%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: draft.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips draft.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.08.06:1232 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 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b(In)n(tro)r(duction) 262 1650 y FC(One)16 b(of)g(the)h(most)e(prominen)o(t)f(complexit)o(y)h (classes,)i(other)f(than)h(p)q(olynomial)12 b(time,)j(is)262 1699 y(the)j(class)g(NC)g(of)f(functions)h(computable)f(in)g(parallel)g (p)q(olylogarithmi)o(c)f(time)g(with)i(a)262 1749 y(p)q(olynomial)8 b(amoun)o(t)i(of)h(hardw)o(are.)h(This)g(class)g(has)g(sev)o(eral)g (natural)f(c)o(haracterizations)262 1799 y(in)h(terms)h(of)g(circuits,) g(alternating)g(T)m(uring)f(mac)o(hines,)g(or)h(parallel)f(random)f (access)k(ma-)262 1849 y(c)o(hines)g(as)h(used)g(in)e(this)h(w)o(ork.)g (It)g(can)g(b)q(e)h(argued)f(that)h(NC)f(is)g(the)g(class)h(of)f (e\016cien)o(tly)262 1899 y(parallalizable)f(problems,)h(just)h(as)h(p) q(olynomial)c(time)i(is)h(generally)g(considered)i(as)f(the)262 1949 y(correct)e(formalization)c(of)i(feasible)h(sequen)o(tial)g (computation.)324 1999 y(Mac)o(hine-indep)q(enden)o(t)i(c)o (haracterizations)h(of)d(computational)f(complexit)o(y)g(classes)262 2049 y(are)20 b(not)g(only)f(of)g(theoretical,)i(but)f(recen)o(tly)h (also)e(of)h(increasing)g(practical)g(in)o(terest.)262 2099 y(Besides)h(indicating)e(the)i(robustness)h(and)e(naturalness)h (of)e(the)i(classes)g(in)f(question,)262 2149 y(they)14 b(also)f(pro)o(vide)g(guidance)h(for)g(the)g(dev)o(elopmen)o(t)f(of)g (programming)d(languages)j([11].)p 262 2190 237 2 v 269 2217 a Fs(?)300 2233 y Fx(Supp)q(orted)i(b)o(y)e(the)g(DF)o(G)g (Graduiertenk)o(ol)q(leg)j(\\Logik)f(in)e(der)h(Informatik")254 2262 y Fs(??)300 2278 y Fx(Supp)q(orted)h(b)o(y)e(the)g(DF)o(G)g(Emm)o (y)h(No)q(ether-Programme)f(under)h(gran)o(t)g(No.)e(Jo)h(291/2-1)223 2308 y Fs(?)7 b(?)g(?)300 2324 y Fx(The)18 b(hospitalit)o(y)j(of)d(the) h(Mittag-Le\017er)g(Institute)g(in)g(the)g(spring)h(of)e(2001)g(is)h (gratefully)300 2370 y(ac)o(kno)o(wledged.)271 2401 y Fr(y)300 2417 y Fx(Supp)q(orted)12 b(b)o(y)g(a)e(Marie)i(Curie)g(fello) o(wship)h(of)d(the)h(Europ)q(ean)h(Union)g(under)g(gran)o(t)f(no.)g (ERB-)300 2462 y(FMBI-CT98-3248)p 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y(Bellan)o(toni)g([2],)f(in)i(all)f(\014nite)h(t)o(yp)q(es.)g (More)h(precisely)m(,)f(our)g(system)g(is)g(a)g(t)o(yp)q(ed)h(lam)o(b)q (da)262 1477 y(calculus)f(whic)o(h)h(allo)o(ws)e(t)o(w)o(o)h(kinds)g (of)g(function)h(t)o(yp)q(es,)g(denoted)g Fp(\033)g Fo(\()e Fp(\034)20 b FC(and)c Fp(\033)f Fn(!)f Fp(\034)5 b FC(,)262 1527 y(and)16 b(t)o(w)o(o)g(sorts)h(of)e(v)n(ariables)h(of)g(the)h (ground)f(t)o(yp)q(e)h Fp(\023)p FC(,)e(the)i Fq(c)n(omplete)j FC(ones)d(in)f(addition)262 1577 y(to)d(the)i(usual)f(ones,)g(whic)o(h) g(are)h(called)f(incomplete)f(for)h(emphasis.)e(A)j(function)e(of)h(t)o (yp)q(e)262 1627 y Fp(\033)h Fn(!)g Fp(\034)20 b FC(can)c(only)f(b)q(e) h(applied)g(to)f(complete)g(terms)h(of)f(t)o(yp)q(e)h Fp(\033)q FC(,)g(i.e.,)e(terms)i(con)o(taining)262 1677 y(only)d(complete)g(free)i(v)n(ariables.)324 1746 y(It)h(features)i(t)o (w)o(o)d(recursion)j(op)q(erators)f Fm(LR)g FC(and)f Fm(CR)o FC(,)g(the)h(latter)f(corresp)q(onding)i(to)262 1795 y(Clote's)h([8])g(concatenation)h(recursion)h(on)e(notation,)f (whic)o(h)i(can)g(naturally)f(only)g(b)q(e)262 1845 y(applied)12 b(to)h(\014rst-order)h(functions.)f(The)g(former)f(is)h(a)g(form)e(of)i (recursion)h(of)e(logarithmic)262 1895 y(length)h(c)o(haracteristic)h (of)f(all)e(function)i(algebra)g(represen)o(tations)i(of)d(NC,)h(and)g (here)h(can)262 1945 y(b)q(e)d(applied)g(to)f(functions)h(of)g(all)f (linear)g(t)o(yp)q(es,)i(i.e.,)d(t)o(yp)q(es)j(only)e(built)h(up)g (using)f Fp(\023)h FC(and)g Fo(\()p FC(.)262 1995 y(The)16 b(function)f(b)q(eing)h(iterated,)f(as)h(w)o(ell)f(as)h(the)g(n)o (umerical)e(argumen)o(t)h(b)q(eing)g(recurred)262 2044 y(on)g(ha)o(v)o(e)g(to)g(b)q(e)h(complete,)e(i.e.,)g(the)i(t)o(yp)q(e)f (of)g Fm(LR)h FC(is)f Fp(\033)g Fo(\()f FC(\()p Fp(\023)f Fn(!)h Fp(\033)h Fo(\()f Fp(\033)q FC(\))g Fn(!)f Fp(\023)h Fn(!)f Fp(\033)k FC(for)262 2094 y(linear)c Fp(\033)q FC(.)324 2163 y(Our)18 b(analysis)g(clearly)f(rev)o(eals)i(the)g 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y(set)d(of)g(base)g(functions)g(is)g(sligh)o (tly)e(extended.)324 1490 y(Finally)m(,)g(in)i(order)h(to)g(b)q(e)g (able)f(to)h(handle)f(n)o(umerals)g(in)g(parallel)f(logarithmic)f (time,)262 1540 y(w)o(e)g(use)g(a)g(tree)h(data)f(structure)i(to)d (store)i(n)o(umerals)e(during)g(the)i(computation.)c(Whereas)262 1589 y(trees)i(are)f(used)h(as)f(the)g(principal)f(data)h(structure)i (in)d(other)i(c)o(haracterizations)f(of)g(parallel)262 1639 y(complexit)o(y)k(classes)19 b([15,)5 b(16],)15 b(our)i(system)g(w)o(orks)g(with)f(usual)h(binary)f(n)o(umerals,)g(and) 262 1689 y(trees)f(are)f(only)f(used)i(in)f(the)g(implemen)o(tation.) 262 1847 y Ft(2)56 b(Clote's)17 b(F)-5 b(unction)19 b(Algebra)g(for)f (NC)262 1972 y FC(Clote)g([8])f(giv)o(es)i(a)f(function)g(algebra)g(c)o (haracterization)h(of)f(NC)h(using)f(t)o(w)o(o)g(recursion)262 2022 y(sc)o(hemes.)d(The)g(class)h Fl(A)f FC(is)g(de\014ned)h(as)f(the) h(least)f(class)g(of)g(functions)g(that)g(con)o(tain)g(the)262 2072 y(constan)o(t)j(0,)f(pro)r(jections)i Fp(\031)726 2057 y FA(n)725 2082 y(j)749 2072 y FC(\()p Fp(x)789 2078 y FB(1)807 2072 y Fp(;)7 b(:)g(:)g(:)12 b(;)7 b(x)931 2078 y FA(n)953 2072 y FC(\))18 b(=)g Fp(x)1061 2078 y FA(j)1079 2072 y FC(,)f(the)i(binary)e(successors)k Fp(s)1536 2078 y FB(0)1555 2072 y FC(,)c Fp(s)1603 2078 y FB(1)1622 2072 y FC(,)h(bit)262 2129 y(test)e Fp(bit)p FC(,)f(binary)g(length)h Fn(j)o Fp(x)p Fn(j)e FC(:=)g Fn(d)p FC(log)870 2139 y FB(2)888 2129 y FC(\()p Fp(x)c FC(+)f(1\))p Fn(e)p FC(,)15 b(and)h(#)f(where)h Fp(x)p FC(#)p Fp(y)g FC(=)e(2)1477 2114 y Fz(j)p FA(x)p Fz(j\001j)o FA(y)q Fz(j)1565 2129 y FC(,)h(and)h(is)262 2179 y(closed)e(under)h (comp)q(osition)d(and)h(the)i(follo)o(wing)c(t)o(w)o(o)j(forms)e(of)h (recursion:)324 2234 y(A)h(function)f Fp(f)18 b FC(is)c(de\014ned)h(b)o (y)e Fq(c)n(onc)n(atenation)j(r)n(e)n(cursion)e(on)h(notation)j FC(\(CRN\))13 b(from)262 2284 y(functions)g Fp(g)q(;)7 b(h)504 2290 y FB(0)523 2284 y Fp(;)g(h)566 2290 y FB(1)598 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b(!)996 371 y Fp(x)19 b FC(\))683 434 y Fp(F)6 b FC(\()p Fp(s)751 440 y FA(i)765 434 y FC(\()p Fp(y)q FC(\))p Fp(;)837 420 y Fn(\000)-27 b(!)842 434 y Fp(x)18 b FC(\))11 b(=)h Fp(h)979 440 y FA(i)993 434 y FC(\()p Fp(y)q(;)1049 420 y Fn(\000)-27 b(!)1053 434 y Fp(x)18 b(;)7 b(F)f FC(\()p Fp(y)q(;)1203 420 y Fn(\000)-28 b(!)1207 434 y Fp(x)18 b FC(\)\))748 496 y Fp(F)6 b FC(\()p Fp(y)q(;)837 482 y Fn(\000)-27 b(!)842 496 y Fp(x)18 b FC(\))11 b Fn(\024)h Fp(r)q FC(\()p Fp(y)q(;)1031 482 y Fn(\000)-27 b(!)1036 496 y Fp(x)17 b FC(\))756 558 y Fp(f)t FC(\()p Fp(y)q(;)836 545 y Fn(\000)-26 b(!)842 558 y Fp(x)18 b FC(\))11 b(=)h Fp(F)6 b FC(\()p Fn(j)o Fp(y)q Fn(j)h Fp(;)1074 545 y Fn(\000)-28 b(!)1078 558 y Fp(x)18 b FC(\))12 b Fp(:)262 659 y Fl(Theorem)i(1)21 b(\(Clote)15 b([8]\).)h Fq(A)f(numb)n(er-the)n (or)n(etic)f(function)h Fp(f)20 b Fq(is)15 b(in)g Fj(A)g Fq(if)f(and)i(only)f(if)262 709 y Fp(f)k Fq(is)c(in)g(NC.)324 797 y FC(It)i(is)f(easy)i(to)f(see)h(that)f(the)g(recursion)h(sc)o 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1223 128 2 v 589 1262 a FC(2)610 1250 y Fz(dj)p FA(n)p Fz(j)p FA(=)p FB(2)o Fz(e)721 1187 y Fi(k)762 1233 y FC(has)f(length)g(ab)q(out)g(half)g(the)h(length)f(of)g Fp(n)p FC(.)f(Both)i(forms)262 1297 y(of)14 b(recursion)i(pro)q(duce)g (log)7 b Fn(j)o Fp(y)q Fn(j)15 b FC(iterations)g(of)g(the)g(step)h (function.)e(W)m(e)h(shall)f(denote)i(the)262 1347 y(function)d (algebra)h(with)f(b)q(ounded)i(logarithmi)o(c)d(recursion)j(b)o(y)f Fl(A)g FC(as)g(w)o(ell.)262 1495 y Ft(3)56 b(F)-5 b(ormal)17 b(De\014nition)h(of)h(the)f(System)262 1610 y FC(W)m(e)11 b(use)j(simple)d(t)o(yp)q(es)i(with)f(t)o(w)o(o)g(forms)f(of)g (abstraction)i(o)o(v)o(er)f(a)h(single)f(base)g(t)o(yp)q(e)h Fp(\023)p FC(,)f(i.e.,)262 1660 y(our)h(t)o(yp)q(es)i(are)g(giv)o(en)e (b)o(y)h(the)g(grammar)742 1758 y Fp(\033)o(;)7 b(\034)16 b FC(::=)11 b Fp(\023)g Fn(j)g Fp(\033)i Fo(\()f Fp(\034)k Fn(j)11 b Fp(\033)i Fn(!)e Fp(\034)16 b(;)262 1855 y FC(and)d(w)o(e)h(call)f(the)i(t)o(yp)q(es)g(that)f(are)g(built)f(up)h (from)e 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y(no)q(des)15 b(ha)o(v)o(e)e (out-degree)j(one,)d(and)h(w)o(e)g(denote)h(the)f(p)q(oin)o(ter)g(to)g (its)g(c)o(hild)f(b)o(y)h Fe(succ)o FC(.)287 322 y Fl({)21 b FC(F)m(or)10 b(eac)o(h)h(constan)o(t)g Fp(c)p FC(,)f(there)i(are)f (no)q(des)g(represen)o(ting)i(the)e(constan)o(t)g Fp(c)p FC(.)f(These)i(no)q(des)332 372 y(ha)o(v)o(e)i(out-degree)h(0.)287 420 y Fl({)21 b FC(Application)12 b(no)q(des)h(@)g(represen)o(ting)h (the)g(application)d(of)h(t)o(w)o(o)g(terms.)g(The)h(ob)o(vious)332 470 y(t)o(yping)k(constrain)o(ts)h(ha)o(v)o(e)f(to)h(b)q(e)g (ful\014lled.)e(W)m(e)h(denote)h(the)g(p)q(oin)o(ters)g(to)g(the)g(t)o (w)o(o)332 520 y(c)o(hildren)c(of)f(an)h(application)f(no)q(de)h(b)o(y) g Fe(left)f FC(and)g Fe(right)o FC(.)287 568 y Fl({)21 b FC(Auxiliary)15 b(no)q(des)j Fp(\024)662 574 y FA(i)693 568 y FC(represen)o(ting)g(the)f(comp)q(osition)e(of)i(t)o(yp)q(e)g (one.)g(These)h(no)q(des)332 618 y(are)13 b(lab)q(eled)g(with)f(a)g (natural)h(n)o(um)o(b)q(er)f Fp(i)p 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1390 y(A)h(tree)h(is)f(called) f(a)h Fq(numer)n(al)k FC(if)14 b(the)h(ro)q(ot)g(is)g(a)f(n)o(umeral)g (no)q(de,)h(all)e(lea)o(v)o(es)i(ha)o(v)o(e)g(the)262 1439 y(same)10 b(distance)i(to)f(the)h(ro)q(ot)g(and)f(the)h(lab)q(el)f Fp(i)h FC(of)e(ev)o(ery)j Fp(\024)1173 1445 y FA(i)1198 1439 y FC(no)q(de)f(is)f(the)h(n)o(um)o(b)q(er)f(of)f(lea)o(v)o(es)262 1489 y(reac)o(hable)15 b(from)e(that)h(no)q(de.)h(By)g(standard)g(op)q (erations)g(on)g(2/3-trees)g(it)f(is)h(p)q(ossible)g(in)262 1539 y(sequen)o(tial)e(logarithmic)f(time)g(to)287 1615 y Fl({)21 b FC(split)14 b(a)f(n)o(umeral)g(at)g(a)h(giv)o(en)f(p)q (osition)g Fp(i)p FC(.)287 1663 y Fl({)21 b FC(\014nd)14 b(out)g(the)g Fp(i)p FC('th)g(bit)g(of)f(the)i(n)o(umeral.)287 1711 y Fl({)21 b FC(concatenate)15 b(t)o(w)o(o)f(n)o(umerals.)262 1788 y(So)g(using)h Fp(\024)454 1773 y Fz(0)480 1788 y FC(and)g Fp(\024)586 1794 y Fz(\001)612 1788 y FC(no)q(des)h(is)e (just)h(a)g(w)o(a)o(y)f(of)g(implemen)o(ting)d(\\no)q(des")16 b(lab)q(eled)e(with)h(a)262 1838 y(n)o(umeral)f(allo)o(wing)f(all)i (the)h(standard)g(op)q(erations)h(on)e(n)o(umerals)g(in)g(logarithmic)e (time.)262 1888 y(Note)g(that)g(the)h(length)f(of)f(the)i(lab)q(el)f Fp(i)g FC(\(co)q(ded)h(in)f(binary\))f(of)h(a)g Fp(\024)1323 1894 y FA(i)1349 1888 y FC(no)q(de)h(is)f(b)q(ounded)h(b)o(y)262 1938 y(the)g(logarithm)d(of)j(the)g(n)o(um)o(b)q(er)f(of)g(no)q(des.) 262 2056 y Fl(Normalization)f(Algorithms)h(and)i(Their)g(Complexit)o(y) 262 2133 y(Lemma)g(2.)21 b Fq(L)n(et)11 b Fp(t)h Fq(b)n(e)g(a)f (simple,)g(line)n(ar)h(term)f(of)g(typ)n(e)h Fp(\023)f Fq(and)1247 2120 y Fn(\000)-28 b(!)1251 2133 y Fp(x)18 b FC(;)1312 2120 y Fn(\000)-28 b(!)1315 2133 y Fp(n)29 b Fq(a)12 b(c)n(ontext)g(such)g(that)262 2183 y(al)r(l)k(fr)n(e)n(e)g (variables)h(in)g Fp(t)f Fq(ar)n(e)h(among)h(the)943 2169 y Fn(\000)-28 b(!)947 2183 y Fp(x)18 b Fq(.)f(Then)g(the)g(normal) g(form)f(of)h Fp(t)p FC([)1525 2169 y Fn(\000)-27 b(!)1529 2183 y Fp(x)33 b FC(:=)1645 2169 y Fn(\000)-28 b(!)1648 2183 y Fp(n)18 b FC(])262 2233 y Fq(c)n(an)d(b)n(e)g(c)n(ompute)n(d)g (in)g(time)g Fp(O)q FC(\()p Fn(j)o Fp(t)p Fn(j)9 b(\001)g FC(log)d Fn(j)906 2219 y(\000)-28 b(!)909 2233 y Fp(n)18 b Fn(j)o FC(\))d Fq(by)g Fp(O)q FC(\()p Fn(j)p Fp(t)p Fn(j)9 b(\001)f(j)1177 2219 y(\000)-27 b(!)1181 2233 y Fp(n)17 b Fn(j)p FC(\))e Fq(pr)n(o)n(c)n(essors.)262 2313 y(Pr)n(o)n(of.)20 b FC(W)m(e)11 b(start)i(one)g(pro)q(cessor)h (for)e(eac)o(h)h(of)e(the)i(no)q(des)g(of)f(the)h(parse-tree)h(of)e Fp(t)p FC(,)g(with)g(a)262 2363 y(p)q(oin)o(ter)g(to)f(this)h(no)q(de)h (in)e(its)h(lo)q(cal)f(register.)i(The)f(registers)h(asso)q(ciated)g (to)f(the)g(v)n(ariables)262 2399 y Fn(\000)-28 b(!)266 2412 y Fp(x)27 b FC(in)9 b(the)h(con)o(text)g(con)o(tain)f(p)q(oin)o (ters)h(to)f(the)h(resp)q(ectiv)o(e)h(n)o(umerals)1342 2399 y Fn(\000)-28 b(!)1345 2412 y Fp(n)18 b FC(,)9 b(and)g(the)h (registers)262 2462 y(asso)q(ciated)k(to)g(all)f(other)h(v)n(ariables)f (are)i(initialized)d(with)h(a)h Fe(NULL)f FC(p)q(oin)o(ter.)p eop %%Page: 12 12 12 11 bop 324 224 a FC(The)13 b(program)e(op)q(erates)j(in)e(rounds,)h (where)h(the)g(next)f(round)g(starts)g(once)h(all)d(activ)o(e)262 274 y(pro)q(cessors)j(ha)o(v)o(e)f(completed)e(the)j(curren)o(t)g (round.)e(The)h(only)e(pro)q(cessors)k(that)e(will)e(ev)o(er)262 324 y(do)h(something)g(are)h(those)h(at)e(the)i(application)d(or)i(v)n (ariable)f(no)q(des.)h(Th)o(us)h(all)d(pro)q(cessors)262 374 y(where)j Fe(cont)i Fp(=)-26 b Fn(2)11 b(f)p FC(@)p Fp(;)c(x;)g Fm(d)p Fn(g)13 b FC(can)h(halt)f(immediately)l(.)d(Pro)q (cessors)17 b(at)c Fm(d)h FC(no)q(des)h(do)e(not)h(halt)262 423 y(b)q(ecause)h(they)f(will)f(b)q(e)h(con)o(v)o(erted)h(to)f(v)n (ariable)f(no)q(des)h(in)g(the)g(course)h(of)f(the)g(reduction.)324 473 y(The)j(action)f(of)g(a)g(pro)q(cessor)i(at)e(an)g(application)g (no)q(de)g(in)g(one)h(round)g(dep)q(ends)h(on)262 523 y(the)g(t)o(yp)q(e)g(of)g(its)g(sons.)g(If)f(the)i(righ)o(t)e(son)h(is) g(a)g(dumm)o(y)c(no)q(de,)k(i.e.,)f Fe(right)o Fp(:)p Fe(cont)g FC(=)h Fn(\005)p FC(,)262 573 y(then)c(this)g(dumm)o(y)e(is)h (eliminated)g(b)o(y)g(setting)i Fe(right)c FC(:=)g Fe(right)o Fp(:)p Fe(succ)o FC(.)j(Otherwise,)h(the)262 623 y(action)e(dep)q(ends) j(on)d(the)i(t)o(yp)q(e)f(of)f(the)i(left)f(son.)287 700 y Fl({)21 b FC(If)12 b Fe(left)p Fp(:)p Fe(cont)e FC(=)i Fn(\005)p FC(,)g(then)h(eliminate)e(this)i(dumm)o(y)c(b)o(y)k (setting)g Fe(left)e FC(:=)g Fe(left)o Fp(:)p Fe(succ)o FC(.)287 748 y Fl({)21 b FC(If)d Fe(left)o Fp(:)p Fe(cont)f FC(=)i Fp(\025x)p FC(,)f(then)g(this)h Fp(\014)r FC(-redex)g(is)f (partially)f(reduced)j(b)o(y)d(cop)o(ying)h(the)332 798 y(argumen)o(t)12 b Fe(right)g FC(in)o(to)g(the)i(register)g Fe(R)p FC([)p Fp(x)p FC(])d(asso)q(ciated)j(to)f(the)h(v)n(ariable)d Fp(x)p FC(.)i(The)g(sub-)332 848 y(stitution)f(part)f(of)h(the)g Fp(\014)r FC(-reduction)h(is)e(then)i(p)q(erformed)e(b)o(y)h(the)g(pro) q(cessors)i(at)d(v)n(ari-)332 898 y(able)j(no)q(des.)g(Afterw)o(ards,)g (replace)g(the)h(@)f(and)f Fp(\025x)h FC(no)q(des)h(b)o(y)e(dummies)e (b)o(y)j(setting)332 948 y Fe(cont)d FC(:=)g Fn(\005)p FC(,)i Fe(left)o Fp(:)p Fe(cont)e FC(:=)g Fn(\005)j FC(and)f Fe(succ)e FC(:=)g Fe(left)p FC(.)287 996 y Fl({)21 b FC(If)10 b Fe(left)o Fp(:)p Fe(cont)h Fn(2)g(f)p Fm(s)644 1002 y FA(i)657 996 y Fp(;)c Fm(len)p Fp(;)g Fm(half)s Fn(g)j FC(and)g(the)h(righ)o(t)f(son)h(is)f(a)g(n)o(umeral,)e Fe(right)o Fp(:)p Fe(cont)j FC(=)g Fp(\024)1679 981 y Fz(0)1691 996 y FC(,)332 1046 y(then)j(replace)g(the)f(curren)o(t)h(no) q(de)g(b)o(y)f(a)f(dumm)o(y)m(,)d(and)k(let)g Fe(succ)f FC(p)q(oin)o(t)g(to)h(a)g(n)o(umeral)332 1096 y(represen)o(ting)k(the)f (result.)f(In)g(the)h(case)g(of)f Fm(s)1049 1102 y FA(i)1078 1096 y FC(and)g Fm(half)s FC(,)f(this)i(can)f(b)q(e)h(implem)o(en)o (ted)332 1146 y(b)o(y)e(2/3-tree)g(op)q(erations)g(using)g(sequen)o (tial)g(time)e Fp(O)q FC(\(log)7 b Fn(j)1269 1132 y(\000)-27 b(!)1273 1146 y Fp(n)17 b Fn(j)p FC(\).)332 1194 y(In)12 b(the)g(case)h(of)e Fm(len)q FC(,)h(the)g(result)h(is)e(equal)h(to)f (the)i(n)o(um)o(b)q(er)e Fp(i)h FC(of)f(lea)o(v)o(es)h(of)f(the)i(n)o (umeral)332 1244 y(argumen)o(t.)e(This)i(v)n(alue)f(is)g(read)h(o\013)g (the)g(topmost)e Fp(\024)1171 1250 y FA(i)1198 1244 y FC(no)q(de,)h(and)h(a)f(n)o(umeral)f(of)h(that)332 1294 y(v)n(alue)g(is)h(pro)q(duced.)h(Since)g Fp(i)f FC(is)g(a)f(n)o(um)o(b) q(er)h(of)f(length)h Fp(O)q FC(\(log)6 b Fn(j)1324 1280 y(\000)-27 b(!)1328 1294 y Fp(n)17 b Fn(j)p FC(\),)12 b(this)h(can)g(also)g(b)q(e)332 1344 y(done)h(in)g(sequen)o(tial)g (time)e Fp(O)q FC(\(log)7 b Fn(j)889 1330 y(\000)-27 b(!)893 1344 y Fp(n)17 b Fn(j)p FC(\).)287 1392 y Fl({)k FC(If)14 b Fe(left)o Fp(:)p Fe(cont)d FC(=)h(@,)i Fe(left)p Fp(:)p Fe(left)n Fp(:)p Fe(cont)d Fn(2)h(f)p Fm(drop)q Fp(;)7 b Fm(bit)o Fn(g)14 b FC(and)h Fe(right)e FC(and)h Fe(left)o Fp(:)p Fe(right)332 1442 y FC(b)q(oth)i(p)q(oin)o(t)g(to)g(n) o(umerals,)e(then)j(again)e(replace)i(the)g(curren)o(t)g(no)q(de)g(b)o (y)f(a)f(dumm)o(y)m(,)332 1492 y(and)e(let)h Fe(succ)e FC(p)q(oin)o(t)h(to)g(a)g(n)o(umeral)f(represen)o(ting)j(the)f(result,) g(whic)o(h)f(again)f(can)i(b)q(e)332 1542 y(computed)g(b)o(y)f (2/3-tree)h(op)q(erations)h(in)e(time)g Fp(O)q FC(\(log)6 b Fn(j)1207 1528 y(\000)-28 b(!)1210 1542 y Fp(n)18 b Fn(j)o FC(\).)287 1590 y Fl({)j FC(If)15 b Fe(left)o Fp(:)p Fe(cont)d FC(=)i Fe(left)o Fp(:)p Fe(left)o Fp(:)p Fe(cont)e FC(=)h(@,)i Fe(left)o Fp(:)p Fe(left)o Fp(:)p Fe(left)o Fp(:)p Fe(cont)d FC(=)i Fm(sm)f FC(and)i(all)f(of)332 1640 y Fe(right)o FC(,)g Fe(left)o Fp(:)p Fe(right)f FC(and)h Fe(left)o Fp(:)p Fe(left)o Fp(:)p Fe(right)e FC(p)q(oin)o(t)i(to)g(n)o(umerals,)f(then)i(again)e(the)332 1690 y(curren)o(t)i(no)q(de)g(is)f(replaced)g(b)o(y)g(a)g(dumm)o(y)d (with)i Fe(succ)g FC(p)q(oin)o(ting)g(to)h(the)g(result.)332 1739 y(T)m(o)e(compute)h(the)h(result,)f(the)h(lengths)f Fp(i)h FC(and)f Fp(j)i FC(are)f(read)f(o\013)g(the)h(second)g(and)f (third)332 1789 y(argumen)o(t,)h(and)h(m)o(ultipli)o(ed.)e(As)i Fp(i)h FC(and)f Fp(j)i FC(are)f Fp(O)q FC(\(log)6 b Fn(j)1223 1775 y(\000)-28 b(!)1227 1789 y Fp(n)17 b Fn(j)p FC(\))e(bit)g(n)o(um)o (b)q(ers,)f(this)h(can)332 1838 y(b)q(e)g(done)f(in)f(parallel)g(time)f Fp(O)q FC(\(log)7 b(log)f Fn(j)960 1825 y(\000)-27 b(!)964 1838 y Fp(n)17 b Fn(j)p FC(\))d(b)o(y)f Fp(O)q FC(\(log)1208 1820 y FB(3)1233 1838 y Fn(j)1245 1825 y(\000)-27 b(!)1249 1838 y Fp(n)17 b Fn(j)p FC(\))d(man)o(y)e(pro)q(cessors.)332 1887 y(The)18 b(pro)q(duct)g Fp(i)12 b Fn(\001)g Fp(j)20 b FC(is)d(compared)g(to)g(the)h(length)g(of)f(the)h(\014rst)g(argumen)o (t;)e(let)i(the)332 1937 y(maxim)n(um)d(of)j(b)q(oth)i(b)q(e)f Fp(k)q FC(.)g(No)o(w)f(the)i(result)g(is)f(a)f(n)o(umeral)g(consisting) h(of)f(a)h(one)332 1987 y(follo)o(w)o(ed)12 b(b)o(y)h Fp(k)i FC(zero)q(es,)g(whic)o(h)e(can)h(b)q(e)g(pro)q(duced)h(in)e (parallel)f(time)h(log)1495 1968 y FB(2)1521 1987 y Fp(k)h FC(b)o(y)f Fp(O)q FC(\()p Fp(k)q FC(\))332 2036 y(man)o(y)e(pro)q (cessors)k(using)e(the)g(square-and-m)o(ultiply)d(metho)q(d,)i(whic)o (h)h(su\016ces)h(since)332 2086 y Fp(k)f Fn(\024)e Fp(O)q FC(\(log)c Fn(j)531 2072 y(\000)-27 b(!)535 2086 y Fp(n)18 b Fn(j)o FC(\).)287 2135 y Fl({)j FC(Finally)m(,)16 b(if)h Fe(left)o Fp(:)p Fe(cont)g FC(=)i Fm(d)f FC(and)g Fe(right)o Fp(:)p Fe(cont)f FC(=)i Fp(\024)1208 2120 y Fz(0)1219 2135 y FC(,)f(then)g(extract)h(the)g(last)f(bit)332 2185 y Fp(b)e FC(of)g(the)h(n)o(umeral)e(at)i Fe(right)o FC(,)f(and)g (create)i(t)o(w)o(o)e(new)h(v)n(ariables)f Fp(x)1420 2191 y FB(0)1454 2185 y FC(and)h Fp(x)1562 2191 y FB(1)1580 2185 y FC(.)f(Then)332 2234 y(reduce)i(the)e Fm(d)p FC(-redex)i(b)o(y)d (replacing)h(the)g(curren)o(t)i(no)q(de)e(and)g(the)h(righ)o(t)e(son)h (b)o(y)g(ab-)332 2284 y(straction)11 b(no)q(des,)g(and)g(the)h(left)e (son)h(b)o(y)g(a)f(v)n(ariable)g(no)q(de,)h(i.e.,)e(setting)i Fe(cont)g FC(:=)g Fp(\025x)1672 2290 y FB(0)1691 2284 y FC(,)332 2334 y Fe(right)o Fp(:)p Fe(cont)f FC(:=)i Fp(\025x)655 2340 y FB(1)673 2334 y FC(,)i Fe(succ)o Fp(:)p Fe(right)n FC(,)g Fe(succ)o Fp(:)p Fe(succ)c FC(:=)h Fe(left)i FC(and)h Fe(left)o Fp(:)p Fe(cont)c FC(:=)i Fp(x)1643 2340 y FA(b)1659 2334 y FC(.)324 2412 y(A)g(pro)q(cessor)j (with)d Fe(cont)e FC(=)i Fp(x)g FC(only)g(b)q(ecomes)h(activ)o(e)f (when)h Fe(R)p FC([)p Fp(x)p FC(])d Fn(6)p FC(=)i Fe(NULL)o FC(,)g(and)h(what)262 2462 y(it)g(do)q(es)i(then)f(dep)q(ends)i(on)d (the)i(t)o(yp)q(e)f(of)g Fp(x)p FC(.)p eop %%Page: 13 13 13 12 bop 324 224 a FC(If)15 b Fp(x)g FC(is)h(not)f(of)g(ground)h(t)o (yp)q(e,)g(then)g(the)g(v)n(ariable)f Fp(x)g FC(o)q(ccurs)i(only)e(in)g (this)h(place,)f(so)262 274 y(the)e(substitution)h(can)f(b)q(e)h (safely)f(p)q(erformed)g(b)o(y)g(setting)h Fe(cont)d FC(:=)g Fn(\005)i FC(and)g Fe(succ)e FC(:=)g Fe(R)p FC([)p Fp(x)p FC(].)324 324 y(If)17 b Fp(x)h FC(is)g(of)f(t)o(yp)q(e)i Fp(\023)p FC(,)e(the)h(pro)q(cessor)i(w)o(aits)e(un)o(til)f(the)i(con)o (ten)o(t)f(of)g(register)h Fe(R)p FC([)p Fp(x)p FC(])d(has)262 374 y(b)q(een)f(normalized,)e(i.e.,)g(it)h(acts)i(only)d(if)h Fe(R)p FC([)p Fp(x)p FC(])p Fp(:)p Fe(cont)c FC(=)j Fp(\024)1172 359 y Fz(0)1184 374 y FC(.)h(In)h(this)f(case,)i(it)e(replaces)i(the) 262 423 y(v)n(ariable)e(no)q(de)i(b)o(y)f(a)g(dumm)o(y)m(,)c(and)16 b(lets)f Fe(succ)g FC(p)q(oin)o(t)g(to)g(a)g(newly)h(formed)e(cop)o(y)h (of)g(the)262 473 y(n)o(umeral)d(in)i Fe(R)p FC([)p Fp(x)p FC(].)f(This)h(cop)o(y)g(can)h(b)q(e)g(pro)q(duced)h(in)e(parallel)f (time)g Fp(O)q FC(\(log)6 b Fn(j)1487 460 y(\000)-27 b(!)1491 473 y Fp(n)17 b Fn(j)p FC(\))d(b)o(y)g Fn(j)1645 460 y(\000)-28 b(!)1648 473 y Fp(n)18 b Fn(j)262 523 y FC(pro)q(cessors,)d(since)g(the)f(depth)h(of)e(an)o(y)h(n)o(umeral)e (is)i(b)q(ounded)g(b)o(y)g(log)6 b Fn(j)1389 509 y(\000)-27 b(!)1393 523 y Fp(n)17 b Fn(j)p FC(.)324 573 y(Concerning)f (correctness,)j(note)d(that)g(the)h(tree)g(structure)h(is)e(preserv)o (ed,)i(since)f(n)o(u-)262 623 y(merals)12 b(b)q(eing)h(substituted)i (for)e(t)o(yp)q(e)g Fp(\023)g FC(v)n(ariables)g(are)h(explicitly)e (copied,)h(and)g(v)n(ariables)262 672 y(of)f(higher)g(t)o(yp)q(e)i(o)q (ccur)f(at)g(most)e(once.)i(Ob)o(viously)m(,)e(no)i(redex)h(is)e(left)h (when)g(the)g(program)262 722 y(halts.)324 772 y(F)m(or)c(the)h(time)e (b)q(ound,)i(observ)o(e)g(that)g(ev)o(ery)g(pro)q(cessor)i(p)q(erforms) d(at)g(most)g(one)g(prop)q(er)262 822 y(reduction)j(plus)f(p)q(ossibly) g(some)g(dumm)o(y)d(reductions.)k(Ev)o(ery)g(dumm)o(y)c(reduction)13 b(mak)o(es)262 872 y(one)h(dumm)o(y)d(no)q(de)k(unreac)o(hable,)g(so)f (the)h(n)o(um)o(b)q(er)e(of)h(dumm)o(y)d(reductions)16 b(is)e(b)q(ounded)262 922 y(b)o(y)h(the)h(n)o(um)o(b)q(er)f(of)g(dumm)o (y)d(no)q(des)k(generated.)h(Ev)o(ery)f(dumm)o(y)c(used)17 b(to)e(b)q(e)h(a)f(prop)q(er)262 971 y(no)q(de,)h(and)g(the)h(n)o(um)o (b)q(er)f(of)g(no)q(des)h(is)f(at)h(most)e(2)7 b Fn(j)o Fp(t)p Fn(j)p FC(,)16 b(so)g(this)h(n)o(um)o(b)q(er)e(is)i(b)q(ounded)g (b)o(y)262 1021 y(2)7 b Fn(j)o Fp(t)p Fn(j)o FC(.)12 b(Th)o(us)h(at)g(most)e(4)c Fn(j)p Fp(t)p Fn(j)12 b FC(reductions)h (are)g(p)q(erformed,)f(and)h(the)g(program)e(ends)j(after)f(at)262 1071 y(most)d(that)j(man)o(y)d(rounds.)i(As)g(argued)h(ab)q(o)o(v)o(e,) e(ev)o(ery)i(round)f(tak)o(es)h(at)f(most)f Fp(O)q FC(\(log)6 b Fn(j)1629 1057 y(\000)-28 b(!)1632 1071 y Fp(n)18 b Fn(j)o FC(\))262 1121 y(op)q(erations)c(with)f Fp(O)q FC(\()p Fn(j)619 1107 y(\000)-28 b(!)622 1121 y Fp(n)18 b Fn(j)o FC(\))c(man)o(y)e(additional)g(pro)q(cessors.)466 b Fn(u)-28 b(t)324 1204 y FC(The)11 b(next)h(lemma)c(is)i(the)i(k)o(ey) f(to)g(sho)o(w)g(that)g(all)f(terms)g(can)i(b)q(e)f(normalized)f(in)g (NC:)h(it)262 1253 y(sho)o(ws)i(ho)o(w)f(to)g(eliminate)f(the)i (constan)o(ts)h(#,)e Fm(LR)h FC(and)f Fm(CR)p FC(.)g(As)h(men)o(tioned) e(in)i(the)g(in)o(tro-)262 1303 y(duction,)h(w)o(e)h(ha)o(v)o(e)f(to)h (distinguish)f(b)q(et)o(w)o(een)i(the)f(program,)e(i.e.,)g(the)i(term)f (w)o(e)h(wish)g(to)262 1353 y(normalize,)10 b(and)j(its)g(input,)f(giv) o(en)g(as)h(a)g(con)o(text.)g(The)g(run)o(time)f(and)g(length)h(of)f (the)i(out-)262 1403 y(put)f(term)g(ma)o(y)e(dep)q(end)j(p)q (olynomiall)o(y)d(on)i(the)g(former,)f(but)i(only)e(p)q(olylogarithmi)o (cally)262 1453 y(on)h(the)i(latter.)324 1502 y(Since)f(an)g(ordinary)f Fp(O)q FC(\()p Fn(\001)p FC(\)-analysis)g(is)h(to)q(o)f(coarse)i(for)f (the)g(inductiv)o(e)g(argumen)o(t,)e(w)o(e)262 1552 y(need)17 b(a)e(more)g(re\014ned)j(asymptotic)c(analysis.)h(Therefore)i(w)o(e)g (in)o(tro)q(duce)f(the)h(follo)o(wing)262 1602 y(notation:)594 1689 y Fp(f)t FC(\()p Fp(n)p FC(\))12 b Fo(.)g Fp(g)q FC(\()p Fp(n)p FC(\))23 b(:)11 b Fn(\()-7 b(\))23 b Fp(f)t FC(\()p Fp(n)p FC(\))12 b Fn(\024)g FC(\(1)d(+)h Fp(o)p FC(\(1\)\))p Fp(g)q FC(\()p Fp(n)p FC(\))i Fp(;)262 1782 y FC(or)h(equiv)n(alen)o(tly)g(lim)7 b(sup)670 1793 y FA(n)p Fz(!1)771 1762 y FA(f)s FB(\()p FA(n)p FB(\))p 771 1773 66 2 v 772 1797 a FA(g)q FB(\()p FA(n)p FB(\))853 1782 y Fn(\024)12 b FC(1.)262 1870 y Fl(Lemma)j(3.)21 b Fq(L)n(et)c Fp(t)g Fq(b)n(e)h(a)g(line)n(ar)e(term)h(of)h(line)n(ar)f (typ)n(e)g(and)1253 1856 y Fn(\000)-27 b(!)1258 1870 y Fp(x)17 b FC(;)1318 1856 y Fn(\000)-27 b(!)1322 1870 y Fp(n)35 b Fq(a)17 b(c)n(ontext)h(with)f(al)r(l)262 1920 y(fr)n(e)n(e)f(variables)g(of)h Fp(t)p FC([)595 1906 y Fn(\000)-27 b(!)599 1920 y Fp(x)33 b FC(:=)716 1906 y Fn(\000)-28 b(!)719 1920 y Fp(n)18 b FC(])e Fq(inc)n(omplete.)h (Then)h(ther)n(e)e(ar)n(e)h(a)g(term)f FC(simp)o(\()p Fp(t;)1576 1906 y Fn(\000)-28 b(!)1580 1920 y Fp(x)18 b FC(;)1641 1906 y Fn(\000)-28 b(!)1644 1920 y Fp(n)17 b FC(\))262 1970 y Fq(and)c(a)g(c)n(ontext)514 1956 y Fn(\000)-28 b(!)519 1970 y Fp(y)21 b FC(;)579 1956 y Fn(\000)-24 b(!)579 1970 y Fp(m)26 b Fq(such)13 b(that)g FC(simp)n(\()p Fp(t;)951 1956 y Fn(\000)-28 b(!)955 1970 y Fp(x)18 b FC(;)1016 1956 y Fn(\000)-28 b(!)1019 1970 y Fp(n)18 b FC(\)[)1090 1956 y Fn(\000)-27 b(!)1095 1970 y Fp(y)32 b FC(:=)1202 1956 y Fn(\000)-23 b(!)1202 1970 y Fp(m)14 b FC(])f Fq(is)f(simple)g(and)h(e)n(quivalent)262 2020 y(to)h Fp(t)p FC([)338 2006 y Fn(\000)-27 b(!)342 2020 y Fp(x)30 b FC(:=)451 2006 y Fn(\000)-27 b(!)455 2020 y Fp(n)17 b FC(])p Fq(,)d(and)i(which)e(c)n(an)i(b)n(e)f(c)n (ompute)n(d)g(in)g(time)576 2106 y Fp(T)6 b FC(\()p Fn(j)634 2093 y(\000)-27 b(!)638 2106 y Fp(n)17 b Fn(j)p FC(\))k Fo(.)g FC(2)803 2089 y FA(])816 2093 y Fd(LR)844 2089 y FB(\()p FA(t)p FB(\))894 2106 y Fn(\001)9 b(j)o Fp(t)p Fn(j)g(\001)g FC(\(2)1020 2089 y FB(#\()p FA(t)p FB(\))1097 2106 y Fn(\001)f FC(log)f Fn(j)1189 2093 y(\000)-27 b(!)1193 2106 y Fp(n)18 b Fn(j)o FC(\))1263 2089 y FA(])1276 2093 y Fd(LR)1305 2089 y FB(\()p FA(t)p FB(\)+2)262 2193 y Fq(by)487 2255 y Fp(P)6 b FC(\()p Fn(j)547 2241 y(\000)-28 b(!)550 2255 y Fp(n)18 b Fn(j)o FC(\))j Fo(.)g Fn(j)p Fp(t)p Fn(j)9 b(\001)f(j)774 2241 y(\000)-28 b(!)778 2255 y Fp(n)17 b Fn(j)832 2234 y FB(2)849 2221 y Fh(#\()p Fk(t)p Fh(\))907 2234 y FB(\()p FA(])933 2238 y Fd(CR)964 2234 y FB(\()p FA(t)p FB(\)+2\))1069 2255 y Fn(\001)8 b FC(\(log)f Fn(j)1178 2241 y(\000)-28 b(!)1181 2255 y Fp(n)18 b Fn(j)o FC(\))1251 2237 y FB(2)1268 2225 y Fh(#\()p Fk(t)p Fh(\))1327 2237 y FB(\()p FA(])1353 2241 y Fd(LR)1382 2237 y FB(\()p FA(t)p FB(\)+1\))262 2327 y Fq(pr)n(o)n(c)n(essors,)c(such)h(that)352 2435 y Fn(j)o FC(simp)o(\()p Fp(t;)499 2421 y Fn(\000)-28 b(!)503 2435 y Fp(x)17 b FC(;)563 2421 y Fn(\000)-28 b(!)567 2435 y Fp(n)17 b FC(\))p Fn(j)h Fo(.)h Fn(j)o Fp(t)p Fn(j)9 b(\001)774 2389 y Fi(\020)799 2435 y FC(2)820 2418 y FB(#\()p FA(t)p FB(\))896 2435 y Fn(\001)g FC(log)e Fn(j)989 2421 y(\000)-27 b(!)993 2435 y Fp(n)17 b Fn(j)1047 2389 y Fi(\021)1072 2398 y FA(])1085 2402 y Fd(LR)1113 2398 y FB(\()p FA(t)p FB(\))1191 2435 y Fq(and)37 b Fn(j)1305 2421 y(\000)-24 b(!)1305 2435 y Fp(m)14 b Fn(j)k Fo(.)h Fn(j)1447 2421 y(\000)-27 b(!)1451 2435 y Fp(n)17 b Fn(j)1505 2414 y FB(2)1522 2402 y Fh(#\()p Fk(t)p Fh(\))1601 2435 y Fp(:)p eop %%Page: 14 14 14 13 bop 324 224 a FC(The)14 b(pro)q(of)g(is)f(somewhat)g(length)o(y)m (,)g(so)h(w)o(e)g(sk)o(etc)o(h)h(it)e(\014rst:)324 275 y(W)m(e)19 b(start)h(b)o(y)g(describing)g(the)g(algorithm.)c(It)k (searc)o(hes)h(for)f(the)g(head-redex)h(and)262 324 y(reduces)f(it)d (in)h(the)h(ob)o(vious)e(w)o(a)o(y)h(\(and)g(then)h(con)o(tin)o(ues)f (in)g(the)h(same)e(w)o(a)o(y)h(un)o(til)f(the)262 374 y(term)11 b(is)h(normal\):)e(in)i(the)g(case)i(of)d(a)h(ground-t)o(yp)q (e)h Fp(\014)r FC(-redex)g(enlarge)g(the)g(con)o(text,)f(in)g(the)262 424 y(case)k(of)f(a)h(higher)g(t)o(yp)q(e)g Fp(\014)r FC(-redex)h(reduce)h(it)d(in)g(the)i(term;)e(in)g(the)h(case)h(of)e Fm(LR)h FC(the)g(step)262 474 y(term)f(has)h(to)f(b)q(e)i(unfolded)e (only)g(logarithmicall)o(y)e(man)o(y)h(times,)g(so)i(w)o(e)g(can)g (just)g(form)262 524 y(a)e(new)i(term,)e(whereas)i(in)e(the)i(case)g (of)e Fm(CR)g FC(w)o(e)i(ha)o(v)o(e)e(to)h(use)h(parallelism.)c(Ho)o(w) o(ev)o(er,)j(in)262 573 y(this)g(case)h(the)g(result)g(of)f(ev)o(ery)h (pro)q(cessor)h(is)e(just)h(a)f(single)g(bit,)f(so)i(the)g(results)g (can)g(b)q(e)262 623 y(collected)g(e\016cien)o(tly)f(and)g(returned)i (to)e(the)h Fq(c)n(ontext)j FC(\(whereas)e(in)e(the)g(case)i(of)d Fm(LR)i FC(the)262 673 y(result)d(is)f(a)h(term)f(of)g(higher)g(t)o(yp) q(e\).)h(Note)g(the)h(crucial)e(in)o(terpla)o(y)g(b)q(et)o(w)o(een)i (the)f(length)g(of)262 723 y(the)f(term,)f(the)i(size)g(of)e(the)i(con) o(text,)f(the)h(running)f(time)f(and)h(the)g(n)o(um)o(b)q(er)g(of)f (pro)q(cessors)262 773 y(needed;)j(therefore)i(w)o(e)e(ha)o(v)o(e)g(to) f(pro)o(vide)h(all)f(four)g(b)q(ounds)i(sim)o(ultaneously)m(.)324 823 y(After)g(the)h(description)f(of)g(the)g(algorithm,)d(a)j(long)f (and)g(tedious)i(\(but)f(elemen)o(tary\))262 873 y(calculation)10 b(follo)o(ws)g(sho)o(wing)h(that)h(all)f(b)q(ounds)h(indeed)g(hold)f (in)g(ev)o(ery)i(case.)f(The)g(struc-)262 923 y(ture)e(of)f(the)h(pro)q (of)f(is)g(alw)o(a)o(ys)g(the)h(same:)e(in)h(the)h(in)o(teresting)g (cases)h(a)f(n)o(umerical)e(argumen)o(t)262 972 y(has)16 b(to)h(b)q(e)h(ev)n(aluated)e(in)g(order)i(to)f(b)q(e)g(able)g(to)f (reduce)j(the)e(redex)h(\(i.e.,)d(the)j(n)o(umeral)262 1022 y(w)o(e)d(recurse)i(on,)d(or)h(the)h(n)o(umeral)e(to)g(b)q(e)i (put)f(in)g(the)g(con)o(text)h(in)f(the)g(case)h(of)f(a)g(ground)262 1072 y(t)o(yp)q(e)d Fp(\014)r FC(-redex\).)h(Then)g(the)f(induction)g (h)o(yp)q(othesis)h(yields)e(the)i(size)g(of)e(this)h(n)o(umeral)f(and) 262 1122 y(also)g(the)i(amoun)o(t)e(of)g(time)g(and)i(pro)q(cessors)h (needed.)g(Then)e(calculate)h(the)g(length)f(of)g(the)262 1172 y(unfolded)f(term.)g(The)h(induction)f(h)o(yp)q(othesis)i(for)e (this)h(term)f(yields)h(the)h(amoun)o(t)d(of)h(time)262 1222 y(and)17 b(pro)q(cessors)i(needed)g(for)e(the)h(\014nal)f (computation,)e(and)i(also)g(the)h(b)q(ounds)g(for)f(the)262 1271 y(\014nal)d(output.)i(Summing)c(up)k(all)e(the)i(times)f (\(calculation)g(of)g(the)h(n)o(umeral,)d(unfolding,)262 1321 y(\014nal)g(computation\))f(one)i(v)o(eri\014es)h(that)f(the)g (time)f(b)q(ound)h(holds)f(as)h(w)o(ell.)262 1411 y Fq(Pr)n(o)n(of)20 b(\(of)d(lemma)g(3\).)g FC(By)f(induction)g(on)g Fp(])981 1417 y Fc(LR)1020 1411 y FC(\()p Fp(t)p FC(\),)g(with)g(a)g (side-induction)g(on)g Fn(j)p Fp(t)p Fn(j)f FC(sho)o(w)262 1461 y(that)e(the)i(follo)o(wing)c(algorithm)g(do)q(es)k(it:)324 1511 y(By)k(pattern)g(matc)o(hing,)e(determine)h(in)h(time)e Fp(O)q FC(\()p Fn(j)p Fp(t)p Fn(j)o FC(\))i(the)g(form)e(of)h Fp(t)p FC(,)g(and)h(branc)o(h)262 1561 y(according)13 b(to)h(the)h(form.)287 1646 y Fl({)21 b FC(If)11 b Fp(t)h FC(is)g(a)f(v)n(ariable)g(or)h(one)g(of)f(the)h(constan)o(ts)h(0)e(or)h Fm(d)p FC(,)g(then)g(return)h Fp(t)f FC(and)f(lea)o(v)o(e)1592 1633 y Fn(\000)-28 b(!)1596 1646 y Fp(x)18 b FC(;)1657 1633 y Fn(\000)-28 b(!)1660 1646 y Fp(n)332 1696 y FC(unc)o(hanged.)287 1746 y Fl({)21 b FC(If)f Fp(t)f FC(is)h Fp(c)487 1733 y Fn(\000)-27 b(!)493 1746 y Fp(s)21 b FC(,)e(where)i Fp(c)f FC(is)g(one)g(of)f(the)i(constan)o(ts)g Fm(s)1195 1752 y FA(i)1208 1746 y FC(,)f Fm(drop)q FC(,)f Fm(bit)p FC(,)h Fm(len)g FC(or)g Fm(sm)f FC(then)332 1796 y(recursiv)o(ely)h (simplify)706 1783 y Fn(\000)-27 b(!)713 1796 y Fp(s)21 b FC(,)d(giving)912 1772 y Fn(\000)-21 b(!)912 1796 y Fp(s)931 1781 y Fz(\003)983 1796 y FC(and)19 b(con)o(texts)1238 1783 y Fn(\000)-21 b(!)1238 1796 y Fp(y)1258 1802 y FA(j)1290 1796 y FC(;)1309 1783 y Fn(\000)-6 b(!)1309 1796 y Fp(m)1345 1802 y FA(j)1376 1796 y FC(,)19 b(and)g(return)h Fp(c)1650 1772 y Fn(\000)-21 b(!)1650 1796 y Fp(s)1669 1781 y Fz(\003)332 1852 y FC(and)413 1828 y Fn(\000)-14 b(!)413 1839 y(\000)-28 b(!)418 1852 y Fp(y)35 b FC(;)492 1828 y Fn(\000)-10 b(!)492 1839 y(\000)-24 b(!)492 1852 y Fp(m)27 b FC(.)287 1903 y Fl({)21 b FC(If)d Fp(t)g FC(is)h Fm(d)7 b Fp(r)513 1889 y Fn(\000)-28 b(!)519 1903 y Fp(s)21 b FC(,)d(then)h(simplify)d Fp(r)j FC(giving)e Fp(r)1036 1888 y Fz(0)1066 1903 y FC(and)1151 1889 y Fn(\000)-28 b(!)1156 1903 y Fp(y)21 b FC(;)1216 1889 y Fn(\000)-24 b(!)1216 1903 y Fp(m)14 b FC(.)k(Compute)f(the)i(n)o(umeral)332 1952 y Fp(r)352 1937 y Fz(\003)389 1952 y FC(:=)g Fp(r)472 1937 y Fz(0)483 1952 y FC([)495 1939 y Fn(\000)-27 b(!)500 1952 y Fp(y)32 b FC(:=)608 1939 y Fn(\000)-24 b(!)608 1952 y Fp(m)14 b FC(])670 1932 y FB(nf)703 1952 y FC(,)j(and)h(reduce)i(the)e(redex)h Fm(d)7 b Fp(r)1192 1937 y Fz(\003)1212 1952 y FC(,)17 b(giving)f Fp(t)1383 1937 y Fz(0)1395 1952 y FC(,)i(and)f(recursiv)o (ely)332 2002 y(simplify)11 b Fp(t)504 1987 y Fz(0)523 1989 y Fn(\000)-28 b(!)529 2002 y Fp(s)35 b FC(with)13 b(con)o(text)825 1989 y Fn(\000)-28 b(!)829 2002 y Fp(x)18 b FC(;)890 1989 y Fn(\000)-28 b(!)893 2002 y Fp(n)17 b FC(.)287 2053 y Fl({)k FC(If)14 b Fp(t)g FC(is)g(#)7 b Fp(r)14 b FC(then)h(simplify)c Fp(r)k FC(giving)d Fp(r)949 2038 y Fz(0)975 2053 y FC(and)1055 2039 y Fn(\000)-27 b(!)1061 2053 y Fp(y)21 b FC(;)1121 2039 y Fn(\000)-24 b(!)1121 2053 y Fp(m)13 b FC(.)h(Compute)f(the)h(n)o(umeral)f 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Fp(x)f FC(has)h(to)262 609 y(b)q(e)f(incomplete,)e(so)i(w)o (e're)h(en)o(titled)f(to)g(use)g(the)h(induction)e(h)o(yp)q(othesis.) 324 707 y Fl(In)f(the)g(case)h FC(\()p Fp(\025x:r)q FC(\))7 b Fp(s)704 694 y Fn(\000)-27 b(!)711 707 y Fp(s)33 b Fl(with)12 b Fj(sever)n(al)17 b Fl(o)q(ccurrences)12 b(of)g Fp(x)24 b FC(in)11 b Fp(r)h FC(note)g(that)f(due)262 757 y(to)f(the)i(fact)f(that)g Fp(t)g FC(is)g(linear,)g Fp(x)f FC(has)i(to)f(b)q(e)g(of)g(ground)g(t)o(yp)q(e)g(\(since)h (higher)g(t)o(yp)q(e)f(v)n(ariables)262 807 y(are)j(only)f(allo)o(w)o (ed)g(to)g(o)q(ccur)i(once\).)f(The)h(time)d(needed)k(to)d(calculate)h Fp(s)1415 792 y Fz(0)1441 807 y FC(is)g(b)q(ounded)h(b)o(y)707 897 y(2)728 879 y FA(])741 883 y Fd(LR)769 879 y FB(\()p FA(s)p FB(\))820 897 y Fn(j)o Fp(s)p Fn(j)7 b FC(\(2)906 879 y FB(#\()p FA(s)p FB(\))984 897 y FC(log)g Fn(j)1056 883 y(\000)-27 b(!)1060 897 y Fp(n)17 b Fn(j)p FC(\))1130 879 y FA(])1143 883 y Fd(LR)1172 879 y FB(\()p FA(s)p FB(\)+2)262 986 y FC(and)c(the)i(n)o(um)o(b)q(er)e(of)g(pro)q(cessors)j (is)e(not)g(to)q(o)f(high.)g(F)m(or)h(the)g(length)g(of)f Fp(s)1441 971 y Fz(0)1467 986 y FC(w)o(e)h(ha)o(v)o(e)709 1076 y Fn(j)o Fp(s)739 1059 y Fz(0)751 1076 y Fn(j)21 b Fo(.)g Fn(j)o Fp(s)p Fn(j)9 b(\001)g FC(\(2)946 1059 y FB(#\()p FA(s)p FB(\))1024 1076 y FC(log)d Fn(j)1096 1062 y(\000)-27 b(!)1100 1076 y Fp(n)17 b Fn(j)p FC(\))1170 1059 y FA(])1183 1063 y Fd(LR)1212 1059 y FB(\()p FA(s)p FB(\))262 1179 y FC(and)c Fn(j)354 1165 y(\000)-23 b(!)354 1179 y Fp(m)14 b Fn(j)d Fo(.)h Fn(j)482 1165 y(\000)-27 b(!)486 1179 y Fp(n)18 b Fn(j)540 1158 y FB(2)557 1146 y Fh(#\()p Fk(s)p Fh(\))620 1179 y FC(.)13 b(So)h(the)g(time)f(for)h (calculating)e Fp(s)1161 1164 y Fz(\003)1195 1179 y FC(is)i(b)q(ounded) g(b)o(y)557 1269 y Fo(.)24 b Fn(j)o Fp(s)643 1252 y Fz(0)655 1269 y Fn(j)7 b FC(log)f Fn(j)746 1255 y(\000)-28 b(!)750 1269 y Fp(n)17 b(;)811 1255 y Fn(\000)-24 b(!)811 1269 y Fp(m)14 b Fn(j)20 b Fo(.)h Fn(j)p Fp(s)p Fn(j)9 b(\001)g FC(\(2)1056 1252 y FB(#\()p FA(s)p FB(\))1133 1269 y FC(log)e Fn(j)1205 1255 y(\000)-27 b(!)1209 1269 y Fp(n)17 b Fn(j)p FC(\))1279 1252 y FA(])1292 1256 y Fd(LR)1321 1252 y FB(\()p FA(s)p FB(\)+1)262 1358 y FC(F)m(or)c(the)h(length)g(of) g(the)g(n)o(umeral)f Fp(s)833 1343 y Fz(\003)866 1358 y FC(w)o(e)h(ha)o(v)o(e)691 1459 y Fn(j)o Fp(s)721 1442 y Fz(\003)741 1459 y Fn(j)d(\024)h(j)o Fp(s)838 1442 y Fz(0)850 1459 y Fn(j)d FC(+)h Fn(j)924 1446 y(\000)-27 b(!)928 1459 y Fp(n)17 b(;)989 1446 y Fn(\000)-24 b(!)989 1459 y Fp(m)14 b Fn(j)20 b Fo(.)h Fn(j)1136 1446 y(\000)-28 b(!)1139 1459 y Fp(n)18 b Fn(j)1193 1438 y FB(2)1210 1426 y Fh(#\()p Fk(s)p Fh(\))262 1549 y FC(So)13 b(the)i(last)e (computation)f(tak)o(es)j(time)551 1660 y(2)572 1643 y FA(])585 1647 y Fd(LR)614 1643 y FB(\()p FA(r)643 1635 y Fz(\000)-23 b(!)648 1643 y FA(s)15 b FB(\))704 1660 y Fn(\001)8 b(j)p Fp(r)756 1646 y Fn(\000)-27 b(!)762 1660 y Fp(s)21 b Fn(j)820 1614 y Fi(\020)845 1660 y FC(2)866 1643 y FB(#\()p FA(r)922 1635 y Fz(\000)-23 b(!)927 1643 y FA(s)16 b FB(\)+#\()p FA(s)p FB(\))1075 1660 y FC(log)7 b Fn(j)1147 1646 y(\000)-27 b(!)1151 1660 y Fp(n)17 b Fn(j)1205 1614 y Fi(\021)1230 1623 y FA(])1243 1627 y Fd(LR)1272 1623 y FB(\()p FA(r)1301 1614 y Fz(\000)-23 b(!)1305 1623 y FA(s)16 b FB(\)+2)1401 1660 y Fp(:)262 1760 y FC(Summi)o(ng)d(up,)j(the)h(time)e(b)q(ound)i(holds.)e(The)i(n)o (um)o(b)q(er)f(of)f(pro)q(cessors)k(needed)f(for)e(the)262 1810 y(last)d(computation)f(is)i(b)q(ounded)h(b)o(y)352 1929 y Fo(.)j Fn(j)p Fp(r)434 1915 y Fn(\000)-27 b(!)440 1929 y Fp(s)21 b Fn(j)9 b(\001)521 1883 y Fi(\020)546 1929 y Fn(j)558 1915 y(\000)-28 b(!)561 1929 y Fp(n)18 b Fn(j)615 1908 y FB(2)632 1895 y Fh(#\()p Fk(s)p Fh(\))695 1883 y Fi(\021)720 1890 y FB(2)737 1877 y Fh(#\()p Fk(r)786 1872 y Fg(\000)-20 b(!)788 1877 y Fk(s)14 b Fh(\))829 1890 y FB(\()p FA(])855 1894 y Fd(CR)886 1890 y FB(\()p FA(r)915 1882 y Fz(\000)-23 b(!)920 1890 y FA(s)15 b FB(\)+2\))1028 1883 y Fi(\020)1053 1929 y FC(2)1074 1911 y FB(#\()p FA(s)p FB(\))1152 1929 y FC(log)6 b Fn(j)1224 1915 y(\000)-28 b(!)1227 1929 y Fp(n)18 b Fn(j)1281 1883 y Fi(\021)1306 1891 y FB(2)1323 1879 y Fh(#\()p Fk(r)1372 1874 y Fg(\000)-20 b(!)1375 1879 y Fk(s)13 b Fh(\))1415 1891 y FB(\()p FA(])1441 1895 y Fd(LR)1470 1891 y FB(\()p FA(r)1499 1883 y Fz(\000)-23 b(!)1504 1891 y FA(s)16 b FB(\)+1\))352 2021 y Fo(.)i Fn(j)p Fp(t)p Fn(j)9 b(\001)f(j)482 2007 y(\000)-28 b(!)485 2021 y Fp(n)18 b Fn(j)539 2000 y FB(2)556 1988 y Fh(#\()p Fk(s)p Fh(\)+#\()p Fk(r)685 1983 y Fg(\000)-20 b(!)688 1988 y Fk(s)14 b Fh(\))729 2000 y FB(\()p FA(])755 2004 y Fd(CR)786 2000 y FB(\()p FA(r)815 1992 y Fz(\000)-23 b(!)820 2000 y FA(s)15 b FB(\)+2\))928 2021 y FC(\(log)7 b Fn(j)1017 2007 y(\000)-28 b(!)1020 2021 y Fp(n)18 b Fn(j)o FC(\))1090 2000 y FB(2)1107 1988 y Fh(#\()p Fk(s)p Fh(\)+#\()p Fk(r)1236 1983 y Fg(\000)-20 b(!)1239 1988 y Fk(s)13 b Fh(\))1280 2000 y FB(\()p FA(])1306 2004 y Fd(LR)1334 2000 y FB(\()p FA(r)1363 1992 y Fz(\000)-23 b(!)1368 2000 y FA(s)16 b FB(\)+1\))262 2129 y FC(The)e(con)o(text)g (\014nally)f(output)h(is)g(b)q(ounded)g(b)o(y)g Fn(j)1037 2115 y(\000)-28 b(!)1040 2129 y Fp(n)18 b(;)7 b(s)1121 2114 y Fz(\003)1140 2129 y Fn(j)1151 2108 y FB(2)1168 2096 y Fh(#\()p Fk(r)1217 2091 y Fg(\000)-20 b(!)1220 2096 y Fk(s)13 b Fh(\))1274 2129 y Fo(.)f Fn(j)1329 2115 y(\000)-27 b(!)1333 2129 y Fp(n)17 b Fn(j)1387 2108 y FB(2)1404 2096 y Fh(#\()p Fk(s)p Fh(\)+#\()p Fk(r)1533 2091 y Fg(\000)-20 b(!)1536 2096 y Fk(s)13 b Fh(\))1578 2129 y FC(.)324 2227 y Fl(In)i(all)g(other)f(cases)g FC(the)h(b)q(ounds)f(trivially)e(hold.)517 b Fn(u)-28 b(t)324 2313 y FC(W)m(e)10 b(conclude)i(that)f(a)g(term)f(of)h(linear)f (t)o(yp)q(e)i(can)f(b)q(e)h(simpli\014ed)d(b)o(y)i(an)f(NC)i (algorithm,)262 2363 y(where)17 b(the)f(degree)h(of)f(the)g(run)o(time) f(b)q(ound)h(only)f(dep)q(ends)j(on)d(the)i(n)o(um)o(b)q(er)e(of)g(o)q (ccur-)262 2412 y(rences)e(of)e Fm(LR)p FC(,)g(and)g(the)i(degree)f(of) f(the)h(hardw)o(are)g(b)q(ound)g(only)f(dep)q(ends)i(on)e(the)h(n)o(um) o(b)q(er)262 2462 y(of)h(o)q(ccurrences)k(of)c(#)h(and)f Fm(CR)p FC(.)g(More)h(precisely)m(,)g(w)o(e)g(ha)o(v)o(e)g(the)h(follo) o(wing)c(corollary)m(.)p eop %%Page: 20 20 20 19 bop 262 224 a Fl(Corollary)14 b(1.)21 b Fq(The)13 b(term)g FC(simp)o(\()p Fp(t;)849 210 y Fn(\000)-28 b(!)852 224 y Fp(x)18 b FC(;)913 210 y Fn(\000)-28 b(!)916 224 y Fp(n)18 b FC(\))13 b Fq(and)i(the)e(new)h(c)n(ontext)1360 210 y Fn(\000)-28 b(!)1365 224 y Fp(y)21 b FC(;)1425 210 y Fn(\000)-24 b(!)1425 224 y Fp(m)27 b Fq(in)13 b(the)h(ab)n(ove) 262 274 y(lemma)g(c)n(an)i(b)n(e)e(c)n(ompute)n(d)i(in)f(time)701 364 y Fp(T)6 b FC(\()p Fn(j)759 350 y(\000)-27 b(!)763 364 y Fp(n)17 b Fn(j)p FC(\))23 b Fn(\024)g Fp(O)q FC(\(\(log)7 b Fn(j)1048 350 y(\000)-27 b(!)1052 364 y Fp(n)17 b Fn(j)p FC(\))1122 347 y FA(])1135 351 y Fd(LR)1164 347 y FB(\()p FA(t)p FB(\)+2)1247 364 y FC(\))262 454 y Fq(by)e(a)g(numb)n(er)g(of)f 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Fp(t)1511 816 y Fn(\000)-27 b(!)1515 831 y Fl(x)17 b Fp(;)1576 816 y Fn(\000)-28 b(!)1579 831 y Fl(x)18 b FC(;)1641 817 y Fn(\000)-28 b(!)1644 831 y Fp(n)17 b FC(\))262 888 y(and)e(a)g(new)h(con)o(text)616 874 y Fn(\000)-27 b(!)621 888 y Fp(y)21 b FC(;)681 874 y Fn(\000)-24 b(!)681 888 y Fp(m)29 b FC(with)15 b Fn(j)p Fp(t)869 873 y Fz(0)880 888 y Fn(j)f(\024)g FC(\(log)7 b Fn(j)1041 874 y(\000)-28 b(!)1044 888 y Fp(n)18 b Fn(j)o FC(\))1114 873 y FA(O)q FB(\(1\))1200 888 y FC(and)e Fn(j)1294 874 y(\000)-23 b(!)1294 888 y Fp(m)15 b Fn(j)e(\024)i(j)1428 874 y(\000)-27 b(!)1432 888 y Fp(n)17 b Fn(j)1486 867 y FA(O)q FB(\(1\))1572 888 y FC(in)e(time)262 945 y(\(log)6 b Fn(j)350 931 y(\000)-28 b(!)353 945 y Fp(n)18 b Fn(j)o FC(\))423 930 y FA(O)q FB(\(1\))508 945 y FC(b)o(y)c Fn(j)577 931 y(\000)-27 b(!)581 945 y Fp(n)17 b Fn(j)635 924 y FA(O)q FB(\(1\))719 945 y FC(man)o(y)12 b(pro)q(cessors.)324 1001 y(Then)19 b(using)f(Lemma)e(2)i(w)o(e)g(compute)g(the)h(normal)e(form)f Fp(t)1319 986 y Fz(0)1331 1001 y FC([)1343 987 y Fn(\000)-27 b(!)1348 1001 y Fp(y)32 b FC(:=)1456 987 y Fn(\000)-24 b(!)1456 1001 y Fp(m)14 b FC(])1517 980 y FB(nf)1569 1001 y FC(in)k(time)262 1058 y Fp(O)q FC(\()p Fn(j)o Fp(t)337 1043 y Fz(0)349 1058 y Fn(j)8 b(\001)h FC(log)e Fn(j)462 1044 y(\000)-23 b(!)462 1058 y Fp(m)15 b Fn(j)o FC(\))d(=)g(\(log)6 b Fn(j)684 1044 y(\000)-28 b(!)687 1058 y Fp(n)18 b Fn(j)o FC(\))757 1043 y FA(O)q FB(\(1\))842 1058 y FC(b)o(y)c Fp(O)q FC(\()p Fn(j)o Fp(t)975 1043 y Fz(0)987 1058 y Fn(j)6 b(j)1017 1044 y(\000)-23 b(!)1017 1058 y Fp(m)14 b Fn(j)p FC(\))d(=)h Fn(j)1162 1044 y(\000)-28 b(!)1165 1058 y Fp(n)18 b Fn(j)1219 1037 y FA(O)q FB(\(1\))1304 1058 y FC(man)o(y)12 b(pro)q(cessors.)324 1108 y(Hence)18 b(the)f(function)f(denoted)h(b)o(y)f Fp(t)h FC(is)f(computable)f(in)h (p)q(olylogarithmic)e(time)h(b)o(y)262 1158 y(p)q(olynomial)o(ly)10 b(man)o(y)i(pro)q(cessors,)k(and)e(th)o(us)g(is)g(in)f(NC.)508 b Fn(u)-28 b(t)324 1243 y FC(F)m(rom)12 b(Theorems)i(2)f(and)h(3)g(w)o (e)g(imm)o(ediately)d(get)j(our)g(main)e(result:)262 1321 y Fl(Corollary)i(2.)21 b Fq(A)11 b(numb)n(er-the)n(or)n(etic)h (function)g Fp(f)17 b Fq(is)11 b(in)h(NC)f(if)h(and)g(only)g(if)g(it)f (is)h(denote)n(d)262 1371 y(by)j(a)g(line)n(ar)f(term)g(of)h(our)g (system.)262 1503 y Ft(References)281 1597 y Fx(1.)20 b(B.)9 b(Allen.)j(Arithmetizing)g(uniform)e(NC.)g Fb(A)o(nnals)e(of)h (Pur)n(e)h(and)f(Applie)n(d)e(L)n(o)n(gic)p Fx(,)h(53\(1\):1{50,)331 1642 y(1991.)281 1688 y(2.)20 b(S.)14 b(Bellan)o(toni.)23 b Fb(Pr)n(e)n(dic)n(ative)11 b(R)n(e)n(cursion)i(and)g(Computational)e (Complexity)p Fx(.)18 b(PhD)c(thesis,)331 1733 y(Univ)o(ersit)o(y)h(of) e(T)m(oron)o(to,)g(1992.)281 1779 y(3.)20 b(S.)15 b(Bellan)o(toni.)25 b(Characterizing)17 b(parallel)g(time)e(b)o(y)g(t)o(yp)q(e)g(2)g (recursions)i(with)e(p)q(olynomial)331 1825 y(output)g(length.)20 b(In)13 b(D.)g(Leiv)n(an)o(t,)i(editor,)f Fb(L)n(o)n(gic)f(and)g (Computational)e(Complexity)p Fx(,)h(pages)331 1870 y(253{268.)i (Springer)h(LNCS)e(960,)g(1995.)281 1916 y(4.)20 b(S.)12 b(Bellan)o(toni)j(and)d(S.)g(Co)q(ok.)j(A)d(new)f(recursion-theoretic)k (c)o(haracterization)g(of)c(the)h(p)q(oly-)331 1961 y(time)i (functions.)k Fb(Computational)11 b(Complexity)p Fx(,)g(2:97{110,)i (1992.)281 2007 y(5.)20 b(S.)13 b(Bellan)o(toni)q(,)i(K.-H.)d(Niggl,)j (and)f(H.)e(Sc)o(h)o(wic)o(h)o(ten)o(b)q(erg.)20 b(Higher)14 b(t)o(yp)q(e)g(recursion,)g(rami\014-)331 2052 y(cation)h(and)e(p)q (olynomial)k(time.)g Fb(A)o(nnals)11 b(of)i(Pur)n(e)h(and)e(Applie)n(d) f(L)n(o)n(gic)p Fx(,)h(104:17{30,)h(2000.)281 2098 y(6.)20 b(S.)13 b(Bellan)o(toni)j(and)e(I.)e(Oita)o(v)o(em.)18 b(Separating)d(NC)d(along)i(the)g Fa(\016)g Fx(axis.)k(Submitted,)c (2001.)281 2143 y(7.)20 b(S.)e(Blo)q(c)o(h.)31 b(F)m(unction-algebrai)q (c)20 b(c)o(haracterizations)g(of)d(log)h(and)g(p)q(olylog)i(parallel)g (time.)331 2189 y Fb(Computational)11 b(Complexity)p Fx(,)g(4:175{205,)j(1994.)281 2234 y(8.)20 b(P)m(.)15 b(Clote.)22 b(Sequen)o(tial,)17 b(mac)o(hine)f(indep)q(enden)o(t)i(c)o (haracterizations)g(of)c(the)h(parallel)i(com-)331 2280 y(plexit)o(y)c(classes)f Fa(ALog)q(T)5 b(I)s(M)t(E)s(;)16 b(AC)874 2264 y Fs(k)893 2280 y Fa(;)h(N)t(C)986 2264 y Fs(k)1015 2280 y Fx(and)11 b Fa(N)t(C)s Fx(.)i(In)e(S.)f(Buss)h(and)h (P)m(.)e(Scott,)g(editors,)331 2325 y Fb(F)m(e)n(asible)i(Mathematics)p Fx(,)e(pages)k(49{69.)f(Birkh\177)-19 b(auser,)15 b(1990.)281 2371 y(9.)20 b(A.)13 b(Cobham.)k(The)c(in)o(trinsic)j(computational)g (di\016cult)o(y)f(of)e(functions.)18 b(In)13 b Fb(Pr)n(o)n(c)n(e)n(e)n (dings)e(of)331 2417 y(the)16 b(se)n(c)n(ond)e(Internationa)o(l)f (Congr)n(ess)i(on)h(L)n(o)n(gic,)e(Metho)n(dolo)n(gy)g(and)h (Philosophy)e(of)j(Sci-)331 2462 y(enc)n(e)p Fx(,)11 b(pages)j(24{30,)g(1965.)p eop %%Page: 21 21 21 20 bop 262 224 a Fx(10.)20 b(K.)c(G\177)-19 b(odel.)529 215 y(\177)524 224 y(Ub)q(er)16 b(eine)h(bisher)h(no)q(c)o(h)f(nic)o(h) o(t)g(b)q(en)q(\177)-20 b(utzte)18 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