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Fx(\()p Fm(x;)15 b(y)s Fx(\))22 b Fj(^)e Fm(E)5 b Fx(\()p Fm(x;)15 b(z)t Fx(\))21 b Fj(^)f Fx(\()p Fm(E)5 b Fx(\()p Fm(x;)15 b(w)r Fx(\))43 b Fj(!)d Fm(w)28 b Fx(=)d Fm(y)e Fj(_)d Fm(w)28 b Fx(=)d Fm(z)t Fx(\)\))605 2631 y(W)-8 b(e)25 b(next)g(sho)m(w)f(that)h(addition)d(of)j(natural)e(n)m(um)m(b)s (ers)g(is)g(\014rst-order)g(expressible.)456 2739 y(W)-8 b(e)23 b(will)d(see)j(that)g(the)f(\014rst-order)g(queries)f(c)m (haracterize)j(the)e(problems)f(computable)456 2847 y(in)43 b(constan)m(t)k(parallel)c(time.)85 b(Th)m(us)44 b(the)h(follo)m(wing)f (ma)m(y)i(b)s(e)e(though)m(t)i(of)f(as)g(an)456 2955 y(addition)28 b(algorithm)i(that)h(runs)d(in)h(constan)m(t)j(parallel)d (time.)605 3125 y Fi(Pr)n(oposition)34 b Fx(1)p Fi(.)46 b Fr(A)-5 b(ddition)33 b(of)e(natur)-5 b(al)34 b(numb)-5 b(ers,)32 b(r)-5 b(epr)g(esente)g(d)34 b(in)e(binary,)g(is)456 3233 y(\014rst-or)-5 b(der)34 b(expr)-5 b(essible.)605 3402 y Fi(Pr)n(oof.)41 b Fx(W)-8 b(e)29 b(use)f(the)g(w)m(ell-kno)m(wn) 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Fx(\))39 b(holds)e(\(that)j(is,)g(the)456 4153 y(carry)d(is)f(propagated\).)62 b(Let)38 b Fj(\010)f Fx(b)s(e)g(an)g(abbreviation)f(for)h(the)g(comm)m(utativ)m(e)i(and)456 4261 y(asso)s(ciativ)m(e)31 b(\\exclusiv)m(e)f(or")h(op)s(eration.)605 4369 y(W)-8 b(e)32 b(can)e(express)g Fm(')1304 4384 y Fk(add)1449 4369 y Fx(as)h(follo)m(ws,)1270 4529 y Fm(\013)20 b Fj(\010)g Fm(\014)89 b Fj(\021)82 b Fm(\013)26 b Fj($)f(:)p Fm(\014)1199 4664 y(')1258 4679 y Fk(add)1373 4664 y Fx(\()p Fm(x)p Fx(\))84 b Fj(\021)e Fm(A)p Fx(\()p Fm(x)p Fx(\))21 b Fj(\010)f Fm(B)5 b Fx(\()p Fm(x)p Fx(\))21 b Fj(\010)f Fm(')2401 4678 y Fk(car)r(r)r(y)2578 4664 y Fx(\()p Fm(x)p Fx(\))605 4824 y(Note)36 b(that)e(the)h(form)m(ula)e Fm(')1586 4839 y Fk(add)1701 4824 y Fx(\()p Fm(x)p Fx(\))h(has)g(the)h (free)f(v)-5 b(ariable)33 b Fm(x)p Fx(.)52 b(Th)m(us,)34 b Fm(')3155 4839 y Fk(add)3304 4824 y Fx(is)f(a)456 4931 y(description)28 b(of)j Fm(n)e Fx(bits:)40 b(one)31 b(for)f(eac)m(h)h (p)s(ossible)d(v)-5 b(alue)30 b(of)h Fm(x)p Fx(.)p 3378 4931 4 62 v 3382 4873 55 4 v 3382 4931 V 3436 4931 4 62 v 605 5108 a(In)c(second-order)g(logic)g(w)m(e)h(also)f(ha)m(v)m(e)i (v)-5 b(ariables)26 b Fm(X)2419 5122 y Fk(i)2475 5108 y Fx(that)i(range)g(o)m(v)m(er)g(relations)456 5216 y(o)m(v)m(er)j(the) g(univ)m(erse.)40 b(These)30 b(v)-5 b(ariables)29 b(ma)m(y)i(b)s(e)e (quan)m(ti\014ed.)p eop %%Page: 3 3 3 2 bop 605 425 a Fx(A)34 b(second-order)g(existen)m(tial)f(form)m(ula) g(\(SO)p Fj(9)o Fx(\))h(b)s(egins)e(with)h(second)g(order)h(ex-)456 533 y(isten)m(tial)26 b(quan)m(ti\014ers)g(and)h(is)g(follo)m(w)m(ed)g (b)m(y)g(a)h(\014rst-order)f(form)m(ula.)39 b(As)27 b(an)g(example,)456 641 y(the)h(follo)m(wing)f(second-order)h(existen)m(tial)g(sen)m (tence,)i(sa)m(ys)f(that)g(the)g(graph)e(in)g(ques-)456 749 y(tion)k(is)h(three-colorable.)47 b(It)32 b(do)s(es)g(this)f(b)m(y) i(asserting)f(that)h(there)f(are)h(three)f(unary)456 857 y(relations,)42 b(Red)f(\(R\),)g(Y)-8 b(ello)m(w)41 b(\(Y\),)h(and)e(Blue)g(\(B\),)i(de\014ned)d(on)i(the)g(univ)m(erse)e (of)456 965 y(v)m(ertices.)71 b(It)41 b(go)s(es)g(on)g(to)g(sa)m(y)g (that)g(ev)m(ery)g(v)m(ertex)h(has)e(some)h(color)g(and)f(no)g(t)m(w)m (o)456 1073 y(adjacen)m(t)31 b(v)m(ertices)g(ha)m(v)m(e)h(the)e(same)h (color.)728 1386 y(\()p Fj(9)p Fm(R)q Fx(\)\()p Fj(9)p Fm(Y)19 b Fx(\)\()p Fj(9)p Fm(B)5 b Fx(\)\()p Fj(8)p Fm(x)p Fx(\))1480 1285 y Fh(h)1524 1312 y(\000)1565 1386 y Fm(R)q Fx(\()p Fm(x)p Fx(\))21 b Fj(_)f Fm(Y)g Fx(\()p Fm(x)p Fx(\))g Fj(_)g Fm(B)5 b Fx(\()p Fm(x)p Fx(\))2351 1312 y Fh(\001)2428 1386 y Fj(^)35 b Fx(\()p Fj(8)p Fm(y)s Fx(\))2693 1285 y Fh(\020)2748 1386 y Fm(E)5 b Fx(\()p Fm(x;)15 b(y)s Fx(\))52 b Fj(!)1052 1585 y(:)1113 1511 y Fh(\000)1154 1585 y Fm(R)q Fx(\()p Fm(x)p Fx(\))21 b Fj(^)f Fm(R)q Fx(\()p Fm(y)s Fx(\))1636 1511 y Fh(\001)1713 1585 y Fj(^)35 b(:)1870 1511 y Fh(\000)1911 1585 y Fm(Y)20 b Fx(\()p Fm(x)p Fx(\))h Fj(^)f Fm(Y)g Fx(\()p Fm(y)s Fx(\))2399 1511 y Fh(\001)2476 1585 y Fj(^)35 b(:)2633 1511 y Fh(\000)2675 1585 y Fm(B)5 b Fx(\()p Fm(x)p Fx(\))20 b Fj(^)g Fm(B)5 b Fx(\()p Fm(y)s Fx(\))3164 1511 y Fh(\001)3205 1484 y(\021)q(i)605 1807 y Fx(A)27 b Fr(query)h Fx(is)e(a)i(p)s (olynomially)c(b)s(ounded)h(mapping)g(from)i(the)h(set)g(of)f(\014nite) f(struc-)456 1915 y(tures)35 b(of)i(one)f(v)m(o)s(cabulary)g(to)h (those)f(of)g(another)h(v)m(o)s(cabulary)-8 b(.)58 b(F)-8 b(or)37 b(example,)g(the)456 2023 y(form)m(ula)j Fm(')859 2038 y Fk(add)1015 2023 y Fx(from)g(Prop)s(osition)f(1)i(de\014nes)g(a) g(query)f(from)h(pairs)f(of)h(strings)f(to)456 2131 y(strings.)46 b(A)32 b Fr(b)-5 b(o)g(ole)g(an)37 b(query)c Fx(is)e(a)i(set)g(of)g (\014nite)e(structures)h(of)h(some)g(v)m(o)s(cabulary)-8 b(.)47 b(A)456 2239 y(complexit)m(y)33 b(class)g(is)f(a)i(set)g(of)g(b) s(o)s(olean)e(queries:)46 b(those)34 b(queries)e(that)i(can)g(b)s(e)f (rec-)456 2347 y(ognized)h(in)f(a)i(giv)m(en)f(amoun)m(t)h(of)g (computation.)53 b(Descriptiv)m(e)34 b(Complexit)m(y)g(b)s(egan)456 2455 y(with)29 b(F)-8 b(agin's)31 b(descriptiv)m(e)e(c)m (haracterization)i(of)g(the)g(complexit)m(y)f(class)g(NP)-8 b(.)605 2648 y Fi(Theorem)33 b Fx(2)i(\([)p Fs(7)q Fx(]\))p Fi(.)46 b Fr(A)30 b(b)-5 b(o)g(ole)g(an)33 b(query)d Fm(T)43 b Fr(is)31 b(in)f Fx(NP)g Fr(i\013)g(ther)-5 b(e)32 b(exists)f(a)f(se)-5 b(c)g(ond-)456 2756 y(or)g(der)34 b(existential)f(formula,)h Fx(\010)e Fr(such)h(that)59 b Fm(T)64 b Fx(=)2247 2682 y Fh(\010)2300 2756 y Fm(K)2424 2679 y Fh(\014)2424 2733 y(\014)2495 2756 y Fm(K)32 b Fj(j)-15 b Fx(=)25 b(\010)2776 2682 y Fh(\011)2828 2756 y Fr(.)605 2949 y Fx(T)-8 b(o)45 b(capture)g(complexit)m(y)g(classes)f (P)h(and)f(b)s(elo)m(w,)k(\014rst-order)c(logic)g(is)g(more)456 3057 y(appropriate.)55 b(In)35 b(order)g(to)h(c)m(haracterize)h (complexit)m(y)e(classes)h(as)g(in)e(F)-8 b(agin's)36 b(the-)456 3165 y(orem)i(but)f(via)g(\014rst-order)g(logics,)j(one)e(m) m(ust)g(lo)s(ok)f(at)i(the)f(co)s(ding)f(of)h(inputs.)61 b(A)456 3273 y(graph)29 b(or)i(other)f(structure)g(is)g(giv)m(en)g(to)h (a)g(computer)f(in)f(some)i(order,)f(e.g.)42 b(v)m(ertices)456 3381 y Fm(v)500 3395 y Ft(1)539 3381 y Fm(;)15 b(v)623 3395 y Ft(2)663 3381 y Fm(;)g(:)g(:)g(:)32 b(;)15 b(v)924 3395 y Fk(n)971 3381 y Fx(.)38 b(F)-8 b(urthermore,)25 b(algorithms)d(ma)m(y)i(use)f(the)g(ordering,)g(e.g.,)j(searc)m(hing) 456 3489 y(through)g(eac)m(h)i(v)m(ertex)g(in)d(turn.)39 b(T)-8 b(o)27 b(sim)m(ulate)f(the)h(mac)m(hine,)h(the)f(logical)f (languages)456 3597 y(need)33 b(access)i(to)g(the)f(ordering.)50 b(\(This)32 b(w)m(as)j(also)e(necessary)i(for)e(F)-8 b(agin's)35 b(theorem,)456 3705 y(but)29 b(in)g(SO)p Fj(9)g Fx(w)m(e)h(ma)m(y)h(existen)m(tially)e(quan)m(tify)g(a)i(total)g (ordering)d(on)i(the)h(univ)m(erse.\))456 3812 y(F)-8 b(rom)29 b(no)m(w)g(on,)g(w)m(e)g(will)e(assume)h(that)i(all)d (\014rst-order)h(languages)h(include)e(a)i(binary)456 3920 y(relation)35 b(sym)m(b)s(ol)f(\\)p Fj(\024)p Fx(")j(denoting)e(a) h(total)h(ordering)e(on)h(the)g(univ)m(erse.)56 b(With)35 b(this)456 4028 y(pro)m(viso,)41 b(w)m(e)e(can)g(relate)h (computational)f(complexit)m(y)f(to)i(\014rst-order)e(descriptiv)m(e) 456 4136 y(complexit)m(y)-8 b(.)605 4244 y(W)g(e)28 b(will)23 b(see)k(that)g(the)f(set)h(of)g(\014rst-order)e(de\014nable)g(b)s(o)s (olean)g(queries)g(is)h(a)g(w)m(eak)456 4352 y(complexit)m(y)37 b(class.)61 b(One)36 b(w)m(a)m(y)j(to)f(increase)f(the)g(p)s(o)m(w)m (er)g(of)h(\014rst-order)e(logic)h(is)f(b)m(y)456 4460 y(allo)m(wing)42 b(inductiv)m(e)g(de\014nitions.)78 b(This)42 b(is)g(formalized)h(via)g(a)h(least)f(\014xed)g(p)s(oin)m(t)456 4568 y(op)s(erator)30 b(\(LFP\).)605 4676 y(As)g(an)f(example,)h(supp)s (ose)e(that)i(w)m(e)g(w)m(an)m(t)g(to)h(de\014ne)d(the)i(transitiv)m(e) f(closure)g(of)456 4784 y(the)23 b(edge)i(relation)e(of)g(a)h(graph.)38 b(This)22 b(w)m(ould)h(b)s(e)g(useful)e(if)i(w)m(e)h(w)m(ere)g(giv)m (en)g(a)g(directed)456 4892 y(graph)29 b Fm(G)24 b Fx(=)h(\()p Fm(V)1014 4859 y Fk(G)1073 4892 y Fm(;)15 b(E)1185 4859 y Fk(G)1245 4892 y Fm(;)g(s;)g(t)p Fx(\))30 b(and)f(w)m(e)h(w)m(an)m (ted)g(to)g(assert)g(that)g(there)g(is)f(a)h(path)f(from)456 5000 y Fm(s)38 b Fx(to)h Fm(t)p Fx(.)65 b(Let)39 b Fm(E)1022 4967 y Fk(?)1100 5000 y Fx(b)s(e)f(the)h(re\015exiv)m(e,)i(transitiv)m (e)c(closure)h(of)h(the)g(edge)g(relation)f Fm(E)5 b Fx(.)456 5108 y(Giv)m(en)32 b Fm(E)793 5075 y Fk(?)864 5108 y Fx(w)m(e)h(can)f(describ)s(e)f(the)h(reac)m(habilit)m(y)f(prop)s (ert)m(y)h(simply)-8 b(,)30 b(\\)p Fm(E)2975 5075 y Fk(?)3015 5108 y Fx(\()p Fm(s;)15 b(t)p Fx(\)".)47 b(W)-8 b(e)456 5216 y(can)30 b(de\014ne)g Fm(E)957 5183 y Fk(?)1027 5216 y Fx(inductiv)m(ely)e(as)j(follo)m(ws:)p eop %%Page: 4 4 4 3 bop 712 425 a Fm(E)784 388 y Fk(?)824 425 y Fx(\()p Fm(x;)15 b(y)s Fx(\))117 b Fj(\021)f Fm(x)25 b Fx(=)g Fm(y)e Fj(_)d Fm(E)5 b Fx(\()p Fm(x;)15 b(y)s Fx(\))46 b Fj(_)f Fx(\()p Fj(9)p Fm(z)t Fx(\)\()p Fm(E)2368 388 y Fk(?)2409 425 y Fx(\()p Fm(x;)15 b(z)t Fx(\))22 b Fj(^)d Fm(E)2791 388 y Fk(?)2831 425 y Fx(\()p Fm(z)t(;)c(y)s Fx(\)\))258 b(\(3\))605 641 y(Consider)29 b(the)h(follo)m(wing)f(map)h Fm(')c Fx(:)f Fm(R)h Fj(7!)f Fm(')p Fx(\()p Fm(R)q Fx(\))32 b(on)e(binary)e(relations:)911 771 y Fm(')p Fx(\()p Fm(R)q(;)15 b(x;)g(y)s Fx(\))118 b Fj(\021)e Fm(E)5 b Fx(\()p Fm(x;)15 b(y)s Fx(\))21 b Fj(_)f Fx(\()p Fj(9)p Fm(z)t Fx(\)\()p Fm(R)q Fx(\()p Fm(x;)15 b(z)t Fx(\))22 b Fj(^)e Fm(R)q Fx(\()p Fm(z)t(;)15 b(y)s Fx(\)\))457 b(\(4\))605 987 y(Since)23 b Fm(R)h Fx(app)s(ears)f(only)g(p)s(ositiv)m(ely)f(in)g Fm(')p Fx(,)k(this)c(op)s(erator)i(has)f(a)h(least)g(\014xed)f(p)s(oin) m(t)456 1095 y(whic)m(h)29 b(w)m(e)i(tak)m(e)h(as)e(the)h(meaning)e(of) i(the)g(inductiv)m(e)e(de\014nition)f(\(3\).)42 b(Th)m(us,)29 b(w)m(e)i(can)456 1203 y(write,)e Fm(E)785 1170 y Fk(?)876 1203 y Fj(\021)50 b Fx(\(LFP)1210 1217 y Fk(R;x;y)1399 1203 y Fm(')p Fx(\).)605 1311 y(LFP\()p Fm(')p Fx(\))36 b(is)e(the)h(union)e(of)i(the)g(sequence:)50 b Fm(')p Fx(\()p Fj(;)p Fx(\))34 b Fj(\022)e Fm(')p Fx(\()p Fm(')p Fx(\()p Fj(;)p Fx(\)\))k Fj(\022)c Fm(')p Fx(\()p Fm(')p Fx(\()p Fm(')p Fx(\()p Fj(;)p Fx(\)\)\))37 b Fj(\022)456 1421 y(\001)15 b(\001)g(\001)h Fx(.)50 b(Since)33 b(the)h(sequence)g (is)e(monotone,)k(and)d(there)h(are)g(at)g(most)g Fm(n)2942 1388 y Fk(k)3018 1421 y Fx(tuples)f(in)f(a)456 1529 y Fm(k)s Fx(-ary)e(relation,)g(LFP)h(is)e(in)g(fact)i(a)g(p)s(olynomial)d (iterator)j(of)f(form)m(ulas.)605 1637 y(W)-8 b(e)27 b(can)f(transform)g(form)m(ulas)f(suc)m(h)g(as)i Fm(')f Fx(\(4\))h(in)m(to)f(a)g(blo)s(c)m(k)g(of)g(restricted)g(quan-)456 1745 y(ti\014ers.)69 b(Iterating)40 b Fm(')g Fx(then)g(just)g(corresp)s (onds)f(to)i(writing)d(another)i(cop)m(y)h(of)f(this)456 1853 y(quan)m(ti\014er)34 b(blo)s(c)m(k.)54 b(W)-8 b(e)36 b(write)e(F)m(O)q([)p Fm(t)p Fx(\()p Fm(n)p Fx(\)])h(to)h(denote)f (those)h(prop)s(erties)d(expressible)456 1961 y(for)24 b(structures)g(of)h(size)f Fm(n)g Fx(b)m(y)h(form)m(ulas)e(that)j (consist)e(of)h(a)g(blo)s(c)m(k)f(of)g(restricted)h(quan-)456 2069 y(ti\014ers)i(rep)s(eated)h Fm(t)p Fx(\()p Fm(n)p Fx(\))h(times.)39 b(Suc)m(h)28 b(a)h(form)m(ula)e(no)h(matter)h(what)g (the)f(length)g(has)g(a)456 2177 y(\014xed)h(n)m(um)m(b)s(er)g(of)i(v) -5 b(ariables)29 b(|)h(the)h(same)f(v)-5 b(ariables)29 b(that)i(app)s(ear)f(in)f Fm(')p Fx(.)605 2284 y(The)21 b(follo)m(wing)e(theorem)j(sa)m(ys)f(that)h(if)e(w)m(e)i(add)e(to)i (\014rst-order)e(logic)h(the)g(p)s(o)m(w)m(er)h(to)456 2392 y(de\014ne)g(new)h(relations)f(b)m(y)h(induction,)g(then)g(w)m(e)g (can)h(express)f(exactly)h(the)f(prop)s(erties)456 2500 y(that)31 b(are)g(c)m(hec)m(k)-5 b(able)32 b(in)e(p)s(olynomial)e (time.)42 b(This)29 b(is)h(exciting)g(b)s(ecause)h(a)h(v)m(ery)f(nat-) 456 2608 y(ural)c(descriptiv)m(e)h(class)g({)h(\014rst-order)f(logic)h (plus)e(inductiv)m(e)g(de\014nitions)f({)j(captures)456 2716 y(a)f(natural)f(and)g(imp)s(ortan)m(t)g(complexit)m(y)h(class.)40 b(P)m(olynomial)26 b(time)i(is)f(c)m(haracterized)456 2824 y(using)h(only)i(basic)g(logical)g(notions)f(and)h(no)g(men)m (tion)g(of)h(computation.)605 2990 y Fi(Theorem)i Fx(5)i(\([)p Fs(14)q(,)g(15)q(,)g(24)p Fx(]\))p Fi(.)46 b Fr(A)36 b(query)g(is)g(in)h(p)-5 b(olynomial)39 b(time)d(i\013)g(it)g(is)h(de-) 456 3098 y(scrib)-5 b(able)25 b(in)g(\014rst-or)-5 b(der)28 b(lo)-5 b(gic)25 b(with)h(the)g(addition)h(of)e(the)h(le)-5 b(ast)26 b(\014xe)-5 b(d)26 b(p)-5 b(oint)26 b(op)-5 b(er)g(ator.)456 3206 y(This)32 b(is)g(e)-5 b(quivalent)32 b(to)g(b)-5 b(eing)32 b(expr)-5 b(essible)33 b(by)e(a)h(\014rst-or)-5 b(der)34 b(formula)g(iter)-5 b(ate)g(d)33 b(p)-5 b(oly-)456 3314 y(nomial)5 b(ly)34 b(many)f(times.)42 b(In)33 b(symb)-5 b(ols,)34 b Fx(P)50 b(=)h(F)m(O\(LFP)q(\))f(=)h(F)m(O[)p Fm(n)2827 3281 y Fk(O)r Ft(\(1\))2977 3314 y Fx(])p Fr(.)605 3480 y Fx(Theorems)27 b(2)h(and)f(5)g(cast)i(the)f(P)d(=?NP)i(question) f(in)g(a)i(di\013eren)m(t)f(ligh)m(t.)39 b(\(In)27 b(the)456 3588 y(follo)m(wing)c(w)m(e)i(are)g(using)e(the)i(fact)g(that)h(if)d(P) i(w)m(ere)g(equal)f(to)h(NP)-8 b(,)26 b(then)e(NP)h(w)m(ould)e(b)s(e) 456 3696 y(closed)g(under)g(complemen)m(tation.)39 b(It)24 b(w)m(ould)e(then)i(follo)m(w)f(that)i(ev)m(ery)f(second-order)456 3804 y(form)m(ula)29 b(w)m(ould)g(b)s(e)h(equiv)-5 b(alen)m(t)30 b(to)h(a)g(second-order)f(existen)m(tial)g(one.\))605 3970 y Fi(Cor)n(ollar)-6 b(y)34 b Fx(6)p Fi(.)46 b Fx(P)28 b Fr(is)g(e)-5 b(qual)29 b(to)f Fx(NP)g Fr(i\013)g(every)g(se)-5 b(c)g(ond-or)g(der)31 b(expr)-5 b(essible)29 b(pr)-5 b(op-)456 4078 y(erty)35 b(over)h(\014nite,)g(or)-5 b(der)g(e)g(d)38 b(structur)-5 b(es)36 b(is)f(alr)-5 b(e)g(ady)38 b(expr)-5 b(essible)36 b(in)g(\014rst-or)-5 b(der)37 b(lo)-5 b(gic)456 4186 y(using)32 b(inductive)g(de\014nitions.)43 b(In)33 b(symb)-5 b(ols,)34 b Fx(\(P)25 b(=)g(NP\))51 b Fj(,)g Fx(F)m(O\(LFP)q(\))25 b(=)g(SO)o Fr(.)605 4351 y Fx(W)-8 b(e)34 b(men)m(tion)f(t)m(w)m(o)i(other)e(natural)g(op)s(erators)g (that)h(let)f(us)g(capture)g(imp)s(ortan)m(t)456 4459 y(complexit)m(y)40 b(classes.)73 b(Let)41 b Fm(')p Fx(\()p Fm(x)1604 4473 y Ft(1)1644 4459 y Fm(;)15 b(:)g(:)g(:)i(;)e(x)1898 4473 y Fk(r)1936 4459 y Fm(;)g(x)2028 4426 y Fl(0)2028 4484 y Ft(1)2068 4459 y Fm(;)g(:)g(:)g(:)i(x)2282 4426 y Fl(0)2282 4482 y Fk(r)2320 4459 y Fx(\))41 b(b)s(e)f(a)h(form)m(ula)g (with)e(2r)i(free)456 4567 y(v)-5 b(ariables.)41 b(W)-8 b(e)33 b(can)e(think)f(of)h Fm(')h Fx(as)f(represen)m(ting)g(an)g(edge) h(relation)e(o)m(v)m(er)j(a)e(v)m(ertex)456 4675 y(set)44 b Fm(V)69 b Fx(=)48 b Fj(j)p Fm(K)7 b Fj(j)986 4642 y Fk(r)1069 4675 y Fx(consisting)43 b(of)i(all)e Fm(r)s Fx(-tuples)g(from)h(the)g(univ)m(erse)f(of)i Fm(K)7 b Fx(.)82 b(De\014ne)456 4783 y(the)30 b(transitiv)m(e)h(closure)f(op)s (erator,)h(\(TC)1885 4802 y Ft(\026)-39 b Fk(x)o(;)p 1940 4765 40 3 v(x)1980 4779 y Fg(0)2006 4783 y Fm(')p Fx(\),)32 b(denoting)e(the)h(re\015exiv)m(e,)g(transitiv)m(e)456 4892 y(closure)26 b(of)h(the)g(binary)e(relation)h Fm(')p Fx(.)40 b(NSP)-8 b(A)m(CE[log)17 b Fm(n)p Fx(])27 b(is)f(an)h(imp)s (ortan)m(t)f(complexit)m(y)456 5000 y(class)j(that)h(includes)d(most)j (path)g(queries.)39 b(The)29 b(follo)m(wing)f(theorem)i(sa)m(ys)g(that) g(this)456 5108 y(complexit)m(y)g(class)g(is)g(captured)h(b)m(y)f(F)m (O)q(\(TC)o(\).)42 b(\(Note)33 b(that)e(transitiv)m(e)f(closures)g(are) 456 5216 y(a)25 b(sp)s(ecial)e(kind)g(of)i(inductiv)m(e)e (de\014nition,)h(see)h(Equation)f(3.)39 b(Ev)m(ery)25 b(time)g(w)m(e)g(reapply)p eop %%Page: 5 5 5 4 bop 456 425 a Fx(the)30 b(inductiv)m(e)f(equation,)h(the)h(paths)f (considered)f(double)g(in)g(length.)40 b(That)30 b(is)g(wh)m(y)456 533 y(F)m(O\(TC\))25 b Fj(\022)g Fx(F)m(O)q([log)17 b Fm(n)p Fx(].\))605 712 y Fi(Theorem)33 b Fx(7)i(\([)p Fs(16)q(,)g(17)q Fx(]\))p Fi(.)46 b Fr(The)25 b(c)-5 b(omplexity)27 b(class)f(nondeterministic)h(lo)-5 b(garith-)456 820 y(mic)28 b(sp)-5 b(ac)g(e)29 b(is)f(e)-5 b(qual)28 b(to)h(the)f(set)g(of)g(b)-5 b(o)g(ole)g(an)31 b(queries)c(describ)-5 b(able)29 b(in)f(\014rst-or)-5 b(der)30 b(lo)-5 b(gic)456 928 y(with)36 b(the)g(addition)i(of)d(a)h(tr)-5 b(ansitive)37 b(closur)-5 b(e)37 b(op)-5 b(er)g(ator.)54 b(This)36 b(is)f(a)h(sub)-5 b(class)37 b(of)f(the)456 1036 y(set)44 b(of)g(queries)g(describ)-5 b(able)45 b(by)f(\014rst-or)-5 b(der)46 b(formulas)g(iter)-5 b(ate)g(d)45 b Fx(log)17 b Fm(n)44 b Fr(times.)76 b(In)456 1144 y(symb)-5 b(ols,)994 1320 y Fx(NSP)d(A)m(CE[log)17 b Fm(n)p Fx(])118 b(=)g(F)m(O\(TC\))g Fj(\022)g Fx(F)m(O)q([log)17 b Fm(n)p Fx(])605 1499 y(Supp)s(ose)27 b(w)m(e)j(are)f(giv)m(en)h(a)f(\014rst-order)f(form)m(ula)h Fm(')p Fx(\()p Fm(R)q(;)15 b(x)2567 1513 y Ft(1)2607 1499 y Fm(;)g(:)g(:)g(:)i(x)2821 1513 y Fk(r)2858 1499 y Fx(\),)30 b(where)f Fm(R)h Fx(is)e(a)456 1606 y(new)i(relation)g(v)-5 b(ariable,)30 b(but)g(in)f(whic)m(h)h Fm(R)h Fx(need)g(not)g(o)s(ccur)g (only)f(p)s(ositiv)m(ely)-8 b(.)40 b(Then)456 1714 y(the)j(least)g (\014xed)f(p)s(oin)m(t)g(of)h Fm(')g Fx(ma)m(y)g(not)g(exist.)78 b(Ho)m(w)m(ev)m(er,)48 b(w)m(e)43 b(ma)m(y)h(describ)s(e)d(its)456 1822 y(\\partial)e(\014xed)h(p)s(oin)m(t")f(\(PFP\))i(whic)m(h)e(is)g (equal)h(to)h(the)f(\014rst)g(\014xed)f(p)s(oin)m(t)g(in)g(the)456 1930 y(sequence)j Fm(')p Fx(\()p Fj(;)p Fx(\))p Fm(;)15 b(')1117 1897 y Ft(2)1159 1930 y Fx(\()p Fj(;)p Fx(\))p Fm(;)g(:)g(:)g(:)i Fx(,)45 b(or)d Fj(;)g Fx(if)e(there)i(is)f(no)h(suc) m(h)f(\014xed)g(p)s(oin)m(t.)74 b(Since)40 b(this)456 2039 y(sequence)28 b(m)m(ust)f(rep)s(eat)h(after)g(at)g(most)g(2)1915 2006 y Fk(n)1958 1983 y Ff(r)2024 2039 y Fx(steps,)g(PFP)g(ma)m(y)g(b)s (e)f(though)m(t)h(of)g(as)f(an)456 2147 y(exp)s(onen)m(tial)i (iterator.)605 2255 y(The)35 b(follo)m(wing)f(theorem)i(sho)m(ws)f (that)h(the)f(arbitrary)g(iteration)g(of)g(\014rst-order)456 2363 y(form)m(ulas)f({)i(whic)m(h)e(is)h(the)h(same)f(as)h(iterating)f (them)h(exp)s(onen)m(tially)d({)j(allo)m(ws)f(the)456 2471 y(description)26 b(of)i(exactly)h(all)e(prop)s(erties)f (computable)i(using)e(a)j(p)s(olynomial)c(amoun)m(t)456 2579 y(of)30 b(space.)605 2758 y Fi(Theorem)j Fx(8)i(\([)p Fs(13)q(,)g(14)q(,)g(24)p Fx(]\))p Fi(.)46 b Fr(A)33 b(query)f(is)h(in)g(p)-5 b(olynomial)36 b(sp)-5 b(ac)g(e)34 b(i\013)f(it)g(is)g(de-)456 2866 y(scrib)-5 b(able)39 b(in)f(\014rst)h(lo)-5 b(gic)39 b(with)g(the)g(addition)h(of)f(the)g(p) -5 b(artial)40 b(\014xe)-5 b(d)39 b(p)-5 b(oint)40 b(op)-5 b(er)g(ator.)456 2974 y(This)30 b(is)h(e)-5 b(quivalent)30 b(to)h(b)-5 b(eing)30 b(expr)-5 b(essible)31 b(by)g(a)f(\014rst-or)-5 b(der)33 b(formula)e(iter)-5 b(ate)g(d)33 b(exp)-5 b(o-)456 3082 y(nential)5 b(ly,)33 b(or)g(by)g(a)g(se)-5 b(c)g(ond-or)g(der)35 b(formula)f(iter)-5 b(ate)g(d)34 b(p)-5 b(olynomial)5 b(ly.)45 b(In)33 b(symb)-5 b(ols,)686 3274 y Fx(PSP)d(A)m(CE)50 b(=)g(F)m(O)q(\(PFP\))h(=)f(F)m(O)q([2)1968 3236 y Fk(n)2011 3213 y Ff(O)r Fq(\(1\))2146 3274 y Fx(])h(=)f(SO)o(\(TC\))h(=)f(SO)o([) p Fm(n)3039 3236 y Fk(O)r Ft(\(1\))3189 3274 y Fx(])1428 3543 y Fs(3.)j(P)m(arallel)35 b(Complexit)m(y)605 3705 y Fx(Quan)m(ti\014cation)d(is)h(a)g(parallel)f(op)s(eration.)49 b(The)33 b(query)f(\()p Fj(9)p Fm(x)p Fx(\))p Fm(M)10 b Fx(\()p Fm(x)p Fx(\))34 b(can)g(b)s(e)f(ex-)456 3812 y(ecuted)42 b(using)f Fm(n)h Fx(pro)s(cessors)f(in)g(constan)m(t)j (parallel)c(time.)76 b(Pro)s(cessor)42 b Fm(p)3124 3826 y Fk(i)3194 3812 y Fx(c)m(hec)m(ks)456 3920 y(whether)c Fm(M)10 b Fx(\()p Fm(i)p Fx(\))41 b(holds)d(for)h Fm(i)i Fx(=)f(0)p Fm(;)15 b Fx(1)p Fm(;)g(:)g(:)g(:)33 b(;)15 b(n)26 b Fj(\000)g Fx(1.)69 b(An)m(y)39 b Fm(p)2545 3934 y Fk(i)2613 3920 y Fx(for)g(whic)m(h)f Fm(M)10 b Fx(\()p Fm(i)p Fx(\))41 b(do)s(es)456 4028 y(hold)27 b(should)h(write)g(a)h (one)h(in)m(to)f(a)h(sp)s(eci\014ed)d(lo)s(cation)i(in)f(global)g (memory)h(that)h(w)m(as)456 4136 y(originally)d(zero.)605 4244 y(A)21 b Fr(c)-5 b(oncurr)g(ent)25 b(r)-5 b(andom)26 b(ac)-5 b(c)g(ess)24 b(machine)e Fx(\(CRAM\))f(consists)f(of)h(a)g (large)g(n)m(um)m(b)s(er)456 4352 y(of)35 b(pro)s(cessors,)i(all)d (connected)i(to)g(a)g(common,)h(global)e(memory)-8 b(.)56 b(The)35 b(pro)s(cessors)456 4460 y(are)d(iden)m(tical)f(except)i(that) g(they)f(eac)m(h)h(con)m(tain)g(a)f(unique)e(pro)s(cessor)i(n)m(um)m(b) s(er.)44 b(A)m(t)456 4568 y(eac)m(h)35 b(step,)g(an)m(y)f(n)m(um)m(b)s (er)e(of)i(pro)s(cessors)f(ma)m(y)i(read)f(or)f(write)g(an)m(y)i(w)m (ord)e(of)h(global)456 4676 y(memory)-8 b(.)40 b(If)29 b(sev)m(eral)h(pro)s(cessors)e(try)i(to)g(write)e(the)i(same)f(w)m(ord) g(at)h(the)g(same)f(time,)456 4784 y(then)h(the)g(lo)m(w)m(est)h(n)m (um)m(b)s(ered)e(pro)s(cessor)h(succeeds.)605 4892 y(The)35 b(CRAM)h(is)f(a)i(sp)s(ecial)d(case)j(of)f(the)g(concurren)m(t)g(read,) i(concurren)m(t)e(write,)456 5000 y(parallel)e(random)h(access)i(mac)m (hine)f(\(CR)m(CW-PRAM\).)i(Let)e(CRAM[)p Fm(t)p Fx(\()p Fm(n)p Fx(\)])h(b)s(e)e(the)456 5108 y(set)27 b(of)g(queries)f (computable)h(in)f(parallel)f(time)i Fm(O)s Fx(\()p Fm(t)p Fx(\()p Fm(n)p Fx(\))g(b)m(y)g(a)h(CRAM)f(using)e(at)j(most)456 5216 y(p)s(olynomially)f(man)m(y)j(pro)s(cessors.)p eop %%Page: 6 6 6 5 bop 605 425 a Fi(Theorem)33 b Fx(9)p Fi(.)46 b Fx(\([)p Fs(18)r Fx(]\))63 b Fr(F)-7 b(or)34 b(al)5 b(l)33 b(p)-5 b(olynomial)5 b(ly)35 b(b)-5 b(ounde)g(d,)34 b(c)-5 b(onstructible)34 b Fm(t)p Fx(\()p Fm(n)p Fx(\))p Fr(,)1403 576 y Fx(F)m(O)q([)p Fm(t)p Fx(\()p Fm(n)p Fx(\)])119 b(=)e(CRAM)q([)p Fm(t)p Fx(\()p Fm(n)p Fx(\)])25 b Fm(:)605 741 y Fx(This)c(means)h(that)h(the) f(quan)m(ti\014er)f(depth)h(of)g(an)g(optimal)g(description)e(of)i(a)h (query)456 849 y(corresp)s(onds)i(exactly)j(to)f(the)g(parallel)e(time) h(needed)h(to)g(compute)g(the)g(query)-8 b(,)28 b(using)456 957 y(p)s(olynomially)23 b(m)m(uc)m(h)k(hardw)m(are.)39 b(P)m(arallel)25 b(time)i(|)f(one)i(of)e(the)h(most)g(fundamen)m(tal) 456 1065 y(computational)j(resources)g(|)g(is)g(quan)m(ti\014er)f (depth.)605 1173 y(The)35 b(amoun)m(t)i(of)e(hardw)m(are)h(needed)f(to) i(compute)f(a)g(query)f(is)g(closely)g(tied)h(to)456 1281 y(the)29 b(n)m(um)m(b)s(er)g(of)h(v)-5 b(ariables)28 b(needed)h(to)i(describ)s(e)d(the)h(query)-8 b(.)41 b(When)29 b(the)h(amoun)m(t)g(of)456 1389 y(hardw)m(are)g(is)f(measured)h(purely) e(as)j(space,)g(this)e(result)h(is)f(tigh)m(t.)605 1554 y Fi(Theorem)k Fx(10)p Fi(.)47 b Fx([)p Fs(20)q Fx(])63 b Fr(F)-7 b(or)33 b Fm(k)c Fx(=)c(1)p Fm(;)15 b Fx(2)p Fm(;)g(:)g(:)g(:)33 b(;)126 b Fx(DSP)-8 b(A)m(CE[)p Fm(n)2679 1521 y Fk(k)2721 1554 y Fx(])26 b(=)f(V)-10 b(AR[)p Fm(k)24 b Fx(+)c(1])p Fr(.)605 1719 y Fx(The)38 b(fundamen)m(tal)f(m)m (ysteries)h(of)h(complexit)m(y)f(theory)g(are)h(the)g(tradeo\013s)g(b)s (e-)456 1827 y(t)m(w)m(een)h(parallel)d(time)i(and)f(hardw)m(are.)67 b(A)39 b(consequence)h(of)f(Theorems)g(9)g(and)f(10)456 1935 y(is)k(that)j(these)f(m)m(ysteries)f(can)h(all)f(b)s(e)g(understo) s(o)s(d)e(as)j(the)g(descriptiv)m(e)f(tradeo\013)456 2043 y(b)s(et)m(w)m(een)31 b(quan)m(ti\014er)e(depth)h(and)f(n)m(um)m (b)s(er)g(of)i(v)-5 b(ariables.)605 2215 y Fs(Complemen)m(tation.)43 b Fx(One)27 b(of)g(the)g(early)f(successes)i(of)f(descriptiv)m(e)e (complex-)456 2323 y(it)m(y)38 b(w)m(as)g(in)e(resp)s(onse)h(to)i (questions)e(concerning)g(database)i(query)f(languages.)63 b(A)456 2431 y(v)m(ery)42 b(p)s(opular)e(mo)s(del)g(of)i(databases)h (is)e(the)h(relational)f(mo)s(del.)74 b(In)41 b(this)g(mo)s(del,)456 2539 y(databases)23 b(are)f(exactly)i(\014nite)d(logical)h(structures.) 37 b(F)-8 b(urthermore,)24 b(query)e(languages)456 2647 y(are)30 b(based)g(on)h(\014rst-order)e(logic.)605 2755 y(As)38 b(an)f(example,)j(supp)s(ose)c(that)i(w)m(e)g(ha)m(v)m(e)h(an)f (airline)d(database.)64 b(One)37 b(of)h(its)456 2863 y(relations)25 b(migh)m(t)g(b)s(e)g(FLIGHTS,)h(with)e(argumen)m(ts:)39 b(\015igh)m(t)26 b(n)m(um)m(b)s(er,)f(origin,)h(depar-)456 2971 y(ture)h(time,)i(destination,)e(arriv)-5 b(al)27 b(time.)39 b(The)28 b(query)-8 b(,)28 b(\\What)h(are)f(the)h(direct)e (\015igh)m(ts)456 3079 y(from)j(JFK)g(to)h(LAX)f(lea)m(ving)g(in)f(the) i(morning?")39 b(could)30 b(b)s(e)f(phrased)h(as:)963 3230 y(\()p Fj(9)p Fm(t)1082 3245 y Fk(d)1123 3230 y Fm(;)15 b(t)1196 3244 y Fk(a)1237 3230 y Fx(\)\()p Fm(t)1340 3245 y Fk(d)1407 3230 y Fm(<)25 b Fx(12)c Fj(^)f Fx(FLIGHTS)o(\()p Fm(x;)15 b Fx(JFK)q Fm(;)g(t)2477 3245 y Fk(d)2517 3230 y Fm(;)g Fx(LAX)q Fm(;)g(t)2824 3244 y Fk(a)2866 3230 y Fx(\)\))456 3381 y(Note)39 b(that)f(this)e(query)h(has)h(one)g(free)f (v)-5 b(ariable,)39 b Fm(x)p Fx(.)62 b(The)37 b(resp)s(onse)g(to)h(the) g(query)456 3489 y(should)c(b)s(e)h(the)h(set)h(of)f Fm(x)p Fx('s)g(suc)m(h)g(that)h Fm(x)f Fx(is)f(the)h(\015igh)m(t)g(n)m (um)m(b)s(er)f(of)h(a)g(direct)g(\015igh)m(t)456 3597 y(from)30 b(JFK)g(to)h(LAX)f(that)h(lea)m(v)m(es)h(b)s(efore)e(no)s (on.)605 3705 y(Of)i(course)g(not)h(all)e(queries)h(that)h(w)m(e)g (migh)m(t)f(w)m(an)m(t)h(to)g(express)f(are)h(\014rst-order.)456 3812 y(F)-8 b(or)26 b(example,)h(the)f(reac)m(habilit)m(y)f(query)g({)h (is)f(there)h(a)g(route,)i(with)c(no)i(\014xed)f(limit)e(on)456 3920 y(the)j(n)m(um)m(b)s(er)f(of)i(hops,)f(taking)g(me)h(from)f Fm(s)g Fx(to)h Fm(t)p Fx(?)38 b({)27 b(is)f(not)g(\014rst-order.)39 b(F)-8 b(or)27 b(this)e(and)456 4028 y(related)f(reasons,)i(Chandra)e (and)g(Harel)g(prop)s(osed)g(a)h(hierarc)m(h)m(y)f(of)h(query)f (languages)456 4136 y(ab)s(o)m(v)m(e)40 b(\014rst-order)e(logic)h (based)g(on)g(alternating)f(applications)f(of)j(quan)m(ti\014cation,) 456 4244 y(negation,)47 b(and)c(the)h(least)g(\014xed)f(p)s(oin)m(t)g (op)s(erator)h([)p Fs(4)p Fx(].)81 b(It)44 b(w)m(as)g(kno)m(wn)f(that)i (for)456 4352 y(in\014nite)22 b(structures)h(this)h(ga)m(v)m(e)i(a)f (strict)f(hierarc)m(h)m(y)g([)p Fs(22)p Fx(];)j(the)e(same)f(w)m(as)h (conjectured)456 4460 y(for)30 b(\014nite)f(structures.)605 4568 y(There)41 b(is)f(a)h(big)g(di\013erence)f(b)s(et)m(w)m(een)i (in\014nite)d(and)i(\014nite)f(structures.)72 b(Ov)m(er)456 4676 y(an)38 b(in\014nite)f(structure,)k(a)e(least)g(\014xed)f(p)s(oin) m(t)g(migh)m(t)h(nev)m(er)g(reac)m(h)h(a)f(stage)h(of)f(the)456 4784 y(induction)23 b(at)j(whic)m(h)f(it)g(has)h(completed.)39 b(Ov)m(er)25 b(a)i(\014nite)d(structure,)j(there)f(is)e(a)i(\014nite) 456 4892 y(stage)34 b(where)e(the)g(\014xed)g(p)s(oin)m(t)f(is)h (realized.)46 b(F)-8 b(urthermore,)33 b(one)g(can)g(v)m(erify)f(within) 456 5000 y(the)38 b(induction)f(that)i(this)e(stage)j(has)f(b)s(een)f (reac)m(hed.)66 b(Th)m(us,)40 b(b)m(y)e(sa)m(ying)h(that)g(w)m(e)456 5108 y(ha)m(v)m(e)32 b(reac)m(hed)f(the)g(\014nal)e(stage)k(and)d(that) h(tuple)2198 5091 y(\026)2197 5108 y Fm(t)f Fx(is)g(not)h(presen)m(t,)g (w)m(e)h(can)f(express)456 5216 y(the)42 b(negation)h(of)g(\(LFP)15 b Fm(')p Fx(\)\()1479 5199 y(\026)1478 5216 y Fm(t)q Fx(\).)78 b(It)43 b(follo)m(ws)e(that)i(the)g(\\hierarc)m(h)m(y")g (prop)s(osed)e(b)m(y)p eop %%Page: 7 7 7 6 bop 456 425 a Fx(Chandra)30 b(and)i(Harel)g(is)f(not)h(a)h(hierarc) m(h)m(y)e(at)i(all.)45 b(Ev)m(ery)32 b(form)m(ula)f(in)g(F)m(O)q (\(LFP\))h(is)456 533 y(expressible)d(as)k(a)f(single)e(\014xed)h(p)s (oin)m(t)g(of)h(a)h(\014rst-order)e(form)m(ula.)44 b(W)-8 b(e)33 b(sa)m(y)g(that)f(the)456 641 y(\\hierarc)m(h)m(y")e(collapses)g (to)h(its)f(\014rst)g(lev)m(el.)605 802 y Fi(Theorem)j Fx(11)j(\([)p Fs(15)q Fx(]\))p Fi(.)46 b Fr(Every)25 b(formula)i(in)f(\014rst-or)-5 b(der)28 b(lo)-5 b(gic)26 b(with)g(the)g(addition)456 910 y(of)34 b(the)g(le)-5 b(ast)36 b(\014xe)-5 b(d)35 b(p)-5 b(oint)35 b(op)-5 b(er)g(ator)37 b(is)d(e)-5 b(quivalent)35 b(to)f(a)h(single)f(applic)-5 b(ation)37 b(of)d(le)-5 b(ast)456 1018 y(\014xe)g(d)33 b(p)-5 b(oint)34 b(to)f(a)g(\014rst-or)-5 b(der)35 b(formula.)605 1179 y Fx(A)20 b(related)h(question)e(could)g(b)s(e)h(ask)m(ed)h(ab)s (out)f(a)h(prop)s(osed)e(hierarc)m(h)m(y)h(of)g(F)m(O\(TC\).)456 1287 y(In)29 b(fact)j(the)e(transitiv)m(e)g(closure)g(\\hierarc)m(h)m (y")h(also)f(collapses)g(to)h(its)f(\014rst-lev)m(el.)605 1447 y Fi(Theorem)j Fx(12)j(\([)p Fs(17)q Fx(]\))p Fi(.)46 b Fr(Every)d(formula)i(in)e(\014rst-or)-5 b(der)45 b(lo)-5 b(gic)44 b(with)g(the)g(addi-)456 1555 y(tion)31 b(of)g(the)g(tr)-5 b(ansitive)32 b(closur)-5 b(e)32 b(op)-5 b(er)g(ator)34 b(is)c(e)-5 b(quivalent)31 b(to)h(a)f(single)g(applic)-5 b(ation)33 b(of)456 1663 y(tr)-5 b(ansitive)33 b(closur)-5 b(e)34 b(to)f(a)g(\014rst-or)-5 b(der)35 b(formula.)605 1824 y Fx(Deterministic)e(complexit)m(y)g(classes)h(are)h(closed)e (under)g(complemen)m(tation.)51 b(It)456 1932 y(had)34 b(b)s(een)h(long)g(b)s(eliev)m(ed)f(that)j(nondeterministic)32 b(space)k(is)f(a)h(\\one-sided")f(class,)456 2040 y(not)24 b(closed)h(under)e(complemen)m(tation.)39 b(Th)m(us)23 b(the)i(follo)m(wing)e(corollary)h(of)h(Theorem)456 2148 y(12)31 b(w)m(as)g(quite)e(surprising.)605 2309 y Fi(Cor)n(ollar)-6 b(y)34 b Fx(13)i(\([)p Fs(17)q(,)f(23)p Fx(]\))p Fi(.)46 b Fr(F)-7 b(or)42 b Fm(s)p Fx(\()p Fm(n)p Fx(\))e Fj(\025)f Fx(log)17 b Fm(n)p Fr(,)42 b Fx(NSP)-8 b(A)m(CE[)p Fm(s)p Fx(\()p Fm(n)p Fx(\)])41 b Fr(is)f(close)-5 b(d)456 2417 y(under)33 b(c)-5 b(omplementation.)1233 2600 y Fs(4.)53 b(Lo)m(w)m(er)35 b(Bounds)h(and)f(Ordering)605 2762 y Fx(Theorems)43 b(2,)j(5,)h(7,)g(and)42 b(8)i(hold)d(out)i(the)h(prosp)s (ect)e(that)i(w)m(e)f(migh)m(t)g(settle)456 2870 y(questions)e(suc)m(h) h(as)h(P)i(=?NP)d(via)g(logical)g(metho)s(ds.)76 b(In)41 b(particular,)j(there)f(are)456 2978 y(quan)m(ti\014er)35 b(games)j(due)d(to)j(Ehrenfeuc)m(h)m(t)e(and)g(F)-8 b(ra)e(\177)-35 b(\020ss)m(\023)-43 b(e)38 b(that)f(c)m(haracterize)h(the)f(ex-)456 3086 y(pressiv)m(e)e(p)s(o)m(w)m(er)h(of)h(logical)e(languages.)58 b(Ehrenfeuc)m(h)m(t-F)-8 b(ra)e(\177)-35 b(\020ss)m(\023)-43 b(e)38 b(games)f(are)f(pla)m(y)m(ed)456 3194 y(on)e(a)h(pair)f(of)g (structures,)i Fm(A;)15 b(B)5 b Fx(.)53 b(There)34 b(are)i(t)m(w)m(o)g (pla)m(y)m(ers,)f(Samson)g(and)f(Delilah.)456 3302 y(A)m(t)h(eac)m(h)h (mo)m(v)m(e,)h(Samson)e(c)m(ho)s(oses)g(some)g(elemen)m(t)g(of)g(the)g (univ)m(erse)f(of)g(one)h(of)g(the)456 3410 y(structures)29 b(and)h(Delilah)f(m)m(ust)i(resp)s(ond)d(with)i(an)g(elemen)m(t)h(from) f(the)h(other)f(struc-)456 3518 y(ture.)39 b(If)26 b(at)i(the)f(end)f (of)h(the)g(game)h(the)f(map)g(from)f(the)i(elemen)m(ts)f(c)m(hosen)g (from)g Fm(A)g Fx(to)456 3626 y(those)35 b(c)m(hosen)h(from)f Fm(B)40 b Fx(is)34 b(an)h(isomorphism)e(of)i(the)h(induced)d (substructures)h(then)456 3733 y(Delilah)28 b(wins,)g(otherwise)h (Samson)g(wins.)39 b(F)-8 b(or)31 b(most)f(logical)f(languages)h Fm(L)f Fx(there)h(is)456 3841 y(a)k(corresp)s(onding)e(game)k Fm(G)1437 3855 y Fk(L)1523 3841 y Fx(with)c(the)j(follo)m(wing)e (fundamen)m(tal)g(prop)s(ert)m(y)-8 b(.)52 b(W)-8 b(rite)456 3949 y Fm(A)25 b Fj(\021)620 3963 y Fk(L)697 3949 y Fm(B)35 b Fx(to)c(mean)f(that)h(for)g(all)e Fm(')d Fj(2)e Fm(L)p Fx(,)31 b Fm(A)25 b Fj(j)-15 b Fx(=)25 b Fm(')31 b Fx(i\013)f Fm(B)f Fj(j)-15 b Fx(=)26 b Fm(')p Fx(.)516 4076 y(\(Delilah)j(has)h(a) h(winning)c(strategy)32 b(for)e(game)i Fm(G)2245 4090 y Fk(L)2296 4076 y Fx(\()p Fm(A;)15 b(B)5 b Fx(\)\))118 b Fj(,)e Fm(A)25 b Fj(\021)3072 4090 y Fk(L)3149 4076 y Fm(B)65 b Fx(\(14\))605 4292 y(W)-8 b(e)36 b(use)e(Ehrenfeuc)m(h)m (t-F)-8 b(ra)e(\177)-35 b(\020ss)m(\023)-43 b(e)36 b(games)f(to)g(pro)m (v)m(e)h(that)f(certain)f(prop)s(erties)f(are)456 4400 y(not)28 b(expressible)d(in)i(certain)g(logics.)40 b(If)27 b Fm(A)h Fx(and)f Fm(B)32 b Fx(disagree)c(on)g(prop)s(ert)m(y)f(\010)g (and)g(y)m(et)456 4508 y(w)m(e)32 b(can)f(construct)h(a)g(winning)d (strategy)k(for)e(Delilah)f(on)h Fm(G)2586 4522 y Fk(L)2638 4508 y Fx(\()p Fm(A;)15 b(B)5 b Fx(\))32 b(then)f(w)m(e)h(ha)m(v)m(e) 456 4616 y(pro)m(v)m(ed)e(that)h(\010)f(is)g(not)g(expressible)f(in)g Fm(L)p Fx(.)605 4724 y(Consider)j(the)h(follo)m(wing)f(lo)m(w)m(er)i(b) s(ound)d(on)i(quan)m(ti\014er-depth)f(for)h(the)h(reac)m(ha-)456 4832 y(bilit)m(y)28 b(query)i(for)g(and-or)g(graphs)g(\()p Fi(REA)m(CH)2058 4846 y Fk(a)2101 4832 y Fx(\),)h(a)f(natural)g (P-complete)h(query)-8 b(.)605 4993 y Fi(Theorem)33 b Fx(15)j(\([)p Fs(13)q Fx(]\))p Fi(.)46 b(REA)m(CH)1830 5007 y Fk(a)1913 4993 y Fr(is)40 b(describ)-5 b(able)41 b(by)f(line)-5 b(ar-size)40 b(\014rst-or)-5 b(der)456 5100 y(formulas)39 b(that)h(do)f(not)g(include)f(the)g(or)-5 b(dering)40 b(r)-5 b(elation.)60 b(However,)40 b(it)e(is)g(not)h(de-) 456 5216 y(scrib)-5 b(able)33 b(by)f(such)h(formulas)h(of)f(size)g (less)g(than)g Fx(2)2231 5132 y Fl(p)p 2291 5132 146 3 v 2291 5183 a Ft(log)13 b Fk(n)2441 5216 y Fr(.)p eop %%Page: 8 8 8 7 bop 605 425 a Fx(If)20 b(Theorem)g(15)h(could)e(b)s(e)h(sho)m(wn)g (with)f(ordering)g(it)g(w)m(ould)g(follo)m(w)h(that)h Fi(REA)m(CH)3460 439 y Fk(a)456 533 y Fx(is)29 b(not)i(in)e(F)m (O[\(log)17 b Fm(n)p Fx(\)])31 b(and)f(th)m(us)g(that)h(P)f(strictly)f (con)m(tains)i(NSP)-8 b(A)m(CE[log)17 b Fm(n)p Fx(].)605 641 y(The)25 b(Ehrenfeuc)m(h)m(t-F)-8 b(ra)e(\177)-35 b(\020ss)m(\023)-43 b(e)27 b(games)g(b)s(ecome)f(m)m(uc)m(h)g(less)f (useful)f(as)i(a)g(pro)s(of)f(tec)m(h-)456 749 y(nique)32 b(when)h(w)m(orking)f(with)h(ordered)g(graphs.)50 b(This)32 b(is)g(b)s(ecause)i(in)e(a)j(fairly)c(w)m(eak)456 857 y(ordered)26 b(language)i(w)m(e)f(can)g(express)g(the)g(prop)s(ert)m(y) g(of)g(a)g(v)m(ertex)i Fm(v)2775 871 y Fk(i)2830 857 y Fx(that)e(it)g(is)f(v)m(ertex)456 965 y Fm(i)39 b Fx(in)e(the)i (ordering.)65 b(Once)39 b(a)g(language)g(has)g(this)e(strength,)42 b(t)m(w)m(o)e(structures)e(will)456 1073 y(b)s(e)31 b(equiv)-5 b(alen)m(t)31 b(i\013)f(they)i(are)g(iden)m(tical.)43 b(\(If)32 b(w)m(e)g(can)f(assert)h(ab)s(out)g(a)g(graph)f Fm(G)g Fx(that)456 1181 y(v)m(ertex)f(17)g(has)f(an)g(edge)h(to)g(v)m (ertex)h(289,)g(then)e(an)m(y)g(graph)g(that)h(agrees)g(on)f(all)g(suc) m(h)456 1289 y(sen)m(tences)i(is)e(iden)m(tical)h(to)h Fm(G)p Fx(.\))605 1397 y(On)37 b(the)h(other)g(hand,)h(if)e(w)m(e)i (just)e(remo)m(v)m(e)j(the)e(ordering,)h(then)e(Theorems)h(5,)456 1504 y(7,)d(and)f(8)g(all)f(fail.)51 b(F)-8 b(or)35 b(example,)g(it)f (is)f(easy)i(to)g(sho)m(w)f(that)g(without)g(an)g(ordering)456 1612 y(w)m(e)i(cannot)h(coun)m(t.)59 b(In)35 b(fact,)k(if)c(EVEN)h (represen)m(ts)h(the)f(query)-8 b(,)38 b(\\The)e(size)g(of)g(the)456 1720 y(univ)m(erse)29 b(is)g(ev)m(en,")j(then:)605 1866 y Fi(Theorem)h Fx(16)j(\([)p Fs(4)q Fx(]\))p Fi(.)46 b Fr(In)39 b(the)h(absenc)-5 b(e)39 b(of)g(the)h(or)-5 b(dering)41 b(r)-5 b(elation,)42 b(\014rst-or)-5 b(der)456 1974 y(lo)g(gic)26 b(with)h(the)g(addition)h(of)e(the)g(\014xe)-5 b(d)27 b(p)-5 b(oint)28 b(op)-5 b(er)g(ator)29 b(c)-5 b(annot)28 b(describ)-5 b(e)26 b(the)h(pr)-5 b(oblem)456 2081 y(EVEN.)605 2227 y Fx(All)26 b(the)i(graph)f(prop)s(erties)e(that) j(w)m(e)g(w)m(an)m(t)g(to)h(express)e(are)g(order)g(indep)s(enden)m(t.) 456 2335 y(Th)m(us,)e(it)g(w)m(ould)f(b)s(e)g(v)m(ery)i(nice)f(to)h(ha) m(v)m(e)g(a)g(language)f(that)h(captures)g(all)e(p)s(olynomial-)456 2443 y(time)j(computable)g(order-indep)s(enden)m(t)e(prop)s(erties.)38 b(This)26 b(w)m(ould)g(b)s(e)h(analogous)h(to)456 2551 y(Theorem)h(5)h(whic)m(h)f(sa)m(ys)h(that)g(F)m(O)q(\(LFP\))g(captures) g(p)s(olynomial)d(time)i(for)h(ordered)456 2658 y(structures.)605 2766 y(Consider)36 b(the)j(language)f(F)m(O)q Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q Fm(;)15 b(C)7 b Fx(\))38 b(in)f(whic)m(h)g(structures)g(are)i(t)m(w)m(o-)456 2874 y(sorted:)c(their)20 b(univ)m(erse)f(is)g(partitioned)g(in)m(to)h(an)g (unordered)f(domain)g Fm(D)28 b Fx(=)d Fj(f)p Fm(d)3166 2888 y Ft(1)3207 2874 y Fm(;)15 b(d)3294 2888 y Ft(2)3334 2874 y Fm(;)g(:)g(:)g(:)31 b(;)15 b(d)3597 2888 y Fk(n)3645 2874 y Fj(g)456 2982 y Fx(and)33 b(a)h(separate)h(n)m(um)m(b)s(er)d (domain:)46 b Fm(N)41 b Fx(=)31 b Fj(f)p Fx(1)p Fm(;)15 b Fx(2)p Fm(;)g(:)g(:)g(:)33 b(;)15 b(n)p Fj(g)p Fx(.)52 b(W)-8 b(e)34 b(ha)m(v)m(e)h(the)f(database)456 3090 y(predicates)d(de\014ned)g(on)h Fm(D)j Fx(and)c(the)h(standard)g (ordering)e(de\014ned)h(on)h Fm(N)10 b Fx(.)45 b(The)32 b(t)m(w)m(o)456 3198 y(sorts)e(are)h(com)m(bined)f(via)g(coun)m(ting)g (quan)m(ti\014ers,)g(\()p Fj(9)p Fm(i)15 b(x)p Fx(\))p Fm(')p Fx(\()p Fm(x)p Fx(\),)32 b(meaning)e(that)h(there)456 3306 y(exist)g(at)h(least)g Fm(i)g Fx(elemen)m(ts)g Fm(x)g Fx(suc)m(h)f(that)i Fm(')p Fx(\()p Fm(x)p Fx(\).)45 b(Here)32 b Fm(i)g Fx(is)f(a)h(n)m(um)m(b)s(er)e(v)-5 b(ariable)31 b(and)456 3414 y Fm(x)f Fx(is)f(a)i(domain)e(v)-5 b(ariable.)605 3522 y(It)27 b(w)m(as)h(conjectured)f(that)g(F)m(O)q Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q Fm(;)15 b(C)7 b Fx(\))27 b(is)f(equal)g(to)i(order)f(indep)s(enden)m(t)456 3630 y(P)o(.)41 b(Instead,)29 b(in)f([)p Fs(3)p Fx(])i(it)e(w)m(as)i (pro)m(v)m(ed)f(that)h(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q Fm(;)15 b(C)7 b Fx(\))29 b(is)f(strictly)g(con)m (tained)456 3738 y(in)33 b(order-indep)s(enden)m(t)f(P.)54 b(In)33 b(the)i(follo)m(wing)e(lo)m(w)m(er)i(b)s(ound,)f(the)h(graphs)f (are)h(\\al-)456 3846 y(most)30 b(ordered.")40 b(They)29 b(consist)g(of)h Fm(n=)p Fx(4)g(groups)f(of)h(4)g(v)m(ertices)g(eac)m (h)h(and)e(there)h(is)f(a)456 3954 y(giv)m(en)h(total)h(ordering)e(on)i (the)f(groups.)605 4099 y Fi(Theorem)j Fx(17)j(\([)p Fs(3)q Fx(]\))p Fi(.)46 b Fr(Ther)-5 b(e)41 b(is)g(an)g(or)-5 b(der-indep)g(endent)43 b(pr)-5 b(op)g(erty)44 b(of)c(gr)-5 b(aphs,)456 4207 y Fm(T)13 b Fr(,)34 b(that)i(is)f(very)f(e)-5 b(asy)36 b(to)f(c)-5 b(ompute)36 b({)f(in)f(a)h(c)-5 b(omplexity)37 b(class)e(wel)5 b(l)35 b(b)-5 b(elow)35 b Fx(P)g Fr({)g(but,)456 4315 y(in)43 b(the)g(absenc)-5 b(e)44 b(of)g(or)-5 b(dering,)47 b(is)c(not)h(expr)-5 b(essible)44 b(in)g(\014rst-or)-5 b(der)45 b(lo)-5 b(gic)44 b(with)g(the)456 4423 y(addition)34 b(of)f(the)g(\014xe)-5 b(d)33 b(p)-5 b(oint)34 b(op)-5 b(er)g(ator)36 b(and)e(c)-5 b(ounting)33 b(quanti\014ers.)605 4568 y Fx(Some)d(b)s(eautiful)e(w)m (ork)j(b)m(y)f(Martin)g(Grohe)g(and)g(his)f(studen)m(ts)h(has)g(sho)m (wn)g(that)456 4676 y(there)48 b(are)h(large)f(and)g(imp)s(ortan)m(t)g (classes)g(of)g(graphs)g(on)g(whic)m(h)f(the)i(language)456 4784 y(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q Fm(;)15 b(C)7 b Fx(\))30 b(do)s(es)f(capture)h(order)f(indep)s(enden)m (t)e(P)i([)p Fs(11)q(,)34 b(8,)g(10)q Fx(].)41 b(F)-8 b(or)30 b(these)456 4892 y(classes)e(of)g(graphs,)g(isomorphism)e(is)h (testable)i(in)d(p)s(olynomial)g(time)i(using)e(the)j(nat-)456 5000 y(ural)23 b(algorithm)h(for)g(equiv)-5 b(alence)24 b(in)g(a)h(sublanguage)f(of)g(F)m(O)q Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q Fm(;)15 b(C)7 b Fx(\).)39 b(These)456 5108 y(results)24 b(include)f(natural)i(classes)g(of)h(graphs)f(for)g (whic)m(h)g(no)g(p)s(olynomial-time)e(graph)456 5216 y(isomorphism)k(algorithm)j(had)f(b)s(een)h(previously)e(kno)m(wn.)p eop %%Page: 9 9 9 8 bop 605 425 a Fx(Most)41 b(imp)s(ortan)m(t)f(complexit)m(y)g (classes)g(b)s(elo)m(w)g(P)g(ha)m(v)m(e)i(had)e(their)f(languages)456 533 y(without)21 b(ordering,)h(or)h(with)d(some)j(partial)e(orderings,) i(separated)f([)p Fs(13)q(,)j(14)q(,)g(9)q(,)g(6,)h(5)p Fx(].)456 641 y(No)33 b(suc)m(h)f(separation)g(had)g(b)s(een)g(pro)m(v) m(ed)g(for)h(the)f(languages)h(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))p Fx(\(LFP\))g(and)456 749 y(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(PFP)q(\).)41 b(In)30 b(1991)i(Abiteb)s(oul)c (and)i(Vian)m(u)f(explained)g(wh)m(y)h(b)m(y)g(pro)m(ving)605 893 y Fi(Theorem)j Fx(18)j(\([)p Fs(1)q Fx(]\))p Fi(.)46 b Fr(The)33 b(fol)5 b(lowing)33 b(c)-5 b(onditions)35 b(ar)-5 b(e)33 b(e)-5 b(quivalent:)605 1019 y Fx(1.)42 b(F)m(O)q Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP\))51 b(=)g(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(PFP)q(\))605 1127 y(2.)42 b(F)m(O)q(\(LFP\))51 b(=)f(F)m(O)q(\(PFP\))605 1235 y(3.)42 b Fr(P)58 b Fx(=)f Fr(PSP)-7 b(A)n(CE)605 1379 y Fx(Theorem)37 b(18)g(is)f(pro)m(v)m(ed)h(b)m(y)g(sho)m(wing)f (that)h(if)f(t)m(w)m(o)i(structures)e Fm(G)g Fx(and)g Fm(H)44 b Fx(are)456 1487 y(F)m(O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))o Fx(\(LFP)q(\)-equiv)-5 b(alen)m(t,)27 b(then)e(they)g(are)h(F)m (O)p Fe(\(w)n(o)p Fd(\024)p Fe(\))p Fx(\(PFP\)-equiv)-5 b(alen)m(t)26 b(as)f(w)m(ell.)456 1595 y(Th)m(us,)35 b(ordering)g(is)f(not)i(the)f(problem,)h(but)e(Ehrenfeuc)m(h)m(t-F)-8 b(ra)e(\177)-35 b(\020ss)m(\023)-43 b(e)37 b(games)f(are)g(un-)456 1702 y(lik)m(ely)29 b(to)i(help)e(separate)i(P)f(from)g(PSP)-8 b(A)m(CE.)1511 1886 y Fs(5.)53 b(Mo)s(del)35 b(Chec)m(king)605 2048 y Fx(Descriptiv)m(e)21 b(Complexit)m(y)g(has)g(b)s(een)g(applied)f (recen)m(tly)h(to)i(computer-aided)e(v)m(er-)456 2156 y(i\014cation.)61 b(When)38 b(w)m(e)g(design)e(a)i(program,)i (distributed)34 b(proto)s(col,)40 b(or)d(circuit,)i(w)m(e)456 2264 y(do)d(so)h(in)e(a)i(formal)f(language.)60 b(This)35 b(migh)m(t)h(b)s(e)g(a)h(programming)f(language)h(suc)m(h)456 2372 y(as)i(Ja)m(v)-5 b(a)40 b(or)f(a)h(circuit)e(design)g(language)i (suc)m(h)f(as)g(VHDL,)h(or)f(a)h(soft)m(w)m(are)g(design)456 2480 y(language)30 b(suc)m(h)g(as)h(Statec)m(harts.)605 2588 y(Suc)m(h)20 b(a)g(formal)g(design)f(determines)g(a)i(logical)f (structure,)i Fm(K)32 b Fx(=)25 b Fj(h)p Fm(S;)15 b(p)3002 2602 y Ft(1)3041 2588 y Fm(;)g(:)g(:)g(:)32 b(;)15 b(p)3304 2603 y Fk(k)3347 2588 y Fm(;)g(R)q Fj(i)p Fx(,)456 2695 y(called)28 b(a)i(Kripk)m(e)e(structure)h(or)h(transition)d(system.)41 b(Here)30 b Fm(S)k Fx(consists)29 b(of)h(the)f(set)h(of)456 2803 y(p)s(ossible)24 b(global)i(states)j(of)e(the)g(design,)g(the)g (set)h(of)f(all)f(p)s(ossible)e(gate)29 b(v)-5 b(alues)26 b(for)h(the)456 2911 y(circuit,)33 b(or)h(all)e(p)s(ossible)f (assignmen)m(ts)i(of)h(v)-5 b(alues)33 b(to)i(the)e(v)-5 b(ariables)33 b(of)g(a)h(program,)456 3019 y(or)25 b(all)f(p)s(ossible) f(assignmen)m(ts)i(of)h(states)g(to)g(eac)m(h)h(pro)s(cessor)e(in)f (the)h(distributed)e(pro-)456 3127 y(to)s(col.)44 b(Binary)31 b(relation)g Fm(R)h Fx(is)f(the)g(next)h(mo)m(v)m(e)h(relation:)43 b Fm(R)q Fx(\()p Fm(s;)15 b(s)2751 3094 y Fl(0)2774 3127 y Fx(\))31 b(means)h(that)g(the)456 3235 y(mo)m(v)m(e)f(from)f Fm(s)g Fx(to)h Fm(s)1132 3202 y Fl(0)1186 3235 y Fx(is)e(a)i(p)s (ossible)d(atomic)j(transition)e(of)h(the)h(circuit)e(or)h(program.)605 3343 y(The)35 b(size)h(of)g Fm(S)41 b Fx(is)35 b(often)i(exp)s(onen)m (tial)d(in)h(the)h(size)g(of)g(the)g(design.)57 b(This)34 b(phe-)456 3451 y(nomenon)28 b(is)f(called)h(the)h Fr(state)i (explosion)i(pr)-5 b(oblem.)42 b Fx(F)-8 b(or)29 b(this)e(reason,)j (structure)e Fm(K)456 3559 y Fx(is)h(often)i(represen)m(ted)f(sym)m(b)s (olically)-8 b(.)605 3667 y(Once)21 b(w)m(e)g(ha)m(v)m(e)h(our)e (formal)g(design,)i(a)f(represen)m(tation)g(of)g(the)g(transition)e (system)456 3775 y(can)38 b(b)s(e)f(automatically)h(generated.)65 b(W)-8 b(e)40 b(ma)m(y)e(no)m(w)g(wish)f(to)i(write)e(some)h(simple)456 3883 y(correctness)27 b(conditions)f(in)f(a)j(formal)e(language.)40 b(F)-8 b(or)27 b(example,)h(w)m(e)f(migh)m(t)g(w)m(an)m(t)h(to)456 3991 y(express)i(the)g(follo)m(wing:)605 4117 y(1.)42 b(If)30 b(the)h(Restart)g(button)f(is)f(pressed,)h(w)m(e)h(ev)m(en)m (tually)f(restart.)605 4225 y(2.)42 b(Do)s(ors)31 b(are)g(not)f(op)s (ened)g(b)s(et)m(w)m(een)h(stations.)605 4333 y(3.)42 b(Division)29 b(is)g(p)s(erformed)g(correctly)-8 b(.)605 4441 y(4.)42 b(The)30 b(motor)h(is)e(not)i(turned)e(on)i(during)d(main) m(tenance.)605 4568 y(These)e(conditions)f(express)g(the)i(fact)g(that) g(\(1\))g(if)e(w)m(e)i(press)e(the)i(Restart)g(button)456 4676 y(on)43 b(a)h(p)s(ersonal)e(computer)i(it)f(will)e(ev)m(en)m (tually)i(restart)i(|)e(without)g(our)g(ha)m(ving)456 4784 y(to)g(unplug)d(it)h(and)h(plug)f(it)h(in)f(again;)49 b(\(2\))43 b(the)g(sub)m(w)m(a)m(y)f(train)g(con)m(troller)g(meets)456 4892 y(some)31 b(simple)f(safet)m(y)j(conditions;)d(\(3\))j(In)m(tel's) e(new)g(pro)s(cessor)g(do)s(es)g(its)g(arithmetic)456 5000 y(correctly;)k(and)e(\(4\))h(the)f(mixer)g(in)f(a)h(n)m(uclear)g (w)m(aste)i(storage)f(con)m(tainer)g(\(whic)m(h)e(is)456 5108 y(designed)h(to)i(k)m(eep)h(its)e(con)m(ten)m(ts)i(mixed)e(and)g (th)m(us)g(less)g(v)m(olatile\))h(cannot)g(turn)e(on)456 5216 y(\(and)d(th)m(us)g(cause)h(injury\))d(during)g(main)m(tenance.)p eop %%Page: 10 10 10 9 bop 605 425 a Fx(No)m(w)27 b(that)g(w)m(e)g(ha)m(v)m(e)h(formally) d(expressed)h(our)g(design)f(and)h(our)g(statemen)m(t)j(con-)456 533 y(cerning)j(its)g(b)s(eha)m(vior,)h(w)m(e)h(can)f(simply)e(press)h (a)i(button)e(and)h(b)s(e)f(told)h(whether)f(or)456 641 y(not)e Fm(K)i Fj(j)-15 b Fx(=)25 b Fm(')p Fx(,)31 b(that)g(is,)f(do)s (es)g(the)h(design)e(satisfy)h(the)g(desired)f(prop)s(ert)m(y?)605 749 y(This)37 b(is)g(the)h(mo)s(del)g(c)m(hec)m(king)h(problem:)55 b(giv)m(en)38 b Fm(K)45 b Fx(and)38 b Fm(')p Fx(,)j(test)e(whether)f Fm(K)456 857 y Fx(satis\014es)f Fm(')p Fx(.)65 b(Mo)s(del)38 b(c)m(hec)m(k)m(ers)i(are)f(curren)m(tly)e(b)s(eing)g(hailed)f(as)j (great)g(debugging)456 965 y(aides.)79 b(This)42 b(they)h(are.)81 b(W)-8 b(e)45 b(can)e(write)g(a)h(simple)d(prop)s(ert)m(y)i(that)h(our) f(system)456 1073 y(should)27 b(satisfy)h(and)h(press)f(the)h(button.) 40 b(If)29 b(it)f(do)s(es)h(not)g(satisfy)g(the)g(prop)s(ert)m(y)g (then)456 1181 y(w)m(e)f(will)d(b)s(e)i(told)h(so.)40 b(T)m(ypical)26 b(mo)s(del)h(c)m(hec)m(king)i(programs)e(will)e(presen) m(t)j(a)g(coun)m(ter-)456 1289 y(example)g(run)f(of)i(the)g(system,)g (i.e.,)h(they)e(pro)s(duce)g(the)h(explicit)e(bug.)39 b(If)29 b(the)f(mo)s(del)456 1397 y(do)s(es)33 b(satisfy)h(the)g(prop)s (ert)m(y)-8 b(,)35 b(then)e(w)m(e)i(will)c(b)s(e)i(told)h(so.)52 b(Note)35 b(that)f(this)f(is)g(b)s(etter)456 1504 y(than)i(just)g(not)h (\014nding)d(a)j(bug.)56 b(Our)34 b(mo)s(del)h(c)m(hec)m(k)m(er)i(has)f (automatically)f(pro)m(v)m(ed)456 1612 y(that)c(our)e(design)h(has)g(a) h(certain)f(desirable)e(prop)s(ert)m(y)-8 b(.)605 1720 y(The)37 b(connection)g(b)s(et)m(w)m(een)h(mo)s(del)e(c)m(hec)m(king)i (and)f(descriptiv)m(e)f(complexit)m(y)h(is)456 1828 y(clear.)64 b(Since)37 b(the)i(mo)s(dels)d(in)m(v)m(olv)m(ed)i(are)h(h)m(uge,)i(it) c(is)h(useful)e(to)j(write)e(queries)h(in)456 1936 y(languages)26 b(that)g(only)g(express)f(feasible)g(queries.)38 b(A)26 b(p)s(opular)e(and)h(quite)h(expressiv)m(e)456 2044 y(temp)s(oral)36 b(logic)h(for)g(mo)s(del)f(c)m(hec)m(king)j(is)d(CTL)2152 2011 y Fk(?)2190 2044 y Fx(.)62 b(The)37 b(mo)s(del)f(c)m(hec)m(king)i (problem)456 2152 y(for)29 b(CTL)783 2119 y Fk(?)851 2152 y Fx(is)g(PSP)-8 b(A)m(CE)30 b(complete)g(and)f(that)i(is)d(PSP)-8 b(A)m(CE)30 b(in)f(the)g(size)h(of)g(the)g(h)m(uge)456 2260 y(Kripk)m(e)f(structure.)42 b(Ho)m(w)m(ev)m(er,)34 b(it)c(w)m(as)h(sho)m(wn)g(in)e([)p Fs(21)q Fx(])i(that)h(CTL)2796 2227 y Fk(?)2865 2260 y Fx(can)f(b)s(e)g(em)m(b)s(ed-)456 2370 y(ded)j(in)g(linear)g(time)h(in)f(the)i(simple)d(language)j(F)m(O) 2286 2334 y Ft(2)2325 2370 y Fx(\(TC\))g(|)f(the)g(sublanguage)g(of)456 2478 y(F)m(O\(TC\))27 b(in)e(whic)m(h)g(there)i(are)g(at)g(most)g(t)m (w)m(o)h(domain)d(v)-5 b(ariables.)38 b(The)26 b(catc)m(h)i(is)e(that) 456 2586 y(for)40 b(v)m(ery)h(complex)g(form)m(ulas)f(in)f(CTL)1867 2553 y Fk(?)1906 2586 y Fx(,)k(the)e(translation)f(migh)m(t)h(also)f (include)f(a)456 2693 y(linear)27 b(n)m(um)m(b)s(er)h(of)h(b)s(o)s (olean)f(v)-5 b(ariables.)39 b(Ho)m(w)m(ev)m(er,)32 b(in)27 b(practice,)j(most)g(queries)e(that)456 2801 y(p)s(eople)h(seem)i(to)g (w)m(an)m(t)g(to)g(write)f(in)m(v)m(olv)m(e)g(zero)h(or)g(v)m(ery)g (close)f(to)h(zero)h(b)s(o)s(oleans.)605 2909 y(In)g([)p Fs(2)q Fx(],)i(it)f(is)f(sho)m(wn)h(that)h(the)f(em)m(b)s(edding)e(of)j (CTL)2467 2876 y Fk(?)2538 2909 y Fx(in)e(F)m(O)2775 2873 y Ft(2)2814 2909 y Fx(\(TC\))h(lands)f(in)g(a)456 3017 y(fragmen)m(t)c(that)g(can)g(b)s(e)f(e\016cien)m(tly)g(mo)s(del)f (c)m(hec)m(k)m(ed)k(in)c(time)h Fm(O)s Fx(\()p Fj(j)-15 b(j)p Fm(K)7 b Fj(j)-15 b(jj)p Fm(')p Fj(j)p Fx(2)3042 2984 y Fk(n)3085 2996 y Ff(b)3122 3017 y Fx(\):)40 b(linear)456 3125 y(in)28 b(the)i(size)g(of)g(the)g(structure)g(and)f(the)h(form)m (ula,)g(but)f(exp)s(onen)m(tial)g(in)g(the)h(n)m(um)m(b)s(er)456 3233 y(of)g(b)s(o)s(oleans)f(in)g(the)i(form)m(ula.)605 3341 y(This)19 b(is)i(useful,)g(b)s(oth)f(b)s(ecause)i Fm(n)1752 3356 y Fk(b)1807 3341 y Fx(tends)e(to)i(b)s(e)f(tin)m(y)-8 b(,)23 b(and)d(b)s(ecause)i(the)f(language)456 3449 y(in)m(v)m(olv)m (ed)h(is)h(closely)f(tied)h(to)h(reac)m(habilit)m(y)e(queries)g(whic)m (h)g(are)h(the)h(bread)e(and)h(butter)456 3557 y(of)f(mo)s(del)g(c)m (hec)m(king.)39 b(One)22 b(nice)g(feature)h(is)e(that)i(w)m(e)g(can)g (lo)s(ok)f(at)h(the)g(form)m(ula,)h(coun)m(t)456 3665 y(the)37 b(n)m(um)m(b)s(er)e(of)i(b)s(o)s(oleans,)h(and)e (automatically)h(sa)m(y)g(whether)g(the)g(query)f(can)h(b)s(e)456 3773 y(c)m(hec)m(k)m(ed)32 b(e\016cien)m(tly)e(or)g(not.)605 3881 y(In)37 b(summary)-8 b(,)39 b(descriptiv)m(e)e(complexit)m(y)h (rev)m(eals)g(a)g(simple)e(and)h(elegan)m(t)i(view)456 3989 y(of)c(computation.)55 b(New)35 b(insigh)m(ts)e(and)i(ev)m(en)h (practical)e(algorithms)g(ha)m(v)m(e)j(resulted)456 4097 y(from)30 b(this)f(approac)m(h.)1705 4340 y Fs(References)494 4485 y Fn([1])39 b(S.)25 b(Abiteb)r(oul)g(and)g(V.)f(Vian)n(u,)h (\\Generic)h(Computation)f(And)f(Its)g(Complexit)n(y)-6 b(,")25 b Fc(23r)l(d)j(A)n(CM)613 4576 y(STOC)e Fn(\(1991\),)h (209-219.)494 4668 y([2])39 b(N.)c(Alec)n(hina)g(and)f(N.)h(Immerman,)f (\\E\016cien)n(t)i(F)-6 b(ragmen)n(t)34 b(of)h(T)-6 b(ransitiv)n(e)35 b(Closure)h(Logic,")613 4759 y(man)n(uscript,)25 b(1999.)494 4850 y([3])39 b(J.)h(Cai,)j(M.)c(F)r(\177)-41 b(urer,)43 b(N.)38 b(Immerman,)h(\\An)f(Optimal)g(Lo)n(w)n(er)h(Bound)f(on)h(the)f (Num)n(b)r(er)f(of)613 4942 y(V)-6 b(ariables)27 b(for)f(Graph)g(Iden)n (ti\014cation,")g Fc(Combinatoric)l(a,)g Fn(\(12:4\))h(\(1992\),)g (389-410.)494 5033 y([4])39 b(A.)31 b(Chandra)g(and)f(D.)h(Harel,)i (\\Structure)d(and)g(Complexit)n(y)g(of)h(Relational)h(Queries,")h Fc(21st)613 5124 y(Symp.)k(on)g(F)-6 b(oundations)39 b(of)d(Computer)i(Scienc)l(e,)h Fn(1980,)h(\(333-347\).)e(Also)e(app)r (eared)g(in)g(a)613 5216 y(revised)26 b(form)g(in)f Fc(JCSS)h Fb(25)p Fn(,)h(1982,)g(\(99-128\).)p eop %%Page: 11 11 11 10 bop 494 425 a Fn([5])39 b(K.)28 b(Etessami)g(and)f(N.)g (Immerman,)e(\\T)-6 b(ree)28 b(Canonization)h(and)e(T)-6 b(ransitiv)n(e)28 b(Closure,")i Fc(IEEE)613 517 y(Symp.)e(L)l(o)l(gic)g (In)f(Comput.)h(Sci.)e Fn(\(1995\),)i(331-341.)494 608 y([6])39 b(K.)34 b(Etessami)g(and)f(N.)g(Immerman,)f(\\Reac)n(habilit)n (y)i(and)f(the)g(P)n(o)n(w)n(er)h(of)g(Lo)r(cal)h(Ordering,")613 699 y Fc(Eleventh)55 b(Symp.)27 b(The)l(or)l(etic)l(al)i(Asp)l(e)l(cts) h(Comp.)d(Sci.)g Fn(\(1994\),)g(123)g(-)e(135.)494 790 y([7])39 b(R.)18 b(F)-6 b(agin,)20 b(\\Generalized)g(First-Order)d(Sp)r (ectra)h(and)g(P)n(olynomial-Time)g(Recognizable)h(Sets,")613 882 y(in)26 b Fc(Complexity)i(of)f(Computation,)g Fn(\(ed.)e(R.)h (Karp\),)f Fc(SIAM-AMS)j(Pr)l(o)l(c.)g(7,)d Fn(1974,)i(\(27-41\).)494 973 y([8])39 b(M.)58 b(F)-6 b(ric)n(k)56 b(and)h(M.)g(Grohe,)66 b(\\Chec)n(king)57 b(First-Order)g(Prop)r(erties)h(of)f(Lo)r(cally)h(T) -6 b(ree-)613 1064 y(Decomp)r(osable)26 b(Graphs,")h(\(1999\).)494 1156 y([9])39 b(E.)c(Gr\177)-38 b(adel)35 b(and)f(G.)h(McColm,)i(\\On)d (the)g(P)n(o)n(w)n(er)h(of)f(Deterministic)g(T)-6 b(ransitiv)n(e)35 b(Closures,")613 1247 y Fc(Information)28 b(and)g(Computation)h Fn(\(119:1\))e(\(1995\),)g(129-135.)456 1338 y([10])39 b(M)29 b(Grohe)f(and)g(J.)h(Mari)r(~)-41 b(no,)31 b(\\De\014nabilit)n (y)d(and)g(Descriptiv)n(e)g(Complexit)n(y)f(on)h(Databases)i(of)613 1430 y(Bounded)c(T)-6 b(ree-Width,")25 b(to)h(app)r(ear)g(in)g Fc(Intl.)h(Conf.)f(on)i(Datab)l(ase)i(The)l(ory)p Fn(\(1999\))456 1521 y([11])39 b(M.)34 b(Grohe,)j(\\Finite)d(V)-6 b(ariable)34 b(Logics)g(in)g(Descriptiv)n(e)f(Complexit)n(y)g(Theory)-6 b(,")35 b Fc(Bul)t(letin)g(of)613 1612 y(Symb)l(olic)28 b(L)l(o)l(gic)g Fn(,)e(4\(4\))g(\(1998\),)h(345-)g(398.)456 1704 y([12])39 b(N.)27 b(Immerman,)c Fc(Descriptive)30 b(Complexity)p Fn(,)d(1998,)i(Springer)d(Graduate)h(T)-6 b(exts)26 b(in)g(Computer)613 1795 y(Science,)h(New)e(Y)-6 b(ork.)456 1886 y([13])39 b(N.)26 b(Immerman,)c(\\Num)n(b)r(er)i(of)i (Quan)n(ti\014ers)f(is)h(Better)g(than)f(Num)n(b)r(er)e(of)k(T)-6 b(ap)r(e)25 b(Cells,")j Fc(JCSS)613 1978 y Fn(\(22:3\))f(\(1981\),)h (65-72.)456 2069 y([14])39 b(N.)27 b(Immerman,)d(\\Upp)r(er)i(and)g(Lo) n(w)n(er)h(Bounds)f(for)i(First)f(Order)f(Expressibilit)n(y)-6 b(,")27 b Fc(JCSS)g Fb(25)p Fn(,)613 2160 y(No.)f(1)g(\(1982\),)i (76-98.)456 2252 y([15])39 b(N.)31 b(Immerman,)e(\\Relational)j (Queries)f(Computable)f(in)h(P)n(olynomial)g(Time,")h Fc(Information)613 2343 y(and)c(Contr)l(ol,)e Fn(68)h(\(1986\),)g (86-104.)456 2434 y([16])39 b(N.)33 b(Immerman,)f(\\Languages)j(That)e (Capture)g(Complexit)n(y)f(Classes,")38 b Fc(SIAM)33 b(J.)h(Comput.)613 2526 y Fb(16)p Fn(,)27 b(No.)f(4)g(\(1987\),)h (760-778.)456 2617 y([17])39 b(N.)29 b(Immerman,)d(\\Nondeterministic)i (Space)g(is)h(Closed)g(Under)e(Complemen)n(tation,")i Fc(SIAM)613 2708 y(J.)f(Comput.)d Fb(17)p Fn(,)i(No.)f(5)g(\(1988\),)h (935-938.)456 2800 y([18])39 b(N.)d(Immerman,)f(\\Expressibilit)n(y)i (and)e(P)n(arallel)j(Complexit)n(y)-6 b(,")38 b Fc(SIAM)e(J.)g(of)h (Comput.)f Fn(18)613 2891 y(\(1989\),)28 b(625-638.)456 2982 y([19])39 b(N.)18 b(Immerman,)e(\\Descriptiv)n(e)i(and)f (Computational)h(Complexit)n(y)-6 b(,")19 b(in)e Fc(Computational)k (Com-)613 3074 y(plexity)26 b(The)l(ory,)d Fn(ed.)g(J.)g(Hartmanis,)g Fc(Pr)l(o)l(c.)i(Symp.)f(in)g(Applie)l(d)h(Math.,)e Fn(38,)h(American)d (Math-)613 3165 y(ematical)27 b(So)r(ciet)n(y)e(\(1989\),)j(75-91.)456 3258 y([20])39 b(N.)29 b(Immerman,)e(\\DSP)-6 b(A)n(CE[)p Fa(n)1553 3226 y Ff(k)1591 3258 y Fn(])53 b(=)26 b(V)-9 b(AR[)p Fa(k)21 b Fn(+)e(1],")30 b Fc(Sixth)h(IEEE)f(Structur)l(e)j(in) d(Complexity)613 3350 y(The)l(ory)f(Symp.)d Fn(\(1991\),)h(334-340.)456 3441 y([21])39 b(N.)22 b(Immerman)c(and)j(M.)h(V)-6 b(ardi.)22 b(Mo)r(del)g(Chec)n(king)g(and)f(T)-6 b(ransitiv)n(e)22 b(Closure)g(Logic.)h Fc(Pr)l(o)l(c.)i(9th)613 3532 y(Int'l)34 b(Conf.)g(on)g(Computer-A)n(ide)l(d)i(V)-6 b(eri\014c)l(ation)35 b Fn(\(CA)-9 b(V'97\),)36 b(Lecture)d(Notes)g(in)g(Computer)613 3624 y(Science,)27 b(Springer-V)-6 b(erlag)25 b(291)i(-)e(302,)i(1997.) 456 3715 y([22])39 b(Yiannis)33 b(N.)e(Mosc)n(ho)n(v)l(akis,)k Fc(Elementary)g(Induction)f(on)f(A)n(bstr)l(act)j(Structur)l(es,)g Fn(North)31 b(Hol-)613 3806 y(land,)26 b(1974.)456 3898 y([23])39 b(R.)j(Szelep)r(cs)n(\023)-36 b(en)n(yi,)46 b(\\The)c(Metho)r(d)g(of)g(F)-6 b(orced)42 b(En)n(umeration)e(for)j (Nondeterministic)e(Au-)613 3989 y(tomata,")27 b Fc(A)l(cta)h (Informatic)l(a)e Fb(26)g Fn(\(1988\),)h(279-284.)456 4080 y([24])39 b(M.)33 b(V)-6 b(ardi,)33 b(\\Complexit)n(y)e(of)i (Relational)g(Query)e(Languages,")k Fc(14th)f(A)n(CM)f(STOC)g(Symp.,) 613 4171 y Fn(\(1982\),)28 b(137-146.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF