s (ounds)456 2692 y(for)j(random)h(form)m(ulas)f(since)g(it)h(app)s(ears) f(v)m(ery)i(di\016cult)d(to)j(apply)d(Ajtai's)j(program)456 2800 y(to)i(them.)40 b(No)31 b(results)e(are)i(kno)m(w)g(for)f(these)h (ev)m(en)g(on)f(depth)f(2)i(F)-8 b(rege)32 b(systems.)456 2908 y Fa(More)h(general)g(lo)n(wer)g(bound)h(technique)e(f)n(or)i (Cutting)h(Planes?)k Fw(The)456 3015 y(in)m(terp)s(olation)28 b(metho)s(d)i(has)h(b)s(een)f(successfully)e(applied)h(to)i(obtain)f (unconditional)456 3123 y(lo)m(w)m(er)d(b)s(ounds)e(for)j(Cutting)e (Planes)h(pro)s(ofs.)38 b(Ho)m(w)m(ev)m(er,)31 b(w)m(e)c(are)h(quite)f (far)g(from)g(un-)456 3231 y(derstanding)32 b(more)i(generally)f(what)h (t)m(yp)s(es)g(of)g(tautologies)h(are)g(hard)e(for)g(Cutting)456 3339 y(Planes.)39 b(In)26 b(particular,)h(are)g(random)g(form)m(ulas)f (hard?)39 b(Other)27 b(families)e(of)i(tautolo-)456 3447 y(gies)d(p)s(ossibly)f(hard)h(for)g(Cutting)g(Planes)g(are)i(the)f(o)s (dd-c)m(harged)g(graph)f(tautologies.)456 3555 y Fa(Exponential)43 b(bounds)h(f)n(or)g Fm(C)7 b(ount)1904 3522 y Fk(n)1904 3578 y(p)1950 3555 y Fa(/onto-)p Fm(P)13 b(H)7 b(P)58 b Fa(in)45 b(Pol)-6 b(ynomial)44 b(Cal-)456 3667 y(culus)32 b Fw(The)h(lo)m(w)m(er)h(b)s(ounds)d(of)j([)p Fr(57)q Fw(])g(are)g(barely)f(sup)s(er-p)s(olynomial)d(and)j(it)g(seems)456 3775 y(unlik)m(ely)24 b(that)j(the)g(metho)s(ds)f(used)f(can)i(pro)s (duce)f(m)m(uc)m(h)g(larger)h(lo)m(w)m(er)g(b)s(ounds.)37 b(The)456 3883 y(lo)m(w)m(er)21 b(b)s(ounds)e(of)j(Razb)s(oro)m(v)g([)p Fr(76)q Fw(])f(in)m(v)m(olv)m(e)h(a)f(direct)g(computation)g(of)h(the)g (lo)m(w)f(degree)456 3991 y(elemen)m(ts)33 b(of)f(the)h(Gro)s(ebner)f (basis)g(of)g(the)h Fm(P)13 b(H)7 b(P)2214 3958 y Fk(m)2201 4013 y(n)2313 3991 y Fw(p)s(olynomials.)45 b(This)31 b(computa-)456 4099 y(tion)d(relies)f(hea)m(vily)h(on)h(the)g(indep)s (endence)d(of)j(the)g(degree)g(one)g Fm(P)13 b(H)7 b(P)2950 4066 y Fk(m)2937 4121 y(n)3045 4099 y Fw(p)s(olynomi-)456 4207 y(als.)49 b(All)32 b(the)h(other)h(coun)m(ting)f(principles)d(ab)s (o)m(v)m(e)35 b(do)e(not)g(ha)m(v)m(e)i(this)d(prop)s(ert)m(y)h(and)456 4315 y(it)26 b(do)s(es)h(not)h(app)s(ear)e(that)i(the)f(same)h(metho)s (d)e(can)i(b)s(e)e(applied)f(to)j(them.)40 b(Is)27 b(there)g(a)456 4422 y(more)32 b(general)g(metho)s(dology)h(that)g(do)s(es)f(not)g (require)g(direct)f(computation)i(of)f(the)456 4530 y(basis?)456 4638 y Fa(Pr)n(obabilisticall)-6 b(y)30 b(checkable)g(algebraic)h(pr)n (oofs.)38 b Fw(The)28 b(Nullstellensatz)456 4746 y(and)d(P)m(olynomial) f(Calculus)f(pro)s(of)i(systems)h(use)f(the)h(dense)g(represen)m (tation)f(of)h(m)m(ul-)456 4854 y(tiv)-5 b(ariate)28 b(p)s(olynomials.)37 b(What)29 b(happ)s(ens)d(if)h(one)i(mo)s(di\014es) d(these)j(pro)s(of)e(systems)i(to)456 4962 y(manipulate)22 b(p)s(olynomials)f(as)j(straigh)m(t-line)f(programs)g(that)i(compute)f (them)f(rather)456 5070 y(than)d(writing)f(them)h(out)h(explicitly?)35 b(One)20 b(di\016cult)m(y)f(is)g(that)j(testing)e(the)h(equalit)m(y)f (of)456 5178 y(p)s(olynomials)27 b(represen)m(ted)i(this)g(w)m(a)m(y)i (is)d(not)i(kno)m(wn)g(to)g(b)s(e)f(e\016cien)m(tly)h(computable)p eop %%Page: 20 20 20 19 bop 456 425 a Fw(deterministically)-8 b(.)37 b(Ho)m(w)m(ev)m(er,) 30 b(there)e(are)g(e\016cien)m(t)g(probabilistic)c(algorithms)j(to)h (do)456 533 y(this)39 b(c)m(hec)m(k)j([)p Fr(84)q Fw(].)71 b(Suc)m(h)39 b(a)i(pro)s(of)f(system)g(w)m(ould)g(lie)f(outside)h(our)g (pro)s(of)f(system)456 641 y(de\014nition)31 b(since)h(the)i(v)m (eri\014cation)f(predicate)g Fm(S)38 b Fw(w)m(ould)33 b(only)f(b)s(e)h(probabilistically)456 749 y(c)m(hec)m(k)-5 b(able.)41 b(See)31 b([)p Fr(71)q Fw(])f(for)h(more)f(details.)456 857 y Fa(Unsa)-6 b(tisfiability)32 b(threshold)e(f)n(or)j(random)f Fm(k)s Fa(-SA)-8 b(T.)30 b Fw(Although)e(this)h(is)f(not)456 965 y(a)43 b(pro)s(of)f(complexit)m(y)g(question)g Fn(p)-5 b(er)45 b(se)d Fw(it)g(is)g(of)h(in)m(terest)g(in)e(understanding)f (the)456 1073 y(pro)s(of)27 b(complexit)m(y)h(of)g(random)f Fm(k)s Fw(-CNF)i(form)m(ulas.)39 b(There)27 b(ha)m(v)m(e)j(b)s(een)d(a) h(n)m(um)m(b)s(er)f(of)456 1181 y(pap)s(ers)f(analyzing)i(the)g (satis\014abilit)m(y)e(prop)s(erties)g(of)i(these)h(form)m(ulas)e(as)h (a)h(function)456 1289 y(of)35 b(their)f(clause-v)-5 b(ariable)34 b(ratios.)55 b(Recen)m(tly)-8 b(,)37 b(it)e(has)g(b)s(een) f(sho)m(wn)h(that)h(there)f(is)f(a)456 1397 y(sharp)23 b(threshold)g(b)s(eha)m(vior)h(for)g(suc)m(h)g(form)m(ulas)g([)p Fr(43)q Fw(])g(but)g(it)g(is)g(not)h(kno)m(wn)f(precisely)456 1504 y(where)39 b(suc)m(h)h(a)g(threshold)f(lies)f(or)i(ev)m(en)h(if)e (it)g(approac)m(hes)i(some)f(\014xed)g(limit.)67 b(In)456 1612 y(general)38 b(it)f(is)g(kno)m(wn)h(that)h(it)e(lies)g(b)s(et)m(w) m(een)i(2)2142 1579 y Fk(k)2185 1612 y Fm(=k)j Fw(and)37 b(2)2548 1579 y Fk(k)2606 1612 y Fw(ln)14 b(2)39 b([)p Fr(32)q(,)k(31)q Fw(])38 b(and)g(for)456 1720 y Fm(k)28 b Fw(=)d(3)j(it)f(is)f(kno)m(wn)h(to)h(lie)e(b)s(et)m(w)m(een)i(3.003)i (and)c(4.598)k([)p Fr(44,)i(56)p Fw(])c(and)f(is)f(conjectured)456 1828 y(to)i(b)s(e)f(around)f(4.2)j([)p Fr(55)q Fw(].)40 b(A)27 b(related)h(question)e(is)h(whether)g(or)g(not)h(the)g (satis\014abilit)m(y)456 1936 y(problem)g(is)i(easy)h(righ)m(t)f(up)f (to)i(the)g(threshold.)456 2044 y Fa(Na)-6 b(tural)22 b(Pr)n(oofs)g(in)i(Pr)n(oof)g(Complexity?)35 b Fw(In)20 b(circuit)f(complexit)m(y)-8 b(,)22 b(Razb)s(oro)m(v)456 2152 y(and)29 b(Rudic)m(h)f([)p Fr(80)q Fw(])i(suggest)g(that,)h(sub)5 b(ject)29 b(to)i(some)f(plausible)d(cryptographic)i(con-)456 2260 y(jectures,)22 b(curren)m(t)e(tec)m(hniques)g(will)e(b)s(e)i (inadequate)g(for)g(obtaining)f(sup)s(er-p)s(olynomial)456 2368 y(lo)m(w)m(er)39 b(b)s(ounds)d(for)j Fx(TC)1294 2331 y Fs(0)1333 2368 y Fw(-circuits.)66 b(T)-8 b(o)39 b(this)f(p)s(oin)m(t,)i(pro)s(of)e(complexit)m(y)h(has)f(made)456 2476 y(steady)j(progress)g(at)h(matc)m(hing)f(the)g(sup)s(erp)s (olynomial)c(lo)m(w)m(er)k(b)s(ounds)e(curren)m(tly)456 2584 y(kno)m(wn)32 b(in)f(the)i(circuit)f(w)m(orld)g(\(alb)s(eit)f (using)h(di\013eren)m(t)g(tec)m(hniques\),)i(and)e(the)h(ma-)456 2694 y(jor)c(remaining)g(analogous)h(result)f(\(a)i(lo)m(w)m(er)f(b)s (ound)e(for)i Fx(A)m(C)2574 2658 y Fs(0)2614 2694 y Fw([)p Fm(p)p Fw(]-F)-8 b(rege)32 b(pro)s(ofs\))d(also)456 2802 y(ma)m(y)38 b(b)s(e)g(within)e(reac)m(h.)65 b(Unlik)m(e)37 b(the)h(circuit)f(w)m(orld,)i(ho)m(w)m(ev)m(er,)j(there)c(is)g(no)g (ana-)456 2910 y(logue)43 b(of)g(Shannon's)f(coun)m(ting)h(argumen)m(t) g(for)g(size)g(lo)m(w)m(er)g(b)s(ounds)e(for)i(random)456 3018 y(functions)29 b(and)i(there)g(do)s(es)g(not)g(seem)h(an)m(y)f (inheren)m(t)f(reason)i(for)f Fx(TC)2925 2981 y Fs(0)2964 3018 y Fw(-F)-8 b(rege)33 b(to)f(b)s(e)456 3126 y(b)s(ey)m(ond)27 b(curren)m(t)h(tec)m(hniques.)39 b(While)27 b(it)h(is)f(true)h(that)g (one)g(can)h(sho)m(w)f(the)g(failure)e(of)456 3234 y Fx(TC)576 3197 y Fs(0)615 3234 y Fw(-F)-8 b(rege)28 b(in)m(terp)s (olation)d(\(also)i(dep)s(ending)d(on)j(cryptographic)f(conjectures\),) i(this)456 3345 y(applies)34 b(to)j Fx(A)m(C)998 3308 y Fs(0)1038 3345 y Fw(-F)-8 b(rege)38 b(as)e(w)m(ell)g(for)g(whic)m(h)f (w)m(e)i(do)f(ha)m(v)m(e)h(lo)m(w)m(er)g(b)s(ounds.)56 b(Is)36 b(there)456 3453 y(an)m(y)41 b(analogue)g(of)f(natural)g(pro)s (ofs)g(in)f(pro)s(of)h(complexit)m(y?)71 b(\(Razb)s(oro)m(v)42 b([)p Fr(78)q Fw(])f(has)456 3561 y(lo)s(ok)m(ed)28 b(at)h(the)g(quite) f(di\013eren)m(t)g(question)f(of)i(examining)e(particular)g(pro)s(of)g (systems)456 3669 y(and)f(sho)m(wing)g(that)i(an)m(y)f(e\016cien)m(t)h (pro)s(ofs)e(of)h(lo)m(w)m(er)g(b)s(ounds)e(for)i(circuits)e(using)h (suc)m(h)456 3776 y(systems)k(automatically)g(naturalize.\))494 4028 y Fp([1])39 b(M.)25 b(Ajtai.)h(The)e(complexit)n(y)f(of)j(the)e (pigeonhole)h(principle.)g Fi(Combinatoric)l(a)p Fp(,)h (14\(4\):417{433,)613 4120 y(1994.)494 4211 y([2])39 b(M.)31 b(Ajtai.)g(The)g(indep)r(endence)e(of)i(the)f(mo)r(dulo)f Fh(p)h Fp(coun)n(ting)g(principles.)h(In)f Fi(Pr)l(o)l(c)l(e)l(e)l (dings)k(of)613 4302 y(the)27 b(Twenty-Sixth)h(A)n(nnual)e(A)n(CM)g (Symp)l(osium)g(on)g(The)l(ory)h(of)e(Computing)p Fp(,)h(pages)e (402{411,)613 4394 y(Mon)n(tr)n(\023)-36 b(eal,)27 b(Qu)n(\023)-36 b(eb)r(ec,)26 b(Canada,)h(Ma)n(y)f(1994.)494 4485 y([3])39 b(M.)83 b(Ajtai.)f(Symmetric)d(systems)i(of)i(linear)f(equations)g(mo)r (dulo)f Fh(p)p Fp(.)h(T)-6 b(ec)n(hnical)613 4576 y(Rep)r(ort)99 b(TR94-015,)119 b(Electronic)100 b(Collo)r(quium)f(in)g(Computation)f (Complexit)n(y)-6 b(,)613 4668 y Fo(http://www.eccc.uni-trier.d)q(e/ec) q(cc/)p Fp(,)32 b(1994.)494 4759 y([4])39 b(Mikl\023)-38 b(os)24 b(Ajtai.)g(The)f(complexit)n(y)f(of)h(the)g(pigeonhole)h (principle.)f(In)f Fi(29th)k(A)n(nnual)f(Symp)l(osium)613 4850 y(on)37 b(F)-6 b(oundations)38 b(of)e(Computer)h(Scienc)l(e)p Fp(,)i(pages)d(346{355,)41 b(White)35 b(Plains,)k(NY,)34 b(Octob)r(er)613 4942 y(1988.)28 b(IEEE.)494 5033 y([5])39 b(Mikl\023)-38 b(os)24 b(Ajtai.)g(P)n(arit)n(y)f(and)f(the)h (pigeonhole)g(principle.)h(In)e(Sam)n(uel)g(R.)g(Buss)h(and)g(P)-6 b(.)23 b(J.)g(Scott,)613 5124 y(editors,)k Fi(F)-6 b(e)l(asible)27 b(Mathematics)p Fp(,)h(pages)d(1{24,)j(A)c(Mathematical)i(Sciences)g (Institute)e(W)-6 b(ork-)613 5216 y(shop,)26 b(Ithaca,)h(NY,)e(1990.)i (Birkh\177)-38 b(auser.)p eop %%Page: 21 21 21 20 bop 494 425 a Fp([6])39 b(M.)34 b(Alekhno)n(vic)n(h,)g(S.)e (Buss,)k(S.)c(Moran,)k(and)d(T.)g(Pitassi.)i(Minim)n(um)c(Prop)r (ositional)k(pro)r(of)613 517 y(length)26 b(is)g(NP-hard)f(to)h (linearly)g(appro)n(ximate.)g(Man)n(uscript.)g(1998.)494 608 y([7])39 b(E.)e(Allender)e(and)g(U.)h(Hertrampf.)f(Depth)g (reduction)g(for)h(circuits)h(of)f(un)n(b)r(ounded)e(fan-in.)613 699 y Fi(Information)28 b(and)g(Computation)p Fp(,)f(112\(2\):217{238,) j(August)25 b(1994.)494 790 y([8])39 b(S.)21 b(Arora,)h(C.)g(Lund,)e (Ra)t(jeev)h(Mot)n(w)n(ani,)i(M.)f(Sudan,)e(and)h(M\023)-38 b(ari\023)g(o)22 b(Szegedy)-6 b(.)20 b(Pro)r(of)i(v)n(eri\014cation)613 882 y(and)i(hardness)g(of)g(appro)n(ximation)f(problems.)h(In)f Fi(Pr)l(o)l(c)l(e)l(e)l(dings)28 b(33r)l(d)e(A)n(nnual)g(Symp)l(osium)g (on)613 973 y(F)-6 b(oundations)30 b(of)d(Computer)i(Scienc)l(e)p Fp(,)e(Pittsburgh,)f(P)-6 b(A,)25 b(Octob)r(er)h(1992.)h(IEEE.)494 1064 y([9])39 b(P)-6 b(.)32 b(Beame,)h(R.)e(Karp,)i(T.)f(Pitassi,)i (and)d(M.)h(Saks.)f(On)g(the)g(complexit)n(y)f(of)i(unsatis\014abilit)n (y)613 1156 y(of)c(random)f Fh(k)r Fp(-CNF)g(form)n(ulas.)h(In)e Fi(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(of)e(the)g(30th)h(A)n(nnual)f(A)n (CM)g(Symp)l(osium)g(on)613 1247 y(The)l(ory)g(of)e(Computing)p Fp(,)g(Dallas,)g(TX,)f(Ma)n(y)g(1998.)456 1338 y([10])39 b(P)-6 b(.)30 b(Beame)e(and)h(T.)h(Pitassi.)h(Simpli\014ed)c(and)i (impro)n(v)n(ed)e(resolution)j(lo)n(w)n(er)g(b)r(ounds.)f(In)f Fi(37th)613 1430 y(A)n(nnual)c(Symp)l(osium)f(on)h(F)-6 b(oundations)25 b(of)e(Computer)h(Scienc)l(e)p Fp(,)f(pages)f(274{282,) j(Burlington,)613 1521 y(VT,)h(Octob)r(er)g(1996.)i(IEEE.)456 1612 y([11])39 b(P)-6 b(.)18 b(Beame)f(and)g(S.)h(Riis.)g(More)g(on)f (the)g(relativ)n(e)h(strength)g(of)g(coun)n(ting)f(principles.)h(In)f (P)-6 b(.)17 b(Beame)613 1704 y(and)k(S.)g(Buss,)h(editors,)h Fi(Pr)l(o)l(of)h(Complexity)f(and)h(F)-6 b(e)l(asible)23 b(A)n(rithmetics)p Fp(,)h(DIMA)n(CS.)c(American)613 1795 y(Mathematical)27 b(So)r(ciet)n(y)-6 b(,)26 b(1998.)h(T)-6 b(o)26 b(app)r(ear.)456 1886 y([12])39 b(P)n(aul)c(W.)e(Beame,)j (Stephen)d(A.)g(Co)r(ok,)k(Je\013)d(Edmonds,)h(Russell)f(Impagliazzo,)j (and)c(T)-6 b(oni-)613 1978 y(ann)31 b(Pitassi.)j(The)d(relativ)n(e)h (complexit)n(y)f(of)h Fh(N)8 b(P)42 b Fp(searc)n(h)32 b(problems.)f(In)g Fi(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)d(the)613 2069 y(Twenty-Seventh)h(A)n(nnual)d(A)n(CM)g(Symp)l(osium)g(on)g(The)l (ory)g(of)g(Computing)p Fp(,)f(pages)g(303{314,)613 2160 y(Las)e(V)-6 b(egas,)26 b(NV,)f(Ma)n(y)h(1995.)456 2252 y([13])39 b(P)n(aul)j(W.)f(Beame,)k(Russell)c(Impagliazzo,)46 b(Jan)41 b(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek,)47 b(T)-6 b(oniann)41 b(Pitassi,)47 b(and)40 b(P)n(a)n(v)n(el)613 2343 y(Pudl\023)-38 b(ak.)34 b(Lo)n(w)n(er)h(b)r(ounds)e(on)h(Hilb)r (ert's)h(Nullstellensatz)g(and)f(prop)r(ositional)h(pro)r(ofs.)h Fi(Pr)l(o)l(c.)613 2434 y(L)l(ondon)29 b(Math.)f(So)l(c.)p Fp(,)e(73\(3\):1{26,)j(1996.)456 2526 y([14])39 b(P)n(aul)29 b(W.)g(Beame,)h(Russell)f(Impagliazzo,)h(Jan)f(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek,)32 b(T)-6 b(oniann)28 b(Pitassi,)j(P)n(a)n(v)n(el)e(Pudl\023)-38 b(ak,)613 2617 y(and)24 b(Alan)h(W)-6 b(o)r(o)r(ds.)25 b(Exp)r(onen)n(tial)f(lo)n (w)n(er)i(b)r(ounds)e(for)h(the)f(pigeonhole)h(principle.)g(In)f Fi(Pr)l(o)l(c)l(e)l(e)l(d-)613 2708 y(ings)k(of)e(the)i(Twenty-F)-6 b(ourth)30 b(A)n(nnual)d(A)n(CM)f(Symp)l(osium)i(on)f(The)l(ory)h(of)e (Computing)p Fp(,)g(pages)613 2800 y(200{220,)j(Victoria,)e(B.C.,)h (Canada,)f(Ma)n(y)e(1992.)456 2891 y([15])39 b(P)n(aul)18 b(W.)g(Beame)f(and)g(T)-6 b(oniann)17 b(Pitassi.)j(An)c(exp)r(onen)n (tial)i(separation)g(b)r(et)n(w)n(een)f(the)g(matc)n(hing)613 2982 y(principle)29 b(and)e(the)h(pigeonhole)h(principle.)f(In)g Fi(8th)i(A)n(nnual)g(IEEE)f(Symp)l(osium)h(on)g(L)l(o)l(gic)g(in)613 3074 y(Computer)f(Scienc)l(e)p Fp(,)e(pages)g(308{319,)h(Mon)n(treal,)f (Queb)r(ec,)f(June)g(1993.)456 3165 y([16])39 b(S.)23 b(Ben-Da)n(vid)g(and)g(A.)f(Gringauze.)j(On)d(the)h(existence)g(of)h (prop)r(ositional)h(pro)r(of)f(systems)f(and)613 3256 y(oracle-relativized)32 b(prop)r(ositional)g(logic.)f(T)-6 b(ec)n(hnical)31 b(Rep)r(ort)e(TR98-021,)k(Electronic)e(Collo-)613 3348 y(quium)24 b(in)i(Computation)f(Complexit)n(y)-6 b(,)25 b Fo(http://www.eccc.uni-trier.de/e)q(ccc/)q Fp(,)31 b(1998.)456 3439 y([17])39 b(E.)27 b(Ben)e(Sasson)i(and)e(A.)h (Wigderson.)g(Priv)l(ate)g(comm)n(unication.)456 3530 y([18])39 b(M.)31 b(Bonet,)h(S.)e(Buss,)i(and)e(T.)h(Pitassi.)h(Are)d (there)h(hard)g(examples)g(for)h(F)-6 b(rege)30 b(systems?)47 b(In)613 3622 y(P)-6 b(.)28 b(Clote)h(and)e(J.)i(Remmel,)d(editors,)j Fi(F)-6 b(e)l(asible)30 b(Mathematics)h(II)p Fp(,)c(pages)i(30{56.)h (Birkhauser,)613 3713 y(1995.)456 3804 y([19])39 b(M.)26 b(Bonet,)g(C.)f(Domingo,)g(and)g(R.)g(Ga)n(v)l(alda.)g(No)g(feasible)i (in)n(terp)r(olation)e(or)h(automatization)613 3896 y(for)h Fh(AC)849 3864 y Fq(0)883 3896 y Fp(-F)-6 b(rege)26 b(pro)r(of)h (systems.)e(Man)n(uscript.)h(1998.)456 3987 y([20])39 b(M.)25 b(Bonet,)g(T.)f(Pitassi,)i(and)d(R.)h(Raz.)g(Lo)n(w)n(er)g(b)r (ounds)f(for)i(cutting)e(planes)h(pro)r(ofs)h(with)f(small)613 4078 y(co)r(e\016cien)n(ts.)k Fi(Journal)g(of)f(Symb)l(olic)h(L)l(o)l (gic)p Fp(,)e(62\(3\):708{728,)k(Septem)n(b)r(er)24 b(1997.)456 4169 y([21])39 b(M.)28 b(Bonet,)f(T.)h(Pitassi,)h(and)d(R.)g(Raz.)h(No) g(feasible)h(in)n(terp)r(olation)g(for)f Fh(T)11 b(C)2873 4138 y Fq(0)2934 4169 y Fp(frege)28 b(pro)r(ofs.)g(In)613 4261 y Fi(38th)h(A)n(nnual)e(Symp)l(osium)h(on)f(F)-6 b(oundations)29 b(of)e(Computer)h(Scienc)l(e)p Fp(.)f(IEEE,)f(Octob)r (er)f(1997.)456 4352 y([22])39 b(S.)28 b(Buss.)g(On)f(Go)r(del's)h (theorems)f(on)g(lengths)h(of)g(pro)r(ofs)h(I)r(I:)e(Lo)n(w)n(er)h(b)r (ounds)f(for)h(recognizing)613 4443 y Fh(k)33 b Fp(sym)n(b)r(ol)c(pro)n (v)l(abilit)n(y.)h(In)g(P)-6 b(.)30 b(Clote)h(and)f(J.)h(Remmel,)f (editors,)i Fi(F)-6 b(e)l(asible)33 b(mathematics)g(II)p Fp(,)613 4535 y(pages)27 b(57{90.)h(Birkhauser-Boston,)f(1995.)456 4626 y([23])39 b(S.)25 b(Buss,)h(R.)e(Impagliazzo,)i(J.)f(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek,)27 b(P)-6 b(.)25 b(Pudl\023)-38 b(ak,)25 b(A.)g(A.)f(Razb)r(oro)n(v,)h(and)f(J.)i (Sgall.)g(Pro)r(of)613 4717 y(complexit)n(y)36 b(in)g(algebraic)j (systems)c(and)i(b)r(ounded)e(depth)h(Frege)h(systems)f(with)h(mo)r (dular)613 4809 y(coun)n(ting.)26 b Fi(Computation)j(Complexity)p Fp(,)e(6\(3\):256{298,)i(1997.)456 4900 y([24])39 b(S.)25 b(Buss)g(and)f(T.)h(Pitassi.)i(Resolution)e(and)f(the)g(w)n(eak)h (pigeonhole)g(principle.)g(In)f Fi(Pr)l(o)l(c)l(e)l(e)l(dings)613 4991 y(of)39 b(Computer)h(Scienc)l(e)f(L)l(o)l(gic)p Fp(,)j(Lecture)37 b(Notes)h(in)g(Computer)f(Science.)h(Springer-V)-6 b(erlag,)613 5083 y(1997.)p eop %%Page: 22 22 22 21 bop 456 425 a Fp([25])39 b(S.)d(R.)g(Buss.)h(Lo)n(w)n(er)g(b)r (ounds)e(on)h(Nullstellensatz)h(pro)r(ofs)h(via)e(designs.)h(In)e(P)-6 b(.)36 b(W.)g(Beame)613 517 y(and)25 b(S.)g(R.)f(Buss,)i(editors,)g Fi(Pr)l(o)l(of)h(Complexity)g(and)g(F)-6 b(e)l(asible)28 b(A)n(rithmetics)p Fp(,)e(DIMA)n(CS,)e(pages)613 608 y(59{71.)k(American)d(Math.)i(So)r(c,)f(1997.)456 699 y([26])39 b(S.)f(R.)g(Buss)g(and)g(T.)g(Pitassi.)i(Go)r(o)r(d)f(degree) f(b)r(ounds)g(on)f(Nullstellensatz)j(refutations)f(of)613 790 y(the)28 b(induction)g(principle.)g(In)g Fi(Pr)l(o)l(c)l(e)l(e)l (dings)k(of)d(the)h(Eleventh)h(A)n(nnual)f(Confer)l(enc)l(e)h(on)f (Com-)613 882 y(putational)35 b(Complexity)f(\(formerly:)47 b(Structur)l(e)36 b(in)c(Complexity)i(The)l(ory\))p Fp(,)i(pages)c (233{242,)613 973 y(Philadelphia,)c(P)-6 b(A,)25 b(Ma)n(y)h(1996.)h (IEEE.)456 1064 y([27])39 b(S.)f(R.)f(Buss)h(and)f(G.)h(T)-6 b(uran.)38 b(Resolution)g(pro)r(ofs)h(of)f(generalized)h(pigeonhole)f (principles.)613 1156 y Fi(The)l(or)l(etic)l(al)29 b(Computer)g(Scienc) l(e)p Fp(,)e(62\(3\):311{317,)j(Decem)n(b)r(er)25 b(1988.)456 1247 y([28])39 b(Sam)n(uel)24 b(R.)h(Buss.)h Fi(Bounde)l(d)i(A)n (rithmetic)p Fp(.)e(Bibliop)r(olis,)i(Nap)r(oli,)e(1986.)h(V)-6 b(olume)24 b(3)h(of)h(Studies)613 1338 y(in)g(Pro)r(of)h(Theory)-6 b(.)456 1430 y([29])39 b(Sam)n(uel)25 b(R.)g(Buss.)i(P)n(olynomial)f (size)g(pro)r(ofs)h(of)f(the)f(pigeonhole)i(principle.)f Fi(Journal)i(of)f(Sym-)613 1521 y(b)l(olic)h(L)l(o)l(gic)p Fp(,)e(57:916{927,)k(1987.)456 1612 y([30])39 b(V.)24 b(Ch)n(v\023)-38 b(atal.)25 b(Edmonds)e(p)r(olytop)r(es)i(and)f(a)g (hierarc)n(h)n(y)g(of)h(com)n(binatorial)g(problems.)f Fi(Discr)l(ete)613 1704 y(Mathematics)p Fp(,)k(4,)e(1973.)456 1795 y([31])39 b(V.)26 b(Ch)n(v\023)-38 b(atal)26 b(and)g(B.)g(Reed.)g (Mic)n(k)g(gets)g(some)f(\(the)h(o)r(dds)f(are)i(on)f(his)f(side\).)i (In)e Fi(33nd)j(A)n(nnual)613 1886 y(Symp)l(osium)36 b(on)g(F)-6 b(oundations)38 b(of)d(Computer)i(Scienc)l(e)p Fp(,)h(pages)d(620{627,)40 b(Pittsburgh,)d(P)-6 b(A,)613 1978 y(Octob)r(er)26 b(1992.)i(IEEE.)456 2069 y([32])39 b(V.)25 b(Ch)n(v\023)-38 b(atal)24 b(and)g(Endre)g(Szemer)n(\023)-36 b(edi.)24 b(Man)n(y)g(hard)g(examples)g(for)h(resolution.)g Fi(Journal)j(of)d(the)613 2160 y(A)n(CM)p Fp(,)h(35\(4\):759{768,)k (1988.)456 2252 y([33])39 b(M.)33 b(Clegg,)i(J.)d(Edmonds,)h(and)e(R.)h (Impagliazzo.)g(Using)g(the)g(Gr\177)-38 b(obner)32 b(basis)g (algorithm)g(to)613 2343 y(\014nd)19 b(pro)r(ofs)j(of)f (unsatis\014abilit)n(y)-6 b(.)20 b(In)g Fi(Pr)l(o)l(c)l(e)l(e)l(dings)k (of)e(the)i(Twenty-Eighth)g(A)n(nnual)f(A)n(CM)f(Sym-)613 2434 y(p)l(osium)28 b(on)g(The)l(ory)h(of)e(Computing)p Fp(,)f(pages)h(174{183,)i(Philadelphia,)e(P)-6 b(A,)25 b(Ma)n(y)h(1996.)456 2526 y([34])39 b(S.)d(Co)r(ok)h(and)f(D.)g(Mitc)n (hell.)h(Finding)f(hard)g(instances)h(of)g(the)e(satis\014abilit)n(y)i (problem:)55 b(A)613 2617 y(surv)n(ey)-6 b(.)25 b(In)g Fi(DIMA)n(CS)i(Series)h(in)f(The)l(or)l(etic)l(al)i(Computer)g(Scienc)l (e)p Fp(,)e(1997.)456 2708 y([35])39 b(Stephen)e(A.)h(Co)r(ok.)i(The)e (complexit)n(y)f(of)i(theorem)e(pro)n(ving)h(pro)r(cedures.)g(In)g Fi(Confer)l(enc)l(e)613 2800 y(R)l(e)l(c)l(or)l(d)32 b(of)d(Thir)l(d)i(A)n(nnual)f(A)n(CM)f(Symp)l(osium)h(on)h(The)l(ory)g (of)e(Computing)p Fp(,)h(pages)f(151{158,)613 2891 y(Shak)n(er)c(Heigh) n(ts,)h(OH,)g(Ma)n(y)f(1971.)456 2982 y([36])39 b(Stephen)27 b(A.)h(Co)r(ok)g(and)g(Rob)r(ert)g(A.)f(Rec)n(kho)n(w.)h(Time)g(b)r (ounded)e(random)h(access)j(mac)n(hines.)613 3074 y Fi(Journal)f(of)e (Computer)i(and)f(System)g(Scienc)l(es)p Fp(,)g(7\(4\):354{375,)h (1973.)456 3165 y([37])39 b(W.)26 b(Co)r(ok,)h(C.)f(R.)g(Coullard,)h (and)e(G.)i(T)-6 b(uran.)25 b(On)g(the)h(complexit)n(y)e(of)i(cutting)g (plane)g(pro)r(ofs.)613 3256 y Fi(Discr)l(ete)j(Applie)l(d)f (Mathematics)p Fp(,)f(18:25{38,)j(1987.)456 3348 y([38])39 b(M.)26 b(Da)n(vis,)g(G.)g(Logemann,)f(and)g(D.)g(Lo)n(v)n(eland.)g(A)g (mac)n(hine)f(program)h(for)h(theorem)e(pro)n(ving.)613 3439 y Fi(Communic)l(ations)29 b(of)e(the)h(A)n(CM)p Fp(,)e(5:394{397,)j(1962.)456 3530 y([39])39 b(M.)23 b(Da)n(vis)g(and)f(H.)g(Putnam.)f(A)h(computing)f(pro)r(cedure)h(for)h (quan)n(ti\014cation)f(theory)-6 b(.)22 b Fi(Commu-)613 3622 y(nic)l(ations)29 b(of)e(the)h(A)n(CM)p Fp(,)e(7:201{215,)j(1960.) 456 3713 y([40])39 b(U.)27 b(F)-6 b(eige,)28 b(S.)f(Goldw)n(asser,)i (L.)e(Lo)n(v\023)-38 b(asz,)28 b(S.)f(Safra,)h(and)e(M.)i(Szegedy)-6 b(.)26 b(Appro)n(ximating)f(clique)613 3804 y(is)31 b(almost)g Fi(NP)7 b Fp(-complete.)30 b(In)f Fi(Pr)l(o)l(c)l(e)l(e)l(dings)34 b(32nd)f(A)n(nnual)e(Symp)l(osium)h(on)g(F)-6 b(oundations)34 b(of)613 3896 y(Computer)29 b(Scienc)l(e)p Fp(,)e(pages)g(2{12,)g(San)e (Juan,)i(Puerto)f(Rico,)g(Octob)r(er)g(1991.)h(IEEE.)456 3987 y([41])39 b(G.)e(F)-6 b(rege.)37 b Fi(Be)l(gri\013sschrift,)j (eine)d(der)h(arithmetischen)g(nachgebildete)h(F)-6 b(ormelspr)l(ache) 40 b(des)613 4078 y(r)l(einb)l(en)29 b(Denkens)p Fp(.)e(Neb)r(ert,)f (Halle,)h(1879.)456 4169 y([42])39 b(G.)27 b(F)-6 b(rege.)26 b Fi(Grundgesetze)31 b(der)d(A)n(rithmetik)p Fp(.)f(Neb)r(ert,)f (Halle,)g(1893,1901.)456 4261 y([43])39 b(E.)29 b(F)-6 b(riedgut.)28 b(Necessary)h(and)f(su\016cien)n(t)g(conditions)h(for)g (sharp)f(thresholds)h(of)g(graph)f(prop-)613 4352 y(erties,)f(and)f (the)f Fh(k)r Fp(-sat)h(problem.)f(Preprin)n(t,)h(Ma)n(y)g(1997.)456 4443 y([44])39 b(A.)28 b(F)-6 b(rieze)28 b(and)f(S.)g(Suen.)g(Analysis) h(of)h(t)n(w)n(o)f(simple)f(heuristics)h(on)g(a)f(random)g(instance)h (of)g(k-)613 4535 y(SA)-6 b(T.)26 b Fi(Journal)i(of)f(A)n(lgorithms)p Fp(,)g(20\(2\):312{355,)j(1996.)456 4626 y([45])39 b(Xudong)28 b(F)-6 b(u.)28 b Fi(On)i(the)h(c)l(omplexity)g(of)f(pr)l(o)l(of)h (systems)p Fp(.)g(PhD)d(thesis,)i(Univ)n(ersit)n(y)d(of)i(T)-6 b(oron)n(to,)613 4717 y(1995.)456 4809 y([46])39 b(M.)30 b(F)-6 b(urst,)28 b(J.)i(B.)f(Saxe,)g(and)f(M.)h(Sipser.)g(P)n(arit)n (y)-6 b(,)29 b(circuits,)h(and)e(the)g(p)r(olynomial-time)g(hier-)613 4900 y(arc)n(h)n(y)-6 b(.)25 b Fi(Mathematic)l(al)k(Systems)g(The)l (ory)p Fp(,)e(17\(1\):13{27,)i(April)d(1984.)456 4991 y([47])39 b(O.)30 b(Gabb)r(er)f(and)g(Z.)h(Galil.)h(Explicit)e (constructions)h(of)g(linear)h(size)f(sup)r(erconcen)n(trators.)g(In) 613 5083 y Fi(20th)i(A)n(nnual)e(Symp)l(osium)g(on)h(F)-6 b(oundations)31 b(of)f(Computer)h(Scienc)l(e)p Fp(,)g(pages)e(364{370,) j(New)613 5174 y(Y)-6 b(ork,)26 b(NY,)f(1979.)i(IEEE.)p eop %%Page: 23 23 23 22 bop 456 425 a Fp([48])39 b(Z.)29 b(Galil.)i(On)d(the)g(complexit) n(y)f(of)i(regular)h(resolution)g(and)e(the)g(Da)n(vis-Putnam)f(pro)r (cedure.)613 517 y Fi(The)l(or)l(etic)l(al)i(Computer)g(Scienc)l(e)p Fp(,)e(4:23{46,)i(1977.)456 608 y([49])39 b(A.)22 b(Go)r(erdt.)h (Regular)f(resolution)h(v)n(ersus)f(unrestricted)f(resolution.)j Fi(SIAM)f(Journal)i(on)f(Com-)613 699 y(puting)p Fp(,)j (22\(4\):661{683,)j(1993.)456 790 y([50])39 b(R.E.)28 b(Gomory)-6 b(.)27 b(Outline)g(of)h(an)f(algorithm)h(for)g(in)n(teger)f (solutions)i(to)e(linear)h(programs.)g Fi(Bul-)613 882 y(letin)g(of)f(the)i(A)n(meric)l(an)f(Mathematic)l(al)g(So)l(ciety)p Fp(,)f(64:275{278,)j(1958.)456 973 y([51])39 b(A.)20 b(Hak)n(en.)f(The)h(in)n(tractabilit)n(y)g(of)h(resolution.)g Fi(The)l(or)l(etic)l(al)j(Computer)f(Scienc)l(e)p Fp(,)f(39:297{305,) 613 1064 y(1985.)456 1156 y([52])39 b(A.)25 b(Hak)n(en)f(and)h(S.)g(A.) f(Co)r(ok.)i(An)e(exp)r(onen)n(tial)h(lo)n(w)n(er)h(b)r(ound)e(for)i (the)f(size)g(of)h(monotone)e(real)613 1247 y(circuits.)j(Preprin)n(t,) f(1995.)456 1338 y([53])39 b(K.)21 b(Iw)n(ama.)f(Complexit)n(y)f(of)i (\014nding)f(short)g(resolution)h(pro)r(ofs.)h(In)e(I.)g(Priv)l(ara)h (and)e(P)-6 b(.)21 b(Ruzic)n(k)l(a,)613 1430 y(editors,)37 b Fi(L)l(e)l(ctur)l(e)h(Notes)e(in)f(Computer)h(Scienc)l(e)g(1295)p Fp(,)h(pages)e(309{318.)i(Springer-V)-6 b(erlag,)613 1521 y(1997.)456 1612 y([54])39 b(G.)30 b(D.)e(James.)h Fi(The)i(R)l(epr)l(esentation)i(The)l(ory)e(of)f(the)h(Symmetric)g(Gr)l (oups)p Fp(.)f(Num)n(b)r(er)d(682)i(in)613 1704 y(Lecture)d(Notes)g(in) g(Mathematics.)g(Springer-V)-6 b(erlag,)26 b(1978.)456 1795 y([55])39 b(S.)31 b(Kirkpatric)n(k)f(and)g(B.)h(Selman.)f (Critical)i(b)r(eha)n(vior)e(in)h(the)f(satis\014abilit)n(y)h(of)g (random)f(for-)613 1886 y(m)n(ulas.)c Fi(Scienc)l(e)p Fp(,)h(264:1297{1301,)k(Ma)n(y)26 b(1994.)456 1978 y([56])39 b(L.)g(M.)g(Kirousis,)k(E.)c(Kranakis,)j(and)c(D.)g(Krizanc.)h(Appro)n (ximating)e(the)h(unsatis\014abilit)n(y)613 2069 y(threshold)30 b(of)h(random)d(form)n(ulas.)j(In)e Fi(Pr)l(o)l(c)l(e)l(e)l(dings)k(of) e(the)h(F)-6 b(ourth)33 b(A)n(nnual)e(Eur)l(op)l(e)l(an)i(Sym-)613 2160 y(p)l(osium)28 b(on)g(A)n(lgorithms)p Fp(,)f(pages)f(27{38,)i (Barcelona,)g(Spain,)e(Septem)n(b)r(er)e(1996.)456 2252 y([57])39 b(J.)25 b(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek.)25 b(On)e(the)g(degree)h(of)g(ideal)h(mem)n(b)r(ership)c(pro)r (ofs)k(from)e(uniform)g(families)i(of)f(p)r(oly-)613 2343 y(nomials)i(o)n(v)n(er)g(\014nite)f(\014elds.)h(Man)n(uscript,)g (1997.)456 2434 y([58])39 b(J.)23 b(Kra)t(jicek.)h(Lo)n(w)n(er)e(b)r (ounds)g(to)g(the)g(size)g(of)h(constan)n(t-depth)e(prop)r(ositional)j (pro)r(ofs.)f Fi(Journal)613 2526 y(of)28 b(Symb)l(olic)f(L)l(o)l(gic)p Fp(,)g(59\(1\):73{86,)i(Marc)n(h)d(1994.)456 2617 y([59])39 b(J.)28 b(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek.)28 b Fi(Bounde)l(d)i(A)n(rithmetic,)f(Pr)l(op)l(ositional)g(L)l(o)l(gic)g (and)f(Complexity)h(The)l(ory)p Fp(.)f(Cam-)613 2708 y(bridge)e(Univ)n(ersit)n(y)f(Press,)i(1996.)456 2800 y([60])39 b(J.)24 b(Kra)t(jicek)g(and)f(P)-6 b(.)23 b(Pudl\023)-38 b(ak.)23 b(Prop)r(ositional)i(pro)r(of)f(systems,)g(the)e(consistency)i (of)f(\014rst)g(order)613 2891 y(theories)37 b(and)e(the)h(complexit)n (y)e(of)i(computations.)g Fi(J.)h(Symb)l(olic)f(L)l(o)l(gic)p Fp(,)j(54\(3\):1063{1079,)613 2982 y(1989.)456 3074 y([61])g(K.)30 b(Kra)t(jicek)g(and)e(P)-6 b(.)30 b(Pudl\023)-38 b(ak.)29 b(Some)f(consequences)h(of)h(cryptographic)g(conjectures)g(for)g Fh(S)3410 3042 y Fq(1)3406 3087 y(2)613 3165 y Fp(and)c(EF.)h(In)e(D.)h (Leiv)l(an)n(t,)g(editor,)g Fi(L)l(o)l(gic)j(and)f(Computational)h (Complexity)p Fp(,)d(pages)h(210{220.)613 3256 y(Springer-V)-6 b(erlag,)26 b(1995.)456 3348 y([62])39 b(J.)i(Kra)t(j)-9 b(\023)-30 b(\020)n(\024)-36 b(cek,)47 b(P)-6 b(.)40 b(Pudl\023)-38 b(ak,)44 b(and)c(A)g(W)-6 b(o)r(o)r(ds.)41 b(Exp)r(onen)n(tial)f(lo)n(w)n(er)h(b)r(ounds)f(to)h(the)e(size)j(of) 613 3439 y(b)r(ounded)25 b(depth)f(Frege)j(pro)r(ofs)g(of)f(the)f (pigeonhole)i(principle.)f Fi(R)l(andom)i(Structur)l(es)i(and)e(A)n(l-) 613 3530 y(gorithms)p Fp(,)g(7\(1\),)e(1995.)456 3622 y([63])39 b(L.)22 b(Lo)n(v)l(asz)g(and)f(A.)g(Sc)n(hrijv)n(er.)g(Cones) i(of)f(matrices)f(and)g(set-functions)h(and)f(0-1)h(optimization.)613 3713 y Fi(SIAM)27 b(J.)h(Optimization)p Fp(,)e(1\(2\):166{190,)j(1991.) 456 3804 y([64])39 b(A.)22 b(Maciel)i(and)e(T.)h(Pitassi.)h (Non-automatizabilit)n(y)e(of)g(b)r(ounded-depth)e(F)-6 b(rege)23 b(pro)r(ofs.)g(Man-)613 3896 y(uscript.)j(1998.)456 3987 y([65])39 b(A.)31 b(Maciel)h(and)f(T.)g(Pitassi.)i(On)d Fh(AC)5 b(C)1822 3955 y Fq(0)1857 3987 y Fp([)p Fh(p)1917 3955 y Fd(k)1955 3987 y Fp(])31 b(frege)h(pro)r(ofs.)g(In)e Fi(Pr)l(o)l(c)l(e)l(e)l(dings)k(of)e(the)h(Twenty)613 4078 y(Ninth)27 b(A)n(nnual)g(A)n(CM)f(Symp)l(osium)h(on)g(The)l(ory)h (of)e(Computing)p Fp(,)g(pages)g(720{729,)i(Ma)n(y)c(1997.)456 4169 y([66])39 b(J.)26 b(Me\031ner)g(and)e(J.)i(T)-6 b(oran.)26 b(Optimal)e(pro)r(of)i(systems)e(for)i(prop)r(ositional)h (logic)f(and)f(complete)613 4261 y(sets.)d(In)d Fi(15th)24 b(A)n(nnual)f(Symp)l(osium)g(on)g(The)l(or)l(etic)l(al)h(Asp)l(e)l(cts) h(of)d(Computer)i(Scienc)l(e)p Fp(,)e(Lecture)613 4352 y(Notes)k(in)g(Computer)f(Science,)h(P)n(aris,)h(F)-6 b(rance,)26 b(F)-6 b(ebruary)25 b(1998.)i(Springer-V)-6 b(erlag.)456 4443 y([67])39 b(D.)25 b(Mitc)n(hell.)i(Hard)d(problems)g (for)i(csp)f(algorithms.)h(In)e Fi(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(fr)l(om)e(AAAI)f('98)p Fp(,)f(1998.)456 4535 y([68])39 b(D.)31 b(Mundici.)g(A)f(lo)n(w)n(er)i(b)r(ound)d(for)j(the)e (complexit)n(y)f(of)j(Craig's)g(in)n(terp)r(olan)n(ts)f(in)g(sen)n(ten) n(tial)613 4626 y(logic.)21 b Fi(A)n(r)l(chiv)h(fur)f(Mathematische)i (L)l(o)l(gik)e(und)h(Grund)t(lagenforschung)p Fp(,)h(23\(1-2\):27{36,)g (1983.)456 4717 y([69])39 b(J.)d(P)n(aris)f(and)g(A.)f(Wilkie.)h(Coun)n (ting)g(problems)f(in)h(b)r(ounded)e(arithmetic.)i(In)f Fi(Metho)l(ds)j(in)613 4809 y(Mathematic)l(al)e(L)l(o)l(gic:)46 b(Pr)l(o)l(c)l(e)l(e)l(dings)35 b(of)e(the)h(6th)g(L)l(atin)g(A)n (meric)l(an)f(Symp)l(osium)h(on)f(Mathe-)613 4900 y(matic)l(al)i(L)l(o) l(gic)h(1983)p Fp(,)h(v)n(olume)32 b(1130)k(of)e Fi(L)l(e)l(ctur)l(e)j (notes)g(in)d(Mathematics)p Fp(,)k(pages)c(317{340,)613 4991 y(Berlin,)27 b(1985.)h(Springer-V)-6 b(erlag.)p eop %%Page: 24 24 24 23 bop 456 425 a Fp([70])39 b(J.B.)34 b(P)n(aris,)g(A.)e(J.)g (Wilkie,)j(and)c(A.)g(R.)h(W)-6 b(o)r(o)r(ds.)33 b(Pro)n(v)l(abilit)n (y)e(of)i(the)e(pigeonhole)i(principle)613 517 y(and)22 b(the)f(existence)h(of)g(in\014nitely)f(man)n(y)f(primes.)h Fi(Journal)k(of)e(Symb)l(olic)h(L)l(o)l(gic)p Fp(,)f(53:1235{1244,)613 608 y(1988.)456 699 y([71])39 b(T.)g(Pitassi.)g(Algebraic)g(prop)r (ositional)g(pro)r(of)g(systems.)e(In)g Fi(DIMA)n(CS)h(Series)h(in)f (Discr)l(ete)613 790 y(Mathematics)p Fp(,)28 b(v)n(olume)c(31,)j(pages) g(215{243.)h(American)d(Math.)i(So)r(c,)f(1997.)456 882 y([72])39 b(T.)30 b(Pitassi)h(and)e(A.)f(Urquhart.)h(The)g(complexit)n (y)f(of)h(the)g(Ha)t(j\023)-38 b(os)30 b(calculus.)h(In)d Fi(33nd)j(A)n(nnual)613 973 y(Symp)l(osium)36 b(on)g(F)-6 b(oundations)38 b(of)d(Computer)i(Scienc)l(e)p Fp(,)h(pages)d(187{196,) 40 b(Pittsburgh,)d(P)-6 b(A,)613 1064 y(Octob)r(er)26 b(1992.)i(IEEE.)456 1156 y([73])39 b(T)-6 b(oniann)21 b(Pitassi,)j(P)n(aul)e(W.)f(Beame,)h(and)e(Russell)i(Impagliazzo.)g (Exp)r(onen)n(tial)f(lo)n(w)n(er)h(b)r(ounds)613 1247 y(for)27 b(the)e(pigeonhole)i(principle.)f Fi(Computational)j (Complexity)p Fp(,)d(3\(2\):97{140,)j(1993.)456 1338 y([74])39 b(P)-6 b(.)24 b(Pudl\023)-38 b(ak.)23 b(Lo)n(w)n(er)h(b)r (ounds)f(for)h(resolution)g(and)f(cutting)h(plane)f(pro)r(ofs)i(and)e (monotone)f(com-)613 1430 y(putations.)27 b Fi(Journal)h(of)f(Symb)l (olic)h(L)l(o)l(gic)p Fp(,)e(62\(3\):981{998,)k(Septem)n(b)r(er)24 b(1997.)456 1521 y([75])39 b(P)-6 b(.)27 b(Pudl\023)-38 b(ak)27 b(and)f(J.)h(Sgall.)h(Algebraic)g(mo)r(dels)e(of)h(computation) f(and)g(in)n(terp)r(olation)i(for)g(alge-)613 1612 y(braic)g(pro)r(of)g (systems.)f(In)f(P)-6 b(.)27 b(W.)g(Beame)g(and)g(S.)f(R.)h(Buss,)h (editors,)h Fi(Pr)l(o)l(of)g(Complexity)g(and)613 1704 y(F)-6 b(e)l(asible)29 b(A)n(rithmetics)p Fp(,)e(DIMA)n(CS,)e(pages)i (279{295.)h(American)d(Math.)i(So)r(c,)f(1997.)456 1795 y([76])39 b(A.)26 b(A.)f(Razb)r(oro)n(v.)h(Lo)n(w)n(er)h(b)r(ounds)e (for)h(the)g(p)r(olynomial)g(calculus.)g(No)n(v)n(em)n(b)r(er)e(1996.) 456 1886 y([77])39 b(A.)23 b(A.)h(Razb)r(oro)n(v.)f(Lo)n(w)n(er)h(b)r (ounds)f(for)h(the)e(size)i(of)g(circuits)g(with)g(b)r(ounded)e(depth)g (with)h(basis)613 1978 y Ff(f^)p Fh(;)14 b Ff(\010g)p Fp(.)26 b Fi(Mat.)i(Zametki)p Fp(,)e(1987.)456 2069 y([78])39 b(A.)34 b(A.)f(Razb)r(oro)n(v.)g(Bounded)g(arithmetic)g(and)g(lo)n(w)n (er)h(b)r(ounds)f(in)g(Bo)r(olean)i(complexit)n(y)-6 b(.)32 b(In)613 2160 y(P)-6 b(.)21 b(Clote)h(and)e(J.)h(Remmel,)f (editors,)i Fi(F)-6 b(e)l(asible)24 b(Mathematics)g(II)p Fp(,)c(pages)h(344{386.)j(Birkhauser,)613 2252 y(1995.)456 2343 y([79])39 b(A.)20 b(A.)f(Razb)r(oro)n(v.)h(Unpro)n(v)l(abilit)n(y) e(of)i(lo)n(w)n(er)g(b)r(ounds)f(on)g(the)g(circuit)h(size)g(in)f (certain)h(fragmen)n(ts)613 2434 y(of)27 b(b)r(ounded)e(arithmetic.)g Fi(Izvestiiya)k(of)e(the)h(RAN)p Fp(,)e(59:201{224,)k(1995.)456 2526 y([80])39 b(A.)c(A.)f(Razb)r(oro)n(v)h(and)f(S.)g(Rudic)n(h.)g (Natural)g(pro)r(ofs.)j Fi(Journal)f(of)f(Computer)i(and)f(System)613 2617 y(Scienc)l(es)p Fp(,)28 b(55\(1\):24{35,)h(August)c(1997.)456 2708 y([81])39 b(A.)24 b(A.)f(Razb)r(oro)n(v,)h(A.)g(Wigderson,)h(and)e (A.)g(C.)h(Y)-6 b(ao.)24 b(Read-once)f(branc)n(hing)g(programs,)i (rect-)613 2800 y(angular)d(pro)r(ofs)g(of)g(the)e(pigeonhole)i (principle)f(and)f(the)h(transv)n(ersal)g(calculus.)h(In)e Fi(Pr)l(o)l(c)l(e)l(e)l(dings)613 2891 y(of)28 b(the)h(Twenty)g(Ninth)g (A)n(nnual)f(A)n(CM)g(Symp)l(osium)h(on)f(The)l(ory)h(of)f(Computing)p Fp(,)g(pages)f(739{)613 2982 y(748,)h(El)e(P)n(aso,)h(TX,)f(Ma)n(y)g (1997.)456 3074 y([82])39 b(S\034ren)g(Riis.)h Fi(Indep)l(endenc)l(e)i (in)d(Bounde)l(d)j(A)n(rithmetic)p Fp(.)e(PhD)e(thesis,)44 b(Oxford)38 b(Univ)n(ersit)n(y)-6 b(,)613 3165 y(1993.)456 3256 y([83])39 b(J.)29 b(A.)e(Robinson.)g(A)g(mac)n(hine)g(orien)n(ted) g(logic)i(based)f(on)f(the)g(resolution)i(principle.)e Fi(Journal)613 3348 y(of)h(the)g(A)n(CM)p Fp(,)e(12\(1\):23{41,)j (1965.)456 3439 y([84])39 b(J.)20 b(T.)g(Sc)n(h)n(w)n(artz.)f (Probabilistic)i(algorithms)e(for)h(v)n(eri\014cation)f(of)h(p)r (olynomial)f(iden)n(tities.)h Fi(Jour-)613 3530 y(nal)28 b(of)f(the)h(A)n(CM)p Fp(,)e(pages)h(701{717,)h(1980.)456 3622 y([85])39 b(M.)26 b(Sipser.)f(The)g(history)g(and)f(status)h(of)g (the)g(P)g(v)n(ersus)f(NP)g(question.)h(In)f Fi(Pr)l(o)l(c)l(e)l(e)l (dings)29 b(of)d(the)613 3713 y(Twenty-F)-6 b(ourth)37 b(A)n(nnual)c(A)n(CM)h(Symp)l(osium)f(on)h(The)l(ory)h(of)e(Computing)p Fp(,)h(pages)f(603{618,)613 3804 y(Victoria,)27 b(B.C.,)h(Canada,)f(Ma) n(y)f(1992.)456 3896 y([86])39 b(Mic)n(hael)34 b(Sipser.)f(A)g (complexit)n(y)e(theoretic)i(approac)n(h)g(to)g(randomness.)f(In)g Fi(Pr)l(o)l(c)l(e)l(e)l(dings)37 b(of)613 3987 y(the)29 b(Fifte)l(enth)f(A)n(nnual)g(A)n(CM)f(Symp)l(osium)g(on)h(The)l(ory)g (of)g(Computing)p Fp(,)e(Boston,)h(MA,)e(April)613 4078 y(1983.)456 4169 y([87])39 b(Roman)c(Smolensky)-6 b(.)34 b(Algebraic)j(metho)r(ds)e(in)h(the)f(theory)g(of)i(lo)n(w)n(er)g(b)r (ounds)e(for)i(Bo)r(olean)613 4261 y(circuit)i(complexit)n(y)-6 b(.)36 b(In)i Fi(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)d(the)i(Ninete)l(enth) h(A)n(nnual)e(A)n(CM)f(Symp)l(osium)h(on)613 4352 y(The)l(ory)29 b(of)e(Computing)p Fp(,)g(pages)f(77{82,)i(New)e(Y)-6 b(ork,)25 b(NY,)g(Ma)n(y)h(1987.)456 4443 y([88])39 b(G.)25 b(S.)f(Tseitin.)h(On)e(the)g(complexit)n(y)g(of)i(deriv)l(ation)f(in)f (the)h(prop)r(ositional)h(calculus.)g(In)f(A.)f(O.)613 4535 y(Slisenk)n(o,)28 b(editor,)f Fi(Studies)j(in)e(Constructive)j (Mathematics)e(and)g(Mathematic)l(al)h(L)l(o)l(gic,)e(Part)613 4626 y(II)p Fp(.)e(1968.)456 4717 y([89])39 b(A.)26 b(Urquhart.)f(Man)n (uscript,)h(1998.)456 4809 y([90])39 b(A.)20 b(Urquhart.)f(Hard)g (examples)g(for)h(resolution.)h Fi(Journal)i(of)f(the)h(A)n(CM)p Fp(,)c(34\(1\):209{219,)25 b(1987.)456 4900 y([91])39 b(A.)f(Urquhart.)f(The)h(complexit)n(y)f(of)h(prop)r(ositional)i(pro)r (ofs.)f Fi(Bul)t(letin)f(of)h(Symb)l(olic)g(L)l(o)l(gic)p Fp(,)613 4991 y(1\(4\):425{467,)30 b(Decem)n(b)r(er)25 b(1995.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF