%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: beatcs.dvi %%CreationDate: Mon May 05 15:16:08 2003 %%Pages: 39 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips beatcs %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.05:1515 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A 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 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y(In)h(the)g(70's)h(w)m(e)f(sa)m(w)g(the)g(gro)m (wth)h(of)f(complexit)m(y)f(classes)h(as)h(researc)m(hers)f(tried)448 2605 y(to)e(encompass)g(di\013eren)m(t)e(mo)s(dels)g(of)i (computations.)39 b(One)28 b(of)g(those)h(mo)s(dels,)f(prob-)448 2718 y(abilistic)j(computation,)i(started)h(with)d(a)j(probabilistic)29 b(test)34 b(for)e(primalit)m(y)-8 b(,)32 b(led)g(to)448 2831 y(probabilistic)d(complexit)m(y)i(classes)h(and)f(a)h(new)f(kind)f (of)h(in)m(teractiv)m(e)i(pro)s(of)e(system)448 2944 y(that)38 b(itself)f(led)f(to)j(hardness)d(results)g(for)h(appro)m (ximating)g(certain)g Fi(NP)p Fn(-complete)448 3057 y(problems.)49 b(W)-8 b(e)35 b(ha)m(v)m(e)g(also)f(seen)f(strong)h(evidence)g(that)g (w)m(e)g(can)g(remo)m(v)m(e)h(the)f(ran-)448 3170 y(domness)25 b(from)g(computations)g(and)g(most)g(recen)m(tly)h(a)g(deterministic)d (algorithm)h(for)448 3283 y(the)31 b(original)d(primalit)m(y)h (problem.)589 3396 y(In)41 b(the)g(80's)h(w)m(e)g(sa)m(w)f(the)h(rise)e (of)h(\014nite)f(mo)s(dels)g(lik)m(e)g(circuits)g(that)i(capture)448 3509 y(computation)37 b(in)f(an)g(inheren)m(tly)g(di\013eren)m(t)g(w)m (a)m(y)-8 b(.)62 b(A)37 b(new)f(approac)m(h)h(to)h(problems)448 3621 y(lik)m(e)e Fi(P)h Fn(=)f Fi(NP)h Fn(arose)g(from)g(these)g (circuits)e(and)i(though)f(they)h(ha)m(v)m(e)h(had)e(limited)448 3734 y(success)31 b(in)f(separating)h(complexit)m(y)f(classes,)i(this)d (approac)m(h)j(brough)m(t)e(com)m(binato-)448 3847 y(rial)f(tec)m (hniques)g(in)m(to)h(the)g(area)h(and)e(led)g(to)i(a)f(m)m(uc)m(h)g(b)s (etter)g(understanding)e(of)i(the)448 3960 y(limits)e(of)j(these)g (devices.)589 4073 y(In)36 b(the)h(90's)h(w)m(e)f(ha)m(v)m(e)g(seen)g (the)g(study)e(of)i(new)f(mo)s(dels)g(of)g(computation)h(lik)m(e)448 4186 y(quan)m(tum)d(computers)h(and)f(prop)s(ositional)e(pro)s(of)i (systems.)54 b(T)-8 b(o)s(ols)34 b(from)g(the)h(past)448 4299 y(ha)m(v)m(e)d(greatly)f(help)s(ed)d(our)i(understanding)e(of)i (these)h(new)f(areas.)589 4412 y(One)g(cannot)h(in)d(the)i(short)g (space)h(of)f(this)e(article)i(men)m(tion)g(all)f(of)h(the)g(amazing) 448 4525 y(researc)m(h)40 b(in)d(computational)h(complexit)m(y)h (theory)-8 b(.)66 b(W)-8 b(e)40 b(surv)m(ey)e(v)-5 b(arious)38 b(areas)h(in)448 4638 y(complexit)m(y)32 b(c)m(ho)s(osing)h(pap)s(ers)e (more)h(for)g(their)g(historical)e(v)-5 b(alue)32 b(than)g(necessarily) 448 4751 y(the)j(imp)s(ortance)f(of)h(the)g(results.)52 b(W)-8 b(e)36 b(hop)s(e)e(that)h(this)e(giv)m(es)i(an)g(insigh)m(t)e (in)m(to)i(the)448 4863 y(ric)m(hness)29 b(and)h(depth)g(of)g(this)g (still)e(quite)i(y)m(oung)g(\014eld.)p eop %%Page: 3 3 3 2 bop 448 573 a Fo(2)135 b(Early)46 b(History)448 776 y Fn(While)28 b(w)m(e)i(can)g(trace)g(the)f(idea)g(of)h(\\e\016cien)m (t)g(algorithms")e(to)i(the)g(ancien)m(t)f(Greeks,)448 889 y(our)c(story)g(starts)h(with)e(the)h(seminal)f(1965)j(pap)s(er)d (of)h(Hartmanis)g(and)f(Stearns,)i(\\On)448 1002 y(the)31 b(Computational)e(Complexit)m(y)g(of)i(Algorithms")e([HS65)q(].)41 b(This)29 b(pap)s(er)g(laid)g(out)448 1115 y(the)f(de\014nitions)d(of)i (quan)m(ti\014ed)g(time)g(and)f(space)i(complexit)m(y)f(on)h(m)m (ultitap)s(e)e(T)-8 b(uring)448 1227 y(mac)m(hines)25 b(and)f(sho)m(w)m(ed)i(the)f(\014rst)f(results)g(of)i(the)f(form)g(giv) m(en)g(more)g(time)g(\(or)g(space\))448 1340 y(one)31 b(can)g(compute)f(more)h(things.)589 1453 y(A)37 b(m)m(ultitap)s(e)e(T) -8 b(uring)35 b(mac)m(hine)i(consists)f(of)g(some)h(\014xed)f(n)m(um)m (b)s(er)f(of)i(\\tap)s(es")448 1566 y(eac)m(h)42 b(of)e(whic)m(h)f(con) m(tains)i(an)f(in\014nite)e(n)m(um)m(b)s(er)h(of)h(tap)s(e)h(cells.)69 b(The)40 b(con)m(ten)m(ts)i(of)448 1679 y(a)c(tap)s(e)f(cell)g(comes)h (from)e(a)i(\014nite)e(set)i(of)f(sym)m(b)s(ols)f(called)g(the)h(tap)s (e)h(alphab)s(et)e(of)448 1792 y(the)g(T)-8 b(uring)33 b(mac)m(hine.)54 b(All)34 b(the)h(tap)s(es)g(initially)d(con)m(tain)j (only)f(a)i(sp)s(ecial)e(\\blank")448 1905 y(c)m(haracter)42 b(except)f(for)f(the)g(\014nite)f(input)f(written)h(at)i(the)f(b)s (eginning)d(of)k(the)f(\014rst)448 2018 y(tap)s(e.)h(Eac)m(h)29 b(tap)s(e)g(has)f(a)h(tap)s(e)g(head)g(sitting)e(on)i(the)g(\014rst)f (c)m(haracter)i(on)f(eac)m(h)h(tap)s(e.)448 2131 y(The)25 b(T)-8 b(uring)24 b(mac)m(hine)i(also)f(has)g(a)h(\014nite)f(state)i (memory)e(to)i(con)m(trol)f(its)e(op)s(erations.)448 2244 y(In)f(eac)m(h)i(step,)g(it)f(can)g(mo)m(v)m(e)h(eac)m(h)f(tap)s (e)g(indep)s(enden)m(tly)d(one)j(c)m(haracter)h(left)f(or)f(righ)m(t,) 448 2357 y(read)j(and)g(p)s(ossibly)d(c)m(hange)k(the)f(c)m(haracters)h (under)e(eac)m(h)i(head,)g(c)m(hange)g(its)e(curren)m(t)448 2469 y(state)39 b(and)f(decide)f(whether)g(to)h(halt)g(and)f(accept)i (or)f(reject)g(the)g(input.)61 b(Time)37 b(is)448 2582 y(measured)23 b(b)m(y)g(the)h(n)m(um)m(b)s(er)e(of)i(steps)f(b)s(efore) g(halting)f(as)i(a)g(function)e(of)h(the)h(length)f(of)448 2695 y(the)30 b(input.)38 b(Space)30 b(is)f(measured)f(as)i(the)g(n)m (um)m(b)s(er)e(of)i(di\013eren)m(t)f(c)m(haracter)i(lo)s(cations)448 2808 y(touc)m(hed)g(b)m(y)f(the)h(v)-5 b(arious)29 b(heads.)589 2921 y(The)20 b(Hartmanis-Stearns)g(pap)s(er)f(did)g(not)h(dev)m(elop)g (in)f(a)i(v)-5 b(acuum.)37 b(T)-8 b(uring)18 b([T)-8 b(ur36)q(],)448 3034 y(of)45 b(course,)i(dev)m(elop)s(ed)d(his)f (notion)g(of)h(a)h(computational)e(device)h(bac)m(k)h(in)e(1936.)448 3147 y(This)36 b(mac)m(hine)g(mo)s(del)g(did)f(and)i(still)e(do)s(es)h (form)h(the)g(basis)f(for)h(most)g(of)g(compu-)448 3260 y(tational)k(complexit)m(y)-8 b(.)73 b(Sligh)m(tly)39 b(earlier,)k(Y)-8 b(amada)42 b([Y)-8 b(am62)r(])42 b(studied)d (\\real-time)448 3373 y(computable)30 b(functions",)f(Myhill)e([Myh60)q (])j(lo)s(ok)m(ed)g(at)g(linear)f(b)s(ounded)e(automata)448 3486 y(and)37 b(Sm)m(ully)m(an)e([Sm)m(u61])i(considered)f(rudimen)m (tary)g(sets.)61 b(These)36 b(mo)s(dels)g(lo)s(ok)m(ed)448 3599 y(at)44 b(sp)s(eci\014c)e(time)h(and)g(space-b)s(ounded)f(mac)m (hines)g(but)h(did)e(not)j(giv)m(e)f(a)h(general)448 3711 y(approac)m(h)31 b(to)g(measuring)e(complexit)m(y)-8 b(.)589 3824 y(After)43 b(Hartmanis)f(and)g(Stearns)g(dev)m(elop)s(ed)f (the)i(general)f(metho)s(d)g(for)g(mea-)448 3937 y(suring)30 b(computational)h(resources,)h(one)f(can)h(ask)g(ho)m(w)f(the)h (di\013eren)m(t)e(v)-5 b(ariations)31 b(of)448 4050 y(T)-8 b(uring)36 b(mac)m(hines)h(a\013ect)h(the)g(complexit)m(y)f(of)g (problems.)59 b(Rabin)36 b([Rab63)q(])h(sho)m(ws)448 4163 y(problems)e(solv)-5 b(able)35 b(faster)h(b)m(y)g(t)m(w)m(o-tap)s (e)j(mac)m(hine)c(than)h(b)m(y)g(one-tap)s(e)h(mac)m(hines.)448 4276 y(Hennie)27 b(and)f(Stearns)h([HS66)q(])g(sho)m(w)g(that)h(a)f (2-tap)s(e)h(T)-8 b(uring)26 b(mac)m(hine)g(can)i(sim)m(ulate)448 4389 y(an)m(y)j(constan)m(t)h(tap)s(e)e(mac)m(hine)g(taking)h(only)e(a) i(logarithmic)e(factor)i(more)g(time.)589 4502 y(Hartmanis)26 b(and)g(Stearns)g(sho)m(w)g(that)h(giv)m(en)f(space)h(functions)e Fh(s)2843 4516 y Fj(1)2908 4502 y Fn(and)g Fh(s)3123 4516 y Fj(2)3189 4502 y Fn(with)g(a)448 4615 y(\\constructibilit)m(y") 33 b(condition)g(and)g Fh(s)1801 4629 y Fj(1)1840 4615 y Fn(\()p Fh(n)p Fn(\))f(=)g Fh(o)p Fn(\()p Fh(s)2222 4629 y Fj(2)2261 4615 y Fn(\()p Fh(n)p Fn(\)\),)k(i.e.,)f Fh(s)2701 4629 y Fj(1)2740 4615 y Fn(\()p Fh(n)p Fn(\))p Fh(=s)2953 4629 y Fj(2)2993 4615 y Fn(\()p Fh(n)p Fn(\))g(go)s(es)f(to) 448 4728 y(zero,)26 b(then)c(there)h(are)g(problems)e(computable)h(in)g (space)h Fh(s)2469 4742 y Fj(2)2508 4728 y Fn(\()p Fh(n)p Fn(\))g(but)f(not)h(space)h Fh(s)3248 4742 y Fj(1)3287 4728 y Fn(\()p Fh(n)p Fn(\).)448 4841 y(The)39 b(Hennie-Stearns)f (result)h(giv)m(es)g(the)g(b)s(est)g(kno)m(wn)g(time-hierarc)m(h)m(y)-8 b(,)42 b(getting)e(a)448 4953 y(separation)h(if)f Fh(t)1026 4967 y Fj(1)1065 4953 y Fn(\()p Fh(n)p Fn(\))15 b(log)i Fh(t)1371 4967 y Fj(1)1410 4953 y Fn(\()p Fh(n)p Fn(\))44 b(=)e Fh(o)p Fn(\()p Fh(t)1804 4967 y Fj(2)1844 4953 y Fn(\()p Fh(n)p Fn(\)\).)73 b(These)41 b(pro)s(ofs)f(use)g(straigh)m (tforw)m(ard)p eop %%Page: 4 4 4 3 bop 448 573 a Fn(diagonalization)30 b(argumen)m(ts)g(that)h(go)g (bac)m(k)h(to)f(Can)m(tor)g([Can74].)589 686 y(Nondeterministic)23 b(computation)h(allo)m(ws)f(a)i(T)-8 b(uring)23 b(mac)m(hine)g(to)i (mak)m(e)h(a)e(c)m(hoice)448 799 y(of)29 b(sev)m(eral)h(p)s(ossible)c (transitions.)39 b(W)-8 b(e)30 b(sa)m(y)f(the)g(mac)m(hine)g(accepts)h (if)e(an)m(y)h(collection)448 912 y(of)43 b(c)m(hoices)g(leads)f(to)h (an)g(accepting)g(state.)78 b(Nondeterministic)40 b(time)j(and)e(space) 448 1024 y(hierarc)m(hies)23 b(are)h(m)m(uc)m(h)g(tric)m(kier)f(to)i (pro)m(v)m(e)f(b)s(ecause)g(one)g(cannot)h(do)e(straigh)m(tforw)m(ard) 448 1137 y(diagonalization)30 b(on)g(nondeterministic)e(computations.) 589 1250 y(Sa)m(vitc)m(h)39 b([Sa)m(v70)r(])f(sho)m(w)m(ed)h(that)f (problems)f(computable)h(in)f(nondeterministic)448 1363 y(space)21 b Fh(s)p Fn(\()p Fh(n)p Fn(\))f(are)h(computable)f(in)f (deterministic)f(space)j Fh(s)2399 1330 y Fj(2)2438 1363 y Fn(\()p Fh(n)p Fn(\).)37 b(In)20 b(1972,)k(Ibarra)c([Iba72)q(])448 1476 y(using)38 b(translational)g(tec)m(hniques)g(of)i(Rub)m(y)e(and)h (Fisc)m(her)g([RF65)q(])g(used)g(Sa)m(vitc)m(h's)448 1589 y(theorem)32 b(to)g(sho)m(w)e(that)i(there)f(exist)g(problems)e (computable)i(in)f(nondeterministic)448 1702 y(space)45 b Fh(n)760 1669 y Fg(a)845 1702 y Fn(but)e(not)h(space)g Fh(n)1511 1669 y Fg(b)1589 1702 y Fn(for)f Fh(a)48 b(>)f(b)h Ff(\025)f Fn(1.)81 b(Sixteen)44 b(y)m(ears)g(later,)j(Immer-)448 1815 y(man)39 b([Imm88])g(and)f(Szelep)s(cs)m(\023)-43 b(en)m(yi)38 b([Sze88)r(])h(indep)s(enden)m(tly)c(sho)m(w)m(ed)k(that)g (nonde-)448 1928 y(terministic)28 b(space)h(is)g(closed)g(under)f (complemen)m(t.)40 b(The)29 b(Immerman-Szelep)s(cs)m(\023)-43 b(en)m(yi)448 2041 y(result)31 b(immediately)g(giv)m(es)h(a)h (nondeterministic)c(space)k(hierarc)m(h)m(y)f(as)g(tigh)m(t)g(as)h(the) 448 2154 y(the)e(deterministic)d(hierarc)m(h)m(y)-8 b(.)589 2267 y(F)g(or)28 b(nondeterministic)c(time,)k(Co)s(ok)f([Co)s(o73)q(])g (uses)g(a)g(more)g(careful)f(translation)448 2379 y(argumen)m(t)42 b(to)g(sho)m(w)f(problems)e(computable)i(in)f(nondeterministic)f(time)h Fh(n)3218 2346 y Fg(a)3301 2379 y Fn(but)448 2492 y(not)d(time)f Fh(n)884 2459 y Fg(b)954 2492 y Fn(for)g Fh(a)f(>)g(b)h Ff(\025)f Fn(1.)59 b(Seiferas,)37 b(Fisc)m(her)f(and)g(Mey)m(er)h ([SFM78)q(])g(giv)m(e)g(the)448 2605 y(curren)m(t)30 b(b)s(est)g(kno)m(wn)g(nondeterministic)d(time)j(hierarc)m(h)m(y)-8 b(,)31 b(getting)g(a)f(separation)g(if)448 2718 y Fh(t)481 2732 y Fj(1)521 2718 y Fn(\()p Fh(n)20 b Fn(+)g(1\))26 b(=)f Fh(o)p Fn(\()p Fh(t)1036 2732 y Fj(2)1075 2718 y Fn(\()p Fh(n)p Fn(\)\).)589 2831 y(In)35 b(1967,)j(Blum)c([Blu67)q(]) h(had)g(his)e(sp)s(eed-up)h(theorem:)51 b(F)-8 b(or)36 b(an)m(y)g(computable)448 2944 y(un)m(b)s(ounded)f(function)h Fh(r)s Fn(\()p Fh(n)p Fn(\))i(there)g(exists)f(a)h(computable)f (language)h Fh(L)f Fn(suc)m(h)g(that)448 3057 y(for)i(an)m(y)h(T)-8 b(uring)37 b(mac)m(hine)i(accepting)h Fh(L)f Fn(in)f(time)h Fh(t)p Fn(\()p Fh(n)p Fn(\))g(there)g(is)f(another)i(T)-8 b(uring)448 3170 y(mac)m(hine)41 b(accepting)g Fh(L)f Fn(in)f(time)i Fh(r)s Fn(\()p Fh(t)p Fn(\()p Fh(n)p Fn(\)\).)71 b(This)39 b(seems)i(to)g(violate)g(the)g(time)f(hi-)448 3283 y(erarc)m(h)m(y)35 b(men)m(tioned)e(earlier)g(but)g(one)h(m)m(ust) g(realize)f Fh(t)p Fn(\()p Fh(n)p Fn(\))h(will)d(not)j(necessarily)f(b) s(e)448 3396 y(time-constructible.)589 3509 y(Blum's)28 b(sp)s(eed-up)g(theorem)h(holds)e(not)i(only)f(for)h(time)g(but)f(also) h(for)f(space)i(and)448 3621 y(an)m(y)h(other)g(measure)g(ful\014lling) 26 b(a)31 b(small)e(list)g(of)i(axioms,)f(whic)m(h)g(w)m(e)h(no)m(w)f (call)g(Blum)448 3734 y(complexit)m(y)g(measures.)589 3847 y(So)s(on)25 b(after)h(w)m(e)f(sa)m(w)h(t)m(w)m(o)h(other)e(ma)5 b(jor)25 b(results)f(that)i(w)m(e)g(will)c(state)27 b(for)e(time)g(but) 448 3960 y(also)i(hold)f(for)h(all)f(Blum)g(complexit)m(y)h(measures.) 39 b(Indep)s(enden)m(tly)25 b(Boro)s(din)g([Bor72)r(])448 4073 y(and)j(T)-8 b(rakh)m(ten)m(brot)28 b([T)-8 b(ra64)q(])28 b(pro)m(v)m(ed)h(the)f(gap)g(theorem:)40 b(F)-8 b(or)28 b(an)m(y)g(computable)g(un-)448 4186 y(b)s(ounded)h Fh(r)s Fn(\()p Fh(n)p Fn(\))h(there)h(exist)g(a)g(computable)f(time)g(b)s (ound)f Fh(t)p Fn(\()p Fh(n)p Fn(\))i(suc)m(h)f(that)i(an)m(y)f(lan-) 448 4299 y(guage)23 b(computable)e(in)e(time)i Fh(t)p Fn(\()p Fh(n)p Fn(\))h(is)e(also)h(computable)g(in)f(time)h Fh(r)s Fn(\()p Fh(t)p Fn(\()p Fh(n)p Fn(\)\).)38 b(McCreigh)m(t)448 4412 y(and)30 b(Mey)m(er)h([MM69)r(])f(sho)m(w)m(ed)g(the)h(union)d (theorem:)41 b(Giv)m(en)30 b(an)m(y)g(computably)f(pre-)448 4525 y(sen)m(table)i(list)f(of)h(computable)f(time)g(b)s(ounds)f Fh(t)2104 4539 y Fj(1)2143 4525 y Fh(;)15 b(t)2216 4539 y Fj(2)2256 4525 y Fh(;)g(:)g(:)g(:)32 b Fn(suc)m(h)f(that)g Fh(t)2869 4539 y Fg(i)p Fj(+1)3013 4525 y Fh(>)26 b(t)3143 4539 y Fg(i)3202 4525 y Fn(for)k(all)448 4638 y Fh(i)e Fn(then)e(there)i(exist)f(a)g(time)g(b)s(ound)e Fh(t)i Fn(suc)m(h)f(that)i(a)f(problem)f(is)g(computable)h(in)e(time)448 4751 y Fh(t)30 b Fn(if)g(and)g(only)f(if)g(it)h(is)g(computable)g(in)f (time)h Fh(t)2073 4765 y Fg(i)2131 4751 y Fn(for)g(some)h Fh(i)p Fn(.)589 4863 y(In)40 b(1964,)k(Cobham)39 b([Cob64)q(])h(noted)h (that)f(the)g(set)h(of)f(problems)f(computable)448 4976 y(in)27 b(p)s(olynomial)d(time)k(remains)e(indep)s(enden)m(t)f(of)j (the)f(particular)f(deterministic)g(ma-)p eop %%Page: 5 5 5 4 bop 448 573 a Fn(c)m(hine)37 b(mo)s(del.)59 b(He)38 b(also)f(sho)m(w)m(ed)g(that)h(man)m(y)f(common)g(mathematical)g (functions)448 686 y(can)31 b(b)s(e)f(computed)g(in)f(p)s(olynomial)f (time.)589 799 y(In)41 b(1965,)47 b(Edmonds)40 b([Edm65b])i(in)e(his)g (pap)s(er)h(sho)m(wing)g(that)h(the)g(matc)m(hing)448 912 y(problem)20 b(has)i(a)f(p)s(olynomial-time)e(algorithm,)k(argues)f (that)g(p)s(olynomial-time)d(giv)m(es)448 1024 y(a)44 b(go)s(o)s(d)f(formalization)f(of)i(e\016cien)m(t)g(computation.)79 b(He)44 b(noted)f(the)h(wide)e(range)448 1137 y(of)37 b(problems)d(computable)i(in)f(p)s(olynomial)e(time)j(and)g(as)h(w)m (ell)e(the)h(fact)h(that)g(this)448 1250 y(class)28 b(of)g(problems)e (remains)g(the)i(same)g(under)f(man)m(y)h(di\013eren)m(t)f(reasonable)g (mo)s(dels)448 1363 y(of)42 b(computation.)74 b(In)40 b(another)i(pap)s(er,)h(Edmonds)d([Edm65a)q(])h(ga)m(v)m(e)j(an)d (informal)448 1476 y(description)30 b(of)h(nondeterministic)e(p)s (olynomial-time.)41 b(This)29 b(set)j(the)g(stage)h(for)e(the)448 1589 y Fi(P)26 b Fn(=)f Fi(NP)g Fn(question,)g(the)g(most)f(famous)h (problem)d(in)i(theoretical)g(computer)h(science)448 1702 y(that)31 b(w)m(e)g(discuss)e(in)g(Section)h(3.)589 1815 y(Sev)m(eral)g(Russians,)d(notably)i(Barzdin)f(and)g(T)-8 b(rakh)m(ten)m(brot,)31 b(indep)s(enden)m(tly)26 b(de-)448 1928 y(v)m(elop)s(ed)d(sev)m(eral)h(of)f(these)h(notions)f(of)g (complexit)m(y)g(during)e(the)j(sixties)e(though)h(their)448 2041 y(w)m(ork)31 b(w)m(as)g(not)f(kno)m(wn)g(to)h(the)g(W)-8 b(est)31 b(un)m(til)e(the)i(sev)m(en)m(ties.)448 2327 y Fo(3)135 b(NP-completeness)448 2530 y Fn(It)36 b(w)m(as)g(in)e(the)i (early)g(1970's)h(that)f(complexit)m(y)g(theory)g(\014rst)e(\015o)m(w)m (ered,)k(and)d(came)448 2643 y(to)42 b(pla)m(y)e(a)i(cen)m(tral)f(role) g(in)e(computer)i(science.)72 b(It)41 b(did)e(so)j(b)m(y)e(fo)s(cusing) g(on)h(one)448 2756 y(fundamen)m(tal)33 b(concept)i(and)e(on)g(the)h (results)f(and)g(ideas)g(stemming)g(from)g(it.)51 b(This)448 2869 y(concept)46 b(w)m(as)f Fi(NP)p Fn(-completeness)g(and)f(it)g(has) h(pro)m(v)m(ed)g(to)g(b)s(e)f(one)h(of)g(the)g(most)448 2982 y(insigh)m(tful)26 b(and)h(fundamen)m(tal)g(theories)h(in)f(the)i (mathematics)f(of)h(the)f(last)g(half)f(cen-)448 3095 y(tury)-8 b(.)40 b Fi(NP)p Fn(-completeness)28 b(captures)f(the)h(com)m (binatorial)e(di\016cult)m(y)g(of)h(a)h(n)m(um)m(b)s(er)e(of)448 3208 y(cen)m(tral)j(problems)d(whic)m(h)h(resisted)g(e\016cien)m(t)i (solution)e(and)g(pro)m(vides)g(a)i(metho)s(d)e(for)448 3320 y(pro)m(ving)h(that)i(a)f(com)m(binatorial)f(problem)g(is)g(as)h (in)m(tractable)g(as)g(an)m(y)g Fi(NP)g Fn(problem.)589 3433 y(By)d(the)f(late)g(1960's,)k(a)c(sizable)f(class)h(of)g(v)m(ery)g (applicable)e(and)i(signi\014can)m(t)f(prob-)448 3546 y(lems)30 b(whic)m(h)e(resisted)i(p)s(olynomial)d(time)j(solution)f(w)m (as)h(widely)e(recognized.)41 b(These)448 3659 y(problems)32 b(are)i(largely)f(optimization)f(problems)g(suc)m(h)h(as)g(the)h(tra)m (v)m(eling)g(salesman)448 3772 y(problem,)53 b(certain)48 b(sc)m(heduling)f(problems,)53 b(or)48 b(linear)g(programming)g (problems.)448 3885 y(They)39 b(all)f(ha)m(v)m(e)j(a)f(v)m(ery)f(large) h(n)m(um)m(b)s(er)e(of)h(p)s(ossible)e(solution)h(where)h(there)h(is)e (no)448 3998 y(ob)m(vious)28 b(w)m(a)m(y)i(to)f(\014nd)d(an)j(optimal)e (solution)g(other)i(than)f(a)h(brute)e(force)i(searc)m(h.)41 b(As)448 4111 y(time)26 b(passed)f(and)g(m)m(uc)m(h)h(e\013ort)g(w)m (as)g(exp)s(ended)e(on)i(attempts)h(at)f(e\016cien)m(tly)f(solving)448 4224 y(these)36 b(problems,)g(it)f(b)s(egan)g(to)h(b)s(e)f(susp)s (ected)g(that)h(there)g(w)m(as)g(no)f(suc)m(h)g(solution.)448 4337 y(Ho)m(w)m(ev)m(er,)j(there)c(w)m(as)g(no)g(hard)f(evidence)h (that)g(this)f(w)m(as)i(the)f(case)h(nor)e(w)m(as)i(there)448 4450 y(an)m(y)28 b(reason)g(to)g(susp)s(ect)f(that)h(these)f(problems)f (w)m(ere)i(in)e(an)m(y)i(sense)f(di\016cult)f(for)h(the)448 4562 y(same)e(reasons)f(or)g(in)f(the)h(same)g(w)m(a)m(ys.)40 b(The)23 b(theory)i(of)f Fi(NP)p Fn(-completeness)g(pro)m(vided)448 4675 y(precisely)29 b(this)h(evidence.)589 4788 y(Pro)m(ving)h(a)g (problem)e(in)g Fi(NP)i Fn(to)g(b)s(e)g Fi(NP)p Fn(-complete)g(tells)f (us)g(that)h(it)f(is)g(as)h(hard)448 4901 y(to)k(solv)m(e)g(as)f(an)m (y)h(other)f Fi(NP)g Fn(problem.)51 b(Said)32 b(another)j(w)m(a)m(y)-8 b(,)36 b(if)e(there)g(is)f(an)m(y)i Fi(NP)p Fn(-)p eop %%Page: 6 6 6 5 bop 448 573 a Fn(complete)31 b(problem)d(that)j(admits)e(an)g (e\016cien)m(t)i(solution)d(then)i(ev)m(ery)h Fi(NP)f Fn(problem)448 686 y(do)s(es)23 b(so.)39 b(The)23 b(question)f(of)i (whether)f(ev)m(ery)h Fi(NP)f Fn(problem)f(has)h(an)h(e\016cien)m(t)f (solution)448 799 y(has)29 b(resisted)f(the)h(e\013orts)h(of)f (computer)g(scien)m(tists)g(since)f(1970.)42 b(It)29 b(is)f(kno)m(wn)h(as)g(the)448 912 y Fi(P)44 b Fn(v)m(ersus)e Fi(NP)i Fn(problem)d(and)i(is)f(among)h(the)g(most)h(cen)m(tral)f(op)s (en)g(problems)e(of)448 1024 y(mathematics.)51 b(The)33 b(fact)h(that)g(a)g(v)m(ery)g(large)g(n)m(um)m(b)s(er)e(of)h(fundamen)m (tal)g(problems)448 1137 y(ha)m(v)m(e)f(b)s(een)e(sho)m(wn)g(to)i(b)s (e)e Fi(NP)p Fn(-complete)h(and)g(that)g(the)g(problem)e(of)i(pro)m (ving)e(that)448 1250 y Fi(P)36 b Fn(is)e(not)i Fi(NP)f Fn(has)h(pro)m(v)m(ed)f(to)h(b)s(e)f(so)g(di\016cult)f(has)h(made)g (this)f(problem)g(and)h(the)448 1363 y(connected)29 b(theory)g(one)f (of)g(the)g(most)h(celebrated)f(in)f(con)m(temp)s(orary)h(mathematics.) 448 1476 y(The)33 b Fi(P)f Fn(=)e Fi(NP)k Fn(problem)e(is)h(one)h(of)g (the)f(sev)m(en)i(Millennium)29 b(Prize)k(Problems)g(and)448 1589 y(solving)40 b(it)h(brings)e(a)i($1,000,000)46 b(prize)40 b(from)g(the)i(Cla)m(y)f(Mathematics)h(Institute)448 1702 y([Cla00)q(].)589 1815 y(Quite)e(surprisingly)-8 b(,)38 b(one)j(of)f(the)g(earliest)f(discussions)f(of)i(a)g(particular) f Fi(NP)p Fn(-)448 1928 y(complete)30 b(problem)e(and)h(the)h (implications)d(of)j(\014nding)d(an)j(e\016cien)m(t)g(solution)e(came) 448 2041 y(from)36 b(Kurt)g(G\177)-45 b(odel.)60 b(In)35 b(a)i(1956)i(letter)e(to)g(v)m(on)g(Neumann)f([Har86)q(,)h(Sip83)o(])g (G\177)-45 b(odel)448 2154 y(asks)39 b(v)m(on)f(Neumann)f(ab)s(out)h (the)h(complexit)m(y)e(of)i(what)f(is)f(no)m(w)h(kno)m(wn)g(to)h(b)s(e) e(an)448 2267 y Fi(NP)q Fn(-complete)32 b(problem)e(concerning)h(pro)s (ofs)g(in)g(\014rst-order)g(logic)g(and)g(asks)h(if)f(the)448 2379 y(problem)k(can)i(b)s(e)f(solv)m(ed)g(in)f(linear)g(or)i (quadratic)f(time.)58 b(In)36 b(fact,)j(G\177)-45 b(odel)37 b(seemed)448 2492 y(quite)g(optimistic)f(ab)s(out)h(\014nding)e(an)i (e\016cien)m(t)h(solution.)60 b(He)38 b(fully)d(realized)h(that)448 2605 y(doing)30 b(so)g(w)m(ould)g(ha)m(v)m(e)h(signi\014can)m(t)e (consequences.)589 2718 y(It)h(is)f(w)m(orth)g(noting)h(that)g(in)e(ab) s(out)h(the)h(same)g(p)s(erio)s(d)e(there)h(w)m(as)i(considerable)448 2831 y(e\013ort)22 b(b)m(y)f(Russian)e(mathematicians)h(w)m(orking)g (on)h(similar)d(com)m(binatorial)i(problems)448 2944 y(trying)30 b(to)h(pro)m(v)m(e)h(that)f(brute)f(force)h(w)m(as)g (needed)f(to)i(solv)m(e)f(them.)41 b(Sev)m(eral)31 b(of)g(these)448 3057 y(problems)e(ev)m(en)m(tually)h(turned)f(out)i(to)g(b)s(e)f Fi(NP)p Fn(-complete)h(as)g(w)m(ell)e([T)-8 b(ra64)r(].)589 3170 y(The)38 b(existence)g(of)g Fi(NP)p Fn(-complete)g(problems)e(w)m (as)i(pro)m(v)m(ed)g(indep)s(enden)m(tly)d(b)m(y)448 3283 y(Stephen)e(Co)s(ok)h(in)e(the)i(United)e(States)j(and)e(Leonid)f (Levin)h(in)f(the)i(So)m(viet)g(Union.)448 3396 y(Co)s(ok,)54 b(then)48 b(a)i(graduate)f(studen)m(t)g(at)g(Harv)-5 b(ard,)54 b(pro)m(v)m(ed)49 b(that)g(the)g(satis\014abil-)448 3509 y(it)m(y)39 b(problem)e(is)g Fi(NP)q Fn(-complete)i([Co)s(o71)q (].)66 b(Levin,)40 b(a)f(studen)m(t)f(of)h(Kolmogoro)m(v)g(at)448 3621 y(Mosco)m(w)28 b(State)f(Univ)m(ersit)m(y)-8 b(,)27 b(pro)m(v)m(ed)f(that)h(a)f(v)-5 b(arian)m(t)26 b(of)h(the)f(tiling)e (problem)g(is)i Fi(NP)p Fn(-)448 3734 y(complete)31 b([Lev73)r(].)589 3847 y(Researc)m(hers)50 b(stro)m(v)m(e)f(to)g(sho)m(w)g(other)f(in)m (teresting,)53 b(natural)47 b(problems)g Fi(NP)p Fn(-)448 3960 y(complete.)38 b(Ric)m(hard)21 b(Karp,)h(in)e(a)i(tremendously)e (in\015uen)m(tial)e(pap)s(er)i([Kar72)q(],)k(pro)m(v)m(ed)448 4073 y(that)h(eigh)m(t)f(cen)m(tral)h(com)m(binatorial)e(problems)f (are)i(all)f Fi(NP)q Fn(-complete.)39 b(These)24 b(prob-)448 4186 y(lems)30 b(included)e(the)j(the)g(clique)f(problem,)f(the)i (indep)s(enden)m(t)e(set)i(problem,)f(the)g(set)448 4299 y(co)m(v)m(er)i(problem,)d(and)h(the)h(tra)m(v)m(eling)f(salesman)g (problem,)f(among)i(others.)589 4412 y(Karp's)g(pap)s(er)f(presen)m (ted)h(sev)m(eral)g(k)m(ey)h(metho)s(ds)f(to)g(pro)m(v)m(e)h Fi(NP)q Fn(-completeness)448 4525 y(using)24 b(reductions)f(from)h (problems)f(previously)g(sho)m(wn)h(to)i(b)s(e)e Fi(NP)p Fn(-complete.)40 b(It)24 b(set)448 4638 y(up)d(a)h(general)g(framew)m (ork)g(for)f(pro)m(ving)g Fi(NP)p Fn(-completeness)h(results)f(and)g (established)448 4751 y(sev)m(eral)37 b(useful)d(tec)m(hniques)i(for)g (suc)m(h)f(pro)s(ofs.)57 b(In)36 b(the)g(follo)m(wing)e(y)m(ears,)39 b(and)c(con-)448 4863 y(tin)m(uing)c(un)m(til)g(to)s(da)m(y)-8 b(,)35 b(literally)30 b(thousands)i(of)h(problems)e(ha)m(v)m(e)j(b)s (een)e(sho)m(wn)g(to)i(b)s(e)448 4976 y Fi(NP)q Fn(-complete.)45 b(A)32 b(pro)s(of)e(of)i Fi(NP)q Fn(-completeness)g(has)f(come)i(to)f (signify)e(the)h(\(w)m(orst)p eop %%Page: 7 7 7 6 bop 448 573 a Fn(case\))41 b(in)m(tractabilit)m(y)d(of)h(a)h (problem.)65 b(Once)39 b(pro)m(v)m(ed)h Fi(NP)p Fn(-complete,)j (researc)m(hers)448 686 y(turn)28 b(to)h(other)f(w)m(a)m(ys)h(of)g (trying)e(to)i(solv)m(e)g(the)f(problem,)g(usually)e(using)g(appro)m (xima-)448 799 y(tion)i(algorithms)f(to)i(giv)m(e)g(an)f(appro)m (ximate)g(solution)f(or)h(probabilistic)d(metho)s(ds)i(to)448 912 y(solv)m(e)k(the)g(problem)d(in)h(\\most")j(cases.)589 1024 y(Another)f(fundamen)m(tal)f(step)g(w)m(as)h(tak)m(en)h(around)d (1970)k(b)m(y)d(Mey)m(er)i(and)e(Sto)s(c)m(k-)448 1137 y(mey)m(er)37 b([MS72)q(],)i([Sto76)q(].)60 b(They)36 b(de\014ned)f(the)h(p)s(olynomial)e(hierarc)m(h)m(y)i(in)g(analogy)448 1250 y(with)j(the)h(arithmetic)g(hierarc)m(h)m(y)f(of)h(Kleene.)70 b(This)38 b(hierarc)m(h)m(y)h(is)h(de\014ned)e(b)m(y)i(it-)448 1363 y(erating)g(the)f(notion)g(of)h(p)s(olynomial)d(jump,)j(in)e (analogy)i(with)e(the)i(T)-8 b(uring)38 b(jump)448 1476 y(op)s(erator.)j(This)29 b(hierarc)m(h)m(y)g(has)h(pro)m(v)m(en)g (useful)f(in)g(classifying)f(man)m(y)i(hard)f(com)m(bi-)448 1589 y(natorial)k(problems)f(whic)m(h)g(do)h(not)h(lie)e(in)g Fi(NP)p Fn(.)50 b(It)33 b(is)g(explored)f(in)g(more)i(detail)e(in)448 1702 y(Section)f(4.2.)589 1815 y(Of)39 b(course,)j(all)c(problems)g(in) g(the)i(p)s(olynomial)c(hierarc)m(h)m(y)j(are)h(recursiv)m(e)f(and)448 1928 y(in)d(fact)i(v)m(ery)f(simple)e(problems)g(within)f(the)j(v)-5 b(ast)37 b(expanse)g(of)g(all)f(recursiv)m(e)g(sets.)448 2041 y(So)c(are)g(there)f(natural)g(problems)f(whic)m(h)g(are)i (recursiv)m(e)f(and)f(are)i(not)g(captured)f(b)m(y)448 2154 y(the)j(hierarc)m(h)m(y?)47 b(The)33 b(answ)m(ers)g(is)f(y)m(es)h (and)g(results)e(in)h(the)h(exploration)g(of)g(sev)m(eral)448 2267 y(imp)s(ortan)m(t)25 b(larger)g(complexit)m(y)f(classes)i(whic)m (h)d(con)m(tain)j(the)f(p)s(olynomial)e(hierarc)m(h)m(y)-8 b(.)448 2379 y(One)25 b(suc)m(h)g(class)g(is)f Fi(PSP)-9 b(A)m(CE)p Fn(,)27 b(those)e(problems)f(whic)m(h)g(can)h(b)s(e)f(solv)m (ed)h(using)f(w)m(ork)448 2492 y(space)34 b(whic)m(h)d(is)h(of)h(p)s (olynomial)d(length)j(relativ)m(e)g(to)g(the)g(length)f(of)h(the)h (problem's)448 2605 y(input.)45 b(Just)31 b(as)i(with)e Fi(P)h Fn(and)g Fi(NP)p Fn(,)h(the)g(full)d(exten)m(t)j(of)g Fi(PSP)-9 b(A)m(CE)33 b Fn(is)e(not)h(kno)m(wn.)448 2718 y Fi(PSP)-9 b(A)m(CE)26 b Fn(con)m(tains)e Fi(P)h Fn(and)f Fi(NP)q Fn(.)38 b(It)25 b(is)f(not)h(kno)m(wn)f(if)g(either)g(of)h (these)g(conclusions)448 2831 y(are)g(prop)s(er.)38 b(Settling)23 b(these)i(questions)f(w)m(ould)g(again)g(b)s(e)g(signi\014can)m(t)g (steps)h(forw)m(ard)448 2944 y(in)k(this)h(theory)-8 b(.)589 3057 y(The)32 b(notion)f(of)h Fi(PSP)-9 b(A)m(CE)q Fn(-completeness)32 b(is)f(de\014ned)g(v)m(ery)h(similarly)d(to)k Fi(NP)p Fn(-)448 3170 y(completeness,)e(and)e(has)h(b)s(een)f(studies)f (alongside)h(the)h(the)g Fi(NP)q Fn(-completeness)g(no-)448 3283 y(tion.)43 b(Namely)-8 b(,)32 b(a)f(problem)f(C)h(is)f Fi(PSP)-9 b(A)m(CE)p Fn(-complete)32 b(if)e(it)h(is)f(in)g Fi(PSP)-9 b(A)m(CE)32 b Fn(and)448 3396 y(if)25 b(an)m(y)i(other)f Fi(PSP)-9 b(A)m(CE)26 b Fn(problem)f(can)h(b)s(e)g(reduced)f(to)i(it)e (in)g(p)s(olynomial)e(time.)39 b(As)448 3509 y(is)32 b(the)h(case)h(with)e Fi(NP)p Fn(-complete)i(problems,)e Fi(PSP)-9 b(A)m(CE)q Fn(-complete)33 b(problems)f(are)448 3621 y(quite)e(common)h(and)f(often)h(arise)f(quite)g(naturally)-8 b(.)40 b(T)m(ypical)30 b Fi(PSP)-9 b(A)m(CE)p Fn(-complete)448 3734 y(problems)26 b(are)h(or)g(arise)f(from)h(generalized)f(games)i (suc)m(h)e(as)i(hex)e(or)h(c)m(hec)m(k)m(ers)i(pla)m(y)m(ed)448 3847 y(on)k(b)s(oards)e(of)i(un)m(b)s(ounded)d(\014nite)h(size)i(\(see) g([GJ79)q(]\).)48 b(Bey)m(ond)33 b Fi(PSP)-9 b(A)m(CE)33 b Fn(lie)f(the)448 3960 y(exp)s(onen)m(tial)j(time)h(\()p Fi(EXPTIME)p Fn(\))h(and)e(exp)s(onen)m(tial)g(space)i(complexit)m(y)f (classes.)448 4073 y(A)i(small)f(n)m(um)m(b)s(er)g(of)h(natural)f (problems)f(ha)m(v)m(e)k(b)s(een)d(sho)m(wn)g(complete)i(for)f(these) 448 4186 y(classes)e(\(see)g([GJ79)q(]\),)i(and)c(as)i(w)m(ell)e Fi(EXPTIME)h Fn(is)f(the)i(smallest)e(deterministic)448 4299 y(class)c(whic)m(h)g(has)g(b)s(een)f(pro)m(v)m(ed)i(to)g(con)m (tain)g Fi(NP)p Fn(.)448 4585 y Fo(4)135 b(Structural)45 b(Complexit)l(y)448 4788 y Fn(By)35 b(the)f(early)f(1970's,)k(the)d (de\014nitions)e(of)i(time)f(and)h(space-b)s(ounded)e(complexit)m(y)448 4901 y(classes)23 b(w)m(ere)g(precisely)e(established)g(and)g(the)i (imp)s(ort)e(of)i(the)f(class)g Fi(NP)h Fn(and)f(of)h Fi(NP)p Fn(-)p eop %%Page: 8 8 8 7 bop 448 573 a Fn(complete)39 b(problems)e(realized.)64 b(A)m(t)40 b(this)d(p)s(oin)m(t)h(e\013ort)h(turned)e(to)j (understanding)448 686 y(the)28 b(relationships)d(b)s(et)m(w)m(een)j (complexit)m(y)f(classes)h(and)f(the)g(prop)s(erties)f(of)i(problems) 448 799 y(within)35 b(the)j(principal)c(classes.)62 b(In)37 b(particular,)h(atten)m(tion)g(w)m(as)g(fo)s(cused)f(on)h Fi(NP)p Fn(-)448 912 y(complete)32 b(problems)e(and)g(their)g(prop)s (erties)g(and)h(on)g(the)g(structure)g(of)g(complexit)m(y)448 1024 y(classes)37 b(b)s(et)m(w)m(een)h Fi(LOGSP)-9 b(A)m(CE)37 b Fn(and)f Fi(PSP)-9 b(A)m(CE)q Fn(.)61 b(W)-8 b(e)38 b(brie\015y)e(surv)m(ey)g(some)i(of)448 1137 y(these)31 b(studies)e(here.)448 1381 y Fe(4.1)113 b(The)37 b(Isomorphism)f (Conjecture)448 1552 y Fn(In)21 b(the)g(mid-70's,)i(building)18 b(on)j(earlier)f(w)m(ork)h(on)g(G\177)-45 b(odel)22 b(n)m(um)m(b)s (erings)d([HB75)q(,)j(Har82)q(])448 1665 y(and)43 b(in)e(analogy)i (with)f(the)h(w)m(ell-kno)m(wn)e(result)h(of)h(Myhill)d(from)j (computabilit)m(y)448 1778 y(theory)26 b([Myh55)q(],)i(Berman)d(and)h (Hartmanis)f([BH77)r(,)h(HB78)q(])g(form)m(ulated)f(their)g(iso-)448 1891 y(morphism)k(conjecture.)42 b(The)31 b(conjecture)g(stated)h(that) f(all)f Fi(NP)p Fn(-complete)h(sets)h(are)448 2004 y Fi(P)p Fn(-isomorphic)f(\(that)j(is,)e(isomorphic)e(via)i(p)s (olynomial)e(time)i(computable)g(and)g(in-)448 2117 y(v)m(ertible)43 b(isomorphisms\).)77 b(This)41 b(conjecture)j(serv)m(ed)g(as)g(a)f (springb)s(oard)e(for)i(the)448 2230 y(further)35 b(study)g(of)h(the)g (structure)f(of)h Fi(NP)q Fn(-complete)g(sets.)58 b(As)36 b(evidence)f(for)h(their)448 2343 y(conjecture,)26 b(Berman)e(and)g (Hartmanis)f(and)g(others)h([MY85)r(,)g(KMR87)q(])g(w)m(ere)g(able)f (to)448 2456 y(giv)m(e)32 b(simple,)f(easily)f(c)m(hec)m(k)-5 b(able)33 b(prop)s(erties)d(of)i Fi(NP)p Fn(-complete)h(sets)f(whic)m (h)e(implied)448 2569 y(they)e(w)m(ere)h(isomorphic.)37 b(Using)27 b(these,)i(they)f(pro)m(v)m(ed)h(that)f(all)f(of)h(the)g Fi(kno)m(wn)g(NP)p Fn(-)448 2682 y(complete)34 b(sets)g(w)m(ere)g(in)e (fact)i Fi(P)p Fn(-isomorphic.)48 b(This)31 b(conjecture)j(remains)f (an)g(op)s(en)448 2794 y(question)d(to)s(da)m(y)-8 b(.)43 b(A)31 b(p)s(ositiv)m(e)e(resolution)h(of)g(the)h(conjecture)h(w)m (ould)d(imply)f(that)k Fi(P)448 2907 y Fn(is)e(not)i(equal)e(to)i Fi(NP)p Fn(.)43 b(Muc)m(h)31 b(e\013ort)h(w)m(as)f(fo)s(cused)f(on)h (pro)m(ving)f(the)i(con)m(v)m(erse,)g(that)448 3020 y(assuming)23 b Fi(P)h Fn(is)f(not)h Fi(NP)g Fn(then)g(the)g(isomorphism)d (conjecture)k(holds.)37 b(This)22 b(remains)448 3133 y(an)31 b(op)s(en)e(question)h(to)s(da)m(y)-8 b(.)589 3246 y(As)25 b(the)h(n)m(um)m(b)s(er)d(of)j(kno)m(wn)e Fi(NP)q Fn(-complete)h(problems)f(grew)h(during)e(the)i(1970's,)448 3359 y(the)g(structure)e(and)h(prop)s(erties)e(of)i(these)h(problems)d (b)s(egan)i(to)h(b)s(e)e(examined.)38 b(While)448 3472 y(v)m(ery)30 b(disparate,)g(the)g Fi(NP)p Fn(-complete)h(sets)f(ha)m(v) m(e)h(certain)e(common)h(prop)s(erties.)39 b(F)-8 b(or)448 3585 y(example,)37 b(they)e(are)h(all)e(rather)h(dense)g(sets.)55 b(Densit)m(y)36 b(of)f(a)h(set)f(is)g(measured)f(here)448 3698 y(simply)29 b(in)i(the)g(sense)h(of)g(ho)m(w)f(man)m(y)h(string)e (of)i(a)g(giv)m(en)f(length)g(are)h(in)f(the)g(set.)45 b(So)448 3811 y(\(assuming)26 b(a)g(binary)f(enco)s(ding)g(of)i(a)g (set\))g(there)g(are)f(2)2371 3778 y Fg(n)2445 3811 y Fn(di\013eren)m(t)g(strings)f(of)i(length)448 3924 y(n.)53 b(W)-8 b(e)36 b(sa)m(y)g(that)f(set)g(S)f(is)g(sparse)g(if)g(there)h (is)e(a)j(p)s(olynomial)c(p\(n\))i(whic)m(h)f(b)s(ounds)448 4036 y(the)j(n)m(um)m(b)s(er)f(of)h(strings)f(in)g(S)g(of)h(length)f (n,)i(for)f(ev)m(ery)h(n.)57 b(It)36 b(is)f(dense)g(otherwise.)448 4149 y(All)29 b(kno)m(wn)h Fi(NP)p Fn(-complete)i(sets)e(are)h(dense.) 589 4262 y(One)26 b(consequence)h(of)f(the)h(isomorphism)c(conjecture)k (is)e(that)i(no)f Fi(NP)p Fn(-complete)448 4375 y(set)f(can)f(b)s(e)f (sparse.)38 b(As)24 b(with)e(the)i(isomorphism)d(conjecture,)27 b(this)22 b(consequence)j(im-)448 4488 y(plies)h(that)j Fi(P)f Fn(is)f(not)h Fi(NP)g Fn(and)g(so)g(it)f(is)g(unlik)m(ely)f (that)i(a)g(pro)s(of)f(of)h(this)f(consequence)448 4601 y(will)32 b(so)s(on)j(b)s(e)f(forthcoming.)54 b(Berman)35 b(and)f(Hartmanis)g(also)h(conjectured)g(that)g(if)448 4714 y Fi(P)41 b Fn(in)e(not)h(equal)f(to)i Fi(NP)g Fn(there)f(are)g (no)g(sparse)g Fi(NP)p Fn(-complete)h(sets.)70 b(This)38 b(con-)448 4827 y(jecture)31 b(w)m(as)g(settled)g(a\016rmativ)m(ely)f (b)m(y)g(the)h(famous)f(result)f(of)i(Mahaney)g([Mah82)r(].)448 4940 y(Mahaney's)39 b(elegan)m(t)g(pro)s(of)f(used)f(sev)m(eral)i(new)e (coun)m(ting)h(tec)m(hniques)g(and)f(had)h(a)p eop %%Page: 9 9 9 8 bop 448 573 a Fn(lasting)30 b(impact)g(on)g(w)m(ork)h(in)e (structural)g(complexit)m(y)h(theory)-8 b(.)448 816 y Fe(4.2)113 b(The)37 b(P)m(olynomial)e(Hierarc)m(h)m(y)448 988 y Fn(While)40 b(n)m(umerous)g(hard)f(decision)h(problems)f(ha)m(v)m (e)j(b)s(een)e(pro)m(v)m(ed)h Fi(NP)p Fn(-complete,)448 1101 y(a)f(small)d(n)m(um)m(b)s(er)g(are)j(outside)e Fi(NP)h Fn(and)f(ha)m(v)m(e)i(escap)s(ed)f(this)e(classi\014cation.)65 b(An)448 1214 y(extended)33 b(classi\014cation,)f(the)h(p)s(olynomial)c (time)j(hierarc)m(h)m(y)g(\()p Fi(PH)q Fn(\),)i(w)m(as)e(pro)m(vided) 448 1327 y(b)m(y)g(Mey)m(er)h(and)e(Sto)s(c)m(kmey)m(er)i([Sto76)q(].) 45 b(They)31 b(de\014ned)f(the)i(hierarc)m(h)m(y)-8 b(,)32 b(a)g(collection)448 1440 y(of)39 b(classes)f(b)s(et)m(w)m(een)h Fi(P)g Fn(and)e Fi(PSP)-9 b(A)m(CE)q Fn(,)40 b(in)d(analogy)i(with)e (Kleene's)h(arithmetic)448 1552 y(hierarc)m(h)m(y)-8 b(.)589 1665 y(The)31 b(p)s(olynomial)e(time)i(hierarc)m(h)m(y)g(\()p Fi(PH)q Fn(\))g(consists)g(of)h(an)f(in\014nite)e(sequence)j(of)448 1778 y(classes)27 b(within)c Fi(PSP)-9 b(A)m(CE)q Fn(.)39 b(The)26 b(b)s(ottom)g(\(0)2068 1745 y Fg(th)2139 1778 y Fn(\))h(lev)m(el)f(of)g(the)g(hierarc)m(h)m(y)g(is)f(just)h(the)448 1891 y(class)33 b Fi(P)p Fn(.)50 b(The)33 b(\014rst)f(lev)m(el)h(is)g (the)g(class)g Fi(NP)q Fn(.)49 b(The)33 b(second)g(lev)m(el)g(are)h (all)e(problems)448 2004 y(in)h Fi(NP)h Fn(relativ)m(e)g(to)h(an)e Fi(NP)h Fn(oracle,)i(etc.)52 b(Iterating)34 b(this)f(idea)g(to)i(all)e (\014nite)f(lev)m(els)448 2117 y(yields)d(the)h(full)f(hierarc)m(h)m(y) -8 b(.)589 2230 y(If)31 b Fi(P)p Fn(=)p Fi(PSP)-9 b(A)m(CE)32 b Fn(then)f(the)h(whole)f Fi(PH)g Fn(collapses)g(to)h(the)g(class)f Fi(P)p Fn(.)44 b(Ho)m(w)m(ev)m(er,)448 2343 y(quite)33 b(the)g(opp)s(osite)f(is)g(b)s(eliev)m(ed)g(to)i(b)s(e)e(the)h(case,)i (namely)d(that)i(the)f Fi(PH)g Fn(is)f(strict)448 2456 y(in)h(the)h(sense)g(that)h(eac)m(h)g(lev)m(el)f(of)h(the)f(hierarc)m (h)m(y)g(is)f(a)h(prop)s(er)f(subset)g(of)i(the)f(next)448 2569 y(lev)m(el.)44 b(While)30 b(ev)m(ery)j(class)e(in)f(the)i Fi(PH)g Fn(is)e(con)m(tained)i(in)e Fi(PSP)-9 b(A)m(CE)q Fn(,)32 b(the)g(con)m(v)m(erse)448 2682 y(is)f(not)h(true)f(if)f(the)i (hierarc)m(h)m(y)f(is)f(strict.)44 b(In)31 b(this)f(case,)j Fi(PSP)-9 b(A)m(CE)32 b Fn(con)m(tains)g(man)m(y)448 2794 y(problems)f(not)i(in)e(the)h Fi(PH)h Fn(and)f(in)f(fact)i(has)f (a)h(v)m(ery)g(complex)f(structure)g(\(see,)i(for)448 2907 y(example,)d([AS89]\).)448 3151 y Fe(4.3)113 b(Alternation)448 3322 y Fn(Another)26 b(unifying)d(and)i(imp)s(ortan)m(t)g(thread)g(of)h (results)e(whic)m(h)h(also)g(originated)g(dur-)448 3435 y(ing)c(the)h(1970's)i(w)m(as)e(the)g(w)m(ork)g(on)f Fi(alternation)h Fn(initiated)e(out)i(b)m(y)f(Kozen,)j(Chandra)448 3548 y(and)30 b(Sto)s(c)m(kmey)m(er)i([CKS81].)43 b(The)30 b(idea)g(b)s(ehind)e(alternation)i(is)g(to)h(classify)f(com)m(bi-)448 3661 y(natorial)j(problems)e(using)g(an)i(alternating)g(T)-8 b(uring)31 b(mac)m(hine,)j(a)f(generalization)g(of)448 3774 y(a)k(nondeterministic)e(T)-8 b(uring)35 b(mac)m(hine.)59 b(In)m(tuitiv)m(ely)35 b(a)j(nondeterministic)c(T)-8 b(uring)448 3887 y(mac)m(hine)42 b(can)g(b)s(e)f(though)m(t)h(of)g(as)g (ha)m(ving)f(an)h(existen)m(tial)f(acceptance)j(criterion.)448 4000 y(That)32 b(is,)e(an)i(input)d(to)j(the)g(TM)f(is)f(accepted)j(if) d(there)i Fd(exists)f Fn(a)h(computation)f(path)448 4113 y(of)j(the)g(mac)m(hine)f(whic)m(h)f(results)h(in)f(acceptance.)52 b(Similarly)-8 b(,)32 b(w)m(e)i(could)e(consider)h(a)448 4226 y(univ)m(ersal)d(acceptance)k(criterion)c(whereb)m(y)h(a)h (nondeterministic)d(mac)m(hine)j(accepts)448 4339 y(of)g(all)f (computation)h(paths)f(lead)h(to)g(acceptance.)48 b(Restricting)31 b(ourselv)m(es)g(to)i(p)s(oly-)448 4452 y(nomial)38 b(length)h (alternation,)i(w)m(e)e(see)h(that)g Fi(NP)f Fn(can)g(b)s(e)g(c)m (haracterized)h(as)g(those)448 4564 y(problems)28 b(accepted)i(b)m(y)f (nondeterministic)e(TM)i(running)d(in)i(p)s(olynomial)f(time)h(us-)448 4677 y(ing)j(the)h(existen)m(tial)f(acceptance)j(criterion.)43 b(Similarly)-8 b(,)28 b(the)k(univ)m(ersal)e(acceptance)448 4790 y(criterion)d(with)g(the)h(same)h(t)m(yp)s(e)f(of)h(mac)m(hines)e (de\014nes)g(the)i(class)f(co-)p Fi(NP)h Fn(consisting)448 4903 y(of)39 b(problems)d(whose)i(complemen)m(ts)h(are)f(in)f Fi(NP)p Fn(.)65 b(F)-8 b(urthermore,)40 b(w)m(e)f(can)f(iterate)p eop %%Page: 10 10 10 9 bop 448 573 a Fn(these)31 b(t)m(w)m(o)i(acceptance)g(metho)s(ds,)d (for)g(example)h(asking)f(that)h(there)g(exist)g(an)f(path)448 686 y(of)j(a)h(TM)e(suc)m(h)h(that)g(for)g(all)e(paths)i(extending)f (that)h(path)g(there)g(exists)f(an)h(exten-)448 799 y(sion)d(of)h(that) g(path)f(whic)m(h)f(accepts.)43 b(This)28 b(idea)i(giv)m(es)h(a)g(mac)m (hine)f(implemen)m(tation)448 912 y(of)39 b(the)f(notion)g(of)g (alternations)g(of)g(univ)m(ersal)e(and)i(existen)m(tial)g(quan)m (ti\014ers.)63 b(It)38 b(is)448 1024 y(not)31 b(hard)f(to)i(see)f(that) h(\014nitely)d(man)m(y)h(alternations)h(results)e(in)h(the)h(\014nite)f (lev)m(els)g(of)448 1137 y(the)j(p)s(olynomial)e(time)h(hierarc)m(h)m (y)h(and)f(that)i(alternating)e(p)s(olynomial)e(time)j(is)f(the)448 1250 y(same)24 b(thing)f(as)h Fi(PSP)-9 b(A)m(CE)q Fn(.)38 b(Other)24 b(relationship)d(b)s(et)m(w)m(een)j(time)g(and)f(space)h (classes)448 1363 y(de\014ned)k(using)f(alternation)h(can)h(b)s(e)f (found)f(in)g([CKS81],)i(for)g(example,)g(alternating)448 1476 y(log)i(space)g(=)f Fi(P)g Fn(and)g(alternating)g Fi(PSP)-9 b(A)m(CE)26 b Fn(=)f Fi(EXPTIME)o Fn(.)448 1720 y Fe(4.4)113 b(Logspace)448 1891 y Fn(T)-8 b(o)28 b(this)d(p)s(oin)m(t)h(all)g(the)h(complexit)m(y)f(classes)h(w)m(e)g (ha)m(v)m(e)i(considered)c(con)m(tain)i(the)g(class)448 2004 y Fi(P)j Fn(of)g(p)s(olynomial)d(time)i(computable)h(problems.)38 b(F)-8 b(or)31 b(some)f(in)m(teresting)f(problems)448 2117 y(it)20 b(is)g(useful)e(to)j(consider)e(classes)i(within)d Fi(P)i Fn(and)g(particularly)e(the)i(seemingly)g(smaller)448 2230 y(space)25 b(classes)g(of)g(deterministic)d(log)j(space,)i (denoted)d Fi(L)p Fn(,)i(and)e(nondeterministic)e(log)448 2343 y(space,)34 b(denoted)f Fi(NL)p Fn(.)47 b(These)32 b(classes)h(pro)m(vide)f(a)h(measure)f(with)f(whic)m(h)h(to)h(distin-) 448 2456 y(guish)e(b)s(et)m(w)m(een)h(some)h(in)m(teresting)e(problems) f(within)f Fi(P)p Fn(,)k(and)f(presen)m(t)g(in)m(teresting)448 2569 y(issues)d(in)g(their)h(o)m(wn)g(righ)m(t.)589 2682 y(A)m(t)41 b(\014rst)d(glance)i(logarithmic)e(space)i(is)f(a)h (problematic)e(notion)h(at)h(b)s(est.)67 b(An)448 2794 y(input)22 b(of)h(length)g Fh(n)f Fn(tak)m(es)j Fh(n)e Fn(squares)f(b)m(y)i(itself,)f(so)h(ho)m(w)f(can)h(a)f(computation)g (on)g(suc)m(h)448 2907 y(an)g(input)e(tak)m(e)j(only)e(log)h(n)f (space?)38 b(The)23 b(answ)m(er)f(lies)g(in)f(c)m(hanging)i(our)f (computation)448 3020 y(mo)s(del)h(sligh)m(tly)f(to)j(only)e(coun)m(t)h (the)g(space)g(tak)m(en)h(b)m(y)f(the)g(computation)g(and)f(not)h(the) 448 3133 y(space)30 b(of)f(the)g(input.)38 b(F)-8 b(ormally)g(,)29 b(this)f(is)g(done)g(b)m(y)h(considering)e(an)h(\\o\013-line)h(T)-8 b(uring)448 3246 y(mac)m(hine.")82 b(This)42 b(is)h(a)h (\(deterministic)e(or)i(nondeterministic\))d(T)-8 b(uring)42 b(mac)m(hine)448 3359 y(whose)36 b(input)e(is)g(written)h(on)h(a)g(sp)s (ecial)e(read-only)h(input)f(tap)s(e.)57 b(Computation)35 b(is)448 3472 y(carried)g(out)h(on)f(read-write)g(w)m(ork)h(tap)s(es)f (whic)m(h)g(are)h(initially)c(blank.)54 b(The)35 b(space)448 3585 y(complexit)m(y)43 b(of)g(the)h(computation)e(is)h(then)f(tak)m (en)i(to)g(b)s(e)f(the)g(amoun)m(t)g(of)g(space)448 3698 y(used)31 b(on)h(the)f(w)m(ork)h(tap)s(es.)45 b(So)31 b(in)f(particular)g(this)h(space)h(can)g(b)s(e)f(less)g(than)g Fh(n)p Fn(,)h(the)448 3811 y(length)k(of)h(the)f(input)e(to)j(the)g (computation.)58 b(W)-8 b(e)38 b(de\014ne)d(logspace,)k Fi(L)p Fn(,)f(to)f(b)s(e)f(the)448 3924 y(class)i(of)f(languages)h (decided)e(b)m(y)i(deterministic)d(T)-8 b(uring)36 b(mac)m(hines)h (whic)m(h)f(use)h(at)448 4036 y(most)26 b Fh(O)s Fn(\(log)17 b Fh(n)p Fn(\))25 b(tap)s(e)h(squares.)39 b(Similarly)-8 b(,)23 b Fi(NL)i Fn(is)g(de\014ned)f(using)g(nondeterministic)448 4149 y(T)-8 b(uring)29 b(mac)m(hines)h(with)f(the)i(same)f(space)h(b)s (ound.)589 4262 y(It)f(is)e(straigh)m(tforw)m(ard)g(to)i(c)m(hec)m(k)h (that)e Fi(L)c Ff(\022)g Fi(NL)g Ff(\022)g Fi(P)p Fn(,)30 b(and)e(these)i(three)f(classes)448 4375 y(are)h(though)m(t)f(to)h(b)s (e)e(distinct.)39 b(There)28 b(are)h(a)h(n)m(um)m(b)s(er)d(of)i(non)m (trivial)e(problems)g(solv-)448 4488 y(able)32 b(in)f Fi(L)h Fn(\(for)h(example)f(see)h([LZ77)q(]\))f(as)h(w)m(ell)e(as)i (problems)e(kno)m(wn)h(to)h(b)s(e)e(in)g Fi(NL)448 4601 y Fn(whic)m(h)36 b(are)h(not)g(b)s(eliev)m(ed)e(to)i(b)s(e)f(in)g Fi(L)g Fn(\(for)h(example)f(see)h([Sa)m(v73)r(,)g(Jon75])g(\).)60 b(Nu-)448 4714 y(merous)31 b(problems)e(in)g Fi(P)i Fn(are)g(though)m (t)h(to)f(lie)f(outside)g(of)h Fi(L)f Fn(or)h Fi(NL)p Fn(.)42 b(F)-8 b(or)31 b(example,)448 4827 y(one)j(suc)m(h)e(problem)g (is)g(the)h(circuit)f(v)-5 b(alue)32 b(problem,)g(the)i(problem)d(of)i (determining)448 4940 y(the)d(v)-5 b(alue)28 b(of)h(a)h(Bo)s(olean)f (circuit,)f(giv)m(en)h(inputs)e(to)j(the)f(circuit.)39 b(The)29 b(circuit)f(v)-5 b(alue)p eop %%Page: 11 11 11 10 bop 448 573 a Fn(problem)34 b(is)h(one)h(of)g(man)m(y)f(problems) f(in)h Fi(P)h Fn(whic)m(h)e(is)h(kno)m(wn)g(to)h(b)s(e)f Fi(P)h Fn(complete.)448 686 y(These)44 b(are)g(problems)e(in)h Fi(P)h Fn(whic)m(h)f(are)h(pro)m(v)m(ed)g(to)h(b)s(e)e(complete)h(with) f(resp)s(ect)448 799 y(to)31 b(log-space)g(b)s(ounded)d(reductions,)h (reductions)f(de\014ned)h(analogously)g(to)i(p)s(olyno-)448 912 y(mial)26 b(time)h(b)s(ounded)e(reduction)h(in)g(the)h(previous)f (section.)40 b(Pro)m(ving)26 b(a)i Fi(P)p Fn(-complete)448 1024 y(problem)h(is)h(in)f Fi(L)h Fn(w)m(ould)f(imply)f(that)j Fi(L)25 b Fn(=)g Fi(P)p Fn(.)448 1268 y Fe(4.5)113 b(Oracles)448 1440 y Fn(Oracle)44 b(results)e(pla)m(y)h(a)h(unique)d(role)j(in)e (complexit)m(y)h(theory)-8 b(.)81 b(They)43 b(are)h(meta-)448 1552 y(mathematical)i(results)e(delineating)g(the)h(limitations)e(of)j (pro)s(of)f(tec)m(hniques)g(and)448 1665 y(indicating)37 b(what)j(results)e(migh)m(t)g(b)s(e)h(p)s(ossible)e(to)j(ac)m(hiev)m(e) g(and)f(whic)m(h)f(are)i(lik)m(ely)448 1778 y(b)s(ey)m(ond)31 b(our)f(curren)m(t)h(reac)m(h.)43 b(Oracle)30 b(results)g(concern)h (relativized)f(computations.)448 1891 y(W)-8 b(e)28 b(sa)m(y)g(that)g (a)f(computation)g(is)f(carried)g(out)h(\\relativ)m(e)h(to)f(an)g (oracle)g(set)h(O")f(if)f(the)448 2004 y(computation)h(has)f(access)h (to)g(the)g(answ)m(ers)f(to)h(mem)m(b)s(ership)d(queries)h(of)h(O.)h (That)f(is,)448 2117 y(the)31 b(computation)g(can)g(query)g(the)g (oracle)g(O)f(ab)s(out)h(whether)f(or)h(not)g(a)g(string)f(x)h(is)448 2230 y(in)e(O.)i(The)f(computation)g(obtains)g(the)h(answ)m(er)f(\(in)g (one)g(step\))h(and)f(pro)s(ceeds)g(with)448 2343 y(the)h(computation,) f(whic)m(h)g(ma)m(y)h(dep)s(end)d(on)i(the)h(answ)m(er)f(to)h(the)g (oracle)g(query)-8 b(.)589 2456 y(The)35 b(\014rst,)g(and)g(still)d (most)k(fundamen)m(tal)d(oracle)j(results)d(in)h(complexit)m(y)h(w)m (ere)448 2569 y(carried)c(out)g(b)m(y)h(Bak)m(er,)h(Gill)c(and)i(Solo)m (v)-5 b(a)m(y)32 b([BGS75)q(].)44 b(They)31 b(pro)m(v)m(ed)h(that)f (there)h(is)448 2682 y(an)h(oracle)h(reactiv)m(e)h(to)f(whic)m(h)e Fi(P)p Fn(=)p Fi(NP)h Fn(and)g(another)g(oracle)h(relativ)m(e)f(to)h (whic)m(h)e(P)448 2794 y(and)e Fi(NP)h Fn(di\013er.)589 2907 y(What)f(do)e(these)h(results)f(sa)m(y)h(ab)s(out)f(the)h Fi(P)g Fn(vs)f Fi(NP)h Fn(question?)39 b(They)28 b(sa)m(y)h(little)448 3020 y(ab)s(out)e(the)f(actual)h(answ)m(er)f(to)i(this)d(question.)39 b(The)26 b(existence)h(of)f(an)h(oracle)g(making)448 3133 y(a)38 b(statemen)m(t)h(S)d(true)h(is)f(simply)e(a)k(kind)d(of)i (consistency)g(result)f(ab)s(out)h(S.)f(It)h(sa)m(ys)448 3246 y(that)d(the)g(statemen)m(t)h(is)d(true)h(in)f(one)i(particular)e (mo)s(del)g(or)h(\\w)m(orld")g(\(that)h(is,)g(the)448 3359 y(oracle)f(set)g(itself)7 b(\).)47 b(As)32 b(suc)m(h,)h(w)m(e)g (can)g(conclude)f(that)h(a)g(pro)s(of)f(of)g(the)h(negation)g(of)448 3472 y(S)i(will)d(not)j(itself)e(relativize)h(to)i(an)m(y)f(oracle.)54 b(Th)m(us,)35 b(as)g(man)m(y)g(pro)s(of)f(metho)s(ds)g(do)448 3585 y(relativize)20 b(to)h(ev)m(ery)g(oracle,)i(an)e(oracle)f(result)g (pro)m(vides)f(a)i(limitation)d(to)j(the)f(p)s(ossible)448 3698 y(metho)s(ds)34 b(used)f(to)i(pro)m(v)m(e)g(S)f(and)f(hence)h(are) h(evidence)f(that)h(the)f(result)f(is,)h(in)f(this)448 3811 y(sense,)38 b(hard.)56 b(Oracle)36 b(results)e(ha)m(v)m(e)j(b)s (een)f(most)g(useful)e(in)h(delineating)f(theorems)448 3924 y(whic)m(h)g(are)h(di\016cult)e(to)i(pro)m(v)m(e)h(\(i.e.,)g (those)g(whic)m(h)d(do)i(no)f(relativize\),)i(from)f(those)448 4036 y(whic)m(h)k(migh)m(t)i(more)f(lik)m(ely)f(b)s(e)h(settled)g(b)m (y)h(w)m(ell-understo)s(o)s(d,)f(relativizing)f(pro)s(of)448 4149 y(tec)m(hniques.)k(In)30 b(particular,)g(the)h(Bak)m(er,)i(Gill)c (and)h(Solo)m(v)-5 b(a)m(y)32 b(results)e(concerning)h Fi(P)448 4262 y Fn(and)j Fi(NP)g Fn(question)g(indicate)f(that)h(a)h (pro)s(of)e(will)f(b)s(e)h(di\016cult)f(to)j(come)g(b)m(y)-8 b(,)36 b(as)e(has)448 4375 y(indeed)29 b(b)s(een)h(the)g(case.)589 4488 y(Since)g(1978)j(n)m(umerous)c(other)i(oracle)g(results)f(ha)m(v)m (e)i(b)s(een)e(pro)m(v)m(ed.)42 b(T)-8 b(ec)m(hniques)448 4601 y(used)20 b(to)h(ac)m(hiev)m(e)h(these)f(results)e(ha)m(v)m(e)j(b) s(ecome)e(quite)g(sophisticated)g(and)g(strong.)37 b(F)-8 b(or)448 4714 y(instance,)24 b(F)-8 b(enner,)24 b(F)-8 b(ortno)m(w)24 b(and)d(Kurtz)h([FFK94)q(])h(ga)m(v)m(e)h(a)f (relativized)d(w)m(orld)i(where)448 4827 y(the)f(isomorphism)c (conjecture)k(holds)e(where)h(Kurtz,)i(Mahaney)f(and)e(Ro)m(y)m(er)j ([KMR89)q(])448 4940 y(had)k(sho)m(w)m(ed)h(that)h(it)e(fails)f (relativ)m(e)i(to)g(most)g(oracles.)40 b(They)26 b(w)m(ere)i(the)e (culmination)p eop %%Page: 12 12 12 11 bop 448 573 a Fn(of)31 b(a)g(long)f(series)f(of)i(partial)e (results)g(addressing)g(this)g(question.)589 686 y(There)45 b(are)h(a)g(few)f(results)f(in)g(complexit)m(y)h(that)h(do)f(not)h (relativize,)i(mostly)448 799 y(relating)32 b(to)i(in)m(teractiv)m(e)f (pro)s(of)f(systems)h(\(see)h(Section)e(6.1\))j(but)d(these)h(tend)f (to)i(b)s(e)448 912 y(the)d(exception)f(and)g(not)h(the)f(rule.)448 1198 y Fo(5)135 b(Coun)l(ting)46 b(Classes)448 1401 y Fn(Another)31 b(w)m(a)m(y)g(to)h(study)d Fi(NP)i Fn(computations)f(is)g (to)h(ask)g(ho)m(w)g(man)m(y)f(computations)448 1514 y(paths)25 b(in)f(the)h(computation)h(lead)e(to)i(acceptance.)41 b(F)-8 b(or)26 b(example,)h(consider)d(the)h(sat-)448 1627 y(is\014abilit)m(y)20 b(problem.)36 b(Giv)m(en)23 b(an)f(instance)g(of)g(this)g(problem,)g(that)h(is)f(a)h(prop)s (ositional)448 1740 y(form)m(ula,)29 b(instead)g(of)h(asking)f(if)f (the)i(form)m(ula)f(has)g(a)h(solution)e(\(a)i(truth)f(assignmen)m(t) 448 1853 y(making)39 b(the)g(form)m(ula)g(true\),)i(ask)f(ho)m(w)f(man) m(y)g(suc)m(h)g(assignmen)m(ts)g(there)g(are.)68 b(In)448 1966 y(1979,)35 b(V)-8 b(alian)m(t)33 b([V)-8 b(al79)r(])32 b(de\014ned)g(the)g(complexit)m(y)g(class)h(#)p Fi(P)f Fn(as)h(the)g(class)f(of)h(func-)448 2078 y(tions)24 b(computing)f(the)h(n)m(um)m(b)s(er)f(of)h(accepting)g(paths)g(of)g(a)g (nondeterministic)e(T)-8 b(uring)448 2191 y(mac)m(hine.)60 b(Ev)m(ery)36 b(#)p Fi(P)h Fn(function)f(is)f(computable)h(in)g(p)s (olynomial)e(space.)60 b(V)-8 b(alian)m(t)448 2304 y(used)29 b(this)f(class)g(to)i(capture)g(the)f(complexit)m(y)g(of)g(the)g(coun)m (ting)g(v)m(ersion)g(of)g(satis\014a-)448 2417 y(bilit)m(y)22 b(as)h(w)m(ell)f(as)i(other)f(in)m(teresting)g(problems)e(suc)m(h)i(as) h(computing)e(the)i(p)s(ermanen)m(t)448 2530 y(function.)589 2643 y(Coun)m(ting)42 b(complexit)m(y)g(has)g(since)f(pla)m(y)m(ed)h (an)g(imp)s(ortan)m(t)g(role)g(in)e(computa-)448 2756 y(tional)31 b(complexit)m(y)g(theory)h(and)f(theoretical)h(computer)g (science.)44 b(The)31 b(tec)m(hniques)448 2869 y(used)42 b(in)g(coun)m(ting)h(complexit)m(y)g(ha)m(v)m(e)h(signi\014can)m(t)e (applications)f(in)h(circuit)f(com-)448 2982 y(plexit)m(y)29 b(and)g(in)f(the)i(series)f(of)h(recen)m(t)h(results)d(on)i(in)m (teractiv)m(e)g(pro)s(of)f(systems.)40 b(\(See)448 3095 y(the)g(next)f(section.\))67 b(The)39 b(t)m(w)m(o)h(most)g(imp)s(ortan) m(t)e(coun)m(ting)h(function)f(classes)h(are)448 3208 y(#)p Fi(P)q Fn(,)33 b(describ)s(ed)d(ab)s(o)m(v)m(e,)35 b(and)d Fi(GapP)p Fn(.)48 b Fi(GapP)33 b Fn(consists)f(of)h(the)f (class)h(of)f(functions)448 3320 y(whic)m(h)j(compute)h(the)g (di\013erence)g(b)s(et)m(w)m(een)g(the)h(n)m(um)m(b)s(er)d(of)i (accepting)h(paths)f(and)448 3433 y(the)i(n)m(um)m(b)s(er)e(of)i (rejecting)g(paths)f(of)h(a)g(nondeterministic)c(T)-8 b(uring)36 b(mac)m(hine.)62 b(F)-8 b(or)448 3546 y(example,)34 b(the)f(function)e(whic)m(h)h(tells,)h(for)f(an)m(y)i(prop)s(ositional) c(form)m(ula,)j(computes)448 3659 y(the)c(di\013erence)e(of)i(the)f(n)m (um)m(b)s(er)f(of)h(accepting)h(and)f(rejecting)g(truth)f(assignmen)m (ts)h(is)448 3772 y(in)h(the)i(class)f Fi(GapP)q Fn(.)589 3885 y(P)m(erhaps)36 b(the)g(t)m(w)m(o)i(most)e(imp)s(ortan)m(t)f (recen)m(t)i(results)e(in)f(coun)m(ting)i(complexit)m(y)448 3998 y(are)26 b(T)-8 b(o)s(da's)25 b(theorem)h([T)-8 b(o)s(d91])26 b(and)e(the)i(closure)e(theorem)i(of)f(Beigel,)i (Reingold)d(and)448 4111 y(Spielman's)33 b([BRS95)q(].)57 b(T)-8 b(o)s(da's)35 b(theorem)h(asserts)g(that)g(one)g(can)g(reduce)f (an)m(y)h(lan-)448 4224 y(guage)f(in)e(the)h(p)s(olynomial-time)d (hierarc)m(h)m(y)j(to)g(a)g(p)s(olynomial)e(time)h(computation)448 4337 y(whic)m(h)28 b(uses)h(a)h(#)p Fi(P)f Fn(function)f(as)h(an)g (oracle.)41 b(Hence,)31 b(that)e(in)f(terms)h(of)h(complexit)m(y)-8 b(,)448 4450 y(hard)36 b(functions)f(in)g(#)p Fi(P)i Fn(lie)f(ab)s(o)m(v)m(e)i(an)m(y)f(problem)e(in)g(the)i(p)s(olynomial)d (time)i(hier-)448 4562 y(arc)m(h)m(y)-8 b(.)50 b(In)33 b(1994,)j(Beigel,)e(Reingold)e(and)g(Spielman)f([BRS95)q(])i(pro)m(v)m (ed)g(that)h Fi(PP)g Fn(is)448 4675 y(closed)d(under)e(union.)39 b(This)29 b(result)h(solv)m(ed)g(a)h(longstanding)e(op)s(en)h(problem)f (in)g(this)448 4788 y(area,)k(\014rst)d(p)s(osed)g(b)m(y)i(Gill)d(in)h (1977)i([Gil77])g(in)e(the)h(initial)d(pap)s(er)i(on)h(probabilistic) 448 4901 y(classes.)39 b(It)25 b(implies)d(that)k Fi(PP)g Fn(is)e(also)h(closed)f(under)g(in)m(tersection)g(Those)h(in)m (terested)p eop %%Page: 13 13 13 12 bop 448 573 a Fn(in)21 b(further)g(exploring)f(coun)m(ting)i (classes)g(and)f(the)i(p)s(o)m(w)m(er)f(of)g(coun)m(ting)g(in)e (complexit)m(y)448 686 y(theory)31 b(should)d(consult)i(the)h(pap)s (ers)e(of)h(Sc)m(h\177)-45 b(oning)30 b([Sc)m(h90)q(])g(and)g(F)-8 b(ortno)m(w)32 b([F)-8 b(or97)r(].)448 972 y Fo(6)135 b(Probabilistic)46 b(Complexit)l(y)448 1175 y Fn(In)22 b(1977,)27 b(Solo)m(v)-5 b(a)m(y)23 b(and)g(Strassen)f([SS77])h(ga)m(v) m(e)i(a)e(new)g(kind)e(of)i(algorithm)e(for)i(testing)448 1288 y(whether)k(a)h(n)m(um)m(b)s(er)e(is)g(prime.)38 b(Their)26 b(algorithm)g(\015ipp)s(ed)f(coins)h(to)i(help)e(searc)m(h)i (for)448 1401 y(a)40 b(coun)m(terexample)g(to)h(primalit)m(y)-8 b(.)67 b(They)39 b(argued)h(that)g(if)e(the)i(n)m(um)m(b)s(er)f(w)m(as) h(not)448 1514 y(prime)29 b(then)h(with)f(v)m(ery)i(high)e (con\014dence)h(a)h(coun)m(terexample)g(could)e(b)s(e)h(found.)589 1627 y(This)j(algorithm)h(suggested)h(that)g(w)m(e)g(should)e(revise)g (our)h(notion)g(of)h(\\e\016cien)m(t)448 1740 y(computation".)72 b(P)m(erhaps)40 b(w)m(e)h(should)e(no)m(w)i(equate)g(the)g(e\016cien)m (tly)g(computable)448 1853 y(problems)30 b(with)g(the)h(class)g(of)g (problems)f(solv)m(e)h(in)f(probabilistic)e(p)s(olynomial)g(time.)448 1966 y(A)35 b(whole)f(new)g(area)i(of)f(complexit)m(y)f(theory)h(w)m (as)g(dev)m(elop)s(ed)f(to)i(help)d(understand)448 2078 y(the)e(p)s(o)m(w)m(er)f(of)h(probabilistic)c(computation.)589 2191 y(Gill)f([Gil77])h(de\014ned)f(the)i(class)f Fi(BPP)h Fn(to)g(capture)g(this)e(new)h(notion.)39 b(Adleman)448 2304 y(and)33 b(Manders)f([AM77)r(])h(de\014ned)f(the)h(class)f Fi(R)i Fn(that)f(represen)m(ted)g(the)g(set)h(of)f(prob-)448 2417 y(lems)38 b(with)e(one-sided)h(randomness{the)h(mac)m(hine)g(only) f(accepts)i(if)e(the)h(instance)448 2530 y(is)c(guaran)m(teed)j(to)e(b) s(e)g(in)f(the)h(language.)55 b(The)35 b(Solo)m(v)-5 b(a)m(y-Strassen)36 b(algorithm)e(puts)448 2643 y(comp)s(ositeness)c (in)f Fi(R)p Fn(.)589 2756 y(Babai)35 b(in)m(tro)s(duced)e(the)i (concept)h(of)f(a)g(\\Las)h(V)-8 b(egas")37 b(probabilistic)31 b(algorithm)448 2869 y(that)f(alw)m(a)m(ys)f(giv)m(es)g(the)f(correct)i (answ)m(er)f(and)f(runs)f(in)g(exp)s(ected)i(p)s(olynomial)d(time.)448 2982 y(This)44 b(class)h Fi(ZPP)h Fn(is)e(equiv)-5 b(alen)m(t)45 b(to)h(those)g(problems)e(with)g(b)s(oth)g(p)s(ositiv)m(e)h(and)448 3095 y(negativ)m(e)36 b(instances)e(in)f Fi(R)p Fn(.)53 b(Adleman)34 b(and)g(Huang)g([AH87)r(])g(building)d(on)j(w)m(ork)h(of) 448 3208 y(Goldw)m(asser)30 b(and)g(Kilian)e([GK99)q(])j(sho)m(w)f (that)h(primalit)m(y)d(is)i(in)f Fi(R)h Fn(and)g(th)m(us)g Fi(ZPP)q Fn(.)589 3320 y(V)-8 b(ery)30 b(recen)m(tly)-8 b(,)30 b(Agra)m(w)m(al,)g(Ka)m(y)m(al)f(and)f(Saxena)h([AKS02)q(])f(ga) m(v)m(e)j(a)e(deterministic)448 3433 y(p)s(olynomial-time)20 b(algorithm)h(for)g(primalit)m(y)-8 b(.)36 b(If)22 b(this)f(result)f(w) m(as)j(kno)m(wn)e(in)g(the)h(70's,)448 3546 y(p)s(erhaps)29 b(the)i(study)f(of)g(probabilistic)d(algorithms)j(w)m(ould)f(not)i(ha)m (v)m(e)h(progressed)e(as)448 3659 y(quic)m(kly)-8 b(.)589 3772 y(In)24 b(1983,)k(Sipser)22 b([Sip83])j(sho)m(w)m(ed)f(that)h Fi(BPP)h Fn(is)e(con)m(tained)h(in)e(the)h(p)s(olynomial-)448 3885 y(time)34 b(hierarc)m(h)m(y)-8 b(.)53 b(G\023)-45 b(acs)35 b(\(see)g([Sip83)o(]\))g(impro)m(v)m(es)f(this)f(result)g(to)i (sho)m(w)g Fi(BPP)g Fn(is)e(in)448 3998 y(the)23 b(second)f(lev)m(el)g (of)h(the)f(hierarc)m(h)m(y)g(and)g(Lautemann)g([Lau83)q(])g(giv)m(es)h (a)g(simple)d(pro)s(of)448 4111 y(of)31 b(this)e(fact.)589 4224 y(One)j(can)h(also)f(consider)f(probabilistic)e(space)j(classes.) 47 b(Aleliunas,)30 b(Karp,)i(Lip-)448 4337 y(ton,)c(Lo)m(v\023)-45 b(asz)27 b(and)f(Rac)m(k)m(o\013)i([AKL)1641 4304 y Fj(+)1700 4337 y Fn(79)q(])e(sho)m(w)g(that)h(undirected)d(graph)i(connectivit)m (y)448 4450 y(can)33 b(b)s(e)e(computed)i(in)d(one-sided)i(randomized)f (logarithmic)g(space,)i(a)g(class)f(called)448 4562 y Fi(RL)q Fn(.)53 b(Similarly)31 b(one)k(can)g(de\014ne)f(the)h(classes)g Fi(BPL)g Fn(and)f Fi(ZPL)p Fn(.)54 b(Boro)s(din,)35 b(Co)s(ok,)448 4675 y(Dymond,)d(Ruzzo)g(and)g(T)-8 b(ompa)31 b([BCD)1829 4642 y Fj(+)1889 4675 y Fn(89)q(])h(sho)m(w)m(ed)g(that)g(undirected)e (graph)h(non-)448 4788 y(connectivit)m(y)40 b(also)g(sits)e(in)h Fi(RL)g Fn(and)g(th)m(us)g Fi(ZPL)q Fn(.)68 b(Nisan)38 b(and)h(T)-8 b(a-Shma)40 b([NT95)q(])448 4901 y(sho)m(w)m(ed)26 b(that)g(the)g(connectivit)m(y)g(question)f(reduced)g(directly)f(to)i (the)g(nonconnectiv-)p eop %%Page: 14 14 14 13 bop 448 573 a Fn(it)m(y)31 b(question.)448 816 y Fe(6.1)113 b(In)m(teractiv)m(e)36 b(Pro)s(of)h(Systems)448 988 y Fn(One)e(can)g(think)e(of)i(the)g(class)g Fi(NP)g Fn(as)g(a)g Fd(pr)-5 b(o)g(of)39 b(system)p Fn(:)50 b(An)34 b(arbitrarily)e(p)s(o)m(w)m(erful)448 1101 y(pro)m(v)m(er)45 b(giv)m(es)f(a)g(pro)s(of)f(that)i(sa)m(y)g(a)f(form)m(ula)f(is)g (satis\014able.)80 b(One)44 b(can)g(general-)448 1214 y(ize)35 b(this)e(notion)h(of)h(pro)s(of)f(system)h(b)m(y)f(allo)m (wing)g(probabilistic)d(v)m(eri\014cation)j(of)h(the)448 1327 y(pro)s(of.)54 b(This)33 b(yields)g(the)i(complexit)m(y)f(class)h Fi(MA)p Fn(.)54 b(One)35 b(can)g(also)f(consider)g(in)m(ter-)448 1440 y(action)j(where)e(the)i(v)m(eri\014er)e(sends)g(messages)i(based) f(on)g(her)g(random)f(coins.)58 b(The)448 1552 y(b)s(ounded)21 b(round)g(v)m(ersion)h(of)h(this)f(class)g(is)g Fi(AM)h Fn(and)f(the)g(un)m(b)s(ounded)e(round)i(v)m(ersion)448 1665 y(is)37 b Fi(IP)p Fn(.)64 b(The)37 b(incredible)e(p)s(o)m(w)m(er)j (of)g(these)h(in)m(teractiv)m(e)f(pro)s(of)g(systems)g(has)f(led)g(to) 448 1778 y(sev)m(eral)25 b(of)g(the)g(most)g(surprising)d(and)i(imp)s (ortan)m(t)g(recen)m(t)h(results)f(in)f(computational)448 1891 y(complexit)m(y)30 b(theory)-8 b(.)589 2004 y(Babai)42 b([Bab85)q(])g(de\014ned)e(in)m(teractiv)m(e)i(pro)s(of)f(systems)g(to) h(help)e(classify)h(some)448 2117 y(group)g(questions.)74 b(An)41 b(alternativ)m(e)h(in)m(teractiv)m(e)h(pro)s(of)e(system)h(w)m (as)g(de\014ned)e(b)m(y)448 2230 y(Goldw)m(asser,)e(Micali)d(and)h(Rac) m(k)m(o\013)i([GMR89)r(])e(as)h(a)f(basis)f(for)h(the)h(cryptographic) 448 2343 y(class)25 b(zero-kno)m(wledge.)40 b(Zero-kno)m(wledge)26 b(pro)s(of)e(systems)h(ha)m(v)m(e)i(themselv)m(es)e(pla)m(y)m(ed)448 2456 y(a)31 b(ma)5 b(jor)30 b(role)h(in)e(cryptograph)m(y)-8 b(.)589 2569 y(The)36 b(t)m(w)m(o)i(mo)s(dels)d(di\013ered)g(on)h (whether)f(the)i(pro)m(v)m(er)f(could)g(see)g(the)h(v)m(eri\014er's)448 2682 y(random)j(coins,)i(but)d(Goldw)m(asser)h(and)g(Sipser)e([GS89)q (])i(sho)m(w)m(ed)h(the)f(t)m(w)m(o)h(mo)s(dels)448 2794 y(equiv)-5 b(alen)m(t.)40 b(Babai)30 b(and)f(Moran)h([BM88)r(])g(sho)m (w)m(ed)g(that)h(an)m(y)f(b)s(ounded-round)c(pro-)448 2907 y(to)s(col)32 b(needs)e(only)h(one)g(question)f(from)h(the)g(v)m (eri\014er)f(follo)m(w)m(ed)h(b)m(y)g(a)g(resp)s(onse)g(from)448 3020 y(the)21 b(pro)m(v)m(er.)38 b(F)s(\177)-48 b(urer,)23 b(Goldreic)m(h,)f(Mansour,)h(Sipser)c(and)h(Zac)m(hos)i([F)m(GM)2956 2987 y Fj(+)3017 3020 y Fn(89)q(])f(sho)m(w)m(ed)448 3133 y(that)29 b(one)f(can)h(assume)f(that)g(for)g(p)s(ositiv)m(e)f (instances)h(the)g(pro)m(v)m(er)g(can)h(succeed)f(with)448 3246 y(no)j(error.)589 3359 y(Goldreic)m(h,)39 b(Micali)c(and)i (Wigderson)f([GMW91)s(])h(sho)m(w)g(that)g(the)h(set)f(of)g(pairs)448 3472 y(of)47 b(nonisomorphic)e(graphs)h(has)g(a)h(b)s(ounded-round)d (in)m(teractiv)m(e)k(pro)s(of)e(system.)448 3585 y(Boppana,)54 b(H)-11 b(\027)-57 b(astad)50 b(and)e(Zac)m(hos)i([BHZ87)q(])f(sho)m(w) g(that)g(if)f(the)h(complemen)m(t)g(of)448 3698 y(an)m(y)33 b Fi(NP)p Fn(-complete)g(language)g(has)f(b)s(ounded-round)d(in)m (teractiv)m(e)k(pro)s(ofs)f(than)g(the)448 3811 y(p)s(olynomial-time)j (hierarc)m(h)m(y)j(collapses.)62 b(This)36 b(remains)h(the)h(b)s(est)f (evidence)g(that)448 3924 y(the)31 b(graph)f(isomorphism)d(problem)i (is)g(probably)g(not)i Fi(NP)p Fn(-complete.)589 4036 y(In)k(1990,)j(Lund,)c(F)-8 b(ortno)m(w,)37 b(Karlo\013)e(and)f(Nisan)g ([LFKN92)q(])h(sho)m(w)m(ed)g(that)h(the)448 4149 y(complemen)m(ts)47 b(of)h Fi(NP)p Fn(-complete)g(languages)f(ha)m(v)m(e)h(un)m(b)s(ounded) d(round)g(in)m(terac-)448 4262 y(tiv)m(e)e(pro)s(of)e(systems.)75 b(Shamir)40 b([Sha92)q(])i(quic)m(kly)f(extended)h(their)f(tec)m (hniques)g(to)448 4375 y(sho)m(w)32 b(that)g(ev)m(ery)h(language)f(in)e Fi(PSP)-9 b(A)m(CE)33 b Fn(has)e(in)m(teractiv)m(e)i(pro)s(of)e (system.)45 b(F)-8 b(eld-)448 4488 y(man)36 b([F)-8 b(el86)q(])36 b(had)f(earlier)g(sho)m(wn)g(that)h(ev)m(ery)h(language)f(with)e(in)m (teractiv)m(e)j(pro)s(ofs)448 4601 y(lies)29 b(in)g Fi(PSP)-9 b(A)m(CE)q Fn(.)589 4714 y(In)m(teractiv)m(e)31 b(pro)s(ofs)e(are)h (notable)f(in)g(that)h(in)e(general)h(pro)s(ofs)g(concerning)g(them)448 4827 y(do)41 b(not)g(relativize,)h(that)g(is)d(they)i(are)g(not)g(true) f(relativ)m(e)h(to)g(ev)m(ery)h(oracle.)72 b(The)448 4940 y(classi\014cation)29 b(of)h(in)m(teractiv)m(e)h(pro)s(ofs)e (turned)f(out)j(not)f(to)g(b)s(e)g(the)g(end)f(of)h(the)g(story)p eop %%Page: 15 15 15 14 bop 448 573 a Fn(but)36 b(only)g(the)h(b)s(eginning)c(of)k(a)g (rev)m(olution)f(connecting)g(complexit)m(y)h(theory)f(with)448 686 y(appro)m(ximation)k(algorithms.)70 b(F)-8 b(or)42 b(the)f(con)m(tin)m(uation)f(of)h(this)f(story)h(w)m(e)g(turn)e(to)448 799 y(probabilistically)26 b(c)m(hec)m(k)-5 b(able)32 b(pro)s(ofs.)448 1042 y Fe(6.2)113 b(Probabilistically)32 b(Chec)m(k)-6 b(able)38 b(Pro)s(ofs)448 1214 y Fn(In)32 b(1988,)k(Ben-Or,)d(Goldw)m(asser,)g(Kilian)d(and)j(Wigderson)f ([BGKW88)r(])h(dev)m(elop)s(ed)448 1327 y(the)h(m)m(ultipro)m(v)m(er)f (in)m(teractiv)m(e)i(pro)s(of)e(system.)51 b(This)32 b(mo)s(del)h(has)g(m)m(ultiple)f(pro)m(v)m(ers)448 1440 y(who)45 b(cannot)h(comm)m(unicate)g(with)e(eac)m(h)i(other)f(or)h(see) f(the)h(con)m(v)m(ersations)g(eac)m(h)448 1552 y(has)35 b(with)e(the)h(v)m(eri\014er.)53 b(This)32 b(mo)s(del)i(allo)m(ws)f (the)i(v)m(eri\014er)f(to)h(pla)m(y)f(one)h(pro)m(v)m(er)g(o\013)448 1665 y(another.)589 1778 y(F)-8 b(ortno)m(w,)39 b(Romp)s(el)c(and)g (Sipser)f([FRS94)q(])i(sho)m(w)g(this)f(mo)s(del)g(is)g(equiv)-5 b(alen)m(t)35 b(to)448 1891 y(probabilistically)28 b(c)m(hec)m(k)-5 b(able)34 b(pro)s(ofs,)f(where)f(the)g(pro)m(v)m(er)h(writes)f(do)m(wn) g(a)h(p)s(ossibly)448 2004 y(exp)s(onen)m(tially)f(long)g(pro)s(of)h (that)g(the)g(v)m(eri\014er)g(sp)s(ot)f(c)m(hec)m(ks)j(in)c (probabilistic)f(p)s(oly-)448 2117 y(nomial)j(time.)52 b(They)34 b(also)g(sho)m(w)g(that)h(ev)m(ery)g(language)f(accepted)i(b) m(y)e(these)h(pro)s(of)448 2230 y(systems)c(lie)e(in)g Fi(NEXP)p Fn(,)i(nondeterministic)c(exp)s(onen)m(tial)j(time.)589 2343 y(In)20 b(1990,)25 b(Babai,)d(F)-8 b(ortno)m(w)22 b(and)e(Lund)f([BFL91)q(])i(sho)m(w)g(the)f(surprising)d(con)m(v)m (erse{)448 2456 y(that)24 b(ev)m(ery)g(language)g(in)e Fi(NEXP)i Fn(has)f(probabilistically)c(c)m(hec)m(k)-5 b(able)24 b(pro)s(ofs.)38 b(Babai,)448 2569 y(F)-8 b(ortno)m(w,)50 b(Levin)44 b(and)g(Szegedy)h([BFLS91)r(])g(scale)g(this)e(pro)s(of)h (do)m(wn)h(to)g(dev)m(elop)448 2682 y(\\holographic")e(pro)s(ofs)f(for) h Fi(NP)g Fn(where,)i(with)d(a)h(prop)s(erly)e(enco)s(ded)i(input,)h (the)448 2794 y(v)m(eri\014er)28 b(can)h(c)m(hec)m(k)h(the)f (correctness)h(of)f(the)f(pro)s(of)g(in)g(v)m(ery)h(short)f(amoun)m(t)h (of)g(time.)589 2907 y(F)-8 b(eige,)30 b(Goldw)m(asser,)f(Lo)m(v\023) -45 b(asz,)30 b(Safra)e(and)f(Szegedy)i([F)m(GL)2632 2874 y Fj(+)2692 2907 y Fn(96)q(])f(made)g(an)g(amaz-)448 3020 y(ing)46 b(connection)h(b)s(et)m(w)m(een)g(probabilistically)42 b(c)m(hec)m(k)-5 b(able)47 b(pro)s(ofs)f(and)g(the)g(clique)448 3133 y(problem.)83 b(By)46 b(viewing)d(p)s(ossible)f(pro)s(ofs)i(as)i (no)s(des)e(of)h(a)g(graph,)k(they)c(sho)m(w)m(ed)448 3246 y(that)37 b(one)f(cannot)h(appro)m(ximate)f(the)g(size)g(of)g(a)g (clique)f(w)m(ell)g(without)g(unexp)s(ected)448 3359 y(collapses)30 b(in)f(complexit)m(y)h(classes.)589 3472 y(In)e(1992,)j(Arora,)e(Lund,)e(Mot)m(w)m(ani,)j(Sudan)c(and)i(Szegedy) h([ALM)2910 3439 y Fj(+)2970 3472 y Fn(98])g(building)448 3585 y(on)j(w)m(ork)h(of)f(Arora)g(and)g(Safra)g([AS98])h(sho)m(w)m(ed) f(that)h(ev)m(ery)g(language)f(in)f Fi(NP)h Fn(has)448 3698 y(a)c(probabilistically)23 b(c)m(hec)m(k)-5 b(able)28 b(pro)s(of)e(where)h(the)g(v)m(eri\014er)g(uses)f(only)h(a)g (logarithmic)448 3811 y(n)m(um)m(b)s(er)i(of)i(random)f(coins)f(and)h (a)h(constan)m(t)g(n)m(um)m(b)s(er)f(of)g(queries)f(to)i(the)g(pro)s (of.)589 3924 y(The)20 b(Arora)h(et.)g(al.)f(result)f(has)h(tremendous) g(implications)d(for)j(the)h(class)f Fi(MAXSNP)448 4036 y Fn(of)h(appro)m(ximation)f(problems.)36 b(This)19 b(class)i(dev)m (elop)s(ed)f(b)m(y)h(P)m(apadimitriou)e(and)h(Y)-8 b(an-)448 4149 y(nak)j(akis)37 b([PY91)q(])h(has)g(man)m(y)f(in)m(teresting)g (complete)h(problems)e(suc)m(h)h(as)h(max-cut,)448 4262 y(v)m(ertex)44 b(co)m(v)m(er,)k(indep)s(enden)m(t)40 b(set,)47 b(tra)m(v)m(eling)c(salesman)f(on)g(an)h(arbitrary)e(metric) 448 4375 y(space)31 b(and)f(maximizing)e(the)j(n)m(um)m(b)s(er)e(of)i (satis\014able)e(clauses)h(of)g(a)h(form)m(ula.)589 4488 y(Arora)41 b(et.)h(al.)e(sho)m(w)g(that,)k(unless)39 b Fi(P)j Fn(=)g Fi(NP)p Fn(,)h(ev)m(ery)f Fi(MAXSNP)p Fn(-complete)448 4601 y(set)g(do)s(es)e(not)h(ha)m(v)m(e)h(a)f(p)s (olynomial-time)d(appro)m(ximation)i(sc)m(heme.)72 b(F)-8 b(or)42 b(eac)m(h)g(of)448 4714 y(these)k(problems)d(there)i(is)f(some) h(constan)m(t)i Fh(\016)53 b(>)c Fn(1)d(suc)m(h)e(that)i(they)f(cannot) h(b)s(e)448 4827 y(appro)m(ximated)30 b(within)e(a)j(factor)g(of)g Fh(\016)j Fn(unless)29 b Fi(P)c Fn(=)g Fi(NP)q Fn(.)589 4940 y(Since)35 b(these)i(initial)c(w)m(orks)j(on)g(probabilistically) 31 b(c)m(hec)m(k)-5 b(able)38 b(pro)s(ofs,)e(w)m(e)h(ha)m(v)m(e)p eop %%Page: 16 16 16 15 bop 448 573 a Fn(seen)27 b(a)g(large)g(n)m(um)m(b)s(er)e(of)i (outstanding)f(pap)s(ers)f(impro)m(ving)g(the)i(pro)s(of)f(systems)h (and)448 686 y(getting)44 b(stronger)f(hardness)f(of)h(appro)m (ximation)g(results.)77 b(H)-11 b(\027)-57 b(astad)44 b([H)-11 b(\027)-57 b(as97)r(])43 b(gets)448 799 y(tigh)m(t)35 b(results)d(for)i(some)h(appro)m(ximation)e(problems.)50 b(Arora)34 b([Aro98)q(])h(after)f(failing)448 912 y(to)g(ac)m(hiev)m(e) f(lo)m(w)m(er)g(b)s(ounds)d(for)j(tra)m(v)m(eling)f(salesman)g(in)f (the)i(plane,)f(has)h(dev)m(elop)s(ed)448 1024 y(a)e(p)s (olynomial-time)d(appro)m(ximation)h(algorithm)h(for)g(this)f(and)h (related)g(problems.)589 1137 y(A)35 b(series)e(of)i(results)e(due)h (to)h(Cai,)g(Condon,)f(Lipton,)g(Lapidot,)h(Shamir,)f(F)-8 b(eige)448 1250 y(and)30 b(Lo)m(v\023)-45 b(asz)31 b([CCL92,)f(CCL90,)h (CCL91,)f(F)-8 b(ei91)q(,)30 b(LS91)q(,)g(FL92)q(])g(ha)m(v)m(e)h(mo)s (di\014ed)d(the)448 1363 y(proto)s(col)i(of)g(Babai,)h(F)-8 b(ortno)m(w)31 b(and)e(Lund)f([BFL91)r(])i(to)h(sho)m(w)e(that)i(ev)m (ery)g(language)448 1476 y(in)26 b Fi(NEXP)h Fn(has)g(a)g(t)m(w)m (o-pro)m(v)m(er,)j(one-round)d(pro)s(of)f(systems)h(with)f(an)h(exp)s (onen)m(tially)448 1589 y(small)k(error.)44 b(This)30 b(problem)g(remained)h(so)h(elusiv)m(e)f(b)s(ecause)h(running)d(these)j (pro)s(of)448 1702 y(systems)g(in)e(parallel)g(do)s(es)h(not)h(ha)m(v)m (e)h(the)f(exp)s(ected)g(error)f(reduction)f([FRS94)q(].)45 b(In)448 1815 y(1995,)37 b(Raz)e([Raz98)r(])g(sho)m(w)m(ed)f(that)h (the)f(error)g(do)s(es)g(go)h(done)f(exp)s(onen)m(tially)e(when)448 1928 y(these)f(pro)s(ofs)f(systems)g(are)h(run)e(in)g(parallel.)448 2171 y Fe(6.3)113 b(Derandomization)448 2343 y Fn(If)38 b(y)m(ou)g(generate)i(a)e(random)f(n)m(um)m(b)s(er)g(on)h(a)g (computer,)i(y)m(ou)f(do)f(not)g(get)h(a)f(truly)448 2456 y(random)26 b(v)-5 b(alue,)27 b(but)f(a)h(pseudorandom)e(n)m(um)m (b)s(er)g(computed)i(b)m(y)f(some)h(complicated)448 2569 y(function)41 b(on)g(some)h(small,)h(hop)s(efully)c(random)i(seed.)74 b(In)41 b(practice)h(this)e(usually)448 2682 y(w)m(orks)22 b(w)m(ell)e(so)i(p)s(erhaps)d(in)h(theory)i(the)f(same)h(migh)m(t)f(b)s (e)g(true.)37 b(Man)m(y)22 b(of)g(the)g(exciting)448 2794 y(results)28 b(in)g(complexit)m(y)h(theory)g(in)f(the)h(1980's)i (and)e(90's)h(consider)e(this)g(question)g(of)448 2907 y(derandomization{ho)m(w)38 b(to)g(reduce)g(or)g(eliminate)e(the)j(n)m (um)m(b)s(er)d(of)i(truly)f(random)448 3020 y(bits)30 b(to)h(sim)m(ulate)e(probabilistic)e(algorithms.)589 3133 y(The)k(\014rst)g(approac)m(h)h(to)g(this)e(problem)g(came)j(from) e(cryptograph)m(y)-8 b(.)44 b(Blum)31 b(and)448 3246 y(Micali)37 b([BM84)r(])i(\014rst)e(to)i(sho)m(w)f(ho)m(w)g(to)h (create)g(randomness)e(from)h(cryptograph-)448 3359 y(ically)i(hard)g (functions.)72 b(Y)-8 b(ao)43 b([Y)-8 b(ao90)r(])42 b(sho)m(w)m(ed)f (ho)m(w)h(to)g(reduce)f(the)g(n)m(um)m(b)s(er)f(of)448 3472 y(random)d(bits)g(of)h(an)m(y)g(algorithm)f(based)g(on)h(an)m(y)g (cryptographically)f(secure)g(one-)448 3585 y(w)m(a)m(y)c(p)s(erm)m (utation.)44 b(H)-11 b(\027)-57 b(astad,)32 b(Impagliazzo,)h(Levin)d (and)h(Lub)m(y)g([HILL99)q(])h(building)448 3698 y(on)e(tec)m(hniques)f (of)h(Goldreic)m(h)f(and)g(Levin)f([GL89)r(])i(and)f(Goldreic)m(h,)g (Kra)m(w)m(czyk)i(and)448 3811 y(Lub)m(y)f([GKL93)q(])h(sho)m(w)f(that) h(one)g(can)g(get)g(pseudorandomness)e(from)h(an)m(y)g(one-w)m(a)m(y) 448 3924 y(function.)589 4036 y(Nisan)43 b(and)f(Wigderson)h([NW94)r(]) g(tak)m(e)i(a)e(di\013eren)m(t)g(approac)m(h.)80 b(They)42 b(sho)m(w)448 4149 y(ho)m(w)26 b(to)g(get)h(pseudorandomness)d(based)h (on)h(a)g(language)g(hard)e(against)i(non)m(uniform)448 4262 y(computation.)38 b(Impagliazzo)23 b(and)f(Wigderson)h([IW97)q(])g (building)c(on)j(this)g(result)g(and)448 4375 y(Babai,)28 b(F)-8 b(ortno)m(w,)28 b(Nisan)e(and)f(Wigderson)h([BFNW93)r(])h(sho)m (w)f(that)h Fi(BPP)g Fn(equals)f Fi(P)448 4488 y Fn(if)32 b(there)i(exists)e(a)i(language)g(in)d(exp)s(onen)m(tial)h(time)h(that) h(cannot)g(b)s(e)e(computed)h(b)m(y)448 4601 y(an)m(y)e(sub)s(exp)s (onen)m(tial)d(circuit.)589 4714 y(F)-8 b(or)50 b(derandomization)e(of) h(space)h(w)m(e)f(ha)m(v)m(e)h(sev)m(eral)g(unconditional)c(results.) 448 4827 y(Nisan)32 b([Nis92)q(])g(giv)m(es)h(general)g(to)s(ols)g(for) f(derandomizing)f(space-b)s(ounded)g(compu-)448 4940 y(tation.)48 b(Among)33 b(the)f(applications,)g(he)g(gets)i(a)f Fh(O)s Fn(\(log)2383 4903 y Fj(2)2438 4940 y Fh(n)p Fn(\))f(space)h (construction)f(for)p eop %%Page: 17 17 17 16 bop 448 573 a Fn(univ)m(ersal)29 b(tra)m(v)m(ersal)i(sequences)g (for)f(undirected)f(graphs.)589 686 y(Saks)39 b(and)f(Zhou)g([SZ99])h (sho)m(w)f(that)h(ev)m(ery)h(probabilistic)35 b(logarithmic)i(space)448 799 y(algorithm)44 b(can)h(b)s(e)g(sim)m(ulated)e(in)h Fh(O)s Fn(\(log)1969 762 y Fj(3)p Fg(=)p Fj(2)2094 799 y Fh(n)p Fn(\))g(deterministic)f(space.)85 b(Armoni,)448 912 y(T)-8 b(a-Shma,)33 b(Wigderson)f(and)g(Zhou)f([A)-8 b(TWZ97)q(])33 b(building)28 b(on)k(w)m(ork)h(of)f(Nisan,)g(Sze-)448 1024 y(mer)m(\023)-43 b(edi,)31 b(and)e(Wigderson)g([NSW92)r(])h(sho)m (w)g(that)g(one)h(can)f(solv)m(e)g(undirected)f(graph)448 1137 y(connectivit)m(y)i(in)e Fh(O)s Fn(\(log)1292 1101 y Fj(4)p Fg(=)p Fj(3)1417 1137 y Fh(n)p Fn(\))h(space.)448 1424 y Fo(7)135 b(Descriptiv)l(e)46 b(Complexit)l(y)448 1627 y Fn(Man)m(y)32 b(of)f(the)g(fundamen)m(tal)f(concepts)i(and)e (metho)s(ds)g(of)h(complexit)m(y)g(theory)g(ha)m(v)m(e)448 1740 y(their)i(genesis)h(in)e(mathematical)i(logic,)h(and)e(in)g (computabilit)m(y)f(theory)i(in)e(partic-)448 1853 y(ular.)49 b(This)32 b(includes)f(the)i(ideas)g(of)h(reductions,)f(complete)h (problems,)e(hierarc)m(hies)448 1966 y(and)38 b(logical)f (de\014nabilit)m(y)-8 b(.)61 b(It)38 b(is)f(a)i(w)m(ell-understo)s(o)s (d)c(principle)g(of)j(mathematical)448 2078 y(logic)33 b(that)h(the)f(more)g(complex)g(a)h(problem's)d(logical)i(de\014nition) d(\(for)k(example,)f(in)448 2191 y(terms)27 b(of)h(quan)m(ti\014er)e (alternation\))h(the)h(more)f(di\016cult)e(its)i(solv)-5 b(abilit)m(y)d(.)38 b(Descriptiv)m(e)448 2304 y(complexit)m(y)32 b(aims)g(to)h(measure)g(the)f(computational)g(complexit)m(y)g(of)h(a)g (problem)d(in)448 2417 y(terms)37 b(of)g(the)f(complexit)m(y)h(of)f (the)h(logical)f(language)h(needed)f(to)i(de\014ne)e(it.)58 b(As)37 b(is)448 2530 y(often)28 b(the)f(case)i(in)d(complexit)m(y)h (theory)-8 b(,)28 b(the)g(issues)e(here)h(b)s(ecome)g(more)h(subtle)e (and)448 2643 y(the)36 b(measure)g(of)f(the)h(logical)f(complexit)m(y)h (of)g(a)g(problem)e(more)h(in)m(tricate)h(than)f(in)448 2756 y(computabilit)m(y)e(theory)-8 b(.)55 b(Descriptiv)m(e)35 b(complexit)m(y)f(has)g(its)h(b)s(eginnings)c(in)j(the)h(re-)448 2869 y(searc)m(h)30 b(of)g(Jones,)f(Selman,)g(F)-8 b(agin)30 b([JS74,)g(F)-8 b(ag73)r(,)30 b(F)-8 b(ag74)r(])30 b(and)e(others)i(in) e(the)h(early)448 2982 y(1970's.)40 b(More)23 b(recen)m(tly)f (descriptiv)m(e)f(complexit)m(y)g(has)h(had)f(signi\014can)m(t)g (applications)448 3095 y(to)31 b(database)h(theory)e(and)g(to)h (computer-aided)f(v)m(eri\014cation.)589 3208 y(The)38 b(ground)f(breaking)g(theorem)i(of)f(this)f(area)i(is)e(due)h(to)h(F)-8 b(agin)38 b([F)-8 b(ag73)s(].)64 b(It)448 3320 y(pro)m(vided)40 b(the)h(\014rst)f(ma)5 b(jor)41 b(imp)s(etus)d(for)j(the)g(study)f(of)h (descriptiv)m(e)f(complexit)m(y)-8 b(.)448 3433 y(F)g(agin's)33 b(Theorem)g(giv)m(es)g(a)g(logical)f(c)m(haracterization)i(of)f(the)g (class)f Fi(NP)p Fn(.)48 b(It)33 b(states)448 3546 y(that)43 b Fi(NP)g Fn(is)e(exactly)j(the)e(class)h(of)f(problems)f(de\014nable)g (b)m(y)h(existen)m(tial)g(second)448 3659 y(order)j(Bo)s(olean)g(form)m (ulas.)84 b(This)43 b(result,)48 b(and)d(others)g(that)g(follo)m(w,)k (sho)m(w)44 b(that)448 3772 y(natural)30 b(complexit)m(y)g(classes)g (ha)m(v)m(e)i(an)e(in)m(trinsic)e(logical)i(complexit)m(y)-8 b(.)589 3885 y(T)g(o)37 b(get)h(a)f(feel)f(for)g(this)g(imp)s(ortan)m (t)f(idea,)j(consider)e(the)g Fi(NP)p Fn(-complete)i(prob-)448 3998 y(lem)j(of)h(3)h(colorabilit)m(y)d(of)i(a)g(graph.)74 b(F)-8 b(agin's)43 b(theorem)f(sa)m(ys)g(there)g(is)f(a)h(second)448 4111 y(order)g(existen)m(tial)g(form)m(ula)f(whic)m(h)g(holds)g(for)h (exactly)h(those)g(graphs)f(whic)m(h)f(are)448 4224 y(3-colorable.)49 b(This)31 b(form)m(ula)h(can)i(b)s(e)e(written)g(as)h(\()p Ff(9)p Fh(A;)15 b(B)5 b(;)15 b(C)7 b Fn(\)\()p Ff(8)p Fh(v)s Fn(\)[\()p Fh(A)p Fn(\()p Fh(v)s Fn(\))25 b Ff(_)c Fh(B)5 b Fn(\()p Fh(v)s Fn(\))23 b Ff(_)448 4337 y Fh(C)7 b Fn(\()p Fh(v)s Fn(\)\))s Ff(^)s Fn(\()p Ff(8)p Fh(w)r Fn(\)\()p Fh(E)e Fn(\()p Fh(v)s(;)15 b(w)r Fn(\))30 b Ff(!)47 b(:)p Fn(\()p Fh(A)p Fn(\()p Fh(v)s Fn(\))s Ff(^)s Fh(A)p Fn(\()p Fh(w)r Fn(\)\))s Ff(^)s(:)p Fn(\()p Fh(B)5 b Fn(\()p Fh(v)s Fn(\))s Ff(^)s Fh(B)g Fn(\()p Fh(w)r Fn(\)\))s Ff(^)s(:)p Fn(\()p Fh(C)i Fn(\()p Fh(v)s Fn(\))s Ff(^)s Fh(C)g Fn(\()p Fh(w)s Fn(\)\)\))q(])p Fh(:)448 4450 y Fn(In)m(tuitiv)m(ely)34 b(this)h(form)m(ula)f(states)j(that)f (ev)m(ery)g(v)m(ertex)h(is)e(colored)g(b)m(y)g(one)h(of)g(three)448 4562 y(colors)43 b(A,)f(B,)h(or)g(C)f(and)f(no)i(t)m(w)m(o)g(adjacen)m (t)h(v)m(ertices)f(ha)m(v)m(e)h(the)e(same)h(color.)77 b(A)448 4675 y(graph,)31 b(considered)e(as)i(a)g(\014nite)f(mo)s(del,)g (satis\014es)g(this)g(form)m(ula)g(if)f(and)h(only)g(if)g(it)g(is)448 4788 y(3-colorable.)589 4901 y(F)-8 b(agin's)35 b(theorem)g(w)m(as)f (the)h(\014rst)e(in)g(a)i(long)f(line)e(of)i(results)f(whic)m(h)g(pro)m (v)m(e)i(that)p eop %%Page: 18 18 18 17 bop 448 573 a Fn(complexit)m(y)42 b(classes)f(can)h(b)s(e)f(giv)m (en)h(logical)f(c)m(haracterizations,)46 b(often)c(v)m(ery)g(sim-)448 686 y(ply)34 b(and)h(elegan)m(tly)-8 b(.)57 b(Notable)36 b(among)g(these)g(is)e(the)i(theorem)g(of)f(Immerman)g(and)448 799 y(V)-8 b(ardi)39 b([Imm82,)h(V)-8 b(ar82)q(])39 b(whic)m(h)f (captures)h(the)h(complexit)m(y)e(of)i(p)s(olynomial)c(time.)448 912 y(Their)27 b(theorem)h(states)h(that)g(the)f(class)g(of)g(problems) e(de\014nable)h(in)g(\014rst)g(order)g(logic)448 1024 y(with)g(the)i(addition)e(of)h(the)h(least)g(\014xed)f(p)s(oin)m(t)f (op)s(erator)i(is)e(exactly)j(the)e(complexit)m(y)448 1137 y(class)f(P)-8 b(.)28 b(Logspace)h(can)f(b)s(e)e(c)m(haracterized) j(along)e(these)h(same)g(lines,)e(but)h(using)f(the)448 1250 y(transitiv)m(e)h(closure)g(\(TC\))g(op)s(erator)h(rather)f(than)h (least)f(\014xed)g(p)s(oin)m(t.)39 b(That)27 b(is,)g(non-)448 1363 y(deterministic)k(logspace)j(is)f(the)g(class)g(of)h(problems)d (de\014nable)h(in)g(\014rst)h(order)f(logic)448 1476 y(with)c(the)g(addition)f(of)i(TC)e(\(see)j(Immerman)d([Imm88)q(]\).)41 b(And)27 b(if)h(one)g(replaces)h(\014rst)448 1589 y(order)c(logic)h (with)e(TC)h(with)f(second)h(order)g(logic)h(with)e(TC)h(the)g(result)g (is)f Fi(PSP)-9 b(A)m(CE)448 1702 y Fn(\(see)40 b(Immerman)e ([Imm83]\).)67 b(Other,)41 b(analogous)e(results)e(in)h(this)g(\014eld) f(go)j(on)e(to)448 1815 y(c)m(haracterize)d(v)-5 b(arious)31 b(circuit)g(and)h(parallel)f(complexit)m(y)i(classes,)g(the)g(p)s (olynomial)448 1928 y(time)38 b(hierarc)m(h)m(y)-8 b(,)40 b(and)d(other)h(space)g(classes,)i(and)d(ev)m(en)i(yield)d(results)h (concerning)448 2041 y(coun)m(ting)31 b(classes.)589 2154 y(The)g(in)m(tuition)f(pro)m(vided)g(b)m(y)h(lo)s(oking)g(at)h (complexit)m(y)f(theory)h(in)e(this)h(w)m(a)m(y)h(has)448 2267 y(pro)m(v)m(ed)27 b(insigh)m(tful)c(and)i(p)s(o)m(w)m(erful.)38 b(In)26 b(fact,)i(one)e(pro)s(of)f(of)i(the)f(famous)g(Immerman-)448 2379 y(Szelep)s(csen)m(yi)e(Theorem)g([Imm88,)h(Sze88)q(])g(\(that)g(b) m(y)f(Immerman\))g(came)i(from)e(these)448 2492 y(logical)39 b(considerations.)65 b(This)37 b(theorem)j(sa)m(y)f(that)h(an)m(y)f (nondeterministic)d(space)448 2605 y(class)h(whic)m(h)f(con)m(tains)i (logspace)g(is)e(closed)h(under)f(complemen)m(t.)62 b(An)37 b(immediate)448 2718 y(consequence)g(is)f(that)g(the)h(con)m(text)h (sensitiv)m(e)d(languages)i(are)f(closed)h(under)d(com-)448 2831 y(plemen)m(t,)d(answ)m(ering)e(a)i(question)e(whic)m(h)g(had)h(b)s (een)g(op)s(en)g(for)g(ab)s(out)g(25)h(y)m(ears.)589 2944 y(T)-8 b(o)27 b(this)f(p)s(oin)m(t)f(w)m(e)i(ha)m(v)m(e)h (considered)d(sev)m(eral)i(of)g(the)g(most)g(fully)d(dev)m(elop)s(ed)i (and)448 3057 y(fundamen)m(tal)31 b(areas)i(of)f(complexit)m(y)g (theory)-8 b(.)46 b(W)-8 b(e)33 b(no)m(w)f(surv)m(ey)g(a)g(few)g(of)g (the)g(more)448 3170 y(cen)m(tral)42 b(topics)f(in)f(the)i(\014eld)e (dealing)g(with)g(other)i(mo)s(dels)e(of)h(computation)h(and)448 3283 y(their)35 b(complexit)m(y)h(theory)-8 b(.)59 b(These)36 b(include)e(circuit)h(complexit)m(y)-8 b(,)38 b(comm)m(unication)448 3396 y(complexit)m(y)30 b(and)g(pro)s(of)g(complexit)m(y)-8 b(.)448 3682 y Fo(8)135 b(Finite)46 b(Mo)t(dels)448 3888 y Fe(8.1)113 b(Circuit)35 b(Complexit)m(y)448 4060 y Fn(The)27 b(prop)s(erties)f(and)h(construction)f(of)i(e\016cien)m(t)g (Bo)s(olean)f(circuits)f(are)i(of)f(practical)448 4173 y(imp)s(ortance)g(as)h(they)g(are)g(the)f(building)d(blo)s(c)m(k)j(of)h (computers.)40 b(Circuit)25 b(complexit)m(y)448 4286 y(studies)41 b(b)s(ounds)g(on)h(the)h(size)f(and)g(depth)g(of)g (circuits)f(whic)m(h)g(compute)i(a)g(giv)m(en)448 4399 y(Bo)s(olean)27 b(functions.)38 b(Aside)25 b(from)h(their)f(practical)h (v)-5 b(alue,)26 b(suc)m(h)g(b)s(ounds)e(are)j(closely)448 4512 y(tied)j(to)h(imp)s(ortan)m(t)f(questions)f(ab)s(out)h(T)-8 b(uring)29 b(mac)m(hine)h(computations.)589 4624 y(Bo)s(olean)44 b(circuits)d(are)j(directed)e(acyclic)h(graphs)f(whose)h(in)m(ternal)f (no)s(des)g(\(or)448 4737 y(\\gates"\))g(are)e(Bo)s(olean)g(functions,) g(most)f(often)h(the)g(\\standard")f(Bo)s(olean)h(func-)448 4850 y(tions,)27 b Fd(and,)k(or)d Fn(and)e Fd(not)p Fn(.)40 b(In)26 b(a)i(circuit,)e(the)h(no)s(des)f(of)h(in-degree)g(0)g(are)g (called)f(input)448 4963 y(no)s(des)37 b(and)g(lab)s(eled)f(with)g (input)f(v)-5 b(ariables.)61 b(The)37 b(no)s(des)g(with)f(out-degree)j (0)f(are)p eop %%Page: 19 19 19 18 bop 448 573 a Fn(called)24 b(output)h(no)s(des.)38 b(The)24 b(v)-5 b(alue)25 b(of)g(the)g(circuit)e(is)h(computed)h(in)e (the)i(natural)f(w)m(a)m(y)448 686 y(b)m(y)34 b(giving)e(v)-5 b(alues)33 b(to)h(the)g(input)e(v)-5 b(ariables,)33 b(applying)e(the)j (gates)h(to)f(these)g(v)-5 b(alues,)448 799 y(and)30 b(computing)g(the)g(output)g(v)-5 b(alues.)589 912 y(The)37 b Fi(size)p Fh(;)15 b(s)p Fn(\()p Fh(C)7 b Fn(\),)39 b(of)e(a)g(circuit)e Fh(C)43 b Fn(is)36 b(the)h(n)m(um)m(b)s(er)e(of)i (gates)h(it)f(con)m(tains.)60 b(The)448 1024 y Fi(depth)p Fh(;)15 b(d)p Fn(\()p Fh(C)7 b Fn(\),)33 b(of)f(a)g(circuit)e Fh(C)38 b Fn(is)30 b(the)i(length)f(of)h(the)f(longest)h(path)g(from)f (an)g(input)448 1137 y(to)g(an)g(output)f(no)s(de.)589 1250 y(A)41 b(circuit)e(with)h(n)g(inputs)e(can)j(b)s(e)f(though)m(t)h (of)g(as)g(a)g(recognizer)g(of)g(a)g(set)g(of)448 1363 y(strings)36 b(of)h(length)f(n,)i(namely)e(those)h(whic)m(h)e(result)h (in)f(the)i(circuit)e(ev)-5 b(aluating)36 b(to)448 1476 y(1.)52 b(In)33 b(order)g(to)i(consider)d(circuits)h(as)h(recognizing)f (an)h(in\014nite)e(set)i(of)g(strings,)g(w)m(e)448 1589 y(consider)27 b(circuit)f(families)f(whic)m(h)i(are)g(in\014nite)f (collections)h(of)g(circuits,)g Fh(C)3064 1603 y Fg(n)3111 1589 y Fn(,)h(one)g(for)448 1702 y(eac)m(h)g(input)c(length.)39 b(In)26 b(this)f(w)m(a)m(y)j(a)f(circuit)e(family)g(can)i(recognize)g (a)g(language)g(just)448 1815 y(as)k(a)g(T)-8 b(uring)28 b(mac)m(hine)i(can.)589 1928 y(A)c(circuit)f(family)f(is)h(a)i(non)m (uniform)c(mo)s(del,)j(the)g(function)f(taking)h Fh(n)f Fn(to)i Fh(C)3198 1942 y Fg(n)3270 1928 y Fn(ma)m(y)448 2041 y(not)43 b(b)s(e)f(computable.)77 b(A)43 b(non)m(uniform)d (circuit)h(family)g(can)i(recognize)h(noncom-)448 2154 y(putable)35 b(sets.)59 b(W)-8 b(e)37 b(can)g(measure)f(the)g(size)g (and)g(depth)f(of)i(circuit)d(families)h(using)448 2267 y(asymptotic)27 b(notation.)40 b(So,)28 b(for)f(example,)g(w)m(e)g(sa)m (y)h(that)f(a)h(circuit)d(family)g(has)i(p)s(oly-)448 2379 y(nomial)j(size)g(if)g(s\()p Fh(C)1139 2393 y Fg(n)1186 2379 y Fn(\))h(is)e(O\(p\(n\)\),)i(for)g(some)g(p)s(olynomial)d (p\(n\).)41 b(An)m(y)30 b(language)h(in)448 2492 y(P)i(has)f(p)s (olynomial)e(size)j(circuits.)46 b(That)32 b(is,)h(it)f(is)g (recognized)h(b)m(y)f(a)h(circuit)f(family)448 2605 y(whic)m(h)25 b(has)h(p)s(olynomial)d(size.)39 b(And)26 b(so)g(pro)m(ving)f(that)i (some)f Fi(NP)h Fn(problem)d(do)s(es)i(not)448 2718 y(ha)m(v)m(e)h(p)s (olynomial)22 b(size)k(circuits)d(w)m(ould)i(imply)e(that)i Fi(P)h Ff(6)p Fn(=)f Fi(NP)p Fn(.)39 b(Largely)26 b(b)s(ecause)f(of)448 2831 y(man)m(y)32 b(suc)m(h)e(implications)f(for)i(complexit)m(y)g (classes,)g(considerable)f(e\013ort)i(has)f(b)s(een)448 2944 y(dev)m(oted)j(to)g(pro)m(ving)e(circuit)g(lo)m(w)m(er)h(b)s (ounds.)46 b(Ho)m(w)m(ev)m(er,)36 b(to)e(this)d(p)s(oin)m(t)h(this)g (e\013ort)448 3057 y(has)e(met)h(with)e(limited)f(success.)589 3170 y(In)f(an)f(early)h(pap)s(er,)g(Shannon)e([Sha49])i(sho)m(w)m(ed)h (that)f(most)g(Bo)s(olean)g(functions)448 3283 y(require)19 b(exp)s(onen)m(tial)h(size)g(circuits.)36 b(This)18 b(pro)s(of)i(w)m (as)h(nonconstructiv)m(e)f(and)g(pro)m(ving)448 3396 y(b)s(ounds)25 b(on)i(particular)e(functions)h(is)g(more)h(di\016cult.) 38 b(In)26 b(fact,)j(no)e(non-linear)e(lo)m(w)m(er)448 3509 y(b)s(ound)k(is)g(kno)m(wn)h(for)g(the)h(circuit)e(size)h(of)g(a)h (concrete)h(function.)589 3621 y(T)-8 b(o)43 b(get)g(more)f(p)s(ositiv) m(e)g(results)e(one)j(needs)e(to)i(restrict)f(the)g(circuit)f(families) 448 3734 y(b)s(eing)e(considered.)69 b(This)39 b(can)h(b)s(e)g(done)g (b)m(y)g(requiring)d(some)k(uniformit)m(y)d(in)h(the)448 3847 y(function)26 b(mapping)g Fh(n)h Fn(to)h Fh(C)1427 3861 y Fg(n)1474 3847 y Fn(,)h(or)e(it)g(can)h(b)s(e)e(done)i(b)m(y)f (restricting)f(the)i(size)f(or)h(depth)448 3960 y(of)41 b(the)f(circuits)f(themselv)m(es.)71 b(F)-8 b(or)41 b(example,)i(the)d (class)g Fi(A)m(C)2686 3927 y Fc(0)2771 3960 y Fn(consists)g(of)h (those)448 4073 y(languages)d(recognized)f(b)m(y)g(uniform,)g(constan)m (t)h(depth,)g(p)s(olynomial)d(size)h(circuits)448 4186 y(with)f(and,)h(or)g(and)f(not)g(gates)i(whic)m(h)e(allo)m(w)g(un)m(b)s (ounded)d(fan-in.)55 b(One)35 b(early)h(and)448 4299 y(fundamen)m(tal)22 b(results,)h(due)f(to)i(F)-8 b(urst,)24 b(Saxe)f(and)f(Sipser)f([FFS88)r(])i(and)f(Ajtai)g([Ajt83)r(])448 4412 y(is)33 b(that)i(the)f(parit)m(y)g(function)e(is)h(not)h(in)f Fi(A)m(C)2057 4379 y Fc(0)2102 4412 y Fn(,)i(and)e(in)g(fact)i (requires)d(exp)s(onen)m(tial)448 4525 y(size)f Fi(A)m(C)773 4492 y Fc(0)818 4525 y Fn(-t)m(yp)s(e)h(circuits)d([Y)-8 b(ao90)s(].)44 b(This)29 b(immediately)h(implies)e(that)k Fi(A)m(C)3121 4492 y Fc(0)3196 4525 y Fn(di\013ers)448 4638 y(from)41 b(the)g(class)g Fi(A)m(CC)f Fn(of)h(languages)h(whic)m (h)d(ha)m(v)m(e)k(circuit)c(families)g(made)j(from)448 4751 y Fi(A)m(C)600 4718 y Fc(0)674 4751 y Fn(circuits)29 b(with)f(the)i(addition)e(of)i Fh(M)10 b(od)2005 4765 y Fg(m)2102 4751 y Fn(gates,)31 b(with)e(m)g(\014xed)g(for)g(the)h (circuit)448 4863 y(family)-8 b(.)45 b(It)32 b(also)g(can)g(b)s(e)f (sho)m(wn)h(to)g(imply)e(the)i(existence)h(of)f(an)g(oracle)g (separating)448 4976 y(the)f(p)s(olynomial)d(hierarc)m(h)m(y)i(from)f Fi(PSP)-9 b(A)m(CE)q Fn(.)p eop %%Page: 20 20 20 19 bop 589 573 a Fn(It)39 b(is)e(also)h(kno)m(wn)f(that)i(the)f (classes)g Fi(A)m(CC)p Fn(\()p Fi(p)p Fn(\))g(are)h(all)e(distinct,)h (where)g(only)448 686 y Fh(M)10 b(od)637 700 y Fg(p)722 686 y Fn(gates)46 b(are)f(allo)m(w)m(ed,)k(for)44 b(p)g(a)h(prime.)82 b(This)43 b(w)m(as)i(sho)m(wn)f(b)m(y)g(Smolensky)448 799 y([Smo87)q(])d(and)e(Razb)s(oro)m(v)i([Raz98)r(].)70 b Fi(A)m(CC)39 b Fn(itself)g(has)h(resisted)g(all)f(lo)m(w)m(er)h(b)s (ound)448 912 y(tec)m(hniques)30 b(and)g(in)f(fact)i(it)f(is)f(not)h (ev)m(en)h(kno)m(w)g(to)g(b)s(e)e(prop)s(erly)f(con)m(tained)j(in)e Fi(NP)p Fn(.)589 1024 y(Razb)s(oro)m(v)42 b([Raz85b)q(])f(sho)m(w)m(ed) f(that)i(clique)d(do)s(es)h(not)h(ha)m(v)m(e)g(small)e(monotone)448 1137 y(circuits,)22 b(i.e.,)h(just)e(AND)h(and)e(OR)h(gates)h(without)e (negations.)38 b(Ho)m(w)m(ev)m(er,)25 b(this)20 b(result)448 1250 y(sa)m(ys)31 b(more)f(ab)s(out)g(the)g(limitations)d(of)j (monotone)h(circuits)e(as)h(Razb)s(oro)m(v)g([Raz85a)s(])448 1363 y(sho)m(w)m(ed)36 b(that)f(the)g(matc)m(hing)g(problem,)g(kno)m (wn)g(to)g(b)s(e)g(in)e Fi(P)p Fn(,)k(also)e(do)s(es)g(not)g(ha)m(v)m (e)448 1476 y(small)29 b(monotone)i(circuit.)448 1720 y Fe(8.2)113 b(Comm)m(unication)35 b(Complexit)m(y)448 1891 y Fn(Muc)m(h)40 b(of)g(mo)s(dern)f(computer)g(science)h(deals)f (with)g(the)h(sp)s(eed)e(and)i(e\016ciency)f(at)448 2004 y(whic)m(h)29 b(digital)g(comm)m(unication)g(can)i(tak)m(e)h(place.)40 b(Comm)m(unication)29 b(complexit)m(y)h(is)448 2117 y(an)i(attempt)h (to)f(mo)s(del)f(the)h(e\016ciency)f(and)g(in)m(trinsic)f(complexit)m (y)h(of)h(comm)m(unica-)448 2230 y(tion)26 b(b)s(et)m(w)m(een)g (computers.)39 b(It)27 b(studies)d(problems)h(whic)m(h)f(mo)s(del)h(t)m (ypical)h(comm)m(uni-)448 2343 y(cation)j(needs)f(of)g(computations)g (and)g(attempts)h(to)g(determine)e(the)i(b)s(ounds)d(on)i(the)448 2456 y(amoun)m(t)j(of)g(comm)m(unication)f(b)s(et)m(w)m(een)h(pro)s (cessors)e(that)i(these)g(problems)e(require.)589 2569 y(The)g(basic)f(question)g(of)h(comm)m(unication)g(complexit)m(y)g(is,) f(ho)m(w)h(m)m(uc)m(h)g(informa-)448 2682 y(tion)j(do)g(t)m(w)m(o)h (parties)f(need)g(to)g(exc)m(hange)i(in)d(order)g(to)i(carry)f(out)h(a) f(computation?)448 2794 y(W)-8 b(e)32 b(assume)e(b)s(oth)g(parties)f (ha)m(v)m(e)j(unlimited)27 b(computational)j(p)s(o)m(w)m(er.)589 2907 y(F)-8 b(or)29 b(example,)f(consider)e(the)i(case)h(where)e(b)s (oth)g(parties)g(ha)m(v)m(e)i(n)e(input)e(bits)i(and)448 3020 y(they)38 b(w)m(an)m(t)g(to)h(determine)e(if)f(there)i(is)e(a)i(p) s(osition)e(i)h Ff(\024)g Fn(n)g(where)g(the)h(t)m(w)m(o)h(bits)d(in) 448 3133 y(p)s(osition)31 b(i)h(matc)m(h.)49 b(It)33 b(is)f(not)h(hard)f(to)i(see)f(that)h(the)f(comm)m(unication)f (complexit)m(y)448 3246 y(of)d(this)e(problem)f(is)h(n,)i(as)f(the)g(n) g(bits)f(are)h(indep)s(enden)m(t)e(and)i(in)f(the)h(w)m(orst)g(case,)i (all)448 3359 y(n)g(bits)f(of)i(one)g(part)m(y)f(ha)m(v)m(e)i(to)f(b)s (e)f(transmitted)f(to)j(the)e(other.)589 3472 y(No)m(w)41 b(consider)d(the)i(problem)e(of)i(computing)e(the)i(parit)m(y)f(of)h(a) g(string)e(of)i(bits)448 3585 y(where)30 b(1/2)i(of)f(the)g(bits)e(are) i(giv)m(en)g(to)g(part)m(y)g(1)g(and)f(the)h(other)g(half)f(to)h(part)m (y)g(2.)42 b(In)448 3698 y(this)37 b(case,)j(part)m(y)e(1)g(need)f (only)g(compute)h(the)f(parit)m(y)g(of)h(her)f(bits)g(and)f(send)h (this)448 3811 y(parit)m(y)32 b(to)g(part)m(y)g(2)h(who)e(can)h(then)g (compute)g(the)g(parit)m(y)f(of)h(the)g(whole)f(bit)g(string.)448 3924 y(So)g(in)e(this)g(case)i(the)g(comm)m(unication)f(complexit)m(y)g (is)f(a)i(single)e(bit.)589 4036 y(Comm)m(unication)22 b(complexit)m(y)h(has)g(pro)m(vided)e(upp)s(er)g(and)i(lo)m(w)m(er)g(b) s(ounds)e(for)i(the)448 4149 y(complexit)m(y)32 b(of)g(man)m(y)h (fundamen)m(tal)e(comm)m(unication)g(problems.)45 b(It)32 b(has)g(clari\014ed)448 4262 y(the)26 b(role)f(whic)m(h)g(comm)m (unication)g(pla)m(ys)g(in)f(distributed)e(and)j(parallel)f (computation)448 4375 y(as)33 b(w)m(ell)e(as)i(in)e(the)h(p)s (erformance)g(of)h(VLSI)e(circuits.)46 b(It)32 b(also)g(applies)f(and)h (has)g(had)448 4488 y(an)j(impact)g(on)g(the)g(study)g(of)g(in)m (teractiv)m(e)h(proto)s(cols.)54 b(F)-8 b(or)36 b(a)g(go)s(o)s(d)f (surv)m(ey)g(of)g(the)448 4601 y(ma)5 b(jor)31 b(results)e(in)g(this)g (\014eld,)g(consult)h(Nisan)g(and)f(Kushelevitz)g([KN96)q(].)p eop %%Page: 21 21 21 20 bop 448 573 a Fe(8.3)113 b(Pro)s(of)37 b(Complexit)m(y)448 744 y Fn(The)25 b(class)h Fi(NP)f Fn(can)h(b)s(e)f(c)m(haracterized)h (as)g(those)g(problems)e(whic)m(h)g(ha)m(v)m(e)j(short,)f(eas-)448 857 y(ily)33 b(v)m(eri\014ed)f(mem)m(b)s(ership)g(pro)s(ofs.)50 b(Dual)33 b(to)i Fi(NP)p Fn(-complete)g(problems,)e(lik)m(e)g(SA)-8 b(T,)448 970 y(are)44 b Fi(co)p Ff(\000)q Fi(NP)p Fn(-complete)g (problems,)h(suc)m(h)e(as)h(T)-8 b(A)m(UT)44 b(\(the)g(collection)f(of) h(prop)s(o-)448 1083 y(sitional)h(tautologies\).)89 b(T)-8 b(A)m(UT)46 b(is)g(not)g(kno)m(wn)g(to)g(ha)m(v)m(e)i(short,)i(easily) 45 b(v)m(eri\014ed)448 1196 y(mem)m(b)s(ership)39 b(pro)s(ofs,)44 b(and)d(in)f(fact)i(if)f(it)g(did)f(then)h Fi(NP)j Fn(=)f Fi(co)p Ff(\000)q Fi(NP)f Fn(\(see)g(Co)s(ok)448 1309 y(and)e(Rec)m(kho)m(w)h([CR73)q(]\).)70 b(Pro)s(of)40 b(complexit)m(y)g(studies)e(the)j(lengths)e(of)h(pro)s(ofs)f(in)448 1422 y(prop)s(ositional)g(logic)h(and)h(the)g(connections)g(b)s(et)m(w) m(een)g(prop)s(ositional)e(pro)s(ofs)h(and)448 1535 y(computational)c (complexit)m(y)h(theory)-8 b(,)38 b(circuit)e(complexit)m(y)g(and)f (automated)j(theo-)448 1648 y(rem)23 b(pro)m(ving.)38 b(In)22 b(the)h(last)g(decade)h(there)f(ha)m(v)m(e)i(b)s(een)d (signi\014can)m(t)g(adv)-5 b(ances)24 b(in)e(lo)m(w)m(er)448 1761 y(b)s(ounds)h(for)h(prop)s(ositional)f(pro)s(of)h(complexit)m(y)g (as)h(w)m(ell)f(as)h(in)f(the)g(study)g(of)h(new)g(and)448 1874 y(in)m(teresting)30 b(pro)s(of)g(systems.)589 1986 y(Co)s(ok)e(and)f(Rec)m(kho)m(w)i([CR73)q(])f(w)m(ere)g(the)g(\014rst)f (to)h(mak)m(e)h(the)f(notion)f(of)h(a)g(prop)s(o-)448 2099 y(sitional)j(pro)s(of)h(system)g(precise.)46 b(They)32 b(realized)g(that)h(to)g(do)f(this)g(they)g(needed)g(to)448 2212 y(sp)s(ecify)g(exactly)i(what)f(a)h(pro)s(of)e(is)g(and)h(to)h (giv)m(e)g(a)f(general)g(format)h(for)f(presen)m(ting)448 2325 y(and)g(e\016cien)m(tly)h(v)m(erifying)e(a)i(pro)s(of)f Fh(p)p Fn(.)51 b(They)33 b(de\014ned)g(a)h(prop)s(ositional)d(pro)s(of) i(sys-)448 2438 y(tem)k(S)e(to)i(b)s(e)f(a)g(p)s(olynomial-time)e (computable)h(predicate,)j(R,)e(suc)m(h)g(that)g(for)g(all)448 2551 y(prop)s(ositional)30 b(form)m(ulas,)h(F,)i Fh(F)40 b Ff(2)28 b Fh(T)13 b(AU)d(T)68 b Ff(\()-15 b(\))55 b(9)p Fh(p)31 b(S)5 b Fn(\()p Fh(F)s(;)15 b(p)p Fn(\))p Fh(:)33 b Fn(The)e(complexit)m(y)h(of)448 2664 y(S)37 b(is)g(then)h(de\014ned)e (to)j(b)s(e)e(the)h(smallest)f(function)f Fh(f)47 b Fn(:)38 b Fh(N)47 b Ff(\000)-15 b(!)37 b Fh(N)48 b Fn(whic)m(h)36 b(b)s(ounds)448 2777 y(the)30 b(lengths)g(of)g(the)g(pro)s(ofs)f(of)h (S)f(as)i(a)f(function)f(of)h(the)g(lengths)f(of)h(the)g(tautologies) 448 2890 y(b)s(eing)36 b(pro)m(v)m(ed.)63 b(E\016cien)m(t)38 b(pro)s(of)e(systems,)k(those)e(with)e(complexit)m(y)i(b)s(ounded)d(b)m (y)448 3003 y(some)c(p)s(olynomial,)d(are)j(called)e(p)s(olynomial-b)s (ounded)d(pro)s(of)k(systems.)589 3116 y(Sev)m(eral)39 b(natural)f(pro)s(of)h(systems)g(ha)m(v)m(e)h(b)s(een)e(de\014ned)g (and)g(their)g(complexit)m(y)448 3228 y(and)24 b(relationship)d (explored.)38 b(Among)25 b(the)f(most)g(studied)f(are)h(F)-8 b(rege)26 b(and)e(extended-)448 3341 y(F)-8 b(rege)34 b(Pro)s(of)f(systems)f([Urq87)q(])h(and)f([KP89)q(],)h(refutation)f (systems,)i(most)f(notably)448 3454 y(resolution)27 b([Rob65)q(])i(and) f(circuit)f(based)h(pro)s(of)g(systems)h([Ajt83)q(])g(and)f([Bus87)q (].)40 b(W)-8 b(e)448 3567 y(brie\015y)24 b(discuss)f(the)j(complexit)m (y)f(of)g(resolution)f(systems)h(here,)i(but)d(see)i(Beame)h(and)448 3680 y(Pitassi)38 b([BP98)r(])h(for)f(a)i(nice)e(o)m(v)m(erview)i(of)f (results)f(concerning)g(these)i(other)f(pro)s(of)448 3793 y(systems.)589 3906 y(Resolution)26 b(pro)s(of)g(systems)h(are)g (the)g(most)g(w)m(ell-studied)d(mo)s(del.)38 b(Resolution)26 b(is)448 4019 y(a)34 b(v)m(ery)f(restricted)g(pro)s(of)g(system)g(and)g (so)g(has)g(pro)m(vided)f(the)h(setting)g(for)g(the)h(\014rst)448 4132 y(lo)m(w)m(er)22 b(b)s(ound)d(pro)s(ofs.)37 b(Resolution)20 b(pro)s(of)h(systems)h(are)f(refutation)g(systems)h(where)f(a)448 4245 y(statemen)m(t)27 b(D)e(is)e(pro)m(v)m(ed)i(b)m(y)g(assuming)e (its)h(negation)h(and)f(deriving)e(a)j(con)m(tradiction)448 4358 y(from)39 b(this)g(negation.)68 b(In)39 b(a)g(resolution)f(pro)s (of)h(system)h(there)f(is)g(a)h(single)e(rule)g(of)448 4471 y(inference,)29 b(resolution,)e(whic)m(h)h(is)f(a)i(form)f(of)h (cut.)40 b(In)28 b(its)f(prop)s(ositional)f(form)i(it)g(sa)m(y)448 4583 y(that)j(is)f Fh(F)j Ff(_)20 b Fh(x)31 b Fn(and)f Fh(G)20 b Ff(_)g(:)p Fh(x)30 b Fn(are)h(true)f(then)g Fh(F)k Ff(_)20 b Fh(G)29 b Fn(follo)m(ws.)41 b(A)31 b(restricted)f (form)g(of)448 4696 y(resolution,)h(called)g(regular)f(resolution,)h(w) m(as)h(pro)m(v)m(ed)g(to)g(ha)m(v)m(e)g(a)g(sup)s(erp)s(olynomial)448 4809 y(lo)m(w)m(er)41 b(b)s(ound)e(b)m(y)h(Tseitin)f([Tse68)q(])i(on)f (certain)g(tautologies)i(represen)m(ting)e(graph)448 4922 y(prop)s(erties.)d(The)21 b(\014rst)h(sup)s(erp)s(olynomial)c(lo)m (w)m(er)k(b)s(ound)e(for)i(general)g(resolution)f(w)m(as)p eop %%Page: 22 22 22 21 bop 448 573 a Fn(ac)m(hiev)m(ed)30 b(b)m(y)g(Hak)m(en)g([H)-11 b(\027)-57 b(as89)q(])30 b(who)f(in)f(1985)j(pro)m(v)m(ed)e(an)g(exp)s (onen)m(tial)g(lo)m(w)m(er)g(b)s(ound)448 686 y(for)k(the)g(pigeonhole) f(principle.)45 b(Since)32 b(then)h(sev)m(eral)g(other)g(classes)g(of)g (tautologies)448 799 y(ha)m(v)m(e)f(b)s(een)e(sho)m(wn)f(to)i(require)e (sup)s(erp)s(olynomial)e(long)j(resolution)f(pro)s(ofs.)448 1085 y Fo(9)135 b(Quan)l(tum)45 b(Computing)448 1288 y Fn(The)34 b(mark)g(of)g(a)g(go)s(o)s(d)g(scien)m(ti\014c)f(\014eld)g (is)g(its)g(abilit)m(y)g(to)i(adapt)f(to)h(new)e(ideas)h(and)448 1401 y(new)27 b(tec)m(hnologies.)41 b(Computational)26 b(complexit)m(y)h(reac)m(hes)h(this)e(ideal.)39 b(As)27 b(w)m(e)h(ha)m(v)m(e)448 1514 y(dev)m(elop)s(ed)d(new)h(ideas)f(of)h (probabilistic)c(and)j(parallel)f(computation,)j(the)f(complex-)448 1627 y(it)m(y)33 b(comm)m(unit)m(y)g(has)g(not)h(thro)m(wn)e(out)i(the) f(previous)f(researc)m(h,)i(rather)f(they)h(ha)m(v)m(e)448 1740 y(mo)s(di\014ed)29 b(the)j(existing)e(mo)s(dels)g(to)i(\014t)f (these)h(new)f(ideas)f(and)h(ha)m(v)m(e)h(sho)m(wn)f(ho)m(w)g(to)448 1853 y(connect)i(the)f(p)s(o)m(w)m(er)g(of)g(probabilistic)d(and)i (parallel)f(computation)i(to)g(our)f(already)448 1966 y(ric)m(h)j(theory)-8 b(.)53 b(Most)35 b(recen)m(tly)g(complexit)m(y)f (theorists)g(ha)m(v)m(e)h(b)s(egun)e(to)i(analyze)g(the)448 2078 y(computational)30 b(p)s(o)m(w)m(er)h(of)f(mac)m(hines)g(based)g (on)g(quan)m(tum)g(mec)m(hanics.)589 2191 y(In)44 b(1982,)49 b(Ric)m(hard)43 b(F)-8 b(eynman)44 b([F)-8 b(ey82)s(],)47 b(the)e(ph)m(ysicist,)h(noted)e(that)g(curren)m(t)448 2304 y(computer)c(tec)m(hnology)g(could)f(not)g(e\016cien)m(tly)g(sim)m (ulate)g(quan)m(tum)g(systems.)68 b(He)448 2417 y(suggested)27 b(the)g(p)s(ossibilit)m(y)22 b(that)27 b(computers)f(built)e(on)i(quan) m(tum)g(mec)m(hanics)g(migh)m(t)448 2530 y(b)s(e)38 b(able)g(to)h(p)s (erform)e(this)g(task.)66 b(Da)m(vid)38 b(Deutc)m(h)i([Deu85)q(])f(in)e (1985)j(dev)m(elop)s(ed)e(a)448 2643 y(theoretical)29 b(computation)g(mo)s(del)f(based)g(on)g(quan)m(tum)h(mec)m(hanics)f (and)g(suggested)448 2756 y(that)k(suc)m(h)f(a)g(mo)s(del)f(could)g (e\016cien)m(tly)h(compute)g(problems)f(not)h(computable)g(b)m(y)g(a) 448 2869 y(traditional)e(computer.)589 2982 y(Tw)m(o)21 b(quan)m(tum)f(algorithms)f(ha)m(v)m(e)i(receiv)m(ed)g(quite)f(a)g(bit) g(of)g(notice:)36 b(Shor's)19 b([Sho97)q(])448 3095 y(pro)s(cedure)26 b(for)h(factoring)g(in)m(tegers)g(in)f(p)s(olynomial)f(time)h(on)h(a)h (quan)m(tum)e(computer)448 3208 y(and)38 b(Gro)m(v)m(er's)i([Gro96)r(]) e(tec)m(hnique)g(for)h(searc)m(hing)f(a)h(database)g(of)g Fh(n)f Fn(elemen)m(ts)g(in)448 3320 y Fh(O)s Fn(\()555 3255 y Ff(p)p 631 3255 55 4 v 65 x Fh(n)p Fn(\))31 b(time.)589 3433 y(W)-8 b(e)25 b(kno)m(w)e(surprisingly)c(little)i(ab)s(out)i(the)h (computational)e(complexit)m(y)h(of)g(quan-)448 3546 y(tum)33 b(computing.)50 b(Bernstein)33 b(and)f(V)-8 b(azirani)33 b([BV97)r(])h(giv)m(e)g(a)g(formal)f(de\014nition)e(of)448 3659 y(the)k(class)f Fi(BQP)h Fn(of)g(language)g(e\016cien)m(tly)f (computable)g(b)m(y)h(quan)m(tum)f(computers.)448 3772 y(They)29 b(sho)m(w)g(the)g(surprising)c(robustness)j(of)h Fi(BQP)h Fn(whic)m(h)e(remains)f(unscathed)i(un-)448 3885 y(der)42 b(v)-5 b(ariations)41 b(of)i(the)f(mo)s(del)f(suc)m(h)h (as)g(restricting)f(to)i(a)g(small)e(set)h(of)h(rational)448 3998 y(amplitudes,)27 b(allo)m(wing)f(quan)m(tum)h(subroutines)e(and)i (a)h(single)e(measuremen)m(t)j(at)f(the)448 4111 y(end)i(of)h(the)f (computation.)589 4224 y(Bernstein)22 b(and)g(V)-8 b(azirani)21 b(sho)m(w)i(that)f Fi(BQP)h Fn(is)e(con)m(tained)i(in)e Fi(PSP)-9 b(A)m(CE)q Fn(.)37 b(Adle-)448 4337 y(man,)g(DeMarrais)f(and) f(Huang)g([ADH97)r(])h(sho)m(w)f(that)i Fi(BQP)e Fn(is)g(con)m(tained)h (in)e(the)448 4450 y(coun)m(ting)j(class)g Fi(PP)q Fn(.)62 b(Bennett,)40 b(Bernstein,)e(Brassard)f(and)g(V)-8 b(azirani)36 b([BBBV97)s(])448 4562 y(giv)m(e)j(a)f(relativized)f(w)m(orld)f(where)i Fi(NP)g Fn(is)f(not)h(con)m(tained)g(in)e Fi(BQP)q Fn(.)63 b(W)-8 b(e)39 b(do)f(not)448 4675 y(kno)m(w)27 b(an)m(y)h (nonrelativized)d(consequences)i(of)g Fi(NP)h Fn(in)d Fi(BQP)i Fn(or)g(if)f Fi(BQP)i Fn(lies)d(in)h(the)448 4788 y(p)s(olynomial-time)i(hierarc)m(h)m(y)-8 b(.)589 4901 y(What)35 b(ab)s(out)f(quan)m(tum)f(v)-5 b(ariations)33 b(of)h Fi(NP)h Fn(and)e(in)m(teractiv)m(e)i(pro)s(of)e(systems?)p eop %%Page: 23 23 23 22 bop 448 573 a Fn(F)-8 b(enner,)41 b(Green,)g(Homer)d(and)g(Pruim) e([F)m(GHP99)r(])j(consider)e(the)h(class)g(consisting)448 686 y(of)44 b(the)f(languages)g Fh(L)g Fn(suc)m(h)g(that)g(for)g(some)h (p)s(olynomial-time)c(quan)m(tum)j(T)-8 b(uring)448 799 y(mac)m(hine,)28 b Fh(x)f Fn(is)f(in)g Fh(L)g Fn(when)g Fh(M)10 b Fn(\()p Fh(x)p Fn(\))28 b(accepts)g(with)e(p)s(ositiv)m(e)g (probabilit)m(y)-8 b(.)38 b(They)26 b(sho)m(w)448 912 y(the)31 b(equiv)-5 b(alence)30 b(of)g(this)f(class)i(to)g(the)f(coun)m (ting)h(class)f Fi(co)p Ff(\000)p Fi(C)2676 926 y Fj(=)2735 912 y Fi(P)p Fn(.)589 1024 y(W)-8 b(atrous)23 b([W)-8 b(at99)s(])21 b(sho)m(ws)g(that)h(ev)m(ery)g(language)g(in)e Fi(PSP)-9 b(A)m(CE)22 b Fn(has)f(a)h(b)s(ounded-)448 1137 y(round)i(quan)m(tum)g(in)m(teractiv)m(e)h(pro)s(of)f(system.)39 b(Kitaev)25 b(and)f(W)-8 b(atrous)26 b([KW00)q(])f(sho)m(w)448 1250 y(that)41 b(ev)m(ery)g(quan)m(tum)f(in)m(teractiv)m(e)i(pro)s(of)d (system)i(has)f(an)g(equiv)-5 b(alen)m(t)40 b(b)s(ounded-)448 1363 y(round)21 b(pro)s(of)g(system)h(and)f(ev)m(ery)h(suc)m(h)g (language)g(sits)f(in)f(deterministic)g(exp)s(onen)m(tial)448 1476 y(time.)589 1589 y(W)-8 b(e)23 b(ha)m(v)m(e)g(seen)e(quite)g(a)h (bit)e(of)i(progress)f(on)h(quan)m(tum)e(decision)g(tree)j(complexit)m (y)-8 b(.)448 1702 y(In)27 b(this)f(mo)s(del)f(w)m(e)j(coun)m(t)g(the)f (n)m(um)m(b)s(er)e(of)j(queries)e(made)h(to)g(a)h(blac)m(k-b)s(o)m(x)f (database)448 1815 y(of)k(size)f Fh(n)p Fn(.)40 b(Quan)m(tum)30 b(queries)f(can)i(b)s(e)f(made)g(in)f(sup)s(erp)s(osition.)589 1928 y(Deutsc)m(h)35 b(and)e(Jousza)h([DJ92)r(])g(ga)m(v)m(e)i(an)d (early)h(example)f(of)h(a)h(simple)c(function)448 2041 y(that)44 b(can)f(b)s(e)g(solv)m(ed)f(with)g(one)h(query)g(quan)m (tumly)e(but)i(requires)e(\012\()p Fh(n)p Fn(\))i(queries)448 2154 y(deterministically)f(or)j(probabilistically)40 b(with)j(no)i(error.)83 b(Bernstein)44 b(and)g(V)-8 b(azi-)448 2267 y(rani)40 b([BV97)r(])h(giv)m(e)h(the)f(\014rst)g(example)f(of)i (a)f(problem)f(that)h(can)h(b)s(e)e(solv)m(ed)h(with)448 2379 y(p)s(olynomial)d(n)m(um)m(b)s(er)i(of)g(queries)g(quan)m(tumly)f (but)h(requires)f(a)i(sup)s(erp)s(olynomial)448 2492 y(n)m(um)m(b)s(er)24 b(of)h(queries)e(probabilistically)e(with)i(b)s (ounded)g(error.)38 b(Simon)23 b([Sim97])i(giv)m(es)448 2605 y(another)j(example)f(with)f(an)h(exp)s(onen)m(tial)f(gap.)41 b(Brassard)27 b(and)f(H\034y)m(er)j([BH97)q(])f(ga)m(v)m(e)448 2718 y(a)d(zero-error)g(quan)m(tum)f(algorithms)f(for)h(Simon's)f (problem.)37 b(Shor's)24 b(factoring)g(algo-)448 2831 y(rithm)30 b([Sho97])i(can)f(b)s(e)g(view)m(ed)f(as)i(an)f(extension)g (of)g(Simon's)e(problem)h(that)i(\014nds)448 2944 y(the)f(p)s(erio)s(d) d(in)h(a)i(p)s(erio)s(dic)d(blac)m(k-b)s(o)m(x)i(function.)589 3057 y(All)39 b(of)g(these)h(examples)g(require)e(a)i(promise,)h(i.e.,) h(restricting)d(the)g(allo)m(w)m(able)448 3170 y(inputs)21 b(to)j(b)s(e)e(tested.)39 b(F)-8 b(ortno)m(w)24 b(and)e(Rogers)i([FR99) q(])f(and)f(Beals,)j(Buhrman,)f(Clev)m(e,)448 3283 y(Mosca)42 b(and)e(de)g(W)-8 b(olf)41 b([BBC)1501 3250 y Fj(+)1561 3283 y Fn(98)q(])f(sho)m(w)h(that)g(a)f(promise)g(is)f(necessary)i(to)g (get)h(a)448 3396 y(sup)s(erp)s(olynomial)26 b(separation.)448 3682 y Fo(10)135 b(F)-11 b(uture)45 b(Directions)448 3885 y Fn(Despite)23 b(the)g(plethora)f(of)h(exciting)f(results)f(in)g (computational)i(complexit)m(y)f(o)m(v)m(er)i(the)448 3998 y(past)e(fort)m(y)h(y)m(ears,)i(true)d(complexit)m(y)f(class)h (separations)g(ha)m(v)m(e)h(remained)e(b)s(ey)m(ond)h(our)448 4111 y(grasp.)59 b(T)-8 b(ac)m(kling)36 b(these)g(problems,)h(esp)s (ecially)d(sho)m(wing)i(a)g(separation)g(of)h Fi(P)g Fn(and)448 4224 y Fi(NP)q Fn(,)30 b(is)g(our)f(greatest)k(c)m(hallenge) d(for)g(the)h(future.)589 4337 y(Ho)m(w)k(will)c(someone)k(pro)m(v)m(e) g(that)g Fi(P)f Fn(and)f Fi(NP)i Fn(di\013er?)49 b(As)34 b(of)h(this)d(writing,)i(w)m(e)448 4450 y(ha)m(v)m(e)43 b(no)e(serious)f(tec)m(hniques)h(that)h(could)e(help)g(separate)i (these)g(classes.)73 b(What)448 4562 y(kind)35 b(of)h(future)f(ideas)h (could)f(lead)h(us)g(to)g(answ)m(er)h(this)e(di\016cult)f(question?)57 b(Some)448 4675 y(p)s(ossibilities:)585 4863 y Ff(\017)46 b Fn(A)30 b(unexp)s(ected)h(connection)g(to)g(other)g(areas)h(of)f (mathematics)g(suc)m(h)f(as)h(alge-)676 4976 y(braic)e(geometry)j(or)f (higher)e(cohomology)-8 b(.)43 b(P)m(erhaps)31 b(ev)m(en)g(an)f(area)i (of)f(math-)p eop %%Page: 24 24 24 23 bop 676 573 a Fn(ematics)44 b(not)g(y)m(et)h(dev)m(elop)s(ed.)80 b(P)m(erhaps)43 b(someone)i(will)c(dev)m(elop)j(a)g(whole)676 686 y(new)37 b(direction)g(for)h(mathematics)h(in)e(order)h(to)h (handle)e(the)h Fi(P)h Fn(v)m(ersus)f Fi(NP)676 799 y Fn(question.)585 986 y Ff(\017)46 b Fn(New)41 b(tec)m(hniques)g(to)g (pro)m(v)m(er)h(lo)m(w)m(er)f(b)s(ounds)e(for)i(circuits,)i(branc)m (hing)c(pro-)676 1099 y(grams)26 b(and/or)g(pro)s(of)f(systems)h(in)f (mo)s(dels)g(strong)h(enough)g(to)g(giv)m(e)h(complex-)676 1212 y(it)m(y)j(class)g(separations.)585 1400 y Ff(\017)46 b Fn(A)20 b(new)g(c)m(haracterization)h(of)g Fi(P)f Fn(or)h Fi(NP)f Fn(that)h(mak)m(es)g(separation)f(more)h(tractable.)585 1587 y Ff(\017)46 b Fn(A)d(clev)m(er)g(t)m(wist)g(on)f(old-fashioned)g (diagonalization,)j(still)c(the)i(only)f(tec)m(h-)676 1700 y(niques)28 b(that)j(has)f(giv)m(en)h(an)m(y)g(lo)m(w)m(er)f(b)s (ounds)e(on)j(complexit)m(y)f(classes.)589 1888 y(Complexit)m(y)h (theory)i(will)c(progress)j(in)e(areas)j(b)s(ey)m(ond)e(class)h (separation.)45 b(Still,)448 2001 y(quite)26 b(a)h(few)f(in)m (teresting)g(questions)f(remain)g(in)h(man)m(y)g(areas,)i(ev)m(en)f (basic)f(questions)448 2114 y(in)40 b(quan)m(tum)h(computational)g (complexit)m(y)g(remain.)73 b(Complexit)m(y)40 b(theorists)h(will)448 2227 y(con)m(tin)m(ue)h(to)g(forge)g(new)f(ground)g(and)g(\014nd)e(new) i(and)g(exciting)g(results)f(in)h(these)448 2340 y(directions.)589 2452 y(As)34 b(with)e(probabilistic,)f(parallel)h(and)h(quan)m(tum)g (complexit)m(y)-8 b(,)35 b(new)e(mo)s(dels)f(of)448 2565 y(computation)42 b(will)e(b)s(e)h(dev)m(elop)s(ed.)76 b(Computational)41 b(complexit)m(y)h(theorists)g(will)448 2678 y(b)s(e)c(righ)m(t)f(on)h(top)g(of)g(these)g(dev)m(elopmen)m(ts)h (leading)d(the)i(w)m(a)m(y)h(to)g(understand)d(the)448 2791 y(inheren)m(t)30 b(e\016cien)m(t)g(computational)h(p)s(o)m(w)m(er) f(of)g(these)h(mo)s(dels.)589 2904 y(W)-8 b(e)44 b(ha)m(v)m(e)g(seen)e (man)m(y)h(b)s(o)s(oks)e(and)h(p)s(opular)e(news)i(stories)g(ab)s(out)g (the)h(other)448 3017 y(\\complexit)m(y",)24 b(complex)d(systems)f (that)i(o)s(ccur)e(in)g(man)m(y)h(asp)s(ects)g(of)g(so)s(ciet)m(y)h (and)e(na-)448 3130 y(ture)29 b(suc)m(h)g(as)h(\014nancial)d(mark)m (ets,)k(the)f(in)m(ternet,)f(biological)f(systems,)i(the)f(w)m(eather) 448 3243 y(and)35 b(debatably)g(ev)m(en)h(ph)m(ysical)e(systems.)56 b(This)34 b(theory)i(suggests)g(that)g(suc)m(h)f(sys-)448 3356 y(tems)i(ha)m(v)m(e)g(a)g(v)m(ery)g(simple)d(set)j(of)f(rules)f (that)i(when)e(com)m(bined)h(pro)s(duce)f(quite)h(a)448 3469 y(complex)28 b(b)s(eha)m(vior.)39 b(Computer)28 b(programs)g(exhibit)e(a)j(v)m(ery)f(similar)e(b)s(eha)m(vior.)39 b(W)-8 b(e)448 3582 y(will)33 b(see)j(computational)f(complexit)m(y)g (tec)m(hniques)g(used)f(to)i(help)e(understand)g(the)448 3694 y(e\016ciency)d(of)f(the)h(complex)f(b)s(eha)m(vior)f(of)i(these)g (systems.)589 3807 y(Finally)-8 b(,)23 b(computational)f(complexit)m(y) g(will)e(con)m(tin)m(ue)j(to)g(ha)m(v)m(e)h(the)e(Big)h(Surprise.)448 3920 y(No)i(one)g(can)f(predict)f(the)i(next)f(big)f(surprise)f(but)h (it)h(will)e(happ)s(en)g(as)j(it)e(alw)m(a)m(ys)i(do)s(es.)589 4033 y(Let)42 b(us)f(end)g(this)f(surv)m(ey)i(with)e(a)i(quote)g(from)f (Juris)e(Hartmanis')i(noteb)s(o)s(ok)448 4146 y(\(see)32 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