This is a multi-part message in MIME format. ---------------9811101848729 Content-Type: text/plain; name="98-704.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-704.keywords" grammar identification, language change, Gibbs states, entropy maximum likelihood, simulated annealing ---------------9811101848729 Content-Type: application/postscript; name="texas.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="texas.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: texas.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o texas.dvi-ps -D 300 texas %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 1998.11.10:2234 %%BeginProcSet: texc.pro %! /TeXDict 300 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 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y(b)q(et)o(w)o(een)c(structure)h(recognition)g(and)g(language)i(c) o(hange.)d(As)h(an)h(application,)e(w)o(e)h(dis-)83 1915 y(cuss)19 b(a)h(concrete)e(case)i(of)f(language)h(c)o(hange,)f(the)g (one)g(whic)o(h)g(leads)g(from)f(Classical)83 1975 y(to)f(Mo)q(dern)g (Europ)q(ean)g(P)o(ortuguese.)f(Finally)l(,)f(in)h(section)g(5,)h(a)f (sim)o(ulated)f(annealing)83 2035 y(mo)q(del)f(of)h(language)h (acquisition)f(is)f(in)o(tro)q(duced,)h(as)g(an)h(extension)f(of)g(the) g(T)l(riggering)83 2095 y(Learning)i(Algorithm)d(recen)o(tly)h(in)o (tro)q(duced)g(in)h(the)g(linguistic)f(litterature.)p 83 2362 150 2 v 83 2394 a Fn(?)128 2411 y Ft(Expanded)20 b(v)o(ersion)f(of)g(a)f(talk)h(presen)o(ted)h(b)o(y)f(A.G.)f(at)g (StatPh)o(ys20.)f(W)l(ork)h(supp)q(orted)83 2467 y(b)o(y)13 b(F)-5 b(APESP)13 b(\()g(Pro)s(jeto)f(T)l(em\023)-23 b(atico)14 b Fu(R)o(hythmic)h(p)n(atterns,)f(p)n(ar)n(ameter)h(setting) f(and)g(language)83 2523 y(change)s Ft(,)h(gran)o(t)f(98/3382-0\))83 2562 y Fm(1)128 2578 y Ft(W)l(ork)j(partially)i(supp)q(orted)f(CNPq)f (\(gran)o(t)f(301301/79\))f(and)i(FINEP)h(\(N)q(\023)-24 b(ucleo)18 b(de)g(Ex-)83 2635 y(cel)o(^)-21 b(encia)16 b Fu(\\Critic)n(al)g(phenomena)g(in)g(pr)n(ob)n(ability)g(and)g(sto)n (chastic)g(pr)n(o)n(c)n(esses")83 2673 y Fm(2)128 2690 y Ft(W)l(ork)f(partially)h(supp)q(orted)g(b)o(y)f(CNPq)g(\(gran)o(t)f (301086/85\))885 2856 y Fo(2)p eop %%Page: 3 3 3 2 bop 83 42 a Fp(2)56 b(Linguistic)17 b(framew)n(ork)83 248 y Fo(The)i(view)e(of)i(language)h(that)f(has)g(dev)o(elop)q(ed)f (in)g(mo)q(dern)g(linguistics)f(is)h(that)h(there)83 308 y(is)d(a)h(genetic)f(inherited)f Fl(linguistic)k(c)n(ap)n(acity)i Fo(whic)o(h)15 b(mak)o(es)g(c)o(hildren)g(able)h(to)h(learn)f(a)83 368 y(language.)d(This)g(linguistic)e(capacit)o(y)g(is)h(c)o (haracterized)f(b)o(y)h(a)h(\014nite)f(set)g(of)h(constrain)o(ts,)83 428 y(what)i(Chomsky)e(calls)g(the)h Fl(Universal)i(Gr)n(ammar)5 b Fo(.)12 b(An)o(y)h(particular)h(solution)g(of)g(these)83 489 y(constrain)o(ts)d(is)g(called)f(a)h Fl(gr)n(ammar)k Fo(and)d(de\014nes)f(in)f(a)i(precise)e(w)o(a)o(y)g(a)i(natural)f (language.)83 549 y(Moreo)o(v)o(er,)18 b(eac)o(h)g(grammar)g(can)h(b)q (e)g(de\014ned)g(in)g(terms)f(of)h(a)h(\014nite)f(\(and)g(not)h (large\))83 609 y(n)o(um)o(b)q(er)15 b(of)h(parameters)f(eac)o(h)h(of)h (whic)o(h)e(tak)o(es)h(one)h(of)f(a)h(small)d(n)o(um)o(b)q(er)h(of)h(v) m(alues.)83 755 y(As)e(a)h(consequence)f(of)h(this,)f(\014rst)g (language)i(acquisition)e(is)g(a)h(pro)q(cess)g(in)f(whic)o(h)g(these) 83 815 y(parameters)k(are)h(set)f(in)h(a)g(particular)f(w)o(a)o(y)l(.)g (T)l(o)h(do)h(this)e(the)h(learning)f(c)o(hild)g(assigns)83 876 y(syn)o(tactic)12 b(structures)g(to)h(the)g(ordered)g(strings)g(of) g(w)o(ords)g(he)g(is)f(exp)q(osed)h(to)g(as)h(p)q(ositiv)o(e)83 936 y(evidence)i(from)h(the)g(paren)o(tal)h(language.)g(The)f(v)m (alues)h(of)g(the)g(parameters)e(are)i(iden-)83 996 y(ti\014ed)d(when)h 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y(tors)d(in)f(the)h(last)f(three)g(decades.)g(F)l(or)h(an)g (up-to-date)h(presen)o(tation)e(of)h(this)f(approac)o(h)83 2423 y(w)o(e)i(refer)f(the)h(reader)g(to)h([9].)83 2570 y(The)i(presen)o(t)f(pap)q(er)h(is)f(part)h(of)g(a)g(researc)o(h)f(pro) s(ject)g(at)h(the)g(in)o(terface)e(b)q(et)o(w)o(een)g(lin-)83 2630 y(guistics)g(and)h(statistical-ph)o(ysics.)e(More)h(details)g(ab)q (out)i(this)e(pro)s(ject)g(can)h(b)q(e)f(found)83 2690 y(at)g(the)f(URL)g Fj(http://www)o(.i)o(me.)o(usp)o(.b)o(r/~)o(tyc)o (ho)p Fo(.)885 2856 y(3)p eop %%Page: 4 4 4 3 bop 83 42 a Fp(3)56 b(A)17 b(Gibbsian)g(mo)r(del)d(for)j(the)f(in)n (terface)g(b)r(et)n(w)n(een)h(syn)n(tax)f(and)h(proso)r(dy)83 212 y Fo(The)c(mo)q(del)e(describ)q(es)h(the)g(w)o(a)o(y)h(a)g(sp)q (eak)o(er)f(c)o(ho)q(oses)h(a)g(sen)o(tence)f(in)g(a)h(giv)o(en)e (discursiv)o(e)83 272 y(con)o(text.)18 b(This)i(c)o(hoice)f(is)g(made)g (among)g(all)g(the)h(sen)o(tences)e(made)h(a)o(v)m(ailable)g(b)o(y)g (his)83 332 y(grammar.)f(The)h(basic)h(idea)f(is)g(that)h(this)g(c)o (hoice)e(is)h(driv)o(en)g(b)o(y)g(euphonic)g(consider-)83 392 y(ations.)f(In)g(other)g(terms,)f(the)h(b)q(etter)f(the)h(proso)q (dic)h(con)o(tour)g(of)f(a)h(sen)o(tence)e(\014ts)h(the)83 452 y(proso)q(dic)h(pattern)g(of)g(the)f(language,)i(the)e(more)f(lik)o (ely)f(this)j(sen)o(tence)e(is)h(the)h(one)g(to)83 513 y(b)q(e)d(c)o(hosen.)83 623 y(A)11 b(Gibbs)h(state)g(expresses)g(this)f (in)h(a)g(natural)g(w)o(a)o(y)l(.)f(On)h(one)g(hand,)g(eac)o(h)f(sp)q (eci\014c)g(gram-)83 683 y(mar)j(de\014nes)h(a)h(set)f(of)g(p)q (ossible)h(con\014gurations)g(of)g(the)f(system.)e(On)i(the)g(other)h (hand,)83 743 y(proso)q(dy)e(is)e(describ)q(ed)h(b)o(y)f(the)g(thermo)q (dynamical)e(p)q(oten)o(tial,)i(whic)o(h)g(tells)f(whic)o(h)h(con-)83 803 y(\014gurations)17 b(are)g(more)e(lik)o(ely)e(to)k(o)q(ccur,)f (among)g(the)g(p)q(ossible)h(ones.)83 913 y(In)24 b(a)g(realistic)f (linguistic)g(situation,)h(the)g(description)f(of)i(the)f(set)g(of)g 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y(shorthand)d Fl(wor)n(d)k Fo(instead)16 b(of)g(w)o(ord)h(category)l(.)e(The)h(class)h(of)f(grammars)f(w)o(e)g (consider)83 1565 y(are)e(top)q(ologically)g(mark)o(o)o(vian)f(and)i (eac)o(h)e(grammar)g Fi(g)j Fo(can)e(b)q(e)h(iden)o(ti\014ed)d(b)o(y)i (a)h(matrix)83 1625 y(indexed)h(b)o(y)i(\003)f(and)h(with)f(en)o(tries) g(equal)g(to)h(0)g(or)f(1.)h(The)f(set)h(of)g(all)f(the)g(sen)o(tences) g(of)83 1685 y(length)g Fi(n)g Fo(generated)h(b)o(y)e Fi(g)k Fo(is)d(the)g(set)241 1789 y Fi(L)274 1796 y Fh(n)297 1789 y Fo(\()p Fi(g)r Fo(\))e(=)g Fg(f)p Fo(\()p Fi(x)498 1796 y Fm(1)517 1789 y Fi(;)8 b Fg(\001)g(\001)g(\001)h Fi(;)f(x)656 1796 y Fh(n)679 1789 y Fo(\))p Fi(;)21 b(x)761 1796 y Fh(j)793 1789 y Fg(2)14 b Fo(\003)p Fi(;)22 b(g)r Fo(\()p Fi(x)982 1796 y Fh(j)1000 1789 y Fi(;)8 b(x)1050 1796 y Fh(j)r Fm(+1)1113 1789 y Fo(\))14 b(=)f(1)p Fi(;)22 b Fo(1)15 b Fg(\024)e Fi(j)k Fg(\024)d Fi(n)d Fg(\000)g Fo(1)p Fg(g)j Fi(:)60 b Fo(\(1\))83 1943 y(This)19 b(means)f(that)h (the)f(grammaticalit)o(y)e(of)j(a)g(sen)o(tence)e(\()p Fi(x)1229 1950 y Fm(1)1249 1943 y Fi(;)8 b Fg(\001)g(\001)g(\001)g Fi(;)g(x)1387 1950 y Fh(n)1410 1943 y Fo(\),)18 b(according)i(to)83 2003 y(grammar)f Fi(g)r Fo(,)i(dep)q(ends)g(only)f(on)h(the)g(allo)o(w) o(ance)f(of)h(eac)o(h)f(transition)h(\()p Fi(x)1493 2010 y Fh(j)1511 2003 y Fi(;)8 b(x)1561 2010 y Fh(j)r Fm(+1)1624 2003 y Fo(\))20 b(ex-)83 2063 y(pressed)g(b)o(y)f(the)g(fact)h(that)g Fi(g)r Fo(\()p Fi(x)697 2070 y Fh(j)715 2063 y Fi(;)8 b(x)765 2070 y Fh(j)r Fm(+1)828 2063 y Fo(\))20 b(=)g(1.)f(W)l(e)h (will)e(denote)i(b)o(y)f Fg(G)j Fo(the)e(set)f(of)h(all)83 2123 y(suc)o(h)13 b(grammars.)f(It)h(is)g(natural)g(to)h(assume)f(that) h(the)f(grammars)f(in)h Fg(G)j Fo(are)d(irreducible)83 2184 y(and)k(ap)q(erio)q(dic)f(matrices.)83 2294 y(F)l(rom)f(no)o(w)i (on)f(w)o(e)g(shall)g(use)h Fi(x)679 2275 y Fh(n)679 2306 y Fm(1)718 2294 y Fo(as)g(a)g(shorthand)g(for)g(\()p Fi(x)1171 2301 y Fm(1)1190 2294 y Fi(;)8 b Fg(\001)g(\001)g(\001)g Fi(;)g(x)1328 2301 y Fh(n)1351 2294 y Fo(\).)83 2404 y(Let)21 b(us)g(no)o(w)h(supp)q(ose)g(that)f Fi(')g Fo(is)g(a)g(real)f (function)h(on)g(\003)1200 2385 y Fh(m)1255 2404 y Fo(where)f Fi(m)h Fo(is)f(a)i(p)q(ositiv)o(e)83 2464 y(in)o(teger.)c(F)l(or)i(an)o (y)f Fi(n)h(>)f(m)h Fo(w)o(e)f(de\014ne)g(a)h(Gibbs)g(state)g Fp(P)1185 2443 y Fh(';)p Fm(g)1185 2476 y(n)1258 2464 y Fo(on)g Fi(L)1362 2471 y Fh(n)1385 2464 y Fo(\()p Fi(g)r Fo(\))g(in)f(the)g(usual)83 2524 y(w)o(a)o(y)d(b)o(y)460 2660 y Fp(P)498 2639 y Fh(';)p Fm(g)498 2672 y(n)551 2660 y Fo(\()p Fi(x)598 2639 y Fh(n)598 2672 y Fm(1)622 2660 y Fo(\))d(=)711 2623 y(exp[)p Fg(\000)p Fi(H)879 2630 y Fh(')904 2623 y Fo(\()p Fi(x)951 2605 y Fh(n)951 2635 y Fm(1)974 2623 y Fo(\)])1015 2590 y Ff(Q)1054 2603 y Fh(n)p Fe(\000)p Fm(1)1054 2635 y Fh(j)r Fm(=1)1131 2623 y Fi(g)r Fo(\()p Fi(x)1203 2630 y Fh(j)1221 2623 y Fi(;)8 b(x)1271 2630 y Fh(j)r Fm(+1)1334 2623 y Fo(\))p 711 2648 642 2 v 945 2694 a Fi(Z)978 2701 y Fh(n)1002 2694 y Fo(\()p Fi(';)g(g)r Fo(\))1371 2660 y Fi(;)280 b Fo(\(2\))885 2856 y(4)p eop %%Page: 5 5 5 4 bop 83 42 a Fo(where)539 182 y Fi(H)579 189 y Fh(')604 182 y Fo(\()p Fi(x)651 162 y Fh(n)651 195 y Fm(1)674 182 y Fo(\))14 b(=)g Fg(\000)806 129 y Fh(n)p Fe(\000)p Fh(m)p Fm(+1)838 141 y Ff(X)840 232 y Fh(i)p Fm(=1)939 182 y Fi(')p Fo(\()p Fi(x)1018 189 y Fh(i)1032 182 y Fi(;)8 b Fg(\001)g(\001)g(\001)g Fi(;)g(x)1170 189 y Fh(i)p Fm(+)p Fh(m)p Fe(\000)p Fm(1)1288 182 y Fo(\))358 b(\(3\))83 386 y(and)433 534 y Fi(Z)466 541 y Fh(n)490 534 y Fo(\()p Fi(';)8 b(g)r Fo(\))13 b(=)672 493 y Ff(X)682 584 y Fh(y)700 573 y Fd(n)699 596 y Fc(1)740 447 y Ff(8)740 485 y(<)740 559 y(:)777 534 y Fo(exp)8 b([)p Fg(\000)p Fi(H)953 541 y Fh(')978 534 y Fo(\()p Fi(y)1023 514 y Fh(n)1021 547 y Fm(1)1046 534 y Fo(\)])1087 480 y Fh(n)p Fe(\000)p Fm(1)1093 493 y Ff(Y)1089 584 y Fh(j)r Fm(=1)1161 534 y Fi(g)r Fo(\()p Fi(y)1229 541 y Fh(j)1248 534 y Fi(;)g(y)1294 541 y Fh(j)r Fm(+1)1357 534 y Fo(\))1376 447 y Ff(9)1376 485 y(=)1376 559 y(;)1665 534 y Fo(\(4\))83 749 y(is)16 b(the)g(partition)g(function.)83 863 y(With)e(this)f (simple)f(mo)q(del)h(it)g(is)h(p)q(ossible)g(to)g(giv)o(e)f(a)i (precise)d(form)o(ulation)h(to)h(the)g(ques-)83 923 y(tion)i(of)g(ho)o (w)g(proso)q(dy)h(pro)o(vides)e(hin)o(ts)g(to)h(language)h (acquisition.)d(W)l(e)i(are)f(in)o(terested)83 983 y(in)k(the)f (problem)g(of)h(iden)o(tifying)e(a)i(grammar)e(in)i Fg(G)i Fo(giv)o(en)d(a)i(sen)o(tence)d Fi(x)1488 965 y Fh(n)1488 996 y Fm(1)1530 983 y Fo(pro)q(duced)83 1044 y(b)o(y)f(a)g(\014xed)g (but)g(unkno)o(wn)h(grammar.)d(W)l(e)i(assume)f(that)i(the)e(proso)q (dy)l(,)i(giv)o(en)f(b)o(y)f(the)83 1104 y(p)q(oten)o(tial)h Fi(')p Fo(,)g(has)h(b)q(een)f(already)g(acquired)f(and)i(is,)f (therefore,)f(kno)o(wn)h(and)h(\014xed.)83 1218 y(A)d(\014rst)g (natural)h(trial)f(is)g(to)h(treat)f(this)g(as)h(the)f(statistical)g (problem)f(of)h(estimating)f(the)83 1278 y(grammar)j(b)o(y)h(a)h Fl(Maximum)h(Likeliho)n(o)n(d)35 b Fo(pro)q(cedure.)18 b(This)f(amoun)o(ts)h(to)g(lo)q(oking)g(for)83 1338 y(the)c(grammars)e Fi(g)k Fo(whic)o(h)d(maximize)d(the)k(probabilit)o(y)f Fp(P)1149 1320 y Fh(';)p Fm(g)1149 1350 y(n)1202 1338 y Fo(\()p Fi(x)1249 1320 y Fh(n)1249 1350 y Fm(1)1272 1338 y Fo(\).)h(Since)f(in)g(this)h(simple)83 1398 y(mo)q(del)20 b(the)h(hamiltonian)f Fi(H)638 1405 y Fh(')685 1398 y Fo(do)q(es)i(not)g(dep)q(end)f(on)h Fi(g)r Fo(,)f(the)g(maxim)o(um)c (lik)o(eliho)q(o)q(d)83 1458 y(pro)q(cedure)e(amoun)o(ts)f(to)h(minimi) o(zing)d(the)i(partition)h(function)g Fi(Z)1319 1465 y Fh(n)1343 1458 y Fo(\()p Fi(';)8 b(g)r Fo(\).)14 b(Needless)g(to)83 1518 y(sa)o(y)i(that)h(this)f(is)g(a)h(standard)g(statistical)f(ph)o (ysics)g(question.)83 1632 y(Let)22 b(^)-26 b Fi(g)197 1639 y Fh(n)221 1632 y Fo(\()p Fi(x)268 1614 y Fh(n)268 1645 y Fm(1)291 1632 y Fo(\))20 b(b)q(e)f(the)h(matrix)e(de\014ned)h(b) o(y)h(assigning)g(the)g(v)m(alue)f(1)h(to)g(all)g(the)f(en)o(tries)83 1692 y(whic)o(h)13 b(app)q(ear)h(as)g(transitions)g(in)f(the)g(sen)o (tence)g Fi(x)1029 1674 y Fh(n)1029 1705 y Fm(1)1052 1692 y Fo(,)g(and)h(the)f(v)m(alue)g(0)h(to)g(all)f(the)g(other)83 1753 y(en)o(tries.)i(W)l(e)h(no)o(w)g(ha)o(v)o(e)g(the)g(follo)o(wing)g (theorem)83 1866 y Fp(Theorem)h(1)24 b Fl(F)l(or)14 b(any)h Fi(')g Fl(and)g(any)f Fi(g)804 1873 y Fm(0)838 1866 y Fg(2)g(G)s Fl(,)g(the)i(estimator)g Fo(^)-26 b Fi(g)1263 1873 y Fh(n)1287 1866 y Fo(\()p Fi(x)1334 1848 y Fh(n)1334 1879 y Fm(1)1357 1866 y Fo(\))15 b Fl(is)g(the)g(maximum)83 1927 y(likeliho)n(o)n(d)j(estimator)f(of)h(the)g(p)n(ar)n(ental)g(gr)n (ammar,)e(given)j(the)f(sample)g Fi(x)1459 1908 y Fh(n)1459 1939 y Fm(1)1482 1927 y Fl(.)g(Mor)n(e)n(over,)83 1987 y(ther)n(e)f(exists)i Fi(\032)14 b Fg(2)g Fo(\(0)p Fi(;)8 b Fo(1\))18 b Fl(such)g(that)f(for)g(any)g Fi(n)h Fl(lar)n(ge)g(enough) 557 2104 y Fp(P)595 2084 y Fh(';)p Fm(g)646 2089 y Fc(0)595 2117 y Fm(n)673 2104 y Fg(f)p Fi(x)726 2084 y Fh(n)726 2117 y Fm(1)763 2104 y Fo(:)e(^)-26 b Fi(g)814 2111 y Fh(n)838 2104 y Fo(\()p Fi(x)885 2084 y Fh(n)885 2117 y Fm(1)908 2104 y Fo(\))14 b(=)f Fi(g)1015 2111 y Fm(0)1035 2104 y Fg(g)h(\025)g Fo(1)d Fg(\000)g Fi(\032)1237 2084 y Fh(n)1275 2104 y Fi(:)376 b Fo(\(5\))83 2275 y(F)l(or)16 b(the)g(pro)q(of)i(w)o(e)e(refer)f(the)h(reader)g(to)h([10].)83 2389 y(This)k(result)g(is)f(quite)g(satisfactory)i(if)e(all)h(w)o(e)f (need)h(is)g(a)g(statemen)o(t)e(ab)q(out)k(the)d(ro-)83 2449 y(bustness)g(of)f(the)g(acquisition)g(pro)q(cedure.)g(It)f(sa)o (ys)i(that)f(the)g(learning)g(c)o(hild)f(alw)o(a)o(ys)83 2509 y(iden)o(ti\014es)c(the)h(paren)o(tal)g(grammar)f Fi(g)791 2516 y Fm(0)827 2509 y Fo(if)g(he)i(has)g(a)f(large)h(enough)g (sample)e(of)i(p)q(ositiv)o(e)83 2570 y(evidence.)c(Ho)o(w)o(ev)o(er,)g (this)h(is)h(not)h(what)f(happ)q(ens)h(in)f(real)g(life,)e(in)h(whic)o (h)h(from)f(time)f(to)83 2630 y(time)k(a)j(generation)g(of)f(c)o (hildren)f(c)o(ho)q(oses)i(a)g(grammar)e(whic)o(h)g(is)h(di\013eren)o (t)g(from)f(the)83 2690 y(paren)o(tal)f(one.)885 2856 y(5)p eop %%Page: 6 6 6 5 bop 83 42 a Fo(T)l(o)15 b(impro)o(v)o(e)d(the)i(mo)q(del)f(in)h (order)h(to)g(co)o(v)o(er)e(situations)i(of)g(language)g(c)o(hange)g(w) o(e)f(ha)o(v)o(e)83 102 y(t)o(w)o(o)h(p)q(ossibilities.)f(The)i(more)e (realistic)g(one)i(is)f(to)h(add)g(an)g(additional)f(feature)g(to)h (the)83 162 y(mo)q(del,)d(namely)g(the)h(fact)h(that)g(a)g(sen)o(tence) e(is)i(not)g(only)f(an)h(ordered)g(string)g(of)g(w)o(ords,)83 222 y(but)f(it)g(has)h(also)f(a)h(syn)o(tactic)e(structure)h(pro)q (duced)g(b)o(y)g(the)g(grammar.)e(This)i(structure)83 282 y(is)g(not)g(explicit)e(in)h(the)h(sample)e(and)i(m)o(ust)f(b)q(e)h (guessed)g(b)o(y)f(the)h(learning)f(c)o(hild.)f(In)i(this)83 342 y(extended)h(mo)q(del)f(the)i(maxim)o(um)11 b(lik)o(eliho)q(o)q(d)k (pro)q(cedure)h(ma)o(y)e(lead)h(to)i(the)e(c)o(hoice)g(of)83 403 y(a)i(new)f(grammar.)e(An)i(example)e(of)j(this)f(will)f(b)q(e)h (presen)o(ted)g(in)g(the)g(next)f(section.)83 514 y(The)g(other)f(p)q (ossibilit)o(y)g(it)g(is)g(to)h(use)g(a)g(di\013eren)o(t)e(criterion)h (to)g(c)o(ho)q(ose)h(a)g(grammar.)e(In-)83 575 y(stead)j(of)g(minimi)o (zing)d(the)j(partition)g(function)f Fi(Z)1027 582 y Fh(n)1051 575 y Fo(\()p Fi(';)8 b(g)r Fo(\),)15 b(w)o(e)h(shall)g(lo)q (ok)g(for)g(a)g(gram-)83 635 y(mar)f Fi(g)k Fo(whic)o(h)c(minimi)o(zes) e(the)j(en)o(trop)o(y)g(of)g Fp(P)947 617 y Fh(';)p Fm(g)947 647 y(n)1000 635 y Fo(.)83 747 y(Let)g(us)h(de\014ne)f(the)g Fl(entr)n(opy)k Fo(of)d(the)f(Gibbs)g(state)h Fp(P)1075 729 y Fh(';)p Fm(g)1075 759 y(n)1144 747 y Fo(de\014ned)f(on)h Fi(L)1413 754 y Fh(n)1437 747 y Fo(\()p Fi(g)r Fo(\))f(as)520 860 y Fi(h)p Fo(\()p Fp(P)605 840 y Fh(';)p Fm(g)605 872 y(n)658 860 y Fo(\))e(=)f Fg(\000)789 819 y Ff(X)799 910 y Fh(x)819 899 y Fd(n)819 921 y Fc(1)858 860 y Fp(P)896 840 y Fh(';)p Fm(g)896 872 y(n)949 860 y Fo(\()p Fi(x)996 840 y Fh(n)996 872 y Fm(1)1019 860 y Fo(\))8 b(log)i Fp(P)1156 840 y Fh(';)p Fm(g)1156 872 y(n)1209 860 y Fo(\()p Fi(x)1256 840 y Fh(n)1256 872 y Fm(1)1279 860 y Fo(\))k Fi(:)339 b Fo(\(6\))83 1069 y(Giv)o(en)16 b(an)h(ordered)g (string)g(of)h(w)o(ords)f Fi(x)834 1051 y Fh(n)834 1081 y Fm(1)874 1069 y Fo(w)o(e)g(de\014ne)f(the)h(Minim)o(um)c(En)o(trop)o (y)k(Subset)83 1129 y Fg(E)109 1136 y Fh(')134 1129 y Fo(\()p Fi(x)181 1111 y Fh(n)181 1142 y Fm(1)205 1129 y Fo(\))f(b)o(y)289 1243 y Fg(E)315 1250 y Fh(')340 1243 y Fo(\()p Fi(x)387 1222 y Fh(n)387 1255 y Fm(1)411 1243 y Fo(\))d(=)h Fg(f)p Fi(g)i Fg(2)e(G)30 b Fo(:)e Fi(x)736 1222 y Fh(n)736 1255 y Fm(1)773 1243 y Fg(2)14 b Fi(L)853 1250 y Fh(n)876 1243 y Fo(\()p Fi(g)r Fo(\))j(and)33 b Fi(h)p Fo(\()p Fp(P)1152 1222 y Fh(';)p Fm(g)1152 1255 y(n)1205 1243 y Fo(\))16 b(is)g(minimal)29 b Fg(g)22 b Fi(:)109 b Fo(\(7\))83 1408 y(W)l(e)22 b(ma)o(y)e(no)o(w)i(in)o(tro)q (duce)f(the)h Fl(Minimum)g(Entr)n(opy)k Fo(pro)q(cedure.)21 b(Giv)o(en)g Fi(')p Fo(,)g Fi(x)1614 1390 y Fh(n)1614 1420 y Fm(1)1659 1408 y Fo(the)83 1468 y(learning)16 b(c)o(hild)f(c)o(ho)q(oses)i(a)f(grammar)f(b)q(elonging)i(to)g Fg(E)1118 1475 y Fh(')1143 1468 y Fo(\()p Fi(x)1190 1450 y Fh(n)1190 1481 y Fm(1)1213 1468 y Fo(\).)83 1580 y(Let)f(us)h (de\014ne)f(the)g(v)m(ariation)g Fp(v)m(ar)p Fo(\()p Fi(')p Fo(\))g(as)319 1694 y Fp(v)m(ar)o Fo(\()p Fi(')p Fo(\))e(=)g(sup)8 b Fg(fj)p Fi(')p Fo(\()p Fi(x)730 1673 y Fh(m)730 1706 y Fm(1)763 1694 y Fo(\))j Fg(\000)g Fi(')p Fo(\()p Fi(y)920 1673 y Fh(m)918 1706 y Fm(1)953 1694 y Fo(\))p Fg(j)27 b Fo(:)g Fi(x)1082 1673 y Fh(m)1082 1706 y Fm(1)1129 1694 y Fg(2)14 b Fo(\003)1210 1673 y Fh(m)1257 1694 y Fi(;)8 b(y)1305 1673 y Fh(m)1303 1706 y Fm(1)1352 1694 y Fg(2)14 b Fo(\003)1433 1673 y Fh(m)1466 1694 y Fg(g)22 b Fi(:)138 b Fo(\(8\))83 1859 y(Then)16 b(the)g(follo)o(wing)g(theorem)f(holds)83 1970 y Fp(Theorem)i(2)24 b Fl(Ther)n(e)15 b(exists)h(a)f(p)n(ositive)h(r)n(e)n(al)f(numb)n(er)g Fi(r)q Fl(,)h(such)f(that)h(for)f(any)g(p)n(otential)83 2031 y Fi(')i Fl(such)h(that)g Fp(v)m(ar)o Fo(\()p Fi(')p Fo(\))c Fg(\024)g Fi(r)19 b Fl(and)e(any)h(gr)n(ammar)e Fi(g)546 2150 y Fo(lim)521 2177 y Fh(n)p Fe(!)p Fm(+)p Fe(1)648 2150 y Fp(P)686 2130 y Fh(';)p Fm(g)686 2163 y(n)748 2102 y Ff(n)776 2150 y Fi(x)804 2130 y Fh(n)804 2163 y Fm(1)854 2150 y Fo(:)28 b Fg(E)926 2130 y Fh(n)922 2163 y(')950 2150 y Fo(\()p Fi(x)997 2130 y Fh(n)997 2163 y Fm(1)1020 2150 y Fo(\))14 b(=)g Fg(f)p Fi(g)r Fg(g)1180 2102 y Ff(o)1221 2150 y Fo(=)g(1)g Fi(:)340 b Fo(\(9\))83 2329 y(Theorem)13 b(2)h(sa)o(ys)g(that)g(the)g(minim)n (um)c(en)o(trop)o(y)j(pro)q(cedure)h(coincides)e(with)i(the)g(max-)83 2389 y(im)o(um)22 b(lik)o(eliho)q(o)q(d)j(pro)q(cedure)h(and)g(iden)o (ti\014es)f(correctly)f(the)h(paren)o(tal)h(grammar,)83 2449 y(whenev)o(er)d(the)h(proso)q(dy)h(is)f(not)g(to)q(o)h(biased.)f (This)g(accoun)o(ts)g(for)g(the)g(robustness)83 2509 y(of)16 b(the)f(acquisition)g(pro)q(cedure.)g(Ho)o(w)o(ev)o(er,)e(it)i (is)h(p)q(ossible)f(to)h(c)o(ho)q(ose)g(the)g(p)q(oten)o(tial)f Fi(')83 2570 y Fo(in)h(suc)o(h)g(a)g(w)o(a)o(y)g(that)g(the)g(minim)n (um)c(en)o(trop)o(y)j(pro)q(cedure)h(leads)g(to)g(a)h(new)f(grammar.)83 2630 y(It)i(is)f(imp)q(ortan)o(t)g(to)i(emphasize)d(that)i(this)g(new)g (grammar)f(ma)o(y)f(b)q(e)i(strictly)f(greater)83 2690 y(than)22 b(the)f(maternal)f(one.)h(Here)f(w)o(e)g(are)i(de\014ning)f (an)h(order)f(relation)g(on)g Fg(G)j Fo(in)d(the)885 2856 y(6)p eop %%Page: 7 7 7 6 bop 83 42 a Fo(usual)18 b(w)o(a)o(y:)f Fi(g)i(<)e(g)448 23 y Fe(0)477 42 y Fo(means)g(that)h Fi(g)r Fo(\()p Fi(x;)8 b(y)r Fo(\))16 b Fg(\024)h Fi(g)971 23 y Fe(0)982 42 y Fo(\()p Fi(x;)8 b(y)r Fo(\))17 b(for)h(all)g(pair)g(of)g(w)o(ords)g Fi(x)g Fo(and)g Fi(y)83 102 y Fo(and)f(there)f(exists)f(at)i(least)f (one)g(pair)h(\()s(\026)-27 b Fi(x)o(;)11 b Fo(\026)-27 b Fi(y)r Fo(\))16 b(for)g(whic)o(h)g Fi(g)r Fo(\()s(\026)-27 b Fi(x;)11 b Fo(\026)-27 b Fi(y)q Fo(\))14 b Fi(<)g(g)1373 84 y Fe(0)1385 102 y Fo(\()s(\026)-27 b Fi(x)o(;)11 b Fo(\026)-27 b Fi(y)r Fo(\).)83 213 y(This)16 b(is)g(the)g(con)o(ten)o (t)g(of)g(the)g(next)g(theorem.)83 323 y Fp(Theorem)h(3)24 b Fl(F)l(or)17 b(any)h Fi(g)i Fl(and)e Fi(g)721 305 y Fe(0)750 323 y Fl(in)g Fg(G)s Fl(,)f(such)h(that)g Fi(g)f(<)d(g)1203 305 y Fe(0)1215 323 y Fl(,)j(ther)n(e)h(exists)g(a)g(p)n(otential)83 384 y Fi(')f Fl(such)h(that)430 495 y Fo(lim)404 522 y Fh(n)p Fe(!)p Fm(+)p Fe(1)532 495 y Fp(P)570 475 y Fh(';)p Fm(g)570 508 y(n)631 447 y Ff(n)659 495 y Fi(x)687 475 y Fh(n)687 508 y Fm(1)738 495 y Fo(:)27 b Fi(g)22 b(=)-30 b Fg(2)14 b(E)895 475 y Fh(n)891 508 y(')919 495 y Fo(\()p Fi(x)966 475 y Fh(n)966 508 y Fm(1)990 495 y Fo(\))p Fi(;)8 b(g)1056 475 y Fe(0)1081 495 y Fg(2)14 b(E)1158 475 y Fh(n)1154 508 y(')1182 495 y Fo(\()p Fi(x)1229 475 y Fh(n)1229 508 y Fm(1)1253 495 y Fo(\))1272 447 y Ff(o)1313 495 y Fo(=)g(1)g Fi(:)223 b Fo(\(10\))83 674 y(W)l(e)16 b(refer)f(the)h(reader)g(to)h([10])f(for)h(the)f(pro)q (ofs)h(of)g(theorems)e(2)h(and)h(3.)83 784 y(A)j(simple)e(example)g(ma) 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y(It)d(follo)o(ws)h(from)f(this)g(de\014nition)h(that)g(the)f (algorithm)g(stops)i(its)e(searc)o(h)h(when,)f(for)h(the)83 1506 y(\014rst)i(time,)c(none)k(of)g(the)f(grammars)f(in)h(the)g(neigh) o(b)q(orho)q(o)q(d)i(is)e(able)g(to)h(do)g(an)o(y)f(b)q(etter)83 1566 y(than)h(the)e(actual)h(grammar.)e(This)i(happ)q(ens)i(in)d (particular)h(\(but)g(not)g(only\))g(an)o(y)g(time)83 1626 y(the)h(paren)o(tal)g(grammar)f(is)h(reac)o(hed.)83 1740 y(Ev)o(en)h(if)f(this)i(issue)f(is)g(not)g(raised)h(in)f(an)g (explicit)e(form)i(in)g(the)g(linguistic)f(literature,)83 1800 y(the)i(aim)f(of)i(the)f(algorithm)g(is)g(to)h(ac)o(hiev)o(e)e(a)i (maximi)o(zation)d(pro)q(cedure,)i(namely)f(to)83 1860 y(\014nd)23 b(the)f(grammars)e(whic)o(h)i(maximi)o(ze)d(a)k(giv)o(en)e (family)f(of)j Fl(evaluation)k Fo(functions.)83 1921 y(Ho)o(w)o(ev)o(er,)12 b(giv)o(en)i(the)g(w)o(a)o(y)g(the)h(algorithm)e (is)h(de\014ned,)g(the)h(pro)q(cess)g(ma)o(y)e(get)h(trapp)q(ed)83 1981 y(b)o(y)j(grammars)e(whic)o(h)i(are)g(not)g(global)h(maxima)c(of)k (the)f(family)d(of)k(\014tness)f(functions.)83 2041 y(This)g(is)f(a)h (ma)s(jor)f(problem)f(for)i(the)g(T)l(rigger)f(Learning)i(Algorithm)d (since)g(the)i(fact)g(of)83 2101 y(getting)f(trapp)q(ed)h(somewhere)e (is)h(the)g(only)g(w)o(a)o(y)g(it)g(has)h(to)f(c)o(ho)q(ose)h(a)g (grammar.)83 2215 y(T)l(o)22 b(a)o(v)o(oid)e(the)h(problem)e(of)i(the)g (algorithm)f(getting)h(trapp)q(ed)g(in)g(grammars)f(whic)o(h)83 2275 y(are)15 b(not)g(global)g(maxima)e(there)h(are)h(t)o(w)o(o)f(p)q (ossibilities.)g(The)h(\014rst)g(one)g(is)f(to)i(allo)o(w)e(the)83 2335 y(algorithm)f(to)i(p)q(erform)e(jumps)g(b)q(et)o(w)o(een)h (grammars)f(whic)o(h)g(are)i(not)f(neigh)o(b)q(ors.)h(This)83 2396 y(h)o(yp)q(othesis)h(do)q(es)h(not)g(seem)e(to)h(corresp)q(ond)h (to)g(a)g(realistic)e(situation.)83 2509 y(The)23 b(second)g(solution)f (is)h(obtained)g(b)o(y)f(w)o(eak)o(ening)g(the)g(restriction)g(ab)q (out)i(jumps)83 2570 y(whic)o(h)17 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y(Tw)o(o)24 b(grammars)d Fi(g)26 b Fo(and)d Fi(g)609 424 y Fe(0)644 442 y Fo(are)g(said)h(to)f (b)q(e)h Fk(neigh)o(b)q(ors)f Fo(if)g(they)f(ha)o(v)o(e)h(all)f(the)h (pa-)83 502 y(rameters)17 b(set)h(at)g(the)g(same)f(v)m(alues)h(with)g (the)f(exception)g(of)i(one.)e(Giv)o(en)h Fi(g)h Fg(2)e(G)j Fo(and)83 562 y Fi(i)13 b Fg(2)i(f)p Fo(1)p Fi(;)8 b Fo(2)p Fi(;)g Fg(\001)g(\001)g(\001)g Fi(;)g(N)d Fg(g)p Fo(,)15 b(let)g(us)h(denote)g(b)o(y)f Fi(g)845 544 y Fh(i)874 562 y Fo(the)h(grammar)e(obtained)i(b)o(y)f(setting)g(all)g (the)83 623 y(parameters)g(as)i(in)f Fi(g)r Fo(,)g(with)g(the)g (exception)f(of)i(the)f(parameter)f(of)h(index)g Fi(i)p Fo(.)83 733 y(Let)g Fi(g)193 740 y Fm(0)230 733 y Fo(b)q(e)g(the)g (paren)o(tal)g(grammar.)e(Let)i(us)h(call)e Fi(L)p Fo(\()p Fi(g)1107 740 y Fm(0)1128 733 y Fo(\))h(the)g(set)g(of)g(all)g(the)g (sen)o(tences)83 793 y(o\013ered)f(as)h(evidence)e(of)h Fi(g)572 800 y Fm(0)607 793 y Fo(during)h(acquisition.)e(It)g(is)h 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Fo(whic)o(h)f(asso)q(ciates)i(a)g (strictly)e(p)q(ositiv)o(e)g(real)83 1405 y(n)o(um)o(b)q(er)g(to)h(eac) o(h)g(grammar)f Fi(g)g Fg(2)g(G)s Fo(.)83 1515 y(A)e(natural)g(class)h (of)f(ev)m(aluation)h(functions)f(is)g(the)g(follo)o(wing.)f(F)l(or)i (an)o(y)f(\014xed)f(grammar)83 1575 y Fi(g)k Fg(2)e(G)s Fo(,)g(let)h Fi(w)335 1582 y Fh(g)370 1575 y Fo(b)q(e)g(the)g(function) g(whic)o(h)f(asso)q(ciates)i(to)g(eac)o(h)f(ordered)g(string)g(of)g(w)o (ords)83 1636 y Fi(x)111 1617 y Fh(n)111 1648 y Fm(1)156 1636 y Fo(either)22 b(its)g(structure)g(according)g(to)h Fi(g)r Fo(,)e(in)h(case)h(there)e(is)h(one)h(a)o(v)m(ailable)e(,)h(or)g (a)83 1696 y(sp)q(ecial)16 b(sym)o(b)q(ol)f Fg(y)p Fo(,)h(in)g(case)g (there)f(is)i(none.)83 1806 y(A)e(mother)f(o\013ers)i(a)g(sen)o(tence)e Fi(\021)h Fo(=)f(\()p Fi(x)801 1788 y Fh(n)801 1818 y Fm(1)824 1806 y Fi(;)8 b(w)881 1813 y Fh(g)898 1818 y Fc(0)917 1806 y Fo(\()p Fi(x)p Fo(\)\))15 b(from)f Fi(L)p Fo(\()p Fi(g)1206 1813 y Fm(0)1227 1806 y Fo(\))h(to)h(her)f(c)o(hild.) e(Ho)o(w)o(ev)o(er,)83 1866 y(the)j(only)f(explicit)f(data)j(receiv)o (ed)c(b)o(y)j(the)f(learning)h(c)o(hild)e(is)i(the)f(string)i(of)f(w)o (ords)g Fi(x)1690 1848 y Fh(n)1690 1878 y Fm(1)1713 1866 y Fo(.)83 1926 y(Let)g Fi(')g Fo(b)q(e)h(the)f(p)q(oten)o(tial)f (de\014ning)i(the)e(paren)o(tal)h(proso)q(dy)l(.)h(Since)e(the)h (learning)g(c)o(hild)83 1986 y(has)f(already)e(acquired)h Fi(')p Fo(,)f(he)h(can)g(ev)m(aluate)g(a)g(candidate)g(grammar)e Fi(g)k Fo(lo)q(oking)f(at)f(the)83 2046 y(v)m(alue)g(of)274 2034 y(\026)261 2046 y Fi(H)301 2053 y Fh(')326 2046 y Fo(\()p Fi(x)373 2028 y Fh(n)373 2059 y Fm(1)397 2046 y Fi(;)8 b(w)454 2053 y Fh(g)473 2046 y Fo(\()p Fi(x)p Fo(\)\).)14 b(Therefore,)g(it)g(is)g(natural)h(to)f(de\014ne)h Fi(f)1303 2053 y Fh(\021)1338 2046 y Fo(as)g(the)f(Boltzmann-)83 2107 y(Gibbs)j(w)o(eigh)o(t)623 2213 y Fi(f)647 2220 y Fh(\021)682 2213 y Fo(=)c(exp[)p Fg(\000)873 2200 y Fo(\026)861 2213 y Fi(H)901 2220 y Fh(')926 2213 y Fo(\()p Fi(x)973 2192 y Fh(n)973 2225 y Fm(1)996 2213 y Fi(;)8 b(w)1053 2220 y Fh(g)1073 2213 y Fo(\()p Fi(x)p Fo(\)\)])13 b Fi(;)441 b Fo(\(25\))83 2368 y(with)16 b(the)g(con)o(v)o(en)o(tion)f (that)682 2474 y(exp[)p Fg(\000)822 2461 y Fo(\026)810 2474 y Fi(H)850 2481 y Fh(')874 2474 y Fo(\()p Fi(x)921 2453 y Fh(n)921 2486 y Fm(1)945 2474 y Fi(;)8 b Fg(y)p Fo(\)])13 b(=)h(0)g Fi(:)501 b Fo(\(26\))83 2630 y(What)21 b(follo)o(ws)f(do)q(es)i(not)f(dep)q(end)f(on)h(the)f(particular)g(w)o (a)o(y)h(w)o(e)f(de\014ne)g(the)g(class)h(of)83 2690 y(ev)m(aluation)16 b(functions.)872 2856 y(11)p eop %%Page: 12 12 12 11 bop 83 42 a Fo(The)16 b(mo)q(del)f(is)g(de\014ned)h(as)h(follo)o (ws.)e(The)h(mother)f(o\013ers)h(a)h(sequence)d(of)j(sen)o(tences)e(to) 83 102 y(her)h(c)o(hild.)e(Let)i(us)h(supp)q(ose)g(that)g(this)f(c)o (hoice)e(is)i(made)f(at)i(eac)o(h)e(step)h(indep)q(enden)o(tly)83 162 y(of)f(the)g(former)e(c)o(hoices)h(and)h(with)f(the)h(same)f(la)o (w)g Fi(p)p Fo(.)h(The)g(distribution)f(remains)f(\014xed)83 222 y(during)j(the)g(ev)o(olution)e(of)i(the)g(pro)q(cess.)g(The)g (only)f(assumption)h(w)o(e)f(mak)o(e)f(ab)q(out)j Fi(p)f Fo(it)83 282 y(is)g(that)h Fi(p)p Fo(\()p Fi(\021)r Fo(\))d Fi(>)g Fo(0)i(for)h(ev)o(ery)d Fi(\021)i Fg(2)e Fi(L)p Fo(\()p Fi(g)797 289 y Fm(0)817 282 y Fo(\).)83 394 y(Let)i(us)f(supp)q (ose)i(that)f(after)f Fi(t)9 b Fg(\000)h Fo(1)15 b(steps,)h(the)f (algorithm)f(has)j(reac)o(hed)d(the)h(grammar)83 454 y Fi(g)r Fo(.)h(The)g(c)o(hoice)f(of)i(the)f(p)q(osition)h(at)f(time)e Fi(t)i Fo(id)g(made)f(as)i(follo)o(ws.)83 565 y(A)22 b(parameter)g(index)g Fi(i)j Fg(2)g(f)p Fo(1)p Fi(;)8 b Fg(\001)g(\001)g(\001)g Fi(;)g(N)d Fg(g)23 b Fo(is)g(c)o(hosen)g(at)g (random)f(and)i(with)e(uniform)83 625 y(distribution.)14 b(A)o(t)g(time)e Fi(t)c Fo(+)g(1)14 b(the)h(pro)q(cess)g(up)q(dates)h (its)e(v)m(alue)g(to)h Fi(g)1355 607 y Fh(i)1384 625 y Fo(with)g(probabilit)o(y)615 776 y Fi(q)637 783 y Fh(t)651 776 y Fo(\()p Fi(g)r(;)8 b(g)742 756 y Fh(i)757 776 y Fo(\))13 b(=)841 703 y Ff( )874 776 y Fo(1)e(+)970 743 y Fi(f)994 750 y Fh(\021)1012 754 y Fd(t)1028 743 y Fo(\()p Fi(g)r Fo(\))p 963 765 135 2 v 963 810 a Fi(f)987 817 y Fh(\021)1005 821 y Fd(t)1021 810 y Fo(\()p Fi(g)1065 796 y Fh(i)1079 810 y Fo(\))1103 703 y Ff(!)1136 715 y Fh(\014)1156 719 y Fd(t)1194 756 y Fe(0)1640 776 y Fo(\(27\))83 971 y(where)k Fi(\021)247 978 y Fh(t)276 971 y Fo(is)g(the)f(sen)o(tence)g(o\013ered)h(to)g(the)g(learning)f(c)o (hild)g(at)h(time)e Fi(t)h Fo(and)i(\()p Fi(\014)1540 978 y Fm(1)1559 971 y Fi(;)8 b(\014)1609 978 y Fm(2)1628 971 y Fi(;)g Fg(\001)g(\001)g(\001)p Fo(\))83 1031 y(is)16 b(a)h(sequence)e(of)h(p)q(ositiv)o(e)g(real)g(n)o(um)o(b)q(ers)e(div)o (erging)i(su\016cien)o(tly)e(slo)o(wly)i(to)g(+)p Fg(1)p Fo(.)83 1143 y(Let)g(us)h(call)e Fg(f)p Fi(G)385 1150 y Fh(t)400 1143 y Fi(;)8 b(t)14 b Fo(=)f(0)p Fi(;)8 b Fo(1)p Fi(;)g Fg(\001)g(\001)g(\001)q(g)16 b Fo(the)g(non)h (homogeneous)f(Mark)o(o)o(v)f(c)o(hain)h(de\014ned)g(this)83 1203 y(w)o(a)o(y)l(.)f(By)h(construction)g(its)g(transition)h (probabilit)o(y)e(matrix)g(at)h(time)e Fi(t)i Fo(is)555 1335 y Fi(Q)594 1342 y Fh(t)609 1335 y Fo(\()p Fi(g)r(;)8 b(g)700 1315 y Fh(i)714 1335 y Fo(\))14 b(=)814 1301 y(1)p 804 1324 45 2 v 804 1369 a Fi(N)887 1294 y Ff(X)853 1388 y Fh(\021)q Fe(2)p Fh(L)p Fm(\()p Fh(g)951 1393 y Fc(0)967 1388 y Fm(\))989 1335 y Fi(p)p Fo(\()p Fi(\021)r Fo(\))p Fi(q)1099 1342 y Fh(t)1114 1335 y Fo(\()p Fi(g)r(;)8 b(g)1205 1315 y Fh(i)1219 1335 y Fo(\))14 b Fi(;)374 b Fo(\(28\))83 1543 y(for)17 b(an)o(y)f Fi(g)i Fo(and)f(an)o(y)f Fi(i)g Fo(and)617 1690 y Fi(Q)656 1697 y Fh(t)670 1690 y Fo(\()p Fi(g)r(;)8 b(g)r Fo(\))14 b(=)g(1)d Fg(\000)945 1636 y Fh(N)931 1648 y Ff(X)932 1739 y Fh(i)p Fm(=1)999 1690 y Fi(Q)1038 1697 y Fh(t)1053 1690 y Fo(\()p Fi(g)r(;)d(g)1144 1669 y Fh(i)1158 1690 y Fo(\))14 b Fi(:)435 b Fo(\(29\))83 1886 y(Let)16 b(us)h(call)e Fi(\031)350 1893 y Fh(t)381 1886 y Fo(b)q(e)i(its)f(in)o(v)m(arian)o(t)f(probabilit)o(y)g(measure.) 83 1997 y(The)h(existence)f(of)h(the)g(limits)727 2110 y(lim)705 2137 y Fh(t)p Fe(!)p Fm(+)p Fe(1)824 2110 y Fi(\031)852 2117 y Fh(t)867 2110 y Fo(\()p Fi(g)r Fo(\))e(=)g Fi(\031)r Fo(\()p Fi(g)r Fo(\))f Fi(;)524 b Fo(\(30\))83 2287 y(and)659 2400 y(lim)638 2427 y Fh(t)p Fe(!)p Fm(+)p Fe(1)757 2400 y Fp(P)8 b Fg(f)p Fi(G)866 2407 y Fh(t)896 2400 y Fo(=)13 b Fi(g)r Fg(g)h Fo(=)g Fi(\031)r Fo(\()p Fi(g)r Fo(\))f Fi(;)457 b Fo(\(31\))83 2577 y(for)17 b(an)o(y)f Fi(g)g Fg(2)e(G)s Fo(,)h(follo)o(ws)h(in)g(a)h(standard)g(w) o(a)o(y)f(under)g(conditions)h(lik)o(e)774 2690 y Fi(\014)802 2697 y Fh(t)831 2690 y Fg(\024)c Fi(C)f Fo(log)d Fi(t)14 b(;)593 b Fo(\(32\))872 2856 y(12)p eop %%Page: 13 13 13 12 bop 83 42 a Fo(where)23 b Fi(C)30 b(>)d Fo(0)d(is)f(a)h(suitable) g(constan)o(t)g(whic)o(h)f(dep)q(ends)h(only)f(on)i(the)e(family)f(of) 83 102 y(ev)m(aluation)16 b(functions)h(\(cf.)e([16,17]\).)83 212 y(The)g(in)o(teresting)g(issue)g(here)g(is)g(to)h(\014nd)g (necessary)f(and)h(su\016cien)o(t)e(conditions)h(on)h(the)83 272 y(family)9 b Fg(f)p Fi(f)276 279 y Fh(\021)297 272 y Fi(;)f(\021)15 b Fg(2)f Fi(L)p Fo(\()p Fi(g)480 279 y Fm(0)501 272 y Fo(\))p Fg(g)d Fo(assuring)h(that)f(the)g(limit)e (distribution)i Fi(\031)h Fo(is)f(a)h(Dirac)f(measure.)83 332 y(In)19 b(e\013ect,)g(it)g(is)h(reasonable)g(to)g(exp)q(ect)f(that) h(in)g(a)g(giv)o(en)f(comm)o(uni)o(t)o(y)l(,)d(in)j(whic)o(h)g(all)83 392 y(the)24 b(adults)h(ha)o(v)o(e)e(the)h(same)g(grammar)e(and)j (proso)q(dy)l(,)h(all)d(the)h(learning)h(c)o(hildren)83 453 y(will)15 b(con)o(v)o(erge)h(to)h(the)f(same)f(and)i(unique)f (grammar.)e(It)i(is)g(imp)q(ortan)o(t)g(to)h(emphasize)83 513 y(that)g(this)g(unique)f(grammar)f(ma)o(y)g(b)q(e)i(di\013eren)o(t) f(from)g(the)h(grammar)e(sp)q(ok)o(en)i(b)o(y)f(the)83 573 y(generation)g(of)h(the)f(adults.)83 683 y(The)g(follo)o(wing)g (theorem)f(holds)83 793 y Fp(Theorem)i(4)24 b Fl(A)18 b(su\016cient)h(c)n(ondition)f(for)f(the)h(pr)n(o)n(c)n(ess)e Fo(\()p Fi(G)1231 800 y Fh(t)1246 793 y Fo(\))i Fl(to)f(c)n(onver)n(ge) i(in)e(law)i(to)83 853 y(a)f(Dir)n(ac)f(me)n(asur)n(e)g(is)h(that)g (ther)n(e)f(exists)i(a)f(gr)n(ammar)g Fo(\026)-26 b Fi(g)r Fl(,)18 b(a)g(sentenc)n(e)23 b Fo(\026)-27 b Fi(\021)20 b Fl(and)e(an)g(index)79 902 y Fo(\026)83 913 y Fi(i)f Fl(such)h(that)742 1005 y Fi(f)769 1012 y Fm(\026)-21 b Fh(\021)787 1005 y Fo(\()r(\026)-26 b Fi(g)r Fo(\))p 735 1027 123 2 v 735 1073 a Fi(f)762 1080 y Fm(\026)-21 b Fh(\021)780 1073 y Fo(\()r(\026)-26 b Fi(g)821 1051 y Fm(\026)824 1059 y Fh(i)838 1073 y Fo(\))876 1038 y Fi(>)942 1005 y(f)966 1012 y Fh(\020)986 1005 y Fo(\()p Fi(g)r Fo(\))p 933 1027 126 2 v 933 1073 a Fi(f)957 1080 y Fh(\020)977 1073 y Fo(\()p Fi(g)1021 1058 y Fh(j)1039 1073 y Fo(\))1077 1038 y Fi(;)549 b Fo(\(33\))83 1225 y Fl(for)17 b(al)r(l)i Fi(g)d Fg(6)p Fo(=)f(\026)-26 b Fi(g)r Fl(,)18 b(and)g(al)r(l)g Fi(j)j Fl(and)c Fi(\020)t Fl(.)83 1335 y Fo(This)i(section)f(is)h(based)g(on)h([2{4].)e(It)h (will)e(app)q(ear)j(in)f(a)g(more)e(general)i(con)o(text)f(in)g(a)83 1396 y(forthcoming)d(article)g([5].)83 1596 y Fp(Ac)n(kno)n(wledgemen)n (ts)83 1766 y Fo(This)g(article)e(is)h(an)h(extended)f(v)o(ersion)g(of) h(the)f(talk)g(presen)o(ted)g(at)h(StatPh)o(ys20)h(b)o(y)e(one)83 1827 y(of)i(the)f(authors,)h(A.G.)e(W)l(e)h(thank)h(the)f(audience)g (of)h(the)f(talk,)f(in)i(particular)f(Mic)o(hael)83 1887 y(Fisher)h(and)i(Jo)q(el)e(Leb)q(o)o(witz,)h(for)g(in)o(teresting)f (remarks)f(and)j(questions.)e(W)l(e)g(are)h(also)83 1947 y(indebted)f(to)h(Maria)f(Bernadete)f(Abaurre,)h(Rob)q(ert)h(Berwic)o (k,)d(Rob)q(ert)j(F)l(rank,)f(Elisa-)83 2007 y(b)q(eti)f(Kira,)f(An)o (thon)o(y)h(Kro)q(c)o(h)g(and)h(Artur)e(Lop)q(es)j(for)e(man)o(y)f (illuminating)e(discussions)83 2067 y(these)k(last)g(y)o(ears.)83 2261 y Fp(References)83 2427 y Ft([1])34 b(R.)22 b(Berwic)o(k)h(and)f (P)l(.)g(Niy)o(ogy)l(.)g(F)l(ormalizing)h(triggers:)e(a)h(learning)h (mo)q(del)g(for)f(\014nite)166 2484 y(space,)15 b Fu(Linguistic)g (Inquiry)k Fr(27)p Ft(,)c(605-622,)e(1997.)83 2577 y([2])34 b(M.)18 b(Cassandro)g(and)h(A.)f(Galv)o(es.)g(Acquisition)j(et)d(c)o (hangemen)o(t)h(linguistique)i(dans)e(le)166 2633 y(mo)q(d)o(\022)-21 b(ele)16 b(de)g(Gibson)g(et)g(W)l(exler,)g Fu(Col)r(lo)n(que)g(L)n (angues)f(et)i(Gr)n(ammair)n(e)g(2)p Ft(,)f(Univ)o(ersit)o(\023)-21 b(e)15 b(de)166 2690 y(P)o(aris)g(VI)q(I,)h(June)g(8-10,)e(1995.)872 2856 y Fo(13)p eop %%Page: 14 14 14 13 bop 83 42 a Ft([3])34 b(M.)13 b(Cassandro)f(and)h(A.)g(Galv)o (es.)g(Language)g(acquisition)i(and)e(c)o(hange)g(in)h(a)f(generalized) 166 98 y(Gibson-W)l(exler)35 b(mo)q(del,)f Fu(F)m(ourth)g(Me)n(eting)e (on)g(Mathematics)i(of)f(the)h(L)n(anguage)166 154 y(\(MOL4\))27 b Ft(,)h(Univ)o(ersit)o(y)h(of)f(P)o(ennsylv)m(ania,)i(Philadelphi)q (a,)g(Octob)q(er)f(27-28,)e(1995.)166 211 y(T)l(arragona,)13 b(Spain,)j(Ma)o(y)f(2-4,)f(1996.)83 306 y([4])34 b(M.)40 b(Cassandro,)g(A.)g(Galv)o(es)h(and)g(C.)f(Galv)o(es.)g(Structure)h (recognition)h(and)166 362 y(language)20 b(c)o(hange)f(in)h(a)f (generalized)h(GW)f(mo)q(del,)h Fu(II)f(International)f(Confer)n(enc)n (e)g(on)166 419 y(Mathematic)n(al)f(Linguistics)d(\(ICML'96\))p Ft(,)83 514 y([5])34 b(M.)15 b(Cassandro,)f(A.)h(Galv)o(es)g(and)h(C.)f (Galv)o(es.)f(Structure)i(iden)o(ti\014cation)h(and)f(language)166 570 y(c)o(hange)f(in)h(a)f(thermalized)i(GW)d(mo)q(del.)i Fu(Work)h(in)f(pr)n(o)n(gr)n(ess)p Ft(.)83 665 y([6])34 b(J.-R.)18 b(Chazottes,)e(E.)i(Floriani)g(and)g(R.)g(Lima.)f(Relativ)o (e)i(en)o(trop)o(y)e(and)h(iden)o(ti\014cation)166 722 y(of)d(Gibbs)h(measures)f(in)h(dynamical)g(systems)f Fu(J.)h(Statist.)g(Phys.)f Fr(90)p Ft(,)g(697{725,)e(1998.)83 817 y([7])34 b(N.)23 b(Chomsky)l(.)f(Three)h(mo)q(dels)g(for)f(the)h (description)h(of)e(language.)h Fu(IRE)g(T)m(r)n(ans.)e(on)166 873 y(Inform.)16 b(The)n(ory)t Ft(,)f Fr(IT)i(2)p Ft(,)e(113-124,)e (1953.)83 968 y([8])34 b(N.)14 b(Chomsky)l(.)g(F)l(ormal)g(prop)q (erties)h(of)f(grammars,)e(in:)j Fu(Handb)n(o)n(ok)g(of)h(Math.)g (Psych.)p Ft(,)d Fr(2)p Ft(,)166 1024 y(323-418,)g(John)j(Wiley)l(,)g (New)g(Y)l(ork,)e(1963.)83 1119 y([9])34 b(N.)27 b(Chomsky)l(.)f Fu(A)h(minimalist)f(pr)n(o)n(gr)n(am)i(for)f(linguistic)f(the)n(ory)31 b Ft(The)d(MIT)e(Press,)166 1176 y(Cam)o(bridge,)15 b(MA,)g(1995.)83 1271 y([10])c(P)l(.)j(Collet,)h(A.)f(Galv)o(es)g(and)h(A.)f(Lop)q(es.)h (Maxim)o(um)f(lik)o(eliho)q(o)q(d)j(and)e(minim)o(um)g(en)o(trop)o(y) 166 1327 y(iden)o(ti\014cation)i(of)e(grammars,)f Fu(R)n(andom)i(and)h (Computational)g(Dynamics)i Fr(3)p Ft(,)c(241-250,)166 1384 y(1995.)83 1479 y([11])c(R.)j(F)l(rank)g(and)g(S.)g(Kapur.)g(On)h (the)f(use)h(of)e(triggers)h(on)g(parameter)f(setting,)h Fu(Linguistic)166 1535 y(Inquiry)19 b Fr(27)p Ft(,)c(623-660,)e(1997.) 83 1630 y([12])e(C.)i(Galv)o(es,)f(Clitic)j(Placemen)o(t)e(and)g(P)o (arametric)f(Changes)h(in)h(P)o(ortuguese,)e(in:)h Fu(Asp)n(e)n(cts)166 1686 y(of)28 b(r)n(omanc)n(e)g(linguistics,)e(Sele)n(cte)n(d)g(p)n(ap)n (ers)i(fr)n(om)g(the)g(Linguistic)e(Symp)n(osium)i(on)166 1743 y(R)n(omanc)n(e)16 b(languages)g(XXIV)7 b Ft(,)15 b(Georgeto)o(wn)e(Univ)o(ersit)o(y)j(Press,)f(1996.)83 1838 y([13])c(A.)21 b(Galv)o(es)g(and)h(C.)e(Galv)o(es.)h(A)g(case)g (study)h(of)f(proso)q(dy)g(driv)o(en)h(language)f(c)o(hange.)166 1894 y(Preprin)o(t)16 b(\(can)f(b)q(e)h(retriev)o(ed)f(at)g(URL)h Fa(http://www.ime.usp.br/~ty)o(cho)p Ft(\).)83 1989 y([14])11 b(D.)j(Geman.)g(Random)i(\014elds)g(and)f(in)o(v)o(erse)g(problem)g(in) h(imaging,)f(in:)g Fu(18e)i(Ec)n(ole)e(d')g(\023)-22 b(et)o(\023)g(e)166 2046 y(de)17 b(pr)n(ob)n(abilit)o(\023)-22 b(es)16 b(de)h(St)g(Flour)p Ft(,)f(Lecture)g(Notes)g(in)h(Mathematics,) e(Berlin,)i(Heidelb)q(erg,)166 2102 y(New)e(Y)l(ork:)g(Springer,)h (1990.)83 2197 y([15])11 b(E.)k(Gibson)h(and)f(K.)g(W)l(exler.)h(T)l (riggers,)e Fu(Linguistic)h(Inquiry)k Fr(25)p Ft(,)c(407-454,)e(1994.) 83 2292 y([16])e(B.)22 b(Gidas.)h Fu(Metr)n(op)n(olis-typ)n(e)f(Monte)g (Carlo)h(simulation)g(algorithms)g(and)g(simulate)n(d)166 2349 y(anne)n(aling)p Ft(,)13 b(Bro)o(wn)i(Univ)o(ersit)o(y)l(,)g (1991.)83 2444 y([17])c(B.)17 b(Ha)s(jek.)f(Co)q(oling)i(sc)o(hedules)h (for)e(optimal)h(annealing,)g Fu(Math.)g(Op)n(er.)g(R)n(ese)n(ar)n(ch)i Fr(13)p Ft(,)166 2500 y(311-329,)13 b(1988.)83 2595 y([18])e(S.)k (Kullbac)o(k.)i Fu(Information)f(the)n(ory)g(and)h(statistics)h Ft(John)d(Wiley)l(,)h(New)g(Y)l(ork,)e(1959.)83 2690 y([19])d(G.)k(Marcus.)f(Negativ)o(e)h(evidence)i(in)f(language)g (acquisition,)g Fu(Co)n(gnition)h Fr(46)p Ft(,)e(1993.)872 2856 y Fo(14)p eop %%Page: 15 15 15 14 bop 83 42 a Ft([20])11 b(J.)33 b(Morgan.)e Fu(F)m(r)n(om)h (Simple)g(Input)g(to)h(Complex)f(Gr)n(ammar)39 b Ft(The)33 b(MIT)f(Press,)166 98 y(Cam)o(bridge,)15 b(MA,)g(1986.)83 191 y([21])c(M.)24 b(Nesp)q(or,)g(M.T.)f(Guasti)h(and)h(A.)f (Christophe.)g(What)g(can)g(infan)o(ts)h(learn)g(from)166 248 y(proso)q(dic)16 b(constituen)o(ts?)g Fu(18th)h(GLO)o(W)f(Col)r(lo) n(quium)s Ft(,)f(T)l(romso,)f(1995.)83 341 y([22])d(A.)18 b(R)o(\023)-21 b(en)o(y)l(.)18 b(On)h(measures)f(of)g(en)o(trop)o(y)g (and)h(information.)f Fu(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(the)h(F)m (ourth)166 397 y(Berkeley)h(Symp)n(osium)h(on)g(Mathematic)n(al)f (Statistics)g(and)h(Pr)n(ob)n(ability)t Ft(,)e Fr(1)p Ft(,)h(547-561,)166 454 y(1961.)872 2856 y Fo(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9811101848729--