%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: TCS.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o TCS.ps TCS.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1996.11.05:1230 %%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 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Fp(,)g(suc)o(h)g(patterns)g(will)f(b)q(e)h(referred)e(to)j (as)f Fi(er)n(asing)g Fp(or)g Fi(E-p)n(atterns)p Fp(,)60 264 y(in)h(the)h(sequel.)k(If)15 b(the)g(replacemen)o(t)e(of)j(a)g(v)m (ariable)f(with)g(the)h(empt)o(y)d(w)o(ord)j(is)g(forbidden,)60 324 y(then)22 b(the)h(generated)f(language)i(will)d(b)q(e)i(denoted)f (b)o(y)g Fo(L)1174 331 y Fm(N)t(E)r(;)p Fl(\006)1271 324 y Fp(\()p Fo(\013)p Fp(\))h(and)g(w)o(e)f(sp)q(eak)h(of)g(a)60 384 y Fi(non-er)n(asing)17 b Fp(or)g Fi(NE-p)n(attern)p Fp(.)133 444 y(It)12 b(is)h(kno)o(wn)g(that)g(the)g(inclusion)f (problem)f(\(is)h(the)h(language)h(generated)e(b)o(y)h(a)g(pattern)60 504 y(included)f(in)h(the)f(language)j(generated)e(b)o(y)f(another)i (pattern?\))21 b(for)13 b(nonerasing)h(and)f(eras-)60 565 y(ing)j(pattern)g(languages)i(is)d(undecidable;)g(see)h([6].)k(Ho)o (w)o(ev)o(er,)14 b(the)i(language)h(equiv)m(alence)60 625 y(problem)e(\(do)i(t)o(w)o(o)f(patterns)h(generate)f(the)g(same)g (language?\))23 b(for)16 b(nonerasing)i(patterns)60 685 y(is)h(trivially)d(decidable:)26 b(Tw)o(o)19 b(patterns)g Fo(\013)g Fp(and)h Fo(\014)h Fp(generate)d(the)h(same)f(NE-language)i (if)60 745 y(and)c(only)f(if)g(they)g(are)g(iden)o(tical)f(up)h(to)h(v) m(ariable)f(renaming.)k(It)c(is)g(v)o(ery)g(easy)g(to)h(see)f(that)60 805 y(this)i(do)q(es)g(not)g(hold)g(for)g(E-patterns.)23 b(Jiang)18 b Fi(et)g(al.)23 b Fp([6])17 b(conjectured)e(that)j(the)e (language)60 866 y(equiv)m(alence)e(problem)h(for)i(erasing)f(pattern)h (languages)g(is)f(decidable)f(\(cf.)h(also)g([10)q(]\).)133 926 y(In)j(this)h(pap)q(er,)g(w)o(e)f(in)o(v)o(estigate)f(the)h (E-language)i(equiv)m(alence)d(problem)f(and)j(sho)o(w)60 986 y(that)d(the)g(conjecture)e(is)i(true)f(in)h(man)o(y)e(cases.)23 b(Our)16 b(\\decision)h(pro)q(cedure")g(is)f(based)i(on)60 1046 y(a)d(decidable)e(and)h(length-decreasing)g(reduction)g(relation)g Fn(!)1228 1053 y Fm(r)1260 1046 y Fn(\032)28 b Fp(\(\006)7 b Fn([)g Fo(V)k Fp(\))1486 1028 y Fl(+)1522 1046 y Fn(\002)c Fp(\(\006)g Fn([)g Fo(V)k Fp(\))1727 1028 y Fl(+)1756 1046 y Fp(.)60 1106 y(Roughly)21 b(sp)q(eaking,)h(pattern)e Fo(\013)h Fp(reduces)f(to)h(another)g(pattern)g Fo(\013)1342 1088 y Fh(0)1375 1106 y Fp(\(whic)o(h)f(will)f(b)q(e)i(de-)60 1166 y(noted)d(b)o(y)f Fo(\013)f Fn(!)363 1173 y Fm(r)398 1166 y Fo(\013)429 1148 y Fh(0)441 1166 y Fp(\))h(if)g Fo(\013)554 1148 y Fh(0)584 1166 y Fp(can)g(b)q(e)h(obtained)g(from)e (the)i(pattern)f Fo(\013)h Fp(b)o(y)f(deleting)g(certain)60 1227 y(v)m(ariables)j(from)f Fo(\013)h Fp(and)h Fo(\013)g Fp(=)f Fo(\033)r Fp(\()p Fo(\013)725 1209 y Fh(0)736 1227 y Fp(\),)h(where)f Fo(\033)h Fp(is)f(a)h(\(linear\))e(morphism.)30 b(Since)19 b Fn(!)1698 1234 y Fm(r)1737 1227 y Fp(is)60 1287 y(length-decreasing,)e(ev)o(ery)e(reduction)i(sequence)f(starting) h(from)f(a)i(pattern)f Fo(\013)g Fp(m)o(ust)f(end)60 1347 y(in)k(an)h(irreducible)d(pattern,)j(a)f(so-called)h(normal)e (form.)32 b(Moreo)o(v)o(er,)20 b Fn(!)1475 1354 y Fm(r)1514 1347 y Fp(is)g(language-)60 1407 y(preserving,)h(that)g(is,)g(if)f Fo(\013)i Fn(!)644 1414 y Fm(r)684 1407 y Fo(\013)715 1389 y Fh(0)727 1407 y Fp(,)f(then)g Fo(L)911 1414 y Fm(E)r(;)p Fl(\006)976 1407 y Fp(\()p Fo(\013)p Fp(\))g(=)h Fo(L)1159 1414 y Fm(E)r(;)p Fl(\006)1224 1407 y Fp(\()p Fo(\013)1274 1389 y Fh(0)1286 1407 y Fp(\).)34 b(W)l(e)21 b(conjecture)f(that)60 1467 y(reduction)c(to)g(normal)g(form)f(yields)g (a)i(decision)e(pro)q(cedure:)60 1588 y Fg(Conjecture)23 b(1:)28 b Fp(Tw)o(o)21 b(E-patterns)g(de\014ne)f(the)f(same)g(language) i(o)o(v)o(er)f(an)g(alphab)q(et)h(\006)60 1648 y(with)16 b Fo(car)q(d)p Fp(\(\006\))f Fn(\025)g Fp(3)i(if)f(and)h(only)f(if)g (their)g(normal)g(forms)g(with)g(resp)q(ect)g(to)h Fn(!)1547 1655 y Fm(r)1583 1648 y Fp(are)f(iden-)60 1708 y(tical,)f(up)h(to)h (renaming)e(of)i(v)m(ariables.)60 1829 y(Note)24 b(that)g(in)g (Conjecture)f(1,)j(the)e(restriction)f Fo(car)q(d)p Fp(\(\006\))k Fn(\025)g Fp(3)d(is)g(essen)o(tial.)43 b(This)24 b(is)60 1889 y(witnessed)16 b(b)o(y)g(the)g(follo)o(wing)g(example)e(from)h ([5].)60 1979 y Fg(Example)h(1.1)24 b Fp(Let)17 b(\006)e(=)g Fn(f)p Fp(0)p Fo(;)8 b Fp(1)p Fn(g)p Fp(,)17 b Fo(\013)f Fp(=)e Fo(x)p Fp(01)p Fo(y)r Fp(0)p Fo(z)r Fp(,)j(and)h Fo(\014)f Fp(=)e Fo(x)p Fp(0)p Fo(y)r Fp(10)p Fo(z)r Fp(.)23 b(Both)17 b Fo(\013)g Fp(and)h Fo(\014)h Fp(are)60 2039 y(in)e(normal)f(form)g(w.r.t.)g Fn(!)581 2046 y Fm(r)600 2039 y Fp(;)h(see)g(Section)g(4.)24 b(Although)18 b Fo(L)1195 2046 y Fm(E)r(;)p Fl(\006)1260 2039 y Fp(\()p Fo(\013)p Fp(\))d(=)h Fo(L)1431 2046 y Fm(E)r(;)p Fl(\006)1496 2039 y Fp(\()p Fo(\014)s Fp(\),)g Fo(\013)i Fp(and)g Fo(\014)60 2099 y Fp(are)e(not)h(iden)o(tical)d(up)j(to)g(renaming)e (of)h(v)m(ariables.)133 2190 y(W)l(e)f(will)f(pro)o(v)o(e)h(in)g (Section)f(4)i(that)g(the)f(conjecture)f(can)i(b)q(e)f(paraphrased)i (as)f(follo)o(ws:)60 2250 y(If)f Fo(\013;)8 b(\014)17 b Fn(2)d Fp(\(\006)c Fn([)h Fo(V)g Fp(\))419 2232 y Fl(+)464 2250 y Fp(are)16 b(patterns)h(and)f Fo(car)q(d)p Fp(\(\006\))e Fn(\025)g Fp(3,)i(then)g Fo(L)1265 2257 y Fm(E)r(;)p Fl(\006)1330 2250 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fo(L)1498 2257 y Fm(E)r(;)p Fl(\006)1563 2250 y Fp(\()p Fo(\014)s Fp(\))h(if)g(and)60 2310 y(only)f(if)f(there)g(are)h(morphisms)e Fo(f)19 b Fp(:)13 b Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))h Fn(!)f Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))1088 2292 y Fh(\003)1120 2310 y Fp(and)i Fo(g)h Fp(:)d Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))f Fn(!)i Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1643 2292 y Fh(\003)1676 2310 y Fp(suc)o(h)60 2370 y(that)j Fo(f)5 b Fp(\()p Fo(\013)p Fp(\))14 b(=)g Fo(\014)k Fp(and)f Fo(g)r Fp(\()p Fo(\014)s Fp(\))d(=)f Fo(\013)p Fp(.)133 2430 y(Also)g(note)h(that)g(if)f(Conjecture)g(1)h (holds)g(true,)f(then)g(the)h(normal)e(form)g(is)i(the)f Fi(shortest)60 2491 y Fp(pattern)e(that)h(generates)g(a)f(giv)o(en)g (E-pattern)h(language.)20 b(This)12 b(is)f(b)q(ecause)g Fn(!)1511 2498 y Fm(r)1541 2491 y Fp(is)g(a)h(length-)60 2551 y(decreasing)h(reduction)g(relation.)20 b(Hence)12 b(w)o(e)h(conjecture,)g(in)g(fact,)g(that)h(our)g(normal)e(form)60 2611 y(is)k(a)h(minim)o(al)c(description)j(of)g(the)g(corresp)q(onding) i(language.)133 2671 y(W)l(e)d(will)f(pro)o(v)o(e)h(in)g(Sections)g(3)h (and)g(5)g(that)f(the)g(equiv)m(alence)f(problem)g(for)h(E-pattern)60 2731 y(languages)f(can)g(in)f(man)o(y)f(cases)h(b)q(e)h(decided)e(b)o (y)h(this)g(pro)q(cedure,)g(for)h(instance)f(whenev)o(er:)903 2856 y(2)p eop %%Page: 3 3 3 2 bop 133 203 a Fn(\017)24 b Fp(the)15 b(underlying)g(alphab)q(et)h (\006)g(con)o(tains)g(t)o(w)o(o)g(constan)o(ts)g(that)g(do)g(not)g(o)q (ccur)g(in)f(the)182 264 y(patterns)i(\(this)f(is)g(alw)o(a)o(ys)g (true)g(if)f(\006)i(is)f(in\014nite\),)133 365 y Fn(\017)24 b Fo(car)q(d)p Fp(\(\006\))14 b(exceeds)e(the)h(n)o(um)o(b)q(er)e(of)i (terminal)f(segmen)o(ts)f(in)i(one)h(pattern)f(b)o(y)g(at)g(least)182 425 y(t)o(w)o(o,)133 527 y Fn(\017)24 b Fp(the)16 b(normal)f(form)g(of) i(one)f(pattern)h(has)g(indep)q(enden)o(t)e(v)m(ariable)h(segmen)o(ts,) 133 629 y Fn(\017)24 b Fp(the)16 b(normal)f(form)g(of)i(one)f(pattern)h (is)f(linear,)133 731 y Fn(\017)24 b Fp(the)16 b(normal)f(form)g(of)i (one)f(pattern)h(is)f(a)g(one-v)m(ariable)h(pattern,)133 832 y Fn(\017)24 b Fp(the)16 b(normal)f(form)g(of)i(one)f(pattern)h(do) q(es)g(not)g(con)o(tain)f(consecutiv)o(e)e(v)m(ariables.)60 934 y(It)21 b(is)f(in)o(teresting)g(to)i(note)f(that)g(the)g(\014rst)g (t)o(w)o(o)g(sp)q(ecial)f(cases)i(can)f(already)g(b)q(e)g(pro)o(v)o(ed) 60 994 y(b)o(y)h(further)h(dev)o(eloping)f(a)h(pro)q(of)h(tec)o(hnique) e(used)h(in)f([6];)k(see)c(Section)h(3.)41 b(Ho)o(w)o(ev)o(er,)60 1054 y(w)o(e)20 b(will)g(indicate)g(wh)o(y)h(this)f(tec)o(hnique)f(is)i (not)g(su\016cien)o(t)f(to)h(pro)o(v)o(e)f(the)h(general)f(case.)60 1115 y(F)l(urthermore,)c(w)o(e)i(will)e(p)q(ose)j(t)o(w)o(o)f(op)q(en)h (questions.)26 b(An)18 b(a\016rmativ)o(e)d(answ)o(er)j(to)h(one)f(of)60 1175 y(them)d(w)o(ould)h(pro)o(v)o(e)f(decidabilit)o(y)f(of)j (E-language)g(equiv)m(alence.)133 1235 y(F)l(or)j(a)g(nice)f(in)o(tro)q (duction)h(to)g(patterns)g(and)g(reasons)h(wh)o(y)f(they)f(are)h(of)g (in)o(terest)e(in)60 1295 y(formal)g(language)j(theory)l(,)f(the)f (reader)g(is)h(referred)e(to)i([10,)f(11)q(].)31 b(Asp)q(ects)19 b(of)h(inductiv)o(e)60 1355 y(inference)15 b(and)h(the)g(theory)g(of)h (learning)f({)g(although)i(closely)d(related)g(to)i(patterns)g({)f (will)60 1416 y(not)h(b)q(e)f(discussed)g(in)g(this)g(pap)q(er,)h (details)e(can)i(b)q(e)f(found)h(in)f([2,)g(4,)g(7],)g(for)g(instance.) 133 1476 y(The)h(pap)q(er)g(is)f(organized)h(as)g(follo)o(ws.)22 b(In)16 b(Section)g(2,)h(basic)g(de\014nitions)f(are)g(recalled)60 1536 y(and)k(in)g(Section)f(3)h(our)g(\014rst)h(results)e(are)h(presen) o(ted.)31 b(Section)19 b(4)h(explains)f(the)h(normal)60 1596 y(form)i(approac)o(h,)j(while)e(Section)f(5)i(con)o(tains)f (decidabilit)o(y)e(results)i(based)g(on)h(normal)60 1656 y(forms.)c(Finally)l(,)12 b(in)i(Section)g(6)g(it)g(is)g(sho)o(wn)h (that)f(in)g(con)o(trast)g(to)h(what)g(has)f(b)q(een)g(claimed)60 1716 y(in)i([3],)f(the)h(recen)o(t)f(approac)o(h)i(of)g(D\023)-24 b(an)o(yi)16 b(and)g(F)q(\177)-26 b(ul\177)i(op)18 b(do)q(es)f(not)f (solv)o(e)g(the)g(problem.)60 1883 y Fq(2)81 b(Basic)27 b(de\014nitions)60 1992 y Fp(Giv)o(en)16 b(an)i(alphab)q(et)g(\006,)g (the)f(set)h(of)f(all)g(w)o(ords)h(o)o(v)o(er)f(\006)h(is)f(denoted)g (b)o(y)g(\006)1468 1974 y Fh(\003)1488 1992 y Fp(,)h(whereas)f(\006) 1740 1974 y Fl(+)60 2053 y Fp(stands)g(for)f(the)g(set)g(of)g(all)f (nonempt)o(y)g(w)o(ords)h(o)o(v)o(er)f(\006.)22 b(The)16 b(length)f(of)h(a)h(w)o(ord)f Fo(\013)e Fn(2)g Fp(\006)1701 2035 y Fh(\003)1737 2053 y Fp(is)60 2113 y(denoted)i(b)o(y)g Fn(j)p Fo(\013)p Fn(j)p Fp(.)21 b(The)16 b(empt)o(y)e(w)o(ord)i(will)f (b)q(e)i(denoted)f(b)o(y)f Fo(\025)p Fp(,)i(its)f(length)f(is)h(zero.) 21 b(F)l(or)c(an)60 2173 y(arbitrary)f(set)g Fo(S)s Fp(,)g Fo(car)q(d)p Fp(\()p Fo(S)s Fp(\))h(denotes)f(its)g(cardinalit)o(y)l(.) 133 2233 y(In)i(the)f(sequel,)f(\006)i(and)h Fo(V)29 b Fp(are)17 b(t)o(w)o(o)h(disjoin)o(t)f(alphab)q(ets.)26 b(Elemen)o(ts)16 b(of)h(\006)h(are)g(called)60 2293 y Fi(terminals)12 b Fp(\(or)g Fi(c)n(onstants)p Fp(\))g(and)g(elemen)o (ts)c(of)k Fo(V)22 b Fp(are)12 b(called)e Fi(variables)p Fp(.)21 b(Usually)l(,)10 b(0)p Fo(;)e Fp(1)p Fo(;)g(a;)g(b;)g(c)60 2354 y Fp(will)k(denote)h(constan)o(ts,)g(whereas)h Fo(x;)8 b(y)r(;)g(z)r(;)g(x)884 2335 y Fh(0)894 2354 y Fo(;)g(y)942 2335 y Fh(0)953 2354 y Fo(;)g(z)1000 2335 y Fh(0)1011 2354 y Fo(;)g(x)1061 2361 y Fl(1)1080 2354 y Fo(;)g(:)g(:)g(:)k Fp(will)g(stand)i(for)f(v)m(ariables.)20 b(An)o(y)60 2414 y(w)o(ord)c Fo(\013)g Fp(o)o(v)o(er)f(the)h(union)g(\006)10 b Fn([)g Fo(V)27 b Fp(is)16 b(said)g(to)g(b)q(e)g(a)g Fi(p)n(attern)p Fp(.)22 b(The)15 b(n)o(um)o(b)q(er)f(of)i(o)q (ccurrences)60 2474 y(of)f(a)g(letter)f Fo(x)g Fn(2)g Fp(\006)8 b Fn([)g Fo(V)26 b Fp(in)15 b Fo(\013)g Fp(is)g(denoted)f(b)o (y)h Fn(j)p Fo(\013)p Fn(j)967 2481 y Fm(x)989 2474 y Fp(.)20 b(The)15 b(set)g(of)g(v)m(ariables)g(of)g Fo(V)26 b Fp(app)q(earing)60 2534 y(in)16 b Fo(\013)h Fp(will)e(b)q(e)h (denoted)g(b)o(y)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\).)133 2594 y(A)d(mapping)f Fo(h)i Fp(:)f(\(\006)t Fn([)t Fo(V)f Fp(\))603 2576 y Fh(\003)637 2594 y Fn(!)h Fp(\(\006)t Fn([)t Fo(V)g Fp(\))855 2576 y Fh(\003)887 2594 y Fp(with)g Fo(h)p Fp(\()p Fo(\025)p Fp(\))h(=)g Fo(\025)g Fp(and)f Fo(h)p Fp(\()p Fo(w)1370 2601 y Fl(1)1390 2594 y Fo(w)1425 2601 y Fl(2)1444 2594 y Fp(\))h(=)g Fo(h)p Fp(\()p Fo(w)1611 2601 y Fl(1)1630 2594 y Fp(\))p Fo(h)p Fp(\()p Fo(w)1731 2601 y Fl(2)1751 2594 y Fp(\))60 2654 y(is)h(called)e(a)i Fi(morphism)p Fp(.)20 b(A)14 b(morphism)e Fo(h)j Fp(is)g(said)g(to)g(b) q(e)g Fi(terminal-pr)n(eserving)h Fp(if)e Fo(h)p Fp(\()p Fo(a)p Fp(\))g(=)f Fo(a)60 2715 y Fp(for)22 b(ev)o(ery)f Fo(a)i Fn(2)h Fp(\006.)40 b(Ev)o(ery)21 b(mapping)g Fo(\033)j Fp(from)d Fo(V)33 b Fp(to)23 b(\(\006)15 b Fn([)g Fo(V)c Fp(\))1297 2697 y Fh(\003)1339 2715 y Fp(extends)22 b(uniquely)f(to)903 2856 y(3)p eop %%Page: 4 4 4 3 bop 60 203 a Fp(a)20 b(terminal-preserving)e(morphism)f Fo(h)j Fp(:)f(\(\006)14 b Fn([)g Fo(V)d Fp(\))1023 185 y Fh(\003)1063 203 y Fn(!)19 b Fp(\(\006)14 b Fn([)g Fo(V)d Fp(\))1305 185 y Fh(\003)1325 203 y Fp(,)20 b(where)f Fo(h)p Fp(\()p Fo(x)p Fp(\))h(=)g Fo(\033)r Fp(\()p Fo(x)p Fp(\))60 264 y(for)e(ev)o(ery)e Fo(x)f Fn(2)h Fo(V)11 b Fp(.)25 b(Th)o(us,)18 b(whenev)o(er)e(w)o(e)h(sp)q(eak)h(of)g(a)f (morphism)e Fo(\033)j Fp(:)d Fo(V)27 b Fn(!)16 b Fp(\(\006)c Fn([)g Fo(V)f Fp(\))1692 246 y Fh(\003)1729 264 y Fp(in)60 324 y(the)i(sequel,)g(its)g(corresp)q(onding)h(terminal-preserving)d (morphism)g(is)i(mean)o(t.)19 b(Henceforth)60 384 y Fo(\033)o(;)8 b(\027;)g(f)s(;)g(g)r(;)g(h)17 b Fp(will)g(denote)h(morphisms.)24 b(A)17 b(bijectiv)o(e)f(morphism)f Fo(\033)j Fp(:)f Fo(V)28 b Fn(!)16 b Fo(V)29 b Fp(is)18 b(called)f(a)60 444 y Fi(variable)i(r)n(enaming)p Fp(.)k(Tw)o(o)17 b(patterns)h Fo(\013)f Fp(and)g Fo(\014)i 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y(ev)o(ery)10 b(pattern)i Fo(\013)g Fp(has)g(a)h(canonical)e(form)g Fo(\013)866 727 y Fh(0)878 745 y Fp(,)h(and)g(furthermore)e Fo(\013)i Fp(and)g Fo(\013)1423 727 y Fh(0)1447 745 y Fp(are)g(iden)o(tical)e(up)60 805 y(to)j(v)m(ariable)f(renaming.)19 b(Moreo)o(v)o(er,)12 b(the)g(canonical)h(form)e(of)i(a)g(pattern)g(can) g(b)q(e)f(computed)60 866 y(in)k(linear)f(time.)k(Th)o(us,)d(it)f(can)i (b)q(e)f(tested)f(in)h(linear)f(time)f(whether)i(or)g(not)h(t)o(w)o(o)f (patterns)60 926 y(are)g(iden)o(tical)f(up)h(to)h(v)m(ariable)f (renaming.)133 986 y(The)g(language)h Fo(L)469 993 y Fm(E)r(;)p Fl(\006)534 986 y Fp(\()p Fo(\013)p Fp(\))f(generated)g(b)o (y)g(an)g(E-pattern)g Fo(\013)f Fn(2)f Fp(\(\006)c Fn([)h Fo(V)g Fp(\))1457 968 y Fl(+)1502 986 y Fp(is)16 b(de\014ned)g(b)o(y) 226 1083 y Fo(L)259 1090 y Fm(E)r(;)p Fl(\006)324 1083 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fn(f)p Fo(w)h Fn(2)f Fp(\006)616 1062 y Fh(\003)652 1083 y Fn(j)i Fo(w)f Fp(=)f Fo(\033)r Fp(\()p Fo(\013)p Fp(\))i(for)g(some)g(morphism)e Fo(\033)h Fp(:)e Fo(V)25 b Fn(!)14 b Fp(\006)1546 1062 y Fh(\003)1566 1083 y Fn(g)p Fo(:)60 1180 y Fp(The)i(language)h Fo(L)396 1187 y Fm(N)t(E)r(;)p Fl(\006)493 1180 y Fp(\()p Fo(\013)p Fp(\))g(generated)f(b)o(y)g(an)h(NE-pattern)f Fo(\013)e Fn(2)g Fp(\(\006)d Fn([)h Fo(V)f Fp(\))1456 1162 y Fl(+)1502 1180 y Fp(is)16 b(de\014ned)g(as)205 1277 y Fo(L)238 1284 y Fm(N)t(E)r(;)p Fl(\006)335 1277 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fn(f)p Fo(w)h Fn(2)f Fp(\006)627 1256 y Fh(\003)663 1277 y Fn(j)i Fo(w)f Fp(=)f Fo(\033)r Fp(\()p Fo(\013)p Fp(\))i(for)g(some)f(morphism)f Fo(\033)i Fp(:)d Fo(V)25 b Fn(!)14 b Fp(\006)1557 1256 y Fl(+)1587 1277 y Fn(g)p Fo(:)60 1374 y Fp(Giv)o(en)23 b(t)o(w)o(o)h(E-patterns)h Fo(\013)g Fp(and)g Fo(\014)s Fp(,)g(it)e(is)h(not)h(di\016cult)e(to)h(pro)o(v)o(e)g(that)g(if)g (there)g(is)g(a)60 1434 y(terminal-preserving)16 b(morphism)h Fo(h)i Fp(with)g Fo(h)p Fp(\()p Fo(\014)s Fp(\))f(=)g Fo(\013)p Fp(,)h(then)g Fo(L)1266 1441 y Fm(E)r(;)p Fl(\006)1331 1434 y Fp(\()p Fo(\013)p Fp(\))g Fn(\022)f Fo(L)1509 1441 y Fm(E)r(;)p Fl(\006)1574 1434 y Fp(\()p Fo(\014)s Fp(\).)29 b(The)60 1494 y(same)15 b(statemen)o(t)f(holds)j(for)f (NE-patterns.)22 b(These)16 b(observ)m(ations)h(will)e(b)q(e)h(used)h (freely)d(in)60 1555 y(the)i(sequel.)133 1615 y(Let)j Fo(\013)g Fp(b)q(e)g(a)h(pattern.)29 b Fo(\013)19 b Fp(has)h(a)f (unique)f(represen)o(tation)g Fo(\013)1282 1622 y Fl(0)1302 1615 y Fo(u)1330 1622 y Fl(1)1350 1615 y Fo(\013)1381 1622 y Fl(1)1400 1615 y Fo(u)1428 1622 y Fl(2)1456 1615 y Fo(:)8 b(:)g(:)g(\013)1553 1622 y Fm(m)p Fh(\000)p Fl(1)1631 1615 y Fo(u)1659 1622 y Fm(m)1692 1615 y Fo(\013)1723 1622 y Fm(m)1756 1615 y Fp(,)60 1675 y(where)22 b Fo(\013)238 1682 y Fl(0)257 1675 y Fo(;)8 b(\013)310 1682 y Fm(m)367 1675 y Fn(2)24 b Fo(V)464 1657 y Fh(\003)483 1675 y Fp(,)f Fo(\013)551 1682 y Fm(i)589 1675 y Fn(2)h Fo(V)686 1657 y Fl(+)737 1675 y Fp(for)e(1)j Fn(\024)e Fo(i)g(<)h(m)p Fp(,)f(and)g Fo(u)1239 1682 y Fm(i)1276 1675 y Fn(2)h Fp(\006)1368 1657 y Fl(+)1420 1675 y Fp(for)e(1)i Fn(\024)g Fo(i)f Fn(\024)h Fo(m)p Fp(,)60 1735 y Fo(m)17 b Fn(\025)h Fp(0.)28 b(W)l(e)18 b(call)g(the)g(sub)o(w)o(ords)h Fo(u)749 1742 y Fm(i)781 1735 y Fp(the)g Fi(terminal)h(se)n(gments)f Fp(of)g Fo(\013)p Fp(.)28 b(F)l(urthermore,)17 b(the)60 1795 y Fo(\013)91 1802 y Fm(i)125 1795 y Fp(are)i(called)g(the)g Fi(variable)i(se)n(gments)g Fp(of)f Fo(\013)p Fp(.)31 b(By)19 b(imp)q(osing)g(syn)o(tatic)f(restrictions,)h(w)o(e)60 1856 y(obtain)f(sp)q(ecial)f(patterns.)26 b(W)l(e)18 b(sa)o(y)f(that)h Fo(\013)g Fp(has)h Fi(indep)n(endent)h(variable)g(se) n(gments)f Fp(if)e(for)60 1916 y(all)h(1)g Fn(\024)g Fo(i)f(<)h(j)j Fn(\024)c Fo(m)i Fp(w)o(e)f(ha)o(v)o(e)g Fo(v)r(ar)q Fp(\()p Fo(\013)792 1923 y Fm(i)805 1916 y Fp(\))13 b Fn(\\)f Fo(v)r(ar)q Fp(\()p Fo(\013)1007 1923 y Fm(j)1025 1916 y Fp(\))18 b(=)f Fn(;)p Fp(.)29 b(P)o(attern)18 b Fo(\013)h Fp(is)f Fi(line)n(ar)25 b Fp(if)18 b(ev)o(ery)60 1976 y(v)m(ariable)i Fo(x)h Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))f(o)q(ccurs)h(exactly)e(once)h(in)g Fo(\013)p Fp(.)34 b(Ob)o(viously)l(,)20 b(ev)o(ery)f(linear)h(pattern) 60 2036 y(has)g(indep)q(enden)o(t)f(v)m(ariable)h(segmen)o(ts.)30 b(P)o(attern)19 b Fo(\013)i Fp(is)e(said)h(to)g(b)q(e)g Fi(terminal)h(fr)n(e)n(e)e Fp(if)h(no)60 2096 y(terminal)10 b(o)q(ccurs)j(in)f(it.)20 b(P)o(attern)12 b Fo(\013)h Fp(is)f(a)h Fi(one-variable)j(p)n(attern)c Fp(if)g(it)g(con)o(tains)g (o)q(ccurrences)60 2156 y(of)18 b(one)g(v)m(ariable)f(only)l(.)25 b(Finally)l(,)16 b(there)h(is)g(the)h(more)e(seman)o(tic)g(notion)i(of) g(unam)o(biguit)o(y)l(.)60 2217 y(A)i(pattern)g Fo(\013)g Fp(is)g Fi(unambiguous)h Fp(if,)f(for)g(ev)o(ery)e(w)o(ord)j Fo(w)g Fn(2)f Fo(L)1233 2224 y Fm(E)r(;)p Fl(\006)1299 2217 y Fp(\()p Fo(\013)p Fp(\),)g(there)g(is)f(a)i(unique)60 2277 y(terminal-preserving)14 b(morphism)h Fo(\033)j Fp(suc)o(h)f(that)g Fo(\033)r Fp(\()p Fo(\013)p Fp(\))e(=)f Fo(w)q Fp(.)23 b(It)17 b(is)f(decidable)g(whether)h(or)60 2337 y(not)j(a)h(pattern)f(is)f(unam)o(biguous;)i(see)e([8].)32 b(Moreo)o(v)o(er,)19 b(ev)o(ery)f(one-v)m(ariable)i(pattern)g(is)60 2397 y(unam)o(biguous;)c(see)f(also)i([8].)60 2562 y Fq(3)81 b(Preliminary)25 b(results)60 2671 y Fp(On)d(the)f(one)h(hand,) i(Jiang)e Fi(et)h(al.)39 b Fp([6])21 b(ha)o(v)o(e)g(sho)o(wn)i(that)f (the)f(inclusion)h(problem)e(for)60 2731 y(E-pattern)d(languages)h(is)f (in)f(general)h(undecidable.)22 b(On)16 b(the)h(other)g(hand,)g(this)g (problem)903 2856 y(4)p eop %%Page: 5 5 5 4 bop 60 203 a Fp(is)22 b(decidable)g(for)g(terminal)f(free)g (patterns;)26 b(see)c([6].)40 b(Consequen)o(tly)l(,)22 b(the)h(E-language)60 264 y(equiv)m(alence)12 b(problem)h(for)i (terminal)d(free)h(patterns)i(is)f(decidable.)20 b(Despite)13 b(the)h(fact)h(that)60 324 y(the)f(inclusion)f(problem)f(for)j(general) e(E-pattern)i(languages)g(is)f(undecidable,)f(it)h(migh)o(t)e(b)q(e)60 384 y(the)19 b(case)h(that)f(the)g(decidabilit)o(y)e(of)j(the)f(equiv)m (alence)f(problem)g(for)h(general)g(E-pattern)60 444 y(languages)d(can)f(b)q(e)g(pro)o(v)o(ed)f(b)o(y)g(sho)o(wing)h(the)g (decidabilit)o(y)d(of)j(the)g(inclusion)f(problem)f(for)60 504 y(certain)h(E-patterns.)21 b(This)15 b(follo)o(ws)g(from)e(the)i (follo)o(wing)f(theorem)f(o)o(wing)i(to)g(Jiang)g Fi(et)i(al.)60 565 y Fp([5,)f(6].)60 679 y Fg(Theorem)h(3.1)24 b Fp(Let)17 b Fo(\013;)8 b(\014)18 b Fn(2)d Fp(\(\006)d Fn([)g Fo(V)f Fp(\))798 661 y Fl(+)827 679 y Fp(.)24 b(Moreo)o(v)o(er,)16 b(let)g Fo(\013)g Fp(=)f Fo(\013)1296 686 y Fl(0)1316 679 y Fo(u)1344 686 y Fl(1)1363 679 y Fo(\013)1394 686 y Fl(1)1414 679 y Fo(u)1442 686 y Fl(2)1470 679 y Fo(:)8 b(:)g(:)f(\013)1566 686 y Fm(m)p Fh(\000)p Fl(1)1645 679 y Fo(u)1673 686 y Fm(m)1706 679 y Fo(\013)1737 686 y Fm(m)60 739 y Fp(and)22 b Fo(\014)k Fp(=)d Fo(\014)303 746 y Fl(0)322 739 y Fo(v)346 746 y Fl(1)365 739 y Fo(\014)393 746 y Fl(1)413 739 y Fo(v)437 746 y Fl(2)464 739 y Fo(:)8 b(:)g(:)g(\014)558 746 y Fm(n)p Fh(\000)p Fl(1)626 739 y Fo(v)650 746 y Fm(n)673 739 y Fo(\014)701 746 y Fm(n)746 739 y Fp(b)q(e)22 b(their)f(unique)f(represen)o(tations,)j(that)f(is)f (to)i(sa)o(y)l(,)60 799 y Fo(\013)91 806 y Fl(0)111 799 y Fo(;)8 b(\014)161 806 y Fl(0)180 799 y Fo(;)g(\013)233 806 y Fm(m)266 799 y Fo(;)g(\014)316 806 y Fm(n)357 799 y Fn(2)19 b Fo(V)449 781 y Fh(\003)468 799 y Fp(,)h Fo(\013)533 806 y Fm(i)565 799 y Fn(2)f Fo(V)656 781 y Fl(+)705 799 y Fp(for)g(1)g Fn(\024)g Fo(i)f(<)h(m)p Fp(,)g Fo(\014)1079 806 y Fm(j)1115 799 y Fn(2)g Fo(V)1206 781 y Fl(+)1255 799 y Fp(for)g(1)g Fn(\024)f Fo(j)k(<)d(n)p Fp(,)g Fo(u)1621 806 y Fm(i)1653 799 y Fn(2)g Fp(\006)1740 781 y Fl(+)60 859 y Fp(for)h(1)h Fn(\024)f Fo(i)g Fn(\024)g Fo(m)p Fp(,)g Fo(v)439 866 y Fm(j)477 859 y Fn(2)g Fp(\006)565 841 y Fl(+)615 859 y Fp(for)g(1)h Fn(\024)f Fo(j)j Fn(\024)d Fo(n)p Fp(,)h Fo(m;)8 b(n)20 b Fn(\025)g Fp(0.)32 b(If)20 b Fo(L)1292 866 y Fm(E)r(;)p Fl(\006)1357 859 y Fp(\()p Fo(\013)p Fp(\))h(=)f Fo(L)1538 866 y Fm(E)r(;)p Fl(\006)1603 859 y Fp(\()p Fo(\014)s Fp(\))f(and)60 919 y Fo(car)q(d)p Fp(\(\006\))g Fn(\025)f Fp(3,)i(then)f Fo(m)f Fp(=)h Fo(n)g Fp(and)h Fo(u)768 926 y Fm(i)800 919 y Fp(=)f Fo(v)881 926 y Fm(i)913 919 y Fp(for)h(1)f Fn(\024)f Fo(i)g Fn(\024)h Fo(m)p Fp(.)29 b(If)18 b Fo(car)q(d)p Fp(\(\006\))i Fn(\025)e Fp(4,)i(then)e(it)60 980 y(further)e(follo)o (ws)g Fo(L)417 987 y Fm(E)r(;)p Fl(\006)482 980 y Fp(\()p Fo(\013)532 987 y Fm(i)546 980 y Fp(\))e(=)g Fo(L)664 987 y Fm(E)r(;)p Fl(\006)729 980 y Fp(\()p Fo(\014)776 987 y Fm(i)790 980 y Fp(\))i(for)h(0)d Fn(\024)f Fo(i)h Fn(\024)f Fo(m)p Fp(.)133 1094 y(Again,)h(the)f(assumption)g Fo(car)q(d)p Fp(\(\006\))h Fn(\025)f Fp(3)h(is)f(crucial;)g(see)g (Example)e(1.1.)21 b(It)13 b(is)g(unkno)o(wn)60 1154 y(whether)j(the)g(restriction)g Fo(car)q(d)p Fp(\(\006\))f Fn(\025)f Fp(4)i(can)h(b)q(e)g(w)o(eak)o(ened)e(to)i Fo(car)q(d)p Fp(\(\006\))e Fn(\025)f Fp(3)i(in)h(the)f(last)60 1214 y(statemen)o(t)23 b(of)h(Theorem)f(3.1.)45 b(In)24 b(view)g(of)g(the)g(ab)q(o)o(v)o(e)h(theorem,)f(w)o(e)f(in)o(tro)q (duce)h(the)60 1274 y(follo)o(wing)16 b(notion.)60 1389 y Fg(De\014nition)i(3.2)24 b Fp(Tw)o(o)19 b(patterns)f Fo(\013)h Fp(and)g Fo(\014)i Fp(are)e(called)e Fi(similar)h Fp(if)g(and)h(only)g(if)f(in)g(their)60 1449 y(unique)e(represen)o (tations)h(the)g(resp)q(ectiv)o(e)e(terminal)g(segmen)o(ts)g(coincide,) h(that)h(is)g(to)h(sa)o(y)l(,)60 1509 y Fo(\013)c Fp(=)g Fo(\013)188 1516 y Fl(0)208 1509 y Fo(u)236 1516 y Fl(1)255 1509 y Fo(\013)286 1516 y Fl(1)306 1509 y Fo(u)334 1516 y Fl(2)362 1509 y Fo(:)8 b(:)g(:)f(\013)458 1516 y Fm(m)p Fh(\000)p Fl(1)537 1509 y Fo(u)565 1516 y Fm(m)598 1509 y Fo(\013)629 1516 y Fm(m)678 1509 y Fp(and)17 b Fo(\014)f Fp(=)e Fo(\014)897 1516 y Fl(0)916 1509 y Fo(u)944 1516 y Fl(1)964 1509 y Fo(\014)992 1516 y Fl(1)1011 1509 y Fo(u)1039 1516 y Fl(2)1067 1509 y Fo(:)8 b(:)g(:)f(\014)1160 1516 y Fm(m)p Fh(\000)p Fl(1)1238 1509 y Fo(u)1266 1516 y Fm(m)1299 1509 y Fo(\014)1327 1516 y Fm(m)1360 1509 y Fp(.)133 1623 y(With)16 b(these)g(notations,)h(the)f(\014rst)g(op)q (en)h(problem)e(can)h(b)q(e)h(stated)f(as)h(follo)o(ws.)60 1743 y Fg(Op)r(en)h(question)g(1:)j Fp(Is)16 b(the)g(inclusion)g (problem)e(decidable)h(for)i(similar)d(E-patterns?)60 1864 y(If)23 b(the)g(answ)o(er)h(w)o(as)f(p)q(ositiv)o(e)g(in)g(case)g Fo(car)q(d)p Fp(\(\006\))k Fn(\025)e Fp(3,)g(then)f(the)f(decidabilit)o (y)e(of)i(the)60 1924 y(equiv)m(alence)c(problem)g(for)i(general)g (E-pattern)g(languages)h(w)o(ould)f(follo)o(w:)30 b(Giv)o(en)19 b(pat-)60 1984 y(terns)k Fo(\013)j Fp(=)g Fo(\013)341 1991 y Fl(0)361 1984 y Fo(u)389 1991 y Fl(1)408 1984 y Fo(\013)439 1991 y Fl(1)459 1984 y Fo(u)487 1991 y Fl(2)515 1984 y Fo(:)8 b(:)g(:)f(\013)611 1991 y Fm(m)p Fh(\000)p Fl(1)690 1984 y Fo(u)718 1991 y Fm(m)751 1984 y Fo(\013)782 1991 y Fm(m)838 1984 y Fp(and)24 b Fo(\014)k Fp(=)e Fo(\014)1088 1991 y Fl(0)1107 1984 y Fo(v)1131 1991 y Fl(1)1150 1984 y Fo(\014)1178 1991 y Fl(1)1197 1984 y Fo(v)1221 1991 y Fl(2)1249 1984 y Fo(:)8 b(:)g(:)g(\014)1343 1991 y Fm(n)p Fh(\000)p Fl(1)1411 1984 y Fo(v)1435 1991 y Fm(n)1458 1984 y Fo(\014)1486 1991 y Fm(n)1509 1984 y Fp(,)24 b(\014rst)g(c)o(hec)o(k)60 2044 y(whether)18 b Fo(m)f Fp(=)g Fo(n)h Fp(and)h Fo(u)536 2051 y Fm(i)567 2044 y Fp(=)e Fo(v)646 2051 y Fm(i)677 2044 y Fp(for)i(1)e Fn(\024)g Fo(i)g Fn(\024)g Fo(m)p Fp(.)26 b(If)18 b(not,)g(then)g Fo(L)1323 2051 y Fm(E)r(;)p Fl(\006)1389 2044 y Fp(\()p Fo(\013)p Fp(\))f Fn(6)p Fp(=)g Fo(L)1563 2051 y Fm(E)r(;)p Fl(\006)1628 2044 y Fp(\()p Fo(\014)s Fp(\).)26 b(If)60 2105 y(so,)16 b(then)h(decide)e Fo(L)427 2112 y Fm(E)r(;)p Fl(\006)492 2105 y Fp(\()p Fo(\013)p Fp(\))f(=)g Fo(L)660 2112 y Fm(E)r(;)p Fl(\006)725 2105 y Fp(\()p Fo(\014)s Fp(\))h(b)o(y)h(testing)g(inclusion)g(in)g(b)q(oth)h(directions.)133 2165 y(It)i(is)h(imp)q(ortan)o(t)e(to)i(note)g(that)g(the)f (undecidabilit)o(y)e(of)j(the)f(inclusion)g(problem)f(for)60 2225 y(general)h(E-pattern)h(languages)g(do)q(es,)g Fi(a)g(priori)p Fp(,)e(not)i(imply)c(the)j(undecidabilit)o(y)e(of)i(the)60 2285 y(inclusion)g(problem)g(for)i(similar)d(E-patterns.)34 b(This)20 b(is)g(b)q(ecause)g(the)g(patterns)h(used)f(in)60 2345 y(the)14 b(undecidabilit)o(y)e(pro)q(of)j(of)g([6])f(are)g(not)h (similar.)j(And)c(in)g(fact,)g(w)o(e)g(can)g(pro)o(v)o(e)g(that)g(the) 60 2406 y(inclusion)i(problem)e(for)j(similar)d(E-patterns)k(is)e (decidable)f(pro)o(vided)g(that)i(the)f(alphab)q(et)60 2466 y(\006)22 b(con)o(tains)g(t)o(w)o(o)g(constan)o(ts)h(not)g(o)q (ccurring)f(in)g(the)f(patterns.)40 b(This)22 b(can)g(already)g(b)q(e) 60 2526 y(done)17 b(b)o(y)g(using)g(the)g(tec)o(hnique)f(Jiang)h Fi(et)i(al.)25 b Fp([6])16 b(emplo)o(y)o(ed)f(for)i(deciding)f(the)h (language)60 2586 y(equiv)m(alence)d(problem)h(for)i(terminal)d(free)h (E-patterns.)903 2856 y(5)p eop %%Page: 6 6 6 5 bop 60 203 a Fg(De\014nition)18 b(3.3)24 b Fp(Let)14 b Fo(V)25 b Fp(=)14 b Fn(f)p Fo(x)661 210 y Fl(1)680 203 y Fo(;)8 b(:)g(:)g(:)g(;)g(x)818 210 y Fm(n)841 203 y Fn(g)14 b Fp(b)q(e)h(a)f(set)h(of)f(v)m(ariables)g(and)h(\006)g(b)q (e)g(an)f(alphab)q(et)60 264 y(with)h Fo(car)q(d)p Fp(\(\006\))f Fn(\025)f Fp(2.)22 b(F)l(or)14 b(ev)o(ery)g(pair)g(of)i(letters)d Fo(a;)8 b(b)14 b Fp(in)h(\006,)g Fo(a)e Fn(6)p Fp(=)h Fo(b)p Fp(,)g(and)h(an)h(in)o(teger)d Fo(k)j(>)e Fp(0,)60 324 y(w)o(e)i(de\014ne)g(a)g(morphism)e Fo(\034)562 331 y Fm(k)q(;a;b)651 324 y Fp(:)f Fo(V)26 b Fn(!)13 b(f)p Fo(a;)8 b(b)p Fn(g)914 306 y Fh(\003)949 324 y Fp(b)o(y)359 434 y Fo(\034)380 441 y Fm(k)q(;a;b)455 434 y Fp(\()p Fo(x)502 441 y Fm(i)515 434 y Fp(\))14 b(=)g Fo(ab)647 413 y Fm(k)q Fh(\003)p Fm(i)p Fl(+1)742 434 y Fo(aab)815 413 y Fm(k)q Fh(\003)p Fm(i)p Fl(+2)910 434 y Fo(a)8 b(:)g(:)g(:)g(ab)1057 413 y Fm(k)q Fh(\003)p Fl(\()p Fm(i)p Fl(+1\))1180 434 y Fo(a;)35 b Fp(1)14 b Fn(\024)g Fo(i)f Fn(\024)h Fo(n:)60 548 y Fg(Lemma)i(3.4)24 b Fp(Let)17 b(\006)g(b)q(e)h(an)f(alphab)q(et)h(and)g(let)f Fo(\013;)8 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Fp(\006)1060 1011 y Fh(\003)1099 1029 y Fp(suc)o(h)f(that)i Fo(\027)s Fp(\()p Fo(\014)s Fp(\))e(=)g Fo(\034)1511 1036 y Fm(k)q(;a;b)1586 1029 y Fp(\()p Fo(\013)p Fp(\).)30 b(F)l(or)60 1090 y(ev)o(ery)11 b Fo(x)i Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\),)e Fo(\034)464 1097 y Fm(k)q(;a;b)539 1090 y Fp(\()p Fo(x)p Fp(\))g(consists)h(of)f Fo(k)i Fp(segmen)o(ts)d(of)h(the)g(form)f Fo(ab)1376 1072 y Fm(j)1394 1090 y Fo(a)p Fp(.)19 b(Th)o(us,)13 b(for)g(ev)o(ery)60 1150 y Fo(x)88 1157 y Fm(i)116 1150 y Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\),)h(there)h(is)g(at)g (least)g(one)g(segmen)o(t)f Fo(ab)1008 1132 y Fm(j)1022 1137 y Ff(i)1037 1150 y Fo(a)p Fp(,)g Fo(k)e Fn(\003)e Fo(i)g Fp(+)f(1)15 b Fn(\024)e Fo(j)1353 1157 y Fm(i)1381 1150 y Fn(\024)h Fo(k)f Fn(\003)e Fp(\()p Fo(i)f Fp(+)h(1\),)16 b(suc)o(h)60 1210 y(that)i(none)g(of)h(the)e(app)q(earance\(s\))i(of)f (this)g(segmen)o(t)e(in)i Fo(\027)s Fp(\()p Fo(\014)s Fp(\))f(is)h(split)f(b)o(y)g(an)o(y)h(partition)60 1270 y(of)g Fo(\027)s Fp(\()p Fo(\014)s Fp(\))g(in)o(to)g Fo(\027)s Fp(\()p Fo(\014)409 1252 y Fh(0)420 1270 y Fp(\))g(and)h Fo(\027)s Fp(\()p Fo(\014)631 1252 y Fh(00)651 1270 y Fp(\))f(where)g Fo(\014)h Fp(=)e Fo(\014)964 1252 y Fh(0)975 1270 y Fo(\014)1006 1252 y Fh(00)1027 1270 y Fp(.)27 b(F)l(or)18 b(ev)o(ery)e Fo(x)h Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\),)h(w)o(e)g(c)o(ho)q(ose)60 1330 y(one)d(suc)o(h)g(segmen)o(t)e(to)j(b)q(e)f(the)g(anc)o(hor)g(segmen)o (t)f(of)h Fo(x)g Fp(in)f Fo(\034)1169 1337 y Fm(k)q(;a;b)1244 1330 y Fp(\()p Fo(\013)p Fp(\))i(w.r.t.)d Fo(\014)s Fp(.)21 b(W)l(e)14 b(also)i(sa)o(y)60 1391 y(that)h(this)f(segmen)o(t)e(anc)o (hors)j Fo(x)p Fp(.)133 1451 y(Supp)q(ose)22 b(there)f(is)g(a)g(v)m (ariable)g Fo(y)i Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))c(suc)o(h)i(that)h Fo(\027)s Fp(\()p Fo(y)r Fp(\))f(con)o(tains)g (a)g(terminal)60 1511 y(di\013eren)o(t)d(from)g Fo(a)g Fp(and)h Fo(b)p Fp(,)g(sa)o(y)g Fo(c)p Fp(,)f(o)q(ccurring)h(in)g Fo(\014)s Fp(.)28 b(Let)19 b Fo(c)f 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Fp(b)q(e)h(the)g(last)g(letter)f(of)h Fo(u)1662 1759 y Fm(i)1691 1752 y Fp(and)60 1812 y Fo(d)j Fp(the)f(\014rst)h(letter)e (of)i Fo(u)511 1819 y Fm(i)p Fl(+1)588 1812 y Fp(\()p Fo(c)f Fp(=)f Fo(d)i Fp(is)f(p)q(ossible\).)28 b(Let)19 b(the)f(last)g(letter)g(of)g Fo(u)1512 1819 y Fm(i)1545 1812 y Fp(b)q(e)g(the)g Fo(p)p Fp(th)60 1872 y(app)q(earance)d(of)g Fo(c)f Fp(and)h(the)g(\014rst)f(letter)f(of)i Fo(u)887 1879 y Fm(i)p Fl(+1)961 1872 y Fp(b)q(e)f(the)g Fo(q)r Fp(th)g(app)q(earance)h(of)g Fo(d)g Fp(in)f Fo(\014)s Fp(.)20 b(Since)60 1932 y Fo(\027)s Fp(\()p Fo(y)r Fp(\))15 b(do)q(es)i(not)f(con)o(tain)f(an)o(y)g(letter)g(from)f(a)i(terminal)d (segmen)o(t,)h Fo(\027)s Fp(\()p Fo(y)r Fp(\))h(is)g(in)h(b)q(et)o(w)o (een)e(the)60 1993 y Fo(p)p Fp(th)i(app)q(earance)g(of)g Fo(c)g Fp(and)g(the)f Fo(q)r Fp(th)g(app)q(earance)h(of)g Fo(d)g Fp(in)f Fo(\027)s Fp(\()p Fo(\014)s Fp(\).)20 b(This)c(observ)m(ation)g(pla)o(ys)60 2053 y(a)h(k)o(ey)e(r^)-24 b(ole)16 b(in)f(the)h(pro)q(of.)133 2113 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b(a)h(segmen)o(t)e Fo(ab)1015 2423 y Fm(j)1029 2428 y Ff(i)1040 2432 y(s)1060 2441 y Fo(a)i Fp(anc)o(hors)g(a)g(v)m (ariable)g Fo(x)1522 2448 y Fm(i)1534 2452 y Ff(s)1566 2441 y Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\),)60 2501 y(1)h Fn(\024)f Fo(i)169 2508 y Fm(s)201 2501 y Fn(\024)g Fo(n)p Fp(.)22 b(Then)17 b(w)o(e)f(de\014ne)g Fo(h)p Fp(\()p Fo(y)r Fp(\))e(=)h Fo(x)847 2508 y Fm(i)859 2513 y Fe(1)878 2501 y Fo(x)906 2508 y Fm(i)918 2513 y Fe(2)945 2501 y Fo(:)8 b(:)g(:)g(x)1039 2508 y Fm(i)1051 2512 y Ff(r)1070 2501 y Fp(.)22 b(Clearly)l(,)15 b Fo(h)p Fp(\()p Fo(\014)s Fp(\))f(=)g Fo(\013)j Fp(b)q(ecause)g(eac)o(h)60 2561 y(app)q(earance)g(of)g Fo(x)f Fp(in)f Fo(\013)i Fp(has)g(exactly)e(one)h(anc)o(hor)h(segmen)o(t)e(w.r.t.)g Fo(\014)s Fp(.)20 b Fd(2)133 2675 y Fp(No)o(w)c(w)o(e)g(are)g(in)g(a)h (p)q(osition)g(to)f(pro)o(v)o(e)g(the)g(\014rst)g(decidabilit)o(y)e (results.)903 2856 y(6)p eop %%Page: 7 7 7 6 bop 60 203 a Fg(Theorem)17 b(3.5)24 b Fp(Let)13 b(\006)h(b)q(e)g (an)g(alphab)q(et)g(and)g(let)f Fo(\013;)8 b(\014)16 b Fn(2)e Fp(\(\006)6 b Fn([)g Fo(V)11 b Fp(\))1316 185 y Fl(+)1359 203 y Fp(b)q(e)j(similar)d(patterns.)60 264 y(If)17 b(\006)h(con)o(tains)f(t)o(w)o(o)h(distinct)e(constan)o(ts)i Fo(a)f Fp(and)i Fo(b)e Fp(not)h(o)q(ccurring)f(in)g Fo(\013)h Fp(and)g Fo(\014)s Fp(,)f(then)g(the)60 324 y(follo)o(wing)f(statemen)o (ts)f(are)h(equiv)m(alen)o(t:)120 421 y(1.)24 b Fo(L)215 428 y Fm(E)r(;)p Fl(\006)280 421 y Fp(\()p Fo(\013)p Fp(\))14 b Fn(\022)g Fo(L)449 428 y Fm(E)r(;)p Fl(\006)514 421 y Fp(\()p Fo(\014)s Fp(\).)120 522 y(2.)24 b Fo(\034)203 530 y Fh(j)p Fm(\014)r Fh(j)p Fm(;a;b)300 522 y Fp(\()p Fo(\013)p Fp(\))14 b Fn(2)g Fo(L)463 529 y Fm(E)r(;)p Fl(\006)528 522 y Fp(\()p Fo(\014)s Fp(\).)120 622 y(3.)24 b(There)16 b(exists)g(a)g(morphism)e Fo(h)g Fp(:)f Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))g Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1159 604 y Fh(\003)1195 622 y Fp(suc)o(h)j(that)h Fo(h)p Fp(\()p Fo(\014)s Fp(\))c(=)h Fo(\013)p 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y(patterns)c(is)f(decidable)f(\(in)h(fact,)f(it)h(is)g (NP-complete\);)e(see)i([1)o(,)g(5)q(].)k Fd(2)60 1299 y Fg(Corollary)e(3.7)24 b Fp(If)16 b Fo(car)q(d)p Fp(\(\006\))f Fn(\025)g Fp(3,)i(then)f(the)h(equiv)m(alence)e(problem)g(for)i (E-pattern)h(lan-)60 1359 y(guages)h(is)e(decidable)g(whenev)o(er)f (the)i(underlying)f(alphab)q(et)h(\006)g(con)o(tains)f(t)o(w)o(o)h (terminals)60 1419 y(that)f(do)f(not)h(o)q(ccur)g(in)e(the)h(patterns.) 60 1539 y Fg(Pro)r(of:)21 b Fp(This)c(follo)o(ws)f(from)f(Theorem)g (3.1)h(and)h(Corollary)g(3.6.)k Fd(2)133 1648 y Fp(Notice)f(that)h(the) f(inclusion)g(problem)f(\(and)i(hence)f(the)g(equiv)m(alence)f (problem\))g(for)60 1708 y(E-pattern)e(languages)g(is)f(particularly)g (decidable)f(if)g(\006)i(is)f(in\014nite.)133 1768 y(W)l(e)j(next)f (discuss)i(the)e(limits)f(of)i(the)g(ab)q(o)o(v)o(e)g(pro)q(of)h(tec)o (hnique.)28 b(Clearly)l(,)18 b(w)o(e)h(w)o(ould)60 1829 y(lik)o(e)13 b(to)i(get)g(rid)f(of)h(the)g(anno)o(ying)g(condition)f 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Fm(;l)181 2233 y Fe(2)200 2220 y Fp(\()p Fo(\013)p Fp(\))21 b(should)h(b)q(e)g (tak)o(en,)f(i.e.,)g(whic)o(h)g(constan)o(ts)h Fo(l)1154 2227 y Fl(1)1173 2220 y Fo(;)8 b(l)1210 2227 y Fl(2)1252 2220 y Fn(2)22 b Fp(\006)p Fo(;)31 b(l)1402 2227 y Fl(1)1444 2220 y Fn(6)p Fp(=)22 b Fo(l)1519 2227 y Fl(2)1560 2220 y Fp(should)g(b)q(e)60 2280 y(c)o(hosen)c(to)g(enco)q(de)g(the)g(v)m (ariables)g(in)f Fo(\013)p Fp(.)27 b(A)18 b(p)q(ossible)g(solution)g(w) o(ould)g(b)q(e)g(to)g(generalize)60 2340 y(the)h(ab)q(o)o(v)o(e)h (metho)q(d:)27 b(Find)19 b(a)h(\014nite)f(test)h(set)f Fo(F)26 b Fn(\032)20 b Fo(L)1111 2347 y Fm(E)r(;)p Fl(\006)1176 2340 y Fp(\()p Fo(\013)p Fp(\))g(suc)o(h)f(that)h Fo(F)26 b Fn(\032)19 b Fo(L)1636 2347 y Fm(E)r(;)p Fl(\006)1702 2340 y Fp(\()p Fo(\014)s Fp(\))60 2400 y(implies)d(the)i(existence)f (of)i(a)g(terminal-preserving)d(morphism)h Fo(h)i Fp(suc)o(h)f(that)h Fo(h)p Fp(\()p Fo(\014)s Fp(\))e(=)h Fo(\013)p Fp(.)60 2461 y(A)e(v)o(ery)f(natural)i(candidate)f(w)o(ould)g(b)q(e)g(the)g (set)h(of)f(all)g(enco)q(dings,)g(i.e.,)e(the)i(set)526 2566 y Fo(F)k Fp(=)14 b Fn(f)p Fo(\034)676 2574 y Fh(j)p Fm(\014)r Fh(j)p Fm(;l)739 2579 y Fe(1)754 2574 y Fm(;l)775 2579 y Fe(2)794 2566 y Fp(\()p Fo(\013)p Fp(\))j Fn(j)f Fo(l)925 2573 y Fl(1)944 2566 y Fo(;)8 b(l)981 2573 y Fl(2)1014 2566 y Fn(2)14 b Fp(\006)p Fo(;)22 b(l)1147 2573 y Fl(1)1180 2566 y Fn(6)p Fp(=)14 b Fo(l)1247 2573 y Fl(2)1266 2566 y Fn(g)p Fo(:)60 2671 y Fp(But)g(the)h(next)f(example) f(sho)o(ws)j(that)f(this)g(set)g(is)f(not)i(su\016cien)o(t)d(when)i Fo(car)q(d)p Fp(\(\006\))f(=)g(3)h(\(the)60 2731 y(situation)h(migh)o (t)f(b)q(e)h(di\013eren)o(t)g(for)g Fo(car)q(d)p Fp(\(\006\))f Fn(\025)e Fp(4\).)903 2856 y(7)p eop %%Page: 8 8 8 7 bop 60 203 a Fg(Example)16 b(3.8)24 b Fp(Let)15 b(\006)f(=)g Fn(f)p Fo(a;)8 b(b;)g(c)p Fn(g)p Fp(,)13 b Fo(\013)i Fp(=)e Fo(y)884 210 y Fl(1)912 203 y Fo(a)8 b(y)970 210 y Fl(2)998 203 y Fo(b)g(y)1051 210 y Fl(3)1070 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685 y Fp(\()p Fo(y)243 692 y Fm(i)256 685 y Fp(\).)258 775 y(\()p Fo(i)p Fp(\))41 b Fo(\034)375 783 y Fh(j)p Fm(\014)r Fh(j)p Fm(;a;b)472 775 y Fp(\()p Fo(\013)p Fp(\))14 b(=)g Fo(abw)689 782 y Fm(y)706 787 y Fe(1)725 775 y Fo(ba)f(a)h(abw)907 782 y Fm(y)924 787 y Fe(2)943 775 y Fo(ba)f(b)g(abw)1119 782 y Fm(y)1136 787 y Fe(3)1155 775 y Fo(ba)h(abw)1298 782 y Fm(y)1315 787 y Fe(2)1334 775 y Fo(ba)f(abw)1476 782 y Fm(y)1493 787 y Fe(4)1512 775 y Fo(ba:)60 865 y Fp(De\014ne)j(the)g(morphism)e Fo(\027)19 b Fp(b)o(y)d Fo(\027)s Fp(\()p Fo(z)703 872 y Fl(1)723 865 y Fp(\))e(=)f Fo(\027)s Fp(\()p Fo(z)876 872 y Fl(2)896 865 y Fp(\))h(=)g Fo(\025)i Fp(and)369 955 y Fo(\027)s Fp(\()p Fo(z)438 962 y Fl(3)458 955 y Fp(\))e(=)f Fo(w)577 962 y Fm(y)594 967 y Fe(1)614 955 y Fo(ba)g(a)h(abw)796 962 y Fm(y)813 967 y Fe(2)832 955 y Fo(ba)f(b)g(abw)1008 962 y Fm(y)1025 967 y Fe(3)1044 955 y Fo(ba)h(abw)1187 962 y Fm(y)1204 967 y Fe(2)1223 955 y Fo(ba)f(abw)1365 962 y Fm(y)1382 967 y Fe(4)1401 955 y Fo(ba:)60 1045 y Fp(Then)j(clearly)f Fo(\027)s Fp(\()p Fo(\014)s Fp(\))e(=)h Fo(\034)525 1053 y Fh(j)p Fm(\014)r Fh(j)p Fm(;a;b)622 1045 y Fp(\()p Fo(\013)p Fp(\),)i(hence)g Fo(\034)878 1053 y Fh(j)p Fm(\014)r Fh(j)p Fm(;a;b)974 1045 y Fp(\()p Fo(\013)p Fp(\))f Fn(2)f Fo(L)1138 1052 y Fm(E)r(;)p Fl(\006)1203 1045 y Fp(\()p Fo(\014)s Fp(\).)249 1135 y(\()p Fo(ii)p Fp(\))40 b Fo(\034)382 1143 y Fh(j)p Fm(\014)r Fh(j)p Fm(;c;a)479 1135 y Fp(\()p Fo(\013)p Fp(\))14 b(=)g Fo(caw)696 1142 y Fm(y)713 1147 y Fe(1)732 1135 y Fo(ac)g(a)f(caw)914 1142 y Fm(y)931 1147 y Fe(2)951 1135 y Fo(ac)g(b)h(caw)1128 1142 y Fm(y)1145 1147 y Fe(3)1164 1135 y Fo(ac)f(caw)1306 1142 y Fm(y)1323 1147 y Fe(2)1343 1135 y Fo(ac)g(caw)1485 1142 y Fm(y)1502 1147 y Fe(4)1521 1135 y Fo(ac:)60 1225 y Fp(No)o(w)18 b(de\014ne)f(the)g(morphism)f Fo(\027)21 b Fp(b)o(y)c Fo(\027)s Fp(\()p Fo(z)813 1232 y Fl(1)832 1225 y Fp(\))g(=)f Fo(caw)1004 1232 y Fm(y)1021 1237 y Fe(1)1040 1225 y Fo(ac)g(a)g(caw)1227 1232 y Fm(y)1244 1237 y Fe(2)1263 1225 y Fp(,)i Fo(\027)s Fp(\()p Fo(z)1364 1232 y Fl(2)1384 1225 y Fp(\))e(=)g Fo(c)i Fp(and)g Fo(\027)s Fp(\()p Fo(z)1677 1232 y Fl(3)1697 1225 y Fp(\))e(=)60 1285 y Fo(caw)142 1292 y Fm(y)159 1297 y Fe(3)178 1285 y Fo(ac)e(caw)321 1292 y Fm(y)338 1297 y Fe(2)357 1285 y Fo(ac)f(caw)499 1292 y Fm(y)516 1297 y Fe(4)536 1285 y Fo(a)p Fp(.)133 1377 y(Note)19 b(that)h(the)g(condition)f(of)h(ha)o (ving)g(t)o(w)o(o)f(extra)g(constan)o(ts)i(could)e(b)q(e)h(dropp)q(ed)g (in)60 1437 y(Lemma)15 b(3.4,)i(if)f(E-language)i(equiv)m(alence)d(w)o (as)j(preserv)o(ed)d(under)i(alphab)q(et)h(extensions,)60 1497 y(more)d(precisely)l(,)f(if)i(the)g(question)g(b)q(elo)o(w)g(w)o (ould)g(ha)o(v)o(e)f(an)i(answ)o(er)g(in)f(the)g(a\016rmativ)o(e.)60 1617 y Fg(Op)r(en)i(question)g(2:)j Fp(Do)q(es)c(the)f(equiv)m(alence:) 424 1707 y Fo(L)457 1714 y Fm(E)r(;)p Fl(\006)522 1707 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fo(L)690 1714 y Fm(E)r(;)p Fl(\006)755 1707 y Fp(\()p Fo(\014)s Fp(\))55 b Fn(,)g Fo(L)1017 1714 y Fm(E)r(;)p Fl(\006)1080 1705 y Fc(0)1094 1707 y Fp(\()p Fo(\013)p Fp(\))14 b(=)f Fo(L)1261 1714 y Fm(E)r(;)p Fl(\006)1324 1705 y Fc(0)1338 1707 y Fp(\()p Fo(\014)s Fp(\))60 1797 y(hold)j(for)h Fo(car)q(d)p Fp(\(\006\))d Fn(\025)g Fp(3,)i Fo(\013;)8 b(\014)16 b Fn(2)e Fp(\(\006)e Fn([)f Fo(V)g Fp(\))844 1779 y Fl(+)890 1797 y Fp(and)17 b(\006)1020 1779 y Fh(0)1045 1797 y Fp(=)d(\006)e Fn([)f(f)p Fo(a)p Fn(g)p Fp(,)k(where)h Fo(a)d Fn(62)h Fp(\006?)60 1918 y(Again,)j(the)h(condition)f Fo(car)q(d)p Fp(\(\006\))g Fn(\025)f Fp(3)i(is)f(essen)o(tial)g(\(cf.)g(Example)e(1.1\).)26 b(Note)17 b(moreo)o(v)o(er)60 1978 y(that)g(the)f Fi(only)i(if)e Fp(part)g(is)g(the)g(crucial)g(one)g(as)h(the)f Fi(if)g Fp(part)h(holds)f(trivially)l(.)133 2038 y(By)24 b(v)m(ariations)h(of)f (the)g(ab)q(o)o(v)o(e)h(pro)q(of)g(tec)o(hnique,)f(w)o(e)g(are)g(able)g (to)h(sho)o(w)g(that)g(E-)60 2098 y(language)16 b(inclusion)f(\(hence)f (E-language)i(equiv)m(alence\))e(is)h(also)g(decidable)f(in)h(other)g (sp)q(e-)60 2159 y(cial)g(cases.)60 2250 y Fg(Prop)r(osition)j(3.9)24 b Fp(Let)16 b(\006)h(b)q(e)f(an)h(alphab)q(et)g(and)f(let)g Fo(\013;)8 b(\014)16 b Fn(2)e Fp(\(\006)d Fn([)h Fo(V)f Fp(\))1411 2232 y Fl(+)1457 2250 y Fp(b)q(e)16 b(similar)e(pat-)60 2310 y(terns)k(of)g(the)f(form)g Fo(\013)g Fp(=)f Fo(\013)576 2317 y Fl(0)596 2310 y Fo(u)624 2317 y Fl(1)643 2310 y Fo(\013)674 2317 y Fl(1)694 2310 y Fo(u)722 2317 y Fl(2)750 2310 y Fo(:)8 b(:)g(:)f(\013)846 2317 y Fm(m)p Fh(\000)p Fl(1)925 2310 y Fo(u)953 2317 y Fm(m)986 2310 y Fo(\013)1017 2317 y Fm(m)1068 2310 y Fp(and)18 b Fo(\014)h Fp(=)d Fo(\014)1293 2317 y Fl(0)1313 2310 y Fo(u)1341 2317 y Fl(1)1360 2310 y Fo(\014)1388 2317 y Fl(1)1407 2310 y Fo(u)1435 2317 y Fl(2)1463 2310 y Fo(:)8 b(:)g(:)g(\014)1557 2317 y Fm(m)p Fh(\000)p Fl(1)1635 2310 y Fo(u)1663 2317 y Fm(m)1696 2310 y Fo(\014)1724 2317 y Fm(m)1756 2310 y Fp(.)60 2370 y(If)16 b Fo(car)q(d)p Fp(\(\006\))e Fn(\025)g Fo(m)c Fp(+)h(2,)17 b Fo(m)c Fn(\025)h Fp(0,)i(then)g(it)g(is)g (decidable)f(whether)h Fo(L)1307 2377 y Fm(E)r(;)p Fl(\006)1372 2370 y Fp(\()p Fo(\013)p Fp(\))e Fn(\022)g Fo(L)1541 2377 y Fm(E)r(;)p Fl(\006)1606 2370 y Fp(\()p Fo(\014)s Fp(\).)60 2491 y Fg(Pro)r(of:)29 b Fp(Let)20 b Fo(l)347 2498 y Fl(1)387 2491 y Fp(b)q(e)g(a)h(constan)o(t)f(in)g Fo(u)791 2498 y Fl(1)811 2491 y Fp(,)h Fo(l)861 2498 y Fl(2)900 2491 y Fp(b)q(e)f(a)h(constan)o(t)g(in)e Fo(u)1304 2498 y Fl(2)1344 2491 y Fp(\()p Fo(l)1378 2498 y Fl(1)1418 2491 y Fp(=)h Fo(l)1491 2498 y Fl(2)1531 2491 y Fp(is)g(p)q(ossible\)) 60 2551 y(etc.)42 b(Since)23 b Fo(car)q(d)p Fp(\()p Fn(f)p Fo(l)468 2558 y Fl(1)487 2551 y Fo(;)8 b(l)524 2558 y Fl(2)543 2551 y Fo(;)g(:)g(:)g(:)f(;)h(l)667 2558 y Fm(m)700 2551 y Fn(g)p Fp(\))26 b Fn(\024)g Fo(m)p Fp(,)e(there)f(are)g(t)o(w)o (o)h(distinct)e(constan)o(ts)i Fo(a)f Fp(and)60 2611 y Fo(b)f Fp(in)g(the)g(set)g(\006)15 b Fn(n)g(f)p Fo(l)468 2618 y Fl(1)487 2611 y Fo(;)8 b(l)524 2618 y Fl(2)543 2611 y Fo(;)g(:)g(:)g(:)g(;)g(l)668 2618 y Fm(m)700 2611 y Fn(g)p Fp(.)39 b(It)22 b(follo)o(ws)g(as)h(in)e(the)h(pro)q(of)i(of)e (Lemma)e(3.4)j(that)60 2671 y Fo(\034)81 2679 y Fh(j)p Fm(\014)r Fh(j)p Fm(;a;b)178 2671 y Fp(\()p Fo(\013)p Fp(\))15 b Fn(2)f Fo(L)342 2678 y Fm(E)r(;)p Fl(\006)408 2671 y Fp(\()p Fo(\014)s Fp(\))h(if)i(and)g(only)f(if)g(there)g(exists) g(a)h(morphism)e Fo(h)f Fp(:)g Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))f Fn(!)i Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1751 2653 y Fh(\003)60 2731 y Fp(suc)o(h)h(that)h Fo(h)p Fp(\()p Fo(\014)s Fp(\))c(=)h Fo(\013)p Fp(.)21 b(The)16 b(rest)h(of)f(the)g (pro)q(of)h(is)g Fi(verb)n(atim)f Fp(the)g(same)f(as)i(ab)q(o)o(v)o(e.) k Fd(2)903 2856 y Fp(8)p eop %%Page: 9 9 9 8 bop 60 203 a Fg(Corollary)18 b(3.10)24 b Fp(Giv)o(en)c(t)o(w)o(o)i (patterns)g Fo(\013)h Fp(=)g Fo(\013)1021 210 y Fl(0)1040 203 y Fo(u)1068 210 y Fl(1)1088 203 y Fo(\013)1119 210 y Fl(1)1139 203 y Fo(u)1167 210 y Fl(2)1194 203 y Fo(:)8 b(:)g(:)g(\013)1291 210 y Fm(m)p Fh(\000)p Fl(1)1369 203 y Fo(u)1397 210 y Fm(m)1430 203 y Fo(\013)1461 210 y Fm(m)1516 203 y Fp(and)22 b Fo(\014)s Fp(,)g(it)f(is)60 264 y(decidable)15 b(whether)h Fo(L)494 271 y Fm(E)r(;)p Fl(\006)559 264 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fo(L)727 271 y Fm(E)r(;)p Fl(\006)792 264 y Fp(\()p Fo(\014)s Fp(\))i(pro)o(vided)f(that)i Fo(car)q(d)p Fp(\(\006\))d Fn(\025)g Fo(m)d Fp(+)g(2,)16 b Fo(m)d Fn(\025)h Fp(0.)60 384 y Fg(Pro)r(of:)k Fp(If)11 b Fo(\013)g Fp(and)h Fo(\014)h Fp(are)e(similar,)e(then)i Fo(L)831 391 y Fm(E)r(;)p Fl(\006)897 384 y Fp(\()p Fo(\013)p Fp(\))j(=)f Fo(L)1064 391 y Fm(E)r(;)p Fl(\006)1130 384 y Fp(\()p Fo(\014)s Fp(\))d(is)h(decidable)e(b)o(y)i(Prop)q(osition)60 444 y(3.9.)45 b(If)23 b Fo(m)k Fp(=)g(0)d(and)h Fo(\013)f Fp(and)h Fo(\014)h Fp(are)e(not)h(similar,)e(then)h Fo(\013)g Fp(is)g(terminal)e(free)h(and)i Fo(\014)60 504 y Fp(con)o(tains)f (terminals.)43 b(Th)o(us)24 b Fo(L)680 511 y Fm(E)r(;)p Fl(\006)745 504 y Fp(\()p Fo(\013)p Fp(\))j Fn(6)p Fp(=)g Fo(L)939 511 y Fm(E)r(;)p Fl(\006)1005 504 y Fp(\()p Fo(\014)s Fp(\))c(b)q(ecause)h(the)g(empt)o(y)d(w)o(ord)k(is)f(an)60 565 y(elemen)o(t)16 b(of)j Fo(L)334 572 y Fm(E)r(;)p Fl(\006)399 565 y Fp(\()p Fo(\013)p Fp(\))g(but)g(not)g(of)g Fo(L)759 572 y Fm(E)r(;)p Fl(\006)824 565 y Fp(\()p Fo(\014)s Fp(\).)28 b(If)18 b Fo(m)g(>)g Fp(0)h(and)g Fo(\013)g Fp(and)h Fo(\014)h Fp(are)d(not)i(similar,)60 625 y(then)c Fo(L)204 632 y Fm(E)r(;)p Fl(\006)269 625 y Fp(\()p Fo(\013)p Fp(\))e Fn(6)p Fp(=)g Fo(L)437 632 y Fm(E)r(;)p Fl(\006)503 625 y Fp(\()p Fo(\014)s Fp(\))h(b)o(y)h(Theorem)f(3.1.)21 b Fd(2)60 739 y Fg(Prop)r(osition)d(3.11)24 b Fp(Let)14 b(\006)h(b)q(e)f(an)h(alphab)q(et)g(and)g(let)e Fo(\013;)8 b(\014)16 b Fn(2)e Fp(\(\006)7 b Fn([)g Fo(V)12 b Fp(\))1417 721 y Fl(+)1461 739 y Fp(b)q(e)i(similar)e(pat-)60 799 y(terns.)20 b(If)12 b(there)g(are)h Fo(a;)8 b(b)13 b Fn(2)h Fp(\006,)f Fo(a)g Fn(6)p Fp(=)h Fo(b)p Fp(,)f(suc)o(h)f(that)h (ev)o(ery)e(terminal)g(segmen)o(t)g(of)i Fo(\013)g Fp(\(hence)f(of)60 859 y Fo(\014)s Fp(\))i(con)o(tains)g(at)h(least)f(one)g(letter)f (di\013eren)o(t)h(from)f Fo(a)h Fp(and)h Fo(b)p Fp(,)f(then)g(it)g(is)g (decidable)f(whether)60 919 y Fo(L)93 926 y Fm(E)r(;)p Fl(\006)158 919 y Fp(\()p Fo(\013)p Fp(\))h Fn(\022)g Fo(L)327 926 y Fm(E)r(;)p Fl(\006)392 919 y Fp(\()p Fo(\014)s Fp(\).)60 1040 y Fg(Pro)r(of:)21 b Fp(The)16 b(pro)q(of)i(of)e(Theorem) f(3.5)i(applies,)e Fi(mutatis)j(mutandis)p Fp(.)j Fd(2)60 1154 y Fg(Corollary)d(3.12)24 b Fp(Let)14 b Fo(\013)g Fp(=)g Fo(\013)646 1161 y Fl(0)666 1154 y Fo(u)694 1161 y Fl(1)713 1154 y Fo(\013)744 1161 y Fl(1)764 1154 y Fo(u)792 1161 y Fl(2)820 1154 y Fo(:)8 b(:)g(:)g(\013)917 1161 y Fm(m)p Fh(\000)p Fl(1)995 1154 y Fo(u)1023 1161 y Fm(m)1056 1154 y Fo(\013)1087 1161 y Fm(m)1135 1154 y Fp(and)14 b Fo(\014)j Fp(b)q(e)d(in)g(\(\006)7 b Fn([)g Fo(V)12 b Fp(\))1551 1136 y Fl(+)1580 1154 y Fp(.)21 b(If)14 b(there)60 1214 y(are)19 b Fo(a;)8 b(b)17 b Fn(2)h Fp(\006,)h Fo(a)f Fn(6)p Fp(=)g Fo(b)p Fp(,)h(suc)o(h)f(that)i(ev)o (ery)d(terminal)f(segmen)o(t)h Fo(u)1271 1221 y Fm(j)1308 1214 y Fp(of)i Fo(\013)g Fp(con)o(tains)g(at)g(least)60 1274 y(one)14 b(letter)f(di\013eren)o(t)g(from)g Fo(a)g Fp(and)i Fo(b)p Fp(,)e(then)h(it)g(is)f(decidable)g(whether)h Fo(L)1390 1281 y Fm(E)r(;)p Fl(\006)1455 1274 y Fp(\()p Fo(\013)p Fp(\))g(=)g Fo(L)1623 1281 y Fm(E)r(;)p Fl(\006)1688 1274 y Fp(\()p Fo(\014)s Fp(\).)60 1395 y Fg(Pro)r(of:)k Fp(If)11 b Fo(\013)g Fp(and)h Fo(\014)h Fp(are)e(similar,)e(then)i Fo(L)831 1402 y Fm(E)r(;)p Fl(\006)897 1395 y Fp(\()p Fo(\013)p Fp(\))j(=)f Fo(L)1064 1402 y Fm(E)r(;)p Fl(\006)1130 1395 y Fp(\()p Fo(\014)s Fp(\))d(is)h(decidable)e(b)o(y)i(Prop)q (osition)60 1455 y(3.11.)22 b(If)15 b Fo(\013)h Fp(and)g Fo(\014)i Fp(are)d(not)h(similar,)d(then)j Fo(car)q(d)p Fp(\(\006\))e Fn(\025)g Fp(3)h(and)i(hence)d Fo(L)1403 1462 y Fm(E)r(;)p Fl(\006)1469 1455 y Fp(\()p Fo(\013)p Fp(\))g Fn(6)p Fp(=)f Fo(L)1636 1462 y Fm(E)r(;)p Fl(\006)1702 1455 y Fp(\()p Fo(\014)s Fp(\))60 1515 y(according)j(to)h(Theorem)e (3.1.)22 b Fd(2)60 1682 y Fq(4)81 b(V)-7 b(ariable)26 b(elimination)f(and)i(normal)g(forms)60 1791 y Fp(Solving)15 b(the)g(equiv)m(alence)e(problem)h(for)h(NE-pattern)g(languages)h(is)f (easy:)21 b(Tw)o(o)16 b(patterns)60 1851 y(generate)j(the)g(same)g (language)h(if)f(and)h(only)f(if)g(they)f(are)i(iden)o(tical)d(up)j(to) g(renaming)e(of)60 1911 y(v)m(ariables;)27 b(see)c([1].)44 b(F)l(or)24 b(E-patterns,)i(ho)o(w)o(ev)o(er,)e(this)f(is)h(not)g(true) g(at)g(all.)43 b(Jiang)24 b Fi(et)60 1972 y(al.)44 b Fp([5])23 b(write)h(that:)36 b(\\Tw)o(o)25 b(v)o(ery)d(di\013eren)o(t)h (lo)q(oking)h(E-patterns)h(ma)o(y)d(generate)h(the)60 2032 y(same)c(language.)33 b(F)l(or)20 b(instance,)g(if)f(a)h (terminal-free)d(pattern)j Fo(\013)g Fp(con)o(tains)g(exactly)f(one)60 2092 y(o)q(ccurrence)h(of)h(some)f(v)m(ariable,)h(then)g Fo(\013)g Fp(is)f(equiv)m(alen)o(t)g(to)h(the)f(pattern)h Fo(x)p Fp(".)35 b(The)21 b(fact)60 2152 y(that)14 b(an)g(E-pattern)h (ma)o(y)d(con)o(tain)h(man)o(y)g(sup)q(er\015uous)i(v)m(ariables)e (seems)g(to)h(b)q(e)g(the)f(ma)s(jor)60 2212 y(problem)f(in)h(deciding) f(whether)h(t)o(w)o(o)g(E-patterns)h(generate)f(the)g(same)g(language.) 21 b(W)l(e)13 b(call)60 2273 y(a)f(v)m(ariable)f Fo(x)h Fp(in)f(an)h(E-pattern)g Fo(\013)j Fn(2)f Fp(\(\006)r Fn([)r Fo(V)d Fp(\))889 2255 y Fl(+)930 2273 y Fi(sup)n(er\015uous)p Fp(,)g(if)g(the)h(E-pattern)g Fo(\013)1562 2255 y Fh(0)1586 2273 y Fp(obtained)60 2333 y(from)18 b Fo(\013)h Fp(b)o(y)f(deleting)f (all)h(o)q(ccurrences)h(of)f Fo(x)h Fp(still)e(generates)i Fo(L)1270 2340 y Fm(E)r(;)p Fl(\006)1335 2333 y Fp(\()p Fo(\013)p Fp(\).)29 b(W)l(e)18 b(next)g(tac)o(kle)60 2393 y(the)h(equiv)m(alence)f(problem)g(of)i(t)o(w)o(o)g(E-patterns)g Fo(\013)g Fp(and)g Fo(\014)i Fp(b)o(y)e(eliminating)d(sup)q(er\015uous) 60 2453 y(v)m(ariables)j(in)f Fo(\013)h Fp(and)h Fo(\014)s Fp(.)31 b(The)19 b(\014rst)i(di\016cult)o(y)c(one)j(encoun)o(ters)g(in) f(this)h(approac)o(h)g(is)g(of)60 2513 y(course)e(the)g(decidabilit)o (y)d(of)k(whether)e(a)i(v)m(ariable)e(is)h(sup)q(er\015uous)i(or)e (not.)27 b(Since)17 b(w)o(e)h(do)60 2574 y(not)j(kno)o(w)f(ho)o(w)h(to) f(decide)f(this)h(in)g(general,)g(w)o(e)g(b)o(y-pass)h(the)f(problem)f (b)o(y)g(de\014ning)i(a)60 2634 y(reduction)16 b(relation)h Fn(!)507 2641 y Fm(r)540 2634 y Fn(\032)e Fp(\(\006)c Fn([)h Fo(V)f Fp(\))762 2616 y Fl(+)803 2634 y Fn(\002)g Fp(\(\006)h Fn([)g Fo(V)f Fp(\))1022 2616 y Fl(+)1068 2634 y Fp(suc)o(h)17 b(that)g(if)f(pattern)h Fo(\013)g Fp(reduces)g(to)60 2694 y(pattern)f Fo(\013)264 2676 y Fh(0)276 2694 y Fp(,)g(then)g(the)g(erased)h(v)m(ariables)f(are)g (sup)q(er\015uous.)903 2856 y(9)p eop %%Page: 10 10 10 9 bop 60 203 a Fg(De\014nition)18 b(4.1)24 b Fp(The)14 b(reduction)g(relation)g Fn(!)958 210 y Fm(r)991 203 y Fn(\032)f Fp(\(\006)7 b Fn([)g Fo(V)13 b Fp(\))1204 185 y Fl(+)1241 203 y Fn(\002)7 b Fp(\(\006)g Fn([)g Fo(V)13 b Fp(\))1448 185 y Fl(+)1491 203 y Fp(is)i(de\014ned)f(b)o(y:) 60 264 y Fo(\013)g Fn(!)155 271 y Fm(r)188 264 y Fo(\013)219 246 y Fh(0)247 264 y Fp(if)i(and)h(only)f(if)120 364 y(1.)24 b(there)18 b(is)g(a)i(non-empt)o(y)d(prop)q(er)i(subset)g Fo(V)988 371 y Fm(d)1027 364 y Fp(of)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))g(suc)o(h)f(that)h(erasing)g(all)f(o)q(c-)182 424 y(currences)e(of)h(v)m(ariables)g(of)g Fo(V)740 431 y Fm(d)777 424 y Fp(in)g Fo(\013)g Fp(yields)f Fo(\013)1052 406 y Fh(0)1080 424 y Fp(\(that)i(is)e(to)h(sa)o(y)l(,)g Fo(\013)1441 406 y Fh(0)1468 424 y Fp(=)d Fo(d)p Fp(\()p Fo(\013)p Fp(\),)j(where)182 485 y Fo(d)d Fp(=)g Fn(f)p Fo(x)f Fn(7!)h Fo(\025)j Fn(j)f Fo(x)d Fn(2)h Fo(V)594 492 y Fm(d)615 485 y Fn(g)p Fp(\),)i(and)120 586 y(2.)24 b(there)c(is)h(a)g(linear)f(morphism)e Fo(\033)23 b Fp(:)e Fo(v)r(ar)q Fp(\()p Fo(\013)993 568 y Fh(0)1004 586 y Fp(\))g Fn(!)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1260 568 y Fl(+)1309 586 y Fp(suc)o(h)f(that)g Fo(\033)r Fp(\()p Fo(\013)1614 568 y Fh(0)1625 586 y Fp(\))h(=)f Fo(\013)p Fp(.)182 646 y(Here)15 b 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(\014nitely)e(man)o(y)h(linear)g(morphisms)e Fo(\033)22 b Fp(whic)o(h)60 1784 y(satisfy)216 1751 y Fb(P)260 1764 y Fm(n)260 1796 y(i)p Fl(=1)327 1784 y Fn(j)p Fo(\015)366 1791 y Fm(i)380 1784 y Fo(\015)408 1766 y Fh(0)405 1796 y Fm(i)420 1784 y Fn(j)e Fp(=)g Fn(j)p Fo(\013)p Fn(j)14 b(\000)f(j)p Fo(\013)682 1766 y Fh(0)694 1784 y Fn(j)19 b Fp(\(since)870 1751 y Fb(P)913 1764 y Fm(n)913 1796 y(i)p Fl(=1)981 1784 y Fn(j)p Fo(\015)1020 1791 y Fm(i)1034 1784 y Fo(\015)1062 1766 y Fh(0)1059 1796 y Fm(i)1074 1784 y Fn(j)h(6)p Fp(=)g Fn(j)p Fo(\013)p Fn(j)13 b(\000)g(j)p Fo(\013)1335 1766 y Fh(0)1347 1784 y Fn(j)20 b Fp(implies)d Fo(\033)r Fp(\()p Fo(\013)1630 1766 y Fh(0)1642 1784 y Fp(\))j Fn(6)p Fp(=)g Fo(\013)60 1844 y Fp(already\).)g(Hence)13 b(it)h(is)g(also)h(decidable)e(whether)h Fo(\013)h Fp(is)f(reducible)f (b)q(ecause)h(there)g(are)g(only)60 1904 y(\014nitely)h(man)o(y)g (non-empt)o(y)g(prop)q(er)h(subsets)h Fo(V)953 1911 y Fm(d)990 1904 y Fp(of)g Fo(v)r(ar)q Fp(\()p Fo(\013)p 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Fp(and)i Fo(\013)851 2653 y Fh(0)880 2671 y Fn(!)930 2678 y Fm(r)965 2671 y Fo(\013)996 2653 y Fh(00)1018 2671 y Fp(,)f(i.e.,)e(there)h(are)h(deletion)f(morphisms) 60 2731 y Fo(d)85 2713 y Fh(0)97 2731 y Fo(;)8 b(d)144 2713 y Fh(00)183 2731 y Fp(and)18 b(linear)f(morphisms)e Fo(\033)694 2713 y Fh(0)706 2731 y Fo(;)8 b(\033)758 2713 y Fh(00)796 2731 y Fp(suc)o(h)17 b(that)h Fo(d)1039 2713 y Fh(0)1051 2731 y Fp(\()p Fo(\013)p Fp(\))e(=)g Fo(\013)1221 2713 y Fh(0)1233 2731 y Fp(,)i Fo(\033)1295 2713 y Fh(0)1306 2731 y Fp(\()p Fo(\013)1356 2713 y Fh(0)1368 2731 y Fp(\))e(=)g Fo(\013)p Fp(,)i Fo(d)1545 2713 y Fh(00)1566 2731 y Fp(\()p Fo(\013)1616 2713 y Fh(0)1628 2731 y Fp(\))e(=)g Fo(\013)1748 2713 y Fh(00)891 2856 y Fp(10)p eop %%Page: 11 11 11 10 bop 60 203 a Fp(and)22 b Fo(\033)190 185 y Fh(0)201 203 y Fp(\()p Fo(\013)251 185 y Fh(00)272 203 y Fp(\))h(=)f Fo(\013)405 185 y Fh(0)417 203 y Fp(.)36 b(De\014ne)21 b Fo(d)h Fp(=)g Fo(d)755 185 y Fh(00)791 203 y Fn(\016)14 b Fo(d)855 185 y Fh(0)889 203 y Fp(and)21 b Fo(\033)j Fp(=)e Fo(\033)1130 185 y Fh(0)1156 203 y Fn(\016)14 b Fo(\033)1225 185 y Fh(00)1246 203 y Fp(.)36 b(Ob)o(viously)l(,)21 b Fo(d)p Fp(\()p Fo(\013)p Fp(\))h(=)g Fo(\013)1748 185 y Fh(00)60 264 y Fp(and)f Fo(\033)189 246 y Fh(00)209 264 y Fp(\()p Fo(\013)259 246 y Fh(00)281 264 y Fp(\))f(=)g Fo(\013)p Fp(.)33 b(It)20 b(is)g(not)g(to)q(o)h(di\016cult)e(to)h(pro)o (v)o(e)g(that)g Fo(\033)i Fp(is)d(linear,)h(hence)g(transi-)60 324 y(tivit)o(y)e(of)i Fn(!)304 331 y Fm(r)342 324 y Fp(follo)o(ws.)31 b(It)19 b(is)g(ob)o(vious)h(that)g Fo(\013)f Fn(!)1028 331 y Fm(r)1067 324 y Fo(\013)1098 306 y Fh(0)1129 324 y Fp(implies)e Fn(j)p Fo(\013)p Fn(j)i Fo(>)h Fn(j)p Fo(\013)1479 306 y Fh(0)1491 324 y Fn(j)p Fp(,)f(th)o(us)h Fn(!)1699 331 y Fm(r)1737 324 y Fp(is)60 384 y(length-decreasing.)h Fd(2)60 498 y Fg(De\014nition)d(4.5)24 b Fp(A)14 b(pattern)i Fo(\013)g Fp(is)f(said)g(to)h(b)q(e)g(in)e Fi(normal)j(form)e Fp(\(w.r.t.)20 b Fn(!)1503 505 y Fm(r)1522 498 y Fp(\),)15 b(if)f(there)h(is)60 558 y(no)i(pattern)f Fo(\013)332 540 y Fh(0)360 558 y Fp(suc)o(h)g(that)h Fo(\013)d Fn(!)671 565 y Fm(r)704 558 y Fo(\013)735 540 y Fh(0)747 558 y Fp(.)133 672 y(Since)g Fn(!)309 679 y Fm(r)343 672 y Fp(is)g(length-decreasing,)h(the)f(successiv)o(e)f (reduction)i(of)g(a)g(pattern)g(m)o(ust)e(end)60 732 y(in)20 b(a)g(normal)f(form)g(\(an)i(irreducible)d(pattern\).)33 b(Clearly)l(,)19 b(a)i(pattern)f Fo(\013)g Fp(ma)o(y)f(ha)o(v)o(e)g(t)o (w)o(o)60 793 y(distinct)g(normal)f(forms.)31 b(Just)20 b(consider)f Fo(\013)h Fp(=)f Fo(xy)r Fp(:)27 b Fo(\013)20 b Fn(!)1179 800 y Fm(r)1218 793 y Fo(x)f Fp(and)h Fo(\013)g Fn(!)1464 800 y Fm(r)1502 793 y Fo(y)r Fp(.)31 b(Our)20 b(next)60 853 y(goal)g(is)f(to)g(sho)o(w)h(that)g(the)f(normal)f(forms) g(of)i(a)f(pattern)h(are)f(iden)o(tical)e(up)j(to)f(v)m(ariable)60 913 y(renaming.)26 b(The)19 b(follo)o(wing)f(tec)o(hnical)e(lemma)g (essen)o(tially)g(enables)i(us)h(to)g(sho)o(w)g(that)g(if)60 973 y Fo(\013)14 b Fn(2)g Fp(\(\006)s Fn([)s Fo(V)d Fp(\))303 955 y Fl(+)345 973 y Fp(is)h(a)g(pattern)g(in)g(normal)f(form)g(and)i Fo(h)h Fp(:)f Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))h Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1446 955 y Fh(\003)1477 973 y Fp(is)e(a)g(morphism)60 1033 y(suc)o(h)g(that)h Fo(\013)g Fp(is)f(a)h(\014xp)q(oin)o(t)g(of)f Fo(h)h Fp(\(i.e.,)e Fo(h)p Fp(\()p Fo(\013)p Fp(\))j(=)g Fo(\013)p Fp(\),)f(then)f Fo(h)h Fp(m)o(ust)e(b)q(e)i(the)f(iden)o(tit)o(y)e (function)60 1093 y(on)18 b Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\).)25 b(The)18 b(fact)f(that)h(the)f(normal)g(form)g(of)h(a)g (pattern)f(is)h(unique)f(up)g(to)h(v)m(ariable)60 1154 y(renaming)d(follo)o(ws)h(naturally)g(from)f(this)h(prop)q(ert)o(y)l(.) 60 1268 y Fg(Lemma)g(4.6)24 b Fp(Let)17 b Fo(\013)e Fn(2)h Fp(\(\006)11 b Fn([)h Fo(V)f Fp(\))706 1250 y Fl(+)753 1268 y Fp(b)q(e)17 b(a)g(pattern)h(and)f(let)g Fo(h)e Fp(:)f Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))h Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1642 1250 y Fh(\003)1678 1268 y Fp(b)q(e)j(a)60 1328 y(morphism)c(suc)o(h)i(that)h Fo(h)p Fp(\()p Fo(\013)p Fp(\))d(=)g Fo(\013)p Fp(.)21 b(Then)16 b(the)g(follo)o(wing)g(statemen)o(ts)f(hold.)120 1429 y(1.)24 b Fo(\013)18 b Fp(has)g(a)f(partition)h Fo(\013)e Fp(=)g Fo(\014)694 1436 y Fl(1)713 1429 y Fo(\014)741 1436 y Fl(2)768 1429 y Fo(:)8 b(:)g(:)g(\014)862 1436 y Fm(k)900 1429 y Fp(in)o(to)17 b(minimal)d(\014xp)q(oin)o(ts)j(of)h Fo(h)p Fp(,)f(i.e.,)f(for)i(eac)o(h)182 1490 y Fo(i)d Fn(2)g(f)p Fp(1)p Fo(;)8 b(:)g(:)g(:)g(;)g(k)r Fn(g)p Fp(,)17 b Fo(\014)532 1497 y Fm(i)561 1490 y Fn(2)e Fp(\(\006)d Fn([)g Fo(V)f Fp(\))778 1472 y Fl(+)807 1490 y Fp(,)17 b(the)g(equalit)o(y)f Fo(h)p Fp(\()p Fo(\014)1185 1497 y Fm(i)1198 1490 y Fp(\))f(=)h Fo(\014)1314 1497 y Fm(i)1344 1490 y Fp(holds,)i(and)f(for)h(ev)o(ery)182 1550 y(prop)q(er)f(sub)o(w) o(ord)f Fo(\015)k Fp(of)c Fo(\014)658 1557 y Fm(i)688 1550 y Fp(w)o(e)g(ha)o(v)o(e)f Fo(h)p Fp(\()p Fo(\015)s Fp(\))f Fn(6)p Fp(=)g Fo(\015)s Fp(.)120 1651 y(2.)24 b(The)16 b(partition)g(is)g(unique.)120 1753 y(3.)24 b(F)l(or)18 b(ev)o(ery)e Fo(i)g Fn(2)h(f)p Fp(1)p Fo(;)8 b(:)g(:)g(:)g(;)g(k)r Fn(g)p Fp(,)17 b(either)g Fo(\014)894 1760 y Fm(i)924 1753 y Fn(2)g Fp(\006)h(or)g Fo(\014)1116 1760 y Fm(i)1146 1753 y Fn(2)e Fo(V)1235 1735 y Fl(+)1264 1753 y Fp(.)26 b(Moreo)o(v)o(er,)17 b(if)g Fo(\014)1608 1760 y Fm(i)1638 1753 y Fn(2)g Fo(V)1727 1735 y Fl(+)1756 1753 y Fp(,)182 1813 y(then)f(it)g(con)o(tains)g(exactly)f(one)i(v)m (ariable)e Fo(x)h Fp(suc)o(h)g(that)210 1919 y(\()p Fn(\003)p Fp(\))101 b Fo(h)p Fp(\()p Fo(x)p Fp(\))13 b(=)h Fo(\015)558 1926 y Fl(1)578 1919 y Fo(x\015)631 1926 y Fl(2)651 1919 y Fo(;)57 b Fp(where)16 b Fo(\015)888 1926 y Fl(1)908 1919 y Fo(;)8 b(\015)955 1926 y Fl(2)989 1919 y Fn(2)14 b Fp(\()p Fo(v)r(ar)q Fp(\()p Fo(\014)1177 1926 y Fm(i)1190 1919 y Fp(\))d Fn(n)g(f)p Fo(x)p Fn(g)p Fp(\))1353 1901 y Fh(\003)1388 1919 y Fp(and)706 1979 y Fo(\014)734 1986 y Fm(i)761 1979 y Fp(=)j Fo(\016)835 1986 y Fl(1)854 1979 y Fo(\015)879 1986 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b(con)o(tain)i(at)g(least)g(one)g (v)m(ariable)f(satisfying)h(\()p Fn(\003)p Fp(\).)21 b(It)13 b(remains)g(to)h(sho)o(w)g(that)h(there)60 1889 y(is)g(no)g(other)g(v)m(ariable)g(in)f Fo(v)r(ar)q Fp(\()p Fo(\014)659 1896 y Fm(i)672 1889 y Fp(\))h(whic)o(h)f(is)h(distinct)f (from)f Fo(x)1208 1896 y Fm(j)1241 1889 y Fp(and)j(whic)o(h)e (ful\014lls)g(\()p Fn(\003)p Fp(\).)20 b(T)l(o)60 1949 y(this)15 b(end,)h(w)o(e)f(\014rst)g(observ)o(e)g(that)h(a)g(v)m (ariable)g Fo(x)959 1956 y Fm(l)971 1949 y Fp(,)g Fo(l)e Fn(2)g(f)p Fp(1)p Fo(;)8 b(:)g(:)g(:)g(;)g(n)p Fn(g)p Fp(,)15 b(o)q(ccurs)h(in)f Fo(h)p Fp(\()p Fo(x)1600 1956 y Fm(l)1613 1949 y Fp(\))g(if)g(and)60 2009 y(only)h(if)318 2069 y Fn(j)p Fo(h)p Fp(\()p Fo(x)407 2076 y Fl(1)427 2069 y Fo(x)455 2076 y Fl(2)482 2069 y Fo(:)8 b(:)g(:)g(x)576 2076 y Fm(l)p Fh(\000)p Fl(1)634 2069 y Fp(\))p Fn(j)13 b Fo(<)h(l)e Fn(\000)f Fp(1)58 b(and)g Fn(j)p Fo(h)p Fp(\()p Fo(x)1116 2076 y Fl(1)1135 2069 y Fo(x)1163 2076 y Fl(2)1191 2069 y Fo(:)8 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2638 y Fo(x)398 2645 y Fl(2)426 2638 y Fo(:)8 b(:)g(:)f(x)519 2645 y Fm(j)546 2638 y Fo(:)h(:)g(:)f(x)639 2645 y Fm(l)652 2638 y Fp(\))p Fn(j)28 b Fo(>)h(l)q Fp(.)47 b(Clearly)l(,)26 b Fn(j)p Fo(h)p Fp(\()p Fo(x)1136 2645 y Fl(1)1155 2638 y Fo(x)1183 2645 y Fl(2)1211 2638 y Fo(:)8 b(:)g(:)f(x)1304 2645 y Fm(j)1331 2638 y Fo(:)h(:)g(:)f(x)1424 2645 y Fm(l)1437 2638 y Fp(\))p Fn(j)29 b(6)p Fp(=)f Fo(l)e Fp(b)q(ecause)60 2698 y Fo(\014)88 2705 y Fm(i)127 2698 y Fp(is)f(a)g(minimal)d(\014xp)q(oin)o(t.)48 b(So)26 b(supp)q(ose)g(for)g(an)f(indirect)f(pro)q(of)i(that)g(there)f(is)g(an) 891 2856 y(12)p eop %%Page: 13 13 13 12 bop 60 203 a Fo(l)14 b Fn(2)h(f)p Fo(j)5 b Fp(+)r(1)p Fo(;)j(:)g(:)g(:)f(;)h(n)r Fn(\000)r Fp(1)p Fn(g)j Fp(suc)o(h)h(that)g Fn(j)p Fo(h)p Fp(\()p Fo(x)788 210 y Fl(1)807 203 y Fo(x)835 210 y Fl(2)863 203 y Fo(:)c(:)g(:)f(x)956 210 y Fm(j)983 203 y Fo(:)h(:)g(:)f(x)1076 210 y Fm(l)1089 203 y Fp(\))p Fn(j)14 b Fo(<)f(l)q Fp(.)20 b(Since)10 b Fn(j)p Fo(h)p Fp(\()p Fo(x)1448 210 y Fl(1)1468 203 y Fo(x)1496 210 y Fl(2)1524 203 y Fo(:)e(:)g(:)f(x)1617 210 y Fm(j)1635 203 y Fp(\))p Fn(j)14 b Fo(>)f(j)s Fp(,)60 264 y(there)25 b(m)o(ust)f(b)q(e)i(an)g(index)f Fo(p)p Fp(,)j Fo(j)33 b(<)d(p)h Fn(\024)e Fo(l)e Fp(with)e Fn(j)p Fo(h)p Fp(\()p Fo(x)1179 271 y Fl(1)1199 264 y Fo(x)1227 271 y Fl(2)1254 264 y Fo(:)8 b(:)g(:)g(x)1348 271 y Fm(p)p Fh(\000)p Fl(1)1412 264 y Fp(\))p Fn(j)30 b Fo(>)g(p)18 b Fn(\000)f Fp(1)26 b(and)60 324 y Fn(j)p Fo(h)p Fp(\()p Fo(x)149 331 y Fl(1)168 324 y Fo(x)196 331 y Fl(2)224 324 y Fo(:)8 b(:)g(:)g(x)318 331 y Fm(p)337 324 y Fp(\))p Fn(j)25 b Fo(<)g(p)p Fp(.)41 b(This)23 b(is)g(eviden)o(tly)e(imp)q(ossible,)h (so)h(the)g(claim)e(is)h(pro)o(v)o(ed.)40 b(It)60 384 y(further)20 b(follo)o(ws)h(from)e Fo(h)p Fp(\()p Fo(\014)587 391 y Fm(i)601 384 y Fp(\))i(=)h Fo(\014)729 391 y Fm(i)763 384 y Fp(that)f Fo(h)p Fp(\()p Fo(x)948 391 y Fm(n)971 384 y Fp(\))g(=)h Fo(\025)p Fp(.)35 b(So)21 b(there)f(cannot)h(b)q(e)g (a)g(second)60 444 y(v)m(ariable)16 b(in)g Fo(v)r(ar)q Fp(\()p Fo(\014)422 451 y Fm(i)435 444 y Fp(\))g(whic)o(h)f(is)i (di\013eren)o(t)e(from)g Fo(x)994 451 y Fm(j)1028 444 y Fp(and)i(satis\014es)g(\()p Fn(\003)p Fp(\).)60 504 y(\(4\))24 b(W)l(e)f(sho)o(w)h(that)g(if)f Fo(x)g Fp(is)g(a)h(v)m (ariable)f(suc)o(h)g(that)h Fo(h)p Fp(\()p Fo(x)p Fp(\))f(con)o(tains)h Fo(x)p Fp(,)g(then)f(there)g(is)60 565 y(some)18 b Fo(q)j Fn(2)f Fa(I)-7 b(N)q Fp(,)19 b(suc)o(h)h(that)g Fo(h)606 546 y Fm(p)625 565 y Fp(\()p Fo(x)p Fp(\))g(is)f(a)h(prop)q(er)g(sub)o (w)o(ord)g(of)g Fo(h)1249 546 y Fm(p)p Fl(+1)1314 565 y Fp(\()p Fo(x)p Fp(\))f(for)g(all)g Fo(p)h(<)g(q)g Fp(and)60 625 y Fo(h)88 607 y Fm(q)107 625 y Fp(\()p Fo(x)p Fp(\))14 b(=)f Fo(h)266 607 y Fm(q)q Fl(+1)331 625 y Fp(\()p Fo(x)p Fp(\).)20 b(In)c(other)f(w)o(ords,)h Fo(h)799 607 y Fm(q)818 625 y Fp(\()p Fo(x)p Fp(\))g(is)f(a)h(minimal)d(\014xp)q(oin)o(t)i(of)h Fo(h)g Fp(whic)o(h)f(con)o(tains)60 685 y Fo(x)p Fp(.)27 b(The)18 b(assertion)h(follo)o(ws)f(directly)e(from)h(this,)h(for)g(if) g Fo(x)g Fp(is)g(con)o(tained)g(in)g(some)f Fo(\014)1660 692 y Fm(i)1691 685 y Fp(and)60 745 y Fo(\014)88 752 y Fm(j)106 745 y Fp(,)f Fo(i)f Fn(6)p Fp(=)g Fo(j)s Fp(,)h(then)h Fo(\014)414 752 y Fm(i)442 745 y Fp(=)e Fo(h)523 727 y Fm(q)542 745 y Fp(\()p Fo(x)p Fp(\))g(=)g Fo(\014)704 752 y Fm(j)721 745 y Fp(.)24 b(W)l(e)16 b(\014rst)h(sho)o(w)h(b)o(y)e (induction)g(on)i Fo(q)g Fp(that)f Fo(h)1594 727 y Fm(q)1613 745 y Fp(\()p Fo(x)p Fp(\))g(is)g(a)60 805 y(sub)o(w)o(ord)g(of)f Fo(h)336 787 y Fm(q)q Fl(+1)400 805 y Fp(\()p Fo(x)p Fp(\).)60 866 y(Base)g(case)g Fo(q)g Fp(=)d(1:)22 b(By)16 b(assumption,)f Fo(x)h Fp(o)q(ccurs)h(in)f Fo(h)p Fp(\()p Fo(x)p Fp(\).)60 926 y Fo(q)i Fn(!)e Fo(q)e Fp(+)e(1:)24 b(The)18 b(induction)g(h)o(yp)q(othesis)f(states)i(that)f Fo(h)1152 908 y Fm(q)q Fh(\000)p Fl(1)1216 926 y Fp(\()p Fo(x)p Fp(\))g(is)f(a)h(sub)o(w)o(ord)h(of)f Fo(h)1672 908 y Fm(q)1691 926 y Fp(\()p Fo(x)p Fp(\),)60 986 y(so)d Fo(h)146 968 y Fm(q)165 986 y Fp(\()p Fo(x)p Fp(\))f(=)g Fo(v)r(h)351 968 y Fm(q)q Fh(\000)p Fl(1)414 986 y Fp(\()p Fo(x)p Fp(\))p Fo(w)i Fp(for)e(some)g Fo(v)i Fp(and)f Fo(w)q Fp(.)21 b(Therefore,)14 b Fo(h)1189 968 y Fm(q)q Fl(+1)1253 986 y Fp(\()p Fo(x)p Fp(\))g(=)f Fo(h)p Fp(\()p Fo(v)r Fp(\))p Fo(h)1504 968 y Fm(q)1523 986 y Fp(\()p Fo(x)p Fp(\))p Fo(h)p Fp(\()p Fo(w)q Fp(\).)20 b(In)60 1046 y(other)c(w)o(ords,)h Fo(h)369 1028 y Fm(q)388 1046 y Fp(\()p Fo(x)p Fp(\))f(is)g(a)g(sub)o(w)o(ord)h(of)g Fo(h)836 1028 y Fm(q)q Fl(+1)900 1046 y Fp(\()p Fo(x)p Fp(\).)60 1106 y(No)o(w)h(since)f Fo(\013)h Fp(is)f(a)h(\014xp)q(oin)o (t)g(of)g Fo(h)p Fp(,)g Fo(h)762 1088 y Fm(q)781 1106 y Fp(\()p Fo(x)p Fp(\))f(cannot)i(gro)o(w)f(b)q(ey)o(ond)g Fo(\013)p Fp(,)g(so)g(for)g(some)f Fo(q)h Fn(2)e Fa(I)-7 b(N)60 1166 y Fp(the)16 b(equalit)o(y)f Fo(h)358 1148 y Fm(q)377 1166 y Fp(\()p Fo(x)p Fp(\))e(=)h Fo(h)536 1148 y Fm(q)q Fl(+1)600 1166 y Fp(\()p Fo(x)p Fp(\))i(m)o(ust)f(hold.) 21 b Fd(2)60 1257 y Fg(Theorem)c(4.7)24 b Fp(Let)h Fo(\013)k Fn(2)g Fp(\(\006)17 b Fn([)g Fo(V)12 b Fp(\))792 1239 y Fl(+)846 1257 y Fp(b)q(e)25 b(a)h(pattern)f(in)g(normal)f(form)g(and) h(let)g Fo(h)j Fp(:)60 1318 y Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))c Fn(!)f Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))445 1299 y Fh(\003)487 1318 y Fp(b)q(e)f(a)h(morphism)d(suc)o(h)i(that)h Fo(h)p Fp(\()p Fo(\013)p Fp(\))h(=)g Fo(\013)p Fp(.)40 b(Then)22 b Fo(h)g Fp(is)g(the)g(iden-)60 1378 y(tit)o(y)15 b(function)h(on)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\).)60 1498 y Fg(Pro)r(of:)25 b Fp(According)17 b(to)i(Lemma)d(4.6,)i Fo(\013)h Fp(has)g(a)f(unique)g(partition)g Fo(\013)f Fp(=)g Fo(\014)1480 1505 y Fl(1)1499 1498 y Fo(\014)1527 1505 y Fl(2)1555 1498 y Fo(:)8 b(:)g(:)f(\014)1648 1505 y Fm(k)1687 1498 y Fp(in)o(to)60 1558 y(minim)o(al)15 b(\014xp)q(oin)o(ts)i(w.r.t.)f Fo(h)h Fp(suc)o(h)g(that)h(either)e Fo(\014)1012 1565 y Fm(i)1041 1558 y Fn(2)g Fp(\006)h(or)h Fo(\014)1231 1565 y Fm(i)1260 1558 y Fn(2)d Fo(V)1348 1540 y Fl(+)1394 1558 y Fp(where)i Fo(\014)1564 1565 y Fm(i)1595 1558 y Fp(con)o(tains)60 1619 y(exactly)k(one)i(v)m (ariable)g Fo(x)548 1626 y Fm(i)584 1619 y Fp(whic)o(h)f(o)q(ccurs)h (in)f Fo(h)p Fp(\()p Fo(x)1025 1626 y Fm(i)1039 1619 y Fp(\).)41 b(Let)23 b Fo(i)1224 1626 y Fl(1)1243 1619 y Fo(;)8 b(:)g(:)g(:)f(;)h(i)1369 1626 y Fm(p)1411 1619 y Fp(b)q(e)23 b(exactly)f(those)60 1679 y(indices)17 b(for)h(whic)o(h)g Fo(\014)468 1686 y Fm(i)480 1691 y Ff(j)514 1679 y Fn(2)f Fo(V)604 1661 y Fl(+)651 1679 y Fp(and)i(let)e Fo(x)848 1686 y Fm(i)860 1691 y Ff(j)896 1679 y Fp(b)q(e)h(the)g(corresp)q(onding)h(v)m(ariables)f(satisfying)60 1739 y(\()p Fn(\003)p Fp(\).)42 b(Let)24 b Fo(V)302 1746 y Fm(nd)370 1739 y Fp(=)h Fn(f)p Fo(x)486 1746 y Fm(i)498 1751 y Fe(1)517 1739 y Fo(;)8 b(:)g(:)g(:)g(;)g(x)655 1746 y Fm(i)667 1750 y Ff(p)686 1739 y Fn(g)24 b Fp(\(note)f(that)h Fo(x)1010 1746 y Fm(i)1022 1751 y Ff(j)1066 1739 y Fp(=)h Fo(x)1157 1746 y Fm(i)1169 1750 y Ff(q)1188 1739 y Fp(,)g Fo(i)1244 1746 y Fm(j)1288 1739 y Fn(6)p Fp(=)h Fo(i)1369 1746 y Fm(q)1387 1739 y Fp(,)f(is)e(p)q(ossible\))g(and)60 1799 y Fo(V)88 1806 y Fm(d)123 1799 y Fp(=)13 b Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))e Fn(n)g Fo(V)393 1806 y Fm(nd)435 1799 y Fp(.)21 b(Supp)q(ose)c(that)g Fo(V)795 1806 y Fm(d)830 1799 y Fn(6)p Fp(=)c Fn(;)j Fp(and)h(let)f Fo(\013)1119 1781 y Fh(0)1147 1799 y Fp(b)q(e)g(the)g(pattern)h(obtained)f(from)60 1859 y Fo(\013)c Fp(b)o(y)f(deleting)f(all)h(the)g(v)m(ariables)g(from) g Fo(V)823 1866 y Fm(d)843 1859 y Fp(,)h(i.e.,)e Fo(\013)988 1841 y Fh(0)1014 1859 y Fp(=)k Fo(d)p Fp(\()p Fo(\013)p Fp(\))e(where)f Fo(d)j Fp(=)g Fn(f)p Fo(x)g Fn(7!)f Fo(\025)f Fn(j)f Fo(x)j Fn(2)g Fo(V)1711 1866 y Fm(d)1732 1859 y Fn(g)p Fp(.)60 1919 y(Note)i(that)h Fo(v)r(ar)q Fp(\()p Fo(\013)409 1901 y Fh(0)420 1919 y Fp(\))d(=)f Fo(V)532 1926 y Fm(nd)574 1919 y Fp(.)22 b(De\014ne)16 b Fo(g)g Fp(:)d Fo(v)r(ar)q Fp(\()p Fo(\013)952 1901 y Fh(0)963 1919 y Fp(\))h Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1204 1901 y Fh(\003)1239 1919 y Fp(b)o(y)793 2009 y Fo(g)r Fp(\()p Fo(x)865 2016 y Fm(i)877 2021 y Ff(j)895 2009 y Fp(\))g(=)f Fo(\014)1007 2016 y Fm(i)1019 2021 y Ff(j)60 2099 y Fp(According)19 b(to)h(Lemma)d(4.6,)j Fo(g)i Fp(is)d(w)o (ell-de\014ned)f(b)q(ecause)i Fo(x)1210 2106 y Fm(i)1222 2111 y Ff(j)1259 2099 y Fp(=)f Fo(x)1344 2106 y Fm(i)1356 2110 y Ff(q)1395 2099 y Fp(implies)d Fo(\014)1591 2106 y Fm(i)1603 2111 y Ff(j)1640 2099 y Fp(=)k Fo(\014)1726 2106 y Fm(i)1738 2110 y Ff(q)1756 2099 y Fp(.)60 2159 y(F)l(urthermore,)12 b(w)o(e)i(claim)e Fo(g)r Fp(\()p Fo(\013)627 2141 y Fh(0)640 2159 y Fp(\))h(=)h Fo(\013)p Fp(.)21 b(In)14 b(order)h(to)f(pro)o(v)o(e)g(this)g(claim,)e(it)i (su\016ces)g(to)g(sho)o(w)60 2219 y Fo(g)r Fp(\()p Fo(d)p Fp(\()p Fo(\014)176 2226 y Fm(i)188 2231 y Ff(j)206 2219 y Fp(\)\))k(=)h Fo(\014)347 2226 y Fm(i)359 2231 y Ff(j)395 2219 y Fp(for)g(all)g(1)f Fn(\024)g Fo(j)j Fn(\024)d Fo(p)p Fp(.)29 b(W)l(e)19 b(ha)o(v)o(e)f Fo(d)p Fp(\()p Fo(\014)1081 2226 y Fm(i)1093 2231 y Ff(j)1111 2219 y Fp(\))g(=)g Fo(x)1232 2226 y Fm(i)1244 2231 y Ff(j)1281 2219 y Fp(b)q(ecause)h Fo(x)1492 2226 y Fm(i)1504 2231 y Ff(j)1541 2219 y Fp(is)g(the)f(only)60 2279 y(v)m(ariable)e(from)g Fo(V)387 2286 y Fm(nd)445 2279 y Fp(in)h Fo(\014)531 2286 y Fm(i)543 2291 y Ff(j)560 2279 y Fp(.)23 b(Hence)15 b Fo(g)r Fp(\()p Fo(d)p Fp(\()p Fo(\014)858 2286 y Fm(i)870 2291 y Ff(j)888 2279 y Fp(\)\))f(=)h Fo(g)r Fp(\()p Fo(x)1065 2286 y Fm(i)1077 2291 y Ff(j)1095 2279 y Fp(\))f(=)h Fo(\014)1209 2286 y Fm(i)1221 2291 y Ff(j)1255 2279 y Fp(and)i(the)f(claim)f(is)h(pro)o(v)o(ed.)60 2340 y(Since)h Fo(\014)217 2347 y Fm(i)229 2352 y Ff(j)265 2340 y Fp(has)h(a)g (represen)o(tation)f Fo(\016)737 2347 y Fl(1)757 2340 y Fo(\015)782 2347 y Fl(1)802 2340 y Fo(x)830 2347 y Fm(i)842 2352 y Ff(j)860 2340 y Fo(\015)885 2347 y Fl(2)905 2340 y Fo(\016)927 2347 y Fl(2)946 2340 y Fp(,)h(where)f Fo(\015)1145 2347 y Fl(1)1165 2340 y Fo(;)8 b(\015)1212 2347 y Fl(2)1232 2340 y Fo(;)g(\016)1276 2347 y Fl(1)1295 2340 y Fo(;)g(\016)1339 2347 y Fl(2)1375 2340 y Fn(2)16 b Fo(V)1464 2321 y Fh(\003)1452 2352 y Fm(d)1483 2340 y Fp(,)i Fo(g)i Fp(is)e(a)g(linear)60 2400 y(morphism.)24 b(Th)o(us)18 b(it)f(follo)o(ws)h Fo(\013)f Fn(!)747 2407 y Fm(r)783 2400 y Fo(\013)814 2382 y Fh(0)826 2400 y Fp(.)26 b(This,)18 b(ho)o(w)o(ev)o(er,)e(con)o(tradicts)i(the)f(fact)h (that)h Fo(\013)60 2460 y Fp(is)h(in)h(normal)e(form.)33 b(So)21 b Fo(V)591 2467 y Fm(d)633 2460 y Fp(=)h Fn(;)p Fp(.)34 b(It)20 b(further)g(follo)o(ws)h(that)g Fo(\014)1294 2467 y Fm(i)1306 2472 y Ff(j)1345 2460 y Fp(=)g Fo(x)1432 2467 y Fm(i)1444 2472 y Ff(j)1482 2460 y Fp(and)h(therefore)60 2520 y Fo(h)p Fp(\()p Fo(x)135 2527 y Fm(i)147 2532 y Ff(j)165 2520 y Fp(\))14 b(=)g Fo(x)278 2527 y Fm(i)290 2532 y Ff(j)324 2520 y Fp(for)j(1)d Fn(\024)f Fo(j)k Fn(\024)d Fo(p)p Fp(.)21 b(Hence)15 b Fo(h)i Fp(is)f(the)g(iden)o(tit)o (y)e(on)j Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))c(=)h Fn(f)p Fo(x)1471 2527 y Fm(i)1483 2532 y Fe(1)1502 2520 y Fo(;)8 b(:)g(:)g(:)f(;)h(x)1639 2527 y Fm(i)1651 2531 y Ff(p)1671 2520 y Fn(g)p Fp(.)21 b Fd(2)133 2611 y Fp(The)16 b(simple)e(example)f Fo(\013)h Fp(=)g Fo(xy)r Fp(,)h Fo(h)p Fp(\()p Fo(x)p Fp(\))e(=)h Fo(xy)j Fp(and)g Fo(h)p Fp(\()p Fo(y)r Fp(\))c(=)h Fo(\025)i Fp(sho)o(ws)h(that)f(in)f(Theorem)60 2671 y(4.7)i(the)e (normal)h(form)f(requiremen)o(t)e(on)j Fo(\013)h Fp(cannot)g(b)q(e)f (dropp)q(ed)h(b)q(ecause)f Fo(h)p Fp(\()p Fo(\013)p Fp(\))f(=)e Fo(\013)k Fp(but)60 2731 y Fo(h)f Fp(is)g(not)h(the)f(iden)o(tit)o(y)e (on)j Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\).)891 2856 y(13)p eop %%Page: 14 14 14 13 bop 60 203 a Fg(Corollary)18 b(4.8)24 b Fp(If)16 b(a)i(pattern)f Fo(\013)f Fn(2)f Fp(\(\006)d Fn([)g Fo(V)f Fp(\))934 185 y Fl(+)981 203 y Fp(reduces)16 b(to)i(t)o(w)o(o)f (distinct)f(normal)g(forms)60 264 y Fo(\013)91 246 y Fh(0)122 264 y Fp(and)k Fo(\013)251 246 y Fh(00)272 264 y Fp(,)f(then)g(there)f(is)h(a)h(v)m(ariable)e(renaming)g Fo(h)g Fp(:)g Fo(v)r(ar)q Fp(\()p Fo(\013)1247 246 y Fh(0)1259 264 y Fp(\))g Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)1489 246 y Fh(00)1510 264 y Fp(\))1529 246 y Fh(\003)1568 264 y Fp(suc)o(h)h(that)60 324 y Fo(h)p Fp(\()p Fo(\013)138 306 y Fh(0)150 324 y Fp(\))14 b(=)g Fo(\013)266 306 y Fh(00)287 324 y Fp(.)60 444 y Fg(Pro)r(of:)21 b Fp(Consider)c(t)o(w)o (o)f(reductions)g(of)g Fo(\013)h Fp(to)g(normal)e(forms)g Fo(\013)1258 426 y Fh(0)1286 444 y Fp(and)i Fo(\013)1412 426 y Fh(00)1434 444 y Fp(:)204 553 y Fo(\013)235 533 y Fh(0)261 553 y Fp(=)d Fo(\013)344 533 y Fh(0)344 566 y Fm(n)381 553 y Fn( )431 560 y Fm(r)464 553 y Fo(:)8 b(:)g(:)13 b Fn( )585 560 y Fm(r)617 553 y Fo(\013)648 533 y Fh(0)648 566 y Fl(1)682 553 y Fn( )732 560 y Fm(r)765 553 y Fo(\013)796 533 y Fh(0)796 566 y Fl(0)829 553 y Fp(=)h Fo(\013)g Fp(=)g Fo(\013)1009 533 y Fh(00)1009 566 y Fl(0)1044 553 y Fn(!)1094 560 y Fm(r)1127 553 y Fo(\013)1158 533 y Fh(00)1158 566 y Fl(1)1193 553 y Fn(!)1243 560 y Fm(r)1276 553 y Fo(:)8 b(:)g(:)13 b Fn(!)1397 560 y Fm(r)1430 553 y Fo(\013)1461 533 y Fh(00)1461 566 y Fm(m)1508 553 y Fp(=)h Fo(\013)1591 533 y Fh(00)1612 553 y Fo(:)60 663 y Fp(Since)19 b Fn(!)241 670 y Fm(r)280 663 y Fp(is)h(transitiv)o(e,)g(w)o(e)g(obtain)h Fo(\013)832 644 y Fh(0)864 663 y Fn( )914 670 y Fm(r)954 663 y Fo(\013)g Fn(!)1056 670 y Fm(r)1095 663 y Fo(\013)1126 644 y Fh(00)1148 663 y Fp(.)33 b(By)20 b(de\014nition)f(of)i(reduction,)60 723 y(there)12 b(are)h(morphisms)e Fo(d)528 705 y Fh(0)540 723 y Fo(;)d(d)587 705 y Fh(00)621 723 y Fp(and)14 b Fo(\033)743 705 y Fh(0)754 723 y Fo(;)8 b(\033)806 705 y Fh(00)839 723 y Fp(suc)o(h)13 b(that)g Fo(d)1073 705 y Fh(0)1085 723 y Fp(\()p Fo(\013)p Fp(\))h(=)g Fo(\013)1251 705 y Fh(0)1263 723 y Fp(,)f Fo(d)1315 705 y Fh(00)1336 723 y Fp(\()p Fo(\013)p Fp(\))i(=)e Fo(\013)1502 705 y Fh(00)1524 723 y Fp(,)g Fo(\033)1581 705 y Fh(0)1592 723 y Fp(\()p Fo(\013)1642 705 y Fh(0)1654 723 y Fp(\))h(=)g Fo(\013)60 783 y Fp(and)20 b Fo(\033)188 765 y Fh(00)209 783 y Fp(\()p Fo(\013)259 765 y Fh(00)280 783 y Fp(\))g(=)f Fo(\013)p Fp(.)32 b(De\014ne)20 b Fo(f)25 b Fp(:)19 b Fo(v)r(ar)q Fp(\()p Fo(\013)815 765 y Fh(0)826 783 y Fp(\))g Fn(!)h Fo(v)r(ar)q Fp(\()p Fo(\013)1059 765 y Fh(00)1080 783 y Fp(\))1099 765 y Fh(\003)1138 783 y Fp(b)o(y)f Fo(f)25 b Fp(=)19 b Fo(d)1340 765 y Fh(00)1375 783 y Fn(\016)14 b Fo(\033)1444 765 y Fh(0)1455 783 y Fp(.)31 b(Analogously)l(,)60 843 y(de\014ne)17 b Fo(g)h Fp(:)e Fo(v)r(ar)q Fp(\()p Fo(\013)398 825 y Fh(00)419 843 y Fp(\))g Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)645 825 y Fh(0)656 843 y Fp(\))675 825 y Fh(\003)712 843 y Fp(b)o(y)h Fo(g)i Fp(=)d Fo(d)902 825 y Fh(0)926 843 y Fn(\016)11 b Fo(\033)992 825 y Fh(00)1013 843 y Fp(.)25 b(Then)18 b Fo(f)5 b Fp(\()p Fo(\013)1260 825 y Fh(0)1272 843 y Fp(\))16 b(=)g Fo(\013)1392 825 y Fh(00)1432 843 y Fp(and)i Fo(g)r Fp(\()p Fo(\013)1603 825 y Fh(00)1624 843 y Fp(\))e(=)g Fo(\013)1744 825 y Fh(0)1756 843 y Fp(.)60 903 y(Hence)d Fo(f)5 b Fp(\()p Fo(g)r Fp(\()p Fo(\013)326 885 y Fh(00)348 903 y Fp(\)\))14 b(=)g Fo(\013)483 885 y Fh(00)519 903 y Fp(and)i Fo(g)r Fp(\()p Fo(f)5 b Fp(\()p Fo(\013)736 885 y Fh(0)748 903 y Fp(\)\))14 b(=)g Fo(\013)883 885 y Fh(0)895 903 y Fp(.)20 b(By)14 b(Theorem)g(4.7,)h Fo(f)e Fn(\016)8 b Fo(g)17 b Fp(is)d(the)h(iden)o (tit)o(y)e(on)60 964 y Fo(v)r(ar)q Fp(\()p Fo(\013)185 945 y Fh(00)206 964 y Fp(\))k(and)i Fo(g)14 b Fn(\016)e Fo(f)23 b Fp(is)17 b(the)h(iden)o(tit)o(y)d(on)j Fo(v)r(ar)q Fp(\()p Fo(\013)971 945 y Fh(0)983 964 y Fp(\).)25 b(Th)o(us)18 b Fo(f)23 b Fp(and)18 b Fo(g)i Fp(are)e(bijections,)e(or)i(in)60 1024 y(other)e(w)o(ords,)h(v)m(ariable)f(renamings.)k Fd(2)60 1137 y Fp(Th)o(us)12 b(the)g(normal)f(form)g(of)i(pattern)f Fo(\013)g Fp(w.r.t.)f Fn(!)971 1144 y Fm(r)1002 1137 y Fp(is)h(unique)f(\(up)h(to)h(v)m(ariable)e(renamings\);)60 1197 y(from)17 b(no)o(w)i(on)f(it)g(will)f(b)q(e)h(denoted)g(b)o(y)g Fo(\013)p Fn(#)p Fp(.)27 b(Next,)17 b(w)o(e)h(deriv)o(e)f(equiv)m(alen) o(t)f(form)o(ulations)60 1257 y(of)h(Conjecture)e(1)i(of)g(Section)e (1.)60 1371 y Fg(Prop)r(osition)j(4.9)24 b Fp(Let)c Fo(\013;)8 b(\014)22 b Fn(2)f Fp(\(\006)13 b Fn([)h Fo(V)e Fp(\))880 1352 y Fl(+)929 1371 y Fp(b)q(e)20 b(patterns)g(and)h(let)e Fo(car)q(d)p Fp(\(\006\))i Fn(\025)e Fp(3.)33 b(The)60 1431 y(follo)o(wing)16 b(statemen)o(ts)f(are)h(equiv)m(alen)o(t.)120 1532 y(1.)24 b Fo(\013)p Fn(#)16 b Fp(and)h Fo(\014)s Fn(#)f Fp(are)g(iden)o(tical)e(up)j(to)g(v)m(ariable)e(renaming.)120 1633 y(2.)24 b(There)12 b(are)h(morphisms)d Fo(f)669 1615 y Fh(0)695 1633 y Fp(:)j Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)p Fp(\))g Fn(!)h Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)o Fp(\))1136 1615 y Fh(\003)1168 1633 y Fp(and)f Fo(g)1284 1615 y Fh(0)1309 1633 y Fp(:)h Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)n Fp(\))g Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)o Fp(\))1750 1615 y Fh(\003)182 1693 y Fp(suc)o(h)i(that)h Fo(f)427 1675 y Fh(0)439 1693 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))d(=)f Fo(\014)s Fn(#)j Fp(and)h Fo(g)790 1675 y Fh(0)802 1693 y Fp(\()p Fo(\014)s Fn(#)o Fp(\))c(=)h Fo(\013)p Fn(#)q Fp(.)120 1795 y(3.)24 b(There)15 b(are)g(morphisms)e Fo(f)19 b Fp(:)14 b Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))f Fn(!)h Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))1084 1777 y Fh(\003)1117 1795 y Fp(and)i Fo(g)g Fp(:)e Fo(v)r(ar)q Fp(\()p Fo(\014)s Fp(\))e Fn(!)i Fo(v)r(ar)q Fp(\()p Fo(\013)p Fp(\))1642 1777 y Fh(\003)1676 1795 y Fp(suc)o(h)182 1855 y(that)j Fo(f)5 b Fp(\()p Fo(\013)p Fp(\))14 b(=)g Fo(\014)k Fp(and)f Fo(g)r Fp(\()p Fo(\014)s Fp(\))c(=)h Fo(\013)p Fp(.)60 1956 y Fg(Pro)r(of:)36 b Fp(The)23 b(implication)e(\(1\))26 b Fn(\))g Fp(\(2\))e(holds)g(trivially)l(.)41 b(The)23 b(con)o(v)o(erse)g(implication)60 2016 y(\(2\))j Fn(\))g Fp(\(1\))e(is)f(a)h(consequence)e(of)i(Theorem)e(4.7)i(\(cf.)e (also)i(pro)q(of)h(of)f(Corollary)f(4.8\).)60 2076 y(The)e(equiv)m (alence)d(\(2\))k Fn(,)e Fp(\(3\))h(follo)o(ws)g(from)e(the)h (equalities)f Fo(d)p Fp(\()p Fo(\013)p Fp(\))j(=)f Fo(\013)p Fn(#)p Fp(,)g Fo(\033)r Fp(\()p Fo(\013)p Fn(#)p Fp(\))g(=)g Fo(\013)p Fp(,)60 2137 y Fo(d)85 2119 y 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Fo(v)r(ar)q Fp(\()p Fo(\013)1588 210 y Fm(i)1602 203 y Fp(\).)29 b(That)182 264 y(is,)18 b Fo(\034)6 b Fp(\()p Fo(x)p Fp(\))16 b(=)h Fo(\034)432 271 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)546 264 y Fp(\()p Fo(x)p Fp(\))h(if)g Fo(x)e Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)896 271 y Fm(i)910 264 y Fp(\))h(and)h Fo(\034)6 b Fp(\()p Fo(x)p Fp(\))16 b(=)h Fo(\025)h Fp(otherwise.)27 b(Since)17 b Fo(\013)p Fn(#)h Fp(has)182 324 y(indep)q(enden)o(t)d(v)m (ariable)h(segmen)o(ts,)f(it)g(follo)o(ws)550 434 y Fo(\034)6 b Fp(\()p Fo(\013)p Fn(#)p Fp(\))14 b(=)g Fo(u)765 441 y Fl(0)784 434 y Fo(u)812 441 y Fl(1)840 434 y Fo(:)8 b(:)g(:)g(u)934 441 y Fm(i)948 434 y Fo(\034)969 442 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)1083 434 y Fp(\()p Fo(\013)1133 441 y Fm(i)1147 434 y Fp(\))p Fo(u)1194 441 y Fm(i)p Fl(+1)1261 434 y Fo(:)g(:)g(:)g(u)1355 441 y Fm(m)1388 434 y Fo(:)182 544 y Fp(There)19 b(is)f(a)i(morphism)d Fo(\033)j Fp(suc)o(h)f(that)g Fo(\033)r Fp(\()p Fo(\014)s Fn(#)o Fp(\))g(=)f Fo(u)1147 551 y Fl(0)1166 544 y Fo(u)1194 551 y Fl(1)1222 544 y Fo(:)8 b(:)g(:)g(u)1316 551 y Fm(i)1330 544 y Fo(\034)1351 552 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)1465 544 y Fp(\()p Fo(\013)1515 551 y Fm(i)1529 544 y Fp(\))p Fo(u)1576 551 y Fm(i)p Fl(+1)1643 544 y Fo(:)g(:)g(:)g(u)1737 551 y Fm(m)182 604 y Fp(b)q(ecause)20 b Fo(L)399 611 y Fm(E)r(;)p Fl(\006)465 604 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))g(=)h Fo(L)671 611 y Fm(E)r(;)p Fl(\006)736 604 y Fp(\()p Fo(\014)s Fn(#)o Fp(\).)33 b(Since)19 b Fo(\033)j Fp(cannot)f(in)o(tro)q(duce)e(the)h(constan)o (ts)h Fo(c)182 664 y Fp(and)g Fo(d)p Fp(,)h(it)e(follo)o(ws)h Fo(\033)r Fp(\()p Fo(\014)637 671 y Fm(i)650 664 y Fp(\))h(=)f Fo(\034)771 672 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)886 664 y Fp(\()p Fo(\013)936 671 y Fm(i)950 664 y Fp(\))g(and)g Fo(\033)r Fp(\()p Fo(\014)1166 671 y Fm(j)1183 664 y Fp(\))h(=)f Fo(\025)g Fp(for)g(ev)o(ery)f Fo(j)k Fn(6)p Fp(=)e Fo(i)p Fp(.)34 b(As)182 724 y(in)17 b(the)g(pro)q(of)i(of)e (Lemma)e(3.4,)j(w)o(e)f(obtain)h(a)f(morphism)e Fo(g)1298 731 y Fm(i)1329 724 y Fp(:)g Fo(v)r(ar)q Fp(\()p Fo(\014)1480 731 y Fm(i)1493 724 y Fp(\))g Fn(!)h Fo(v)r(ar)q Fp(\()p Fo(\013)1718 731 y Fm(i)1731 724 y Fp(\))1750 706 y Fh(\003)182 785 y Fp(suc)o(h)h(that)g Fo(g)422 792 y Fm(i)436 785 y Fp(\()p Fo(\014)483 792 y Fm(i)497 785 y Fp(\))e(=)g Fo(\013)615 792 y Fm(i)629 785 y Fp(.)23 b(Moreo)o(v)o(er)15 b Fo(g)903 792 y Fm(i)918 785 y Fp(\()p Fo(x)p Fp(\))f(=)h Fo(\025)j Fp(whenev)o(er)d Fo(x)g Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\014)1524 792 y Fm(i)1537 785 y Fp(\))c Fn(\\)h Fo(v)r(ar)q Fp(\()p Fo(\014)1734 792 y Fm(j)1751 785 y Fp(\))182 845 y(for)k(some)g Fo(j)g Fn(6)p Fp(=)e Fo(i)p Fp(.)120 946 y(3.)24 b(Let)e Fo(x)h Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)p Fp(\).)38 b(Since)21 b Fo(\013)p Fn(#)h Fp(has)h(indep)q(enden)o(t)e(v)m(ariable)g(segmen)o(ts,)h(there) f(is)182 1007 y(exactly)h(one)h(index)f Fo(i)h Fp(suc)o(h)g(that)g Fo(x)i Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)1097 1014 y Fm(i)1110 1007 y Fp(\).)42 b(De\014ne)23 b Fo(f)5 b Fp(\()p Fo(x)p Fp(\))25 b(=)h Fo(f)1551 1014 y Fm(i)1565 1007 y Fp(\()p Fo(x)p Fp(\).)41 b(The)182 1067 y(morphism)13 b Fo(f)20 b Fp(obtained)c(in)f(this)g(manner)f(is)h(w)o(ell-de\014ned)f (and)i(moreo)o(v)o(er)d Fo(f)5 b Fp(\()p Fo(\013)p Fn(#)p Fp(\))14 b(=)182 1127 y Fo(\014)s Fn(#)o Fp(.)120 1229 y(4.)24 b(De\014ne)481 1308 y Fo(g)r Fp(\()p Fo(x)p Fp(\))14 b(=)637 1235 y Fb(\()692 1277 y Fo(g)715 1284 y Fm(i)729 1277 y Fp(\()p Fo(x)p Fp(\))57 b(if)16 b Fo(x)e Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\014)1108 1284 y Fm(i)1121 1277 y Fp(\))d Fn(n)1187 1244 y Fb(S)1221 1287 y Fm(j)r Fh(6)p Fl(=)p Fm(i)1287 1277 y Fo(v)r(ar)q Fp(\()p Fo(\014)1409 1284 y Fm(j)1426 1277 y Fp(\))692 1337 y Fo(\025)116 b Fp(otherwise)182 1426 y(The)16 b(morphism)e Fo(g)k Fp(is)f(w)o(ell-de\014ned)d(and)j(moreo)o(v)o(er)d Fo(g)r Fp(\()p Fo(\014)s Fn(#)p Fp(\))f(=)h Fo(\013)p Fn(#)p Fp(.)60 1528 y Fd(2)60 1642 y Fg(Corollary)k(5.2)24 b Fp(Let)f Fo(\013;)8 b(\014)28 b Fn(2)e Fp(\(\006)16 b Fn([)f Fo(V)d Fp(\))844 1624 y Fl(+)896 1642 y Fp(b)q(e)24 b(patterns)f(and)h(let)e Fo(car)q(d)p Fp(\(\006\))k Fn(\025)f Fp(4.)42 b(The)60 1702 y(language)17 b(equiv)m(alence)e(problem)f(is)i (decidable)f(if)120 1804 y(1.)24 b Fo(\013)p Fn(#)16 b Fp(has)h(indep)q(enden)o(t)f(v)m(ariable)g(segmen)o(ts.)120 1905 y(2.)24 b Fo(\013)17 b Fp(has)g(indep)q(enden)o(t)e(v)m(ariable)h (segmen)o(ts.)120 2007 y(3.)24 b Fo(\013)p Fn(#)16 b Fp(is)g(linear.)120 2109 y(4.)24 b Fo(\013)17 b Fp(is)f(linear.)60 2210 y Fg(Pro)r(of:)120 2312 y Fp(1.)24 b(This)16 b(is)g(pro)o(v)o(ed)g (in)g(Prop)q(osition)h(5.1.)120 2414 y(2.)24 b(If)16 b Fo(\013)g Fp(has)h(indep)q(enden)o(t)f(v)m(ariable)g(segmen)o(ts,)e (then)i(so)h(do)q(es)g Fo(\013)p Fn(#)p Fp(.)120 2516 y(3.)24 b(If)16 b Fo(\013)p Fn(#)g Fp(is)g(linear,)f(then)h(it)g(has)h (indep)q(enden)o(t)f(v)m(ariable)f(segmen)o(ts.)120 2617 y(4.)24 b(If)16 b Fo(\013)g Fp(is)g(linear,)f(then)h Fo(\013)p Fn(#)h Fp(is)f(linear.)60 2719 y Fd(2)891 2856 y Fp(16)p eop %%Page: 17 17 17 16 bop 133 203 a Fp(It)19 b(has)h(already)f(b)q(een)h(sho)o(wn)g(b)o (y)f(Jiang)h Fi(et)h(al.)31 b Fp([5])19 b(that)h(the)f(inclusion)g Fo(L)1578 210 y Fm(E)r(;)p Fl(\006)1643 203 y Fp(\()p Fo(\013)p Fp(\))g Fn(\022)60 264 y Fo(L)93 271 y Fm(E)r(;)p Fl(\006)158 264 y Fp(\()p Fo(\014)s Fp(\))h(is)f(decidable)g(if)h Fo(\014)i Fp(is)e(a)g(one-v)m(ariable)g(pattern)h(and)f Fo(car)q(d)p Fp(\(\006\))h Fn(\025)f Fp(2)g(\(the)g(same)60 324 y(result)14 b(for)g(NE-patterns)h(is)f(con)o(tained)f(in)h([1]\).) 20 b(Th)o(us)15 b(our)f(next)g(result)g(is)g(not)g(surprising.)60 384 y(Its)20 b(pro)q(of,)i(ho)o(w)o(ev)o(er,)e(is)g(m)o(uc)o(h)e(more)h (complicated)f(than)j(exp)q(ected)e(b)q(ecause)i(only)f(one)60 444 y(normal)15 b(form)g(is)i(assumed)e(to)i(b)q(e)f(a)h(one-v)m (ariable)f(pattern.)60 558 y Fg(Prop)r(osition)i(5.3)24 b Fp(Let)18 b Fo(\013;)8 b(\014)21 b Fn(2)d Fp(\(\006)13 b Fn([)f Fo(V)g Fp(\))872 540 y Fl(+)920 558 y Fp(b)q(e)18 b(patterns)h(and)h(let)d Fo(car)q(d)p Fp(\(\006\))i Fn(\025)e Fp(3.)29 b(If)18 b Fo(\013)p Fn(#)60 619 y Fp(is)g(a)h(one-v)m(ariable) g(pattern,)g(then)f Fo(L)765 626 y Fm(E)r(;)p Fl(\006)831 619 y Fp(\()p Fo(\013)p Fp(\))g(=)f Fo(L)1006 626 y Fm(E)r(;)p Fl(\006)1072 619 y Fp(\()p Fo(\014)s Fp(\))h(if)g(and)h(only)f(if)g Fo(\013)p Fn(#)h Fp(and)g Fo(\014)s Fn(#)f Fp(are)60 679 y(iden)o(tical)c(up)j(to)g(v)m(ariable)e(renaming.)60 799 y Fg(Pro)r(of:)27 b Fp(Again,)19 b(w)o(e)g(sho)o(w)g Fo(L)629 806 y Fm(E)r(;)p Fl(\006)695 799 y Fp(\()p Fo(\013)p Fp(\))g(=)f Fo(L)872 806 y Fm(E)r(;)p Fl(\006)938 799 y Fp(\()p Fo(\014)s Fp(\))g(if)g(and)i(only)f(if)g(there)f(are)h (morphisms)60 859 y Fo(f)28 b Fp(:)23 b Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)p Fp(\))g Fn(!)f Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)o Fp(\))581 841 y Fh(\003)622 859 y Fp(and)h Fo(g)i Fp(:)d Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)o Fp(\))h Fn(!)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)o Fp(\))1239 841 y Fh(\003)1281 859 y Fp(suc)o(h)e(that)h Fo(f)5 b Fp(\()p Fo(\013)p Fn(#)q Fp(\))23 b(=)g Fo(\014)s Fn(#)60 919 y Fp(and)d Fo(g)r Fp(\()p Fo(\014)s Fn(#)o Fp(\))e(=)h Fo(\013)p Fn(#)p Fp(.)30 b(In)18 b(order)h(to)h(pro)o(v)o(e)e(the)h Fi(only)h(if)f Fp(case,)g(let)g Fo(L)1312 926 y Fm(E)r(;)p Fl(\006)1377 919 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))g(=)f Fo(L)1579 926 y Fm(E)r(;)p Fl(\006)1644 919 y Fp(\()p Fo(\013)p Fp(\))h(=)60 980 y Fo(L)93 987 y Fm(E)r(;)p Fl(\006)158 980 y Fp(\()p Fo(\014)s Fp(\))g(=)g Fo(L)336 987 y Fm(E)r(;)p Fl(\006)401 980 y Fp(\()p Fo(\014)s Fn(#)o Fp(\).)31 b(According)19 b(to)h(Theorem)e(3.1,)i Fo(\013)p Fn(#)g Fp(and)g Fo(\014)s Fn(#)f Fp(are)g(similar,)f(so)i (let)60 1040 y Fo(\013)p Fn(#)h Fp(=)g Fo(\013)227 1047 y Fl(0)246 1040 y Fo(u)274 1047 y Fl(1)294 1040 y Fo(\013)325 1047 y Fl(1)344 1040 y Fo(u)372 1047 y Fl(2)400 1040 y Fo(:)8 b(:)g(:)g(\013)497 1047 y Fm(m)p Fh(\000)p Fl(1)575 1040 y Fo(u)603 1047 y Fm(m)636 1040 y Fo(\013)667 1047 y Fm(m)721 1040 y Fp(and)21 b Fo(\014)s Fn(#)f Fp(=)g Fo(\014)982 1047 y Fl(0)1001 1040 y Fo(u)1029 1047 y Fl(1)1049 1040 y Fo(\014)1077 1047 y Fl(1)1096 1040 y Fo(u)1124 1047 y Fl(2)1152 1040 y Fo(:)8 b(:)g(:)f(\014)1245 1047 y Fm(m)p Fh(\000)p Fl(1)1323 1040 y Fo(u)1351 1047 y Fm(m)1384 1040 y Fo(\014)1412 1047 y Fm(m)1445 1040 y Fp(.)33 b(Let)21 b Fo(x)f Fp(b)q(e)g(the)60 1100 y(\(only\))d(v)m (ariable)h(o)q(ccurring)f(in)h Fo(\013)p Fn(#)p Fp(,)f(and)i(let)e Fo(x)950 1107 y Fl(1)969 1100 y Fo(;)8 b(x)1019 1107 y Fl(2)1038 1100 y Fo(;)g(:)g(:)g(:)g(;)g(x)1176 1107 y Fm(k)1214 1100 y Fp(b)q(e)18 b(the)f(v)m(ariables)h(o)q(ccurring)60 1160 y(in)g Fo(\014)s Fn(#)p Fp(.)29 b(Without)19 b(loss)g(of)g (generalit)o(y)l(,)f(w)o(e)g(ma)o(y)f(assume)i(that)g Fo(\013)1301 1167 y Fl(0)1339 1160 y Fp(is)g(not)g(empt)o(y)l(,)e (i.e.,)g(it)60 1220 y(con)o(tains)h(at)g(least)f(one)h(o)q(ccurrence)f (of)h Fo(x)p Fp(.)25 b(Let)18 b Fo(c)g Fp(b)q(e)f(the)h(\014rst)g (letter)e(of)i Fo(u)1487 1227 y Fl(1)1507 1220 y Fp(.)25 b(W)l(e)18 b(c)o(ho)q(ose)60 1281 y Fo(a;)8 b(b)13 b Fn(2)h Fp(\006)5 b Fn(n)g(f)p Fo(c)p Fn(g)14 b Fp(suc)o(h)f(that)h Fo(a)g Fn(6)p Fp(=)f Fo(b)p Fp(.)20 b(Consider)14 b Fo(\034)922 1288 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1038 1281 y Fp(\()p Fo(\014)s Fn(#)o Fp(\).)20 b Fo(\034)1186 1288 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1302 1281 y Fp(\()p Fo(\014)s Fn(#)o Fp(\))13 b(can)h(b)q(e)f(written)g(as)60 1341 y Fo(w)95 1348 y Fl(0)115 1341 y Fo(u)143 1348 y Fl(1)162 1341 y Fo(w)197 1348 y Fl(1)225 1341 y Fo(:)8 b(:)g(:)f(w)325 1348 y Fm(m)p Fh(\000)p Fl(1)404 1341 y Fo(u)432 1348 y Fm(m)465 1341 y Fo(w)500 1348 y Fm(m)533 1341 y Fp(,)13 b(where)f Fo(w)732 1348 y Fm(i)760 1341 y Fp(=)i Fo(\034)833 1349 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)948 1341 y Fp(\()p Fo(\014)995 1348 y Fm(i)1009 1341 y Fp(\).)20 b(Since)11 b Fo(L)1218 1348 y Fm(E)r(;)p Fl(\006)1284 1341 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))j(=)f Fo(L)1476 1348 y Fm(E)r(;)p Fl(\006)1542 1341 y Fp(\()p Fo(\014)s Fn(#)o Fp(\),)g(there)60 1401 y(is)j(a)h(morphism)d Fo(\033)k Fp(suc)o(h)e(that)h Fo(\033)r Fp(\()p Fo(\013)p Fn(#)p Fp(\))d(=)g Fo(w)865 1408 y Fl(0)884 1401 y Fo(u)912 1408 y Fl(1)932 1401 y Fo(w)967 1408 y Fl(1)995 1401 y Fo(:)8 b(:)g(:)f(w)1095 1408 y Fm(m)p Fh(\000)p Fl(1)1174 1401 y Fo(u)1202 1408 y Fm(m)1235 1401 y Fo(w)1270 1408 y Fm(m)1303 1401 y Fp(.)21 b(Clearly)l(,)16 b Fo(\033)r Fp(\()p Fo(\013)1598 1408 y Fl(0)1617 1401 y Fp(\))e(=)g Fo(w)1737 1408 y Fl(0)1756 1401 y Fp(.)60 1461 y(It)26 b(is)f(not)i(di\016cult)e(to)h(pro)o(v)o(e)f(that)i Fo(\033)r Fp(\()p Fo(x)p Fp(\))i(=)i Fo(\034)1008 1469 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1123 1461 y Fp(\()p Fo(x)1170 1468 y Fm(i)1182 1473 y Fe(1)1202 1461 y Fp(\))p Fo(\034)1242 1469 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1357 1461 y Fp(\()p Fo(x)1404 1468 y Fm(i)1416 1473 y Fe(2)1435 1461 y Fp(\))8 b Fo(:)g(:)g(:)g(\034)1549 1469 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1665 1461 y Fp(\()p Fo(x)1712 1468 y Fm(i)1724 1474 y Ff(l)1737 1461 y Fp(\),)60 1521 y(where)24 b Fo(i)226 1528 y Fl(1)245 1521 y Fo(;)8 b(i)284 1528 y Fl(2)303 1521 y Fo(;)g(:)g(:)g(:)g(i)408 1528 y Fm(l)447 1521 y Fn(2)27 b(f)p Fp(1)p Fo(;)8 b(:)g(:)g(:)g(;)g(k)r Fn(g)p Fp(,)25 b(b)q(ecause)f Fo(\013)976 1528 y Fl(0)1020 1521 y Fp(con)o(tains)g(only)g Fo(x)p Fp('s.)44 b(Supp)q(ose)25 b Fo(\033)r Fp(\()p Fo(\013)1732 1528 y Fl(1)1751 1521 y Fp(\))60 1582 y(con)o(tains)d(a)g(pre\014x)f Fo(w)i Fp(of)f Fo(u)592 1589 y Fl(2)633 1582 y Fp(\()p Fo(w)i Fp(=)f Fo(u)800 1589 y Fl(2)841 1582 y Fp(is)e(p)q(ossible\).)37 b(Then)22 b Fo(w)h Fp(m)o(ust)d(b)q(e)i(of)g(the)f(form)60 1642 y Fo(\034)81 1650 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)197 1642 y Fp(\()p Fo(x)244 1649 y Fm(j)258 1654 y Fe(1)277 1642 y Fp(\))p Fo(\034)317 1650 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)432 1642 y Fp(\()p Fo(x)479 1649 y Fm(j)493 1654 y Fe(2)513 1642 y Fp(\))8 b Fo(:)g(:)g(:)g(\034)627 1650 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)742 1642 y Fp(\()p Fo(x)789 1649 y Fm(j)803 1659 y Ff(l)813 1652 y Fc(0)829 1642 y Fp(\),)17 b(where)g Fo(j)1041 1649 y Fl(1)1061 1642 y Fo(;)8 b(j)1103 1649 y Fl(2)1122 1642 y Fo(;)g(:)g(:)g(:)g(j)1230 1649 y Fm(l)1241 1640 y Fc(0)1270 1642 y Fn(2)16 b(f)p Fp(1)p Fo(;)8 b(:)g(:)g(:)g(;)g(k)r Fn(g)p Fp(.)24 b(This)18 b(con-)60 1702 y(tradicts)24 b(the)f(fact)h(that)g Fn(j)p Fo(\034)590 1710 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)706 1702 y Fp(\()p Fo(x)753 1709 y Fm(j)771 1702 y Fp(\))p Fn(j)i Fo(>)h Fn(j)p Fo(u)937 1709 y Fl(2)956 1702 y Fn(j)d Fp(for)g(ev)o(ery)e Fo(x)1240 1709 y Fm(j)1285 1702 y Fn(2)27 b Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)n Fp(\).)44 b(Rep)q(eated)60 1762 y(application)23 b(of)h(this)f(argumen)o(t)g(yields)f Fo(\033)r Fp(\()p Fo(\013)934 1769 y Fm(i)948 1762 y Fp(\))k(=)g Fo(w)1092 1769 y Fm(i)1132 1762 y Fp(=)g Fo(\034)1217 1770 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1332 1762 y Fp(\()p Fo(\014)1379 1769 y Fm(i)1393 1762 y Fp(\).)43 b(As)23 b(in)g(Lemma)60 1822 y(3.4,)d(w)o(e)e(obtain)i(a)f(morphism)e Fo(f)23 b Fp(:)18 b Fn(f)p Fo(x)p Fn(g)g(!)h Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)n Fp(\))1070 1804 y Fh(\003)1109 1822 y Fp(suc)o(h)g(that)g Fo(f)5 b Fp(\()p Fo(\013)p Fn(#)q Fp(\))18 b(=)g Fo(\014)s Fn(#)p Fp(,)h(namely)60 1883 y Fo(f)5 b Fp(\()p Fo(x)p Fp(\))17 b(=)g Fo(x)255 1890 y Fm(i)267 1895 y Fe(1)286 1883 y Fo(x)314 1890 y Fm(i)326 1895 y Fe(2)353 1883 y Fo(:)8 b(:)g(:)g(x)447 1890 y Fm(i)459 1896 y Ff(l)472 1883 y Fp(.)27 b(Moreo)o(v)o(er,)17 b(if)g Fn(j)p Fo(\013)834 1890 y Fm(i)848 1883 y Fn(j)862 1890 y Fm(x)901 1883 y Fp(=)f Fo(r)q Fp(,)j(then)e Fo(\014)1151 1890 y Fm(i)1182 1883 y Fp(=)g Fo(f)5 b Fp(\()p Fo(x)p Fp(\))j Fo(:)g(:)g(:)f(f)e Fp(\()p Fo(x)p Fp(\),)18 b(where)g Fo(f)5 b Fp(\()p Fo(x)p Fp(\))60 1943 y(o)q(ccurs)15 b Fo(r)i Fp(times)12 b(in)j Fo(\014)460 1950 y Fm(i)474 1943 y Fp(.)20 b(Th)o(us)c(ev)o(ery)d(v)m(ariable)h Fo(x)967 1950 y Fm(j)999 1943 y Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)o Fp(\))h(o)q(ccurs)g(at)g(least)g(as)g(often)g(in)60 2003 y Fo(\014)s Fn(#)f Fp(as)h Fo(x)g Fp(in)f Fo(\013)p Fn(#)p Fp(.)21 b(No)o(w)15 b(consider)f Fo(\034)696 2011 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)811 2003 y Fp(\()p Fo(\013)p Fn(#)p Fp(\).)20 b(Again,)15 b(there)f(is)h(a)g(morphism)d Fo(\027)18 b Fp(suc)o(h)d(that)60 2063 y Fo(\027)s Fp(\()p Fo(\014)s Fn(#)o Fp(\))f(=)g Fo(\034)267 2071 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)381 2063 y Fp(\()p Fo(\013)p Fn(#)q Fp(\))h(and)i(furthermore)d Fo(\027)s Fp(\()p Fo(\014)929 2070 y Fl(0)948 2063 y Fp(\))g(=)g Fo(\013)1064 2070 y Fl(0)1084 2063 y Fp(.)21 b(Let)15 b Fo(x)1233 2070 y Fm(j)1267 2063 y Fp(b)q(e)h(a)g(v)m(ariable)f(of)i Fo(v)r(ar)q Fp(\()p Fo(\014)1733 2070 y Fl(0)1751 2063 y Fp(\))60 2123 y(suc)o(h)g(that)h Fo(\027)s Fp(\()p Fo(x)352 2130 y Fm(j)370 2123 y Fp(\))f(con)o(tains)g(an)h(anc)o(hor)g (segmen)o(t)d(of)j Fo(x)p Fp(.)24 b(It)17 b(follo)o(ws)g(that)g Fn(j)p Fo(\013)p Fn(#)q(j)1545 2130 y Fm(x)1582 2123 y Fp(=)e Fn(j)p Fo(\014)s Fn(#)o(j)1718 2130 y Fm(x)1738 2135 y Ff(j)1756 2123 y Fp(.)60 2183 y(Since)i Fo(\014)s Fn(#)f Fp(is)i(in)f(normal)g(form,)f(w)o(e)h(infer)g Fo(j)i Fp(=)d Fo(k)i Fp(=)e(1,)i(that)g(is)f(to)h(sa)o(y)l(,)g Fo(\014)s Fn(#)f Fp(con)o(tains)g(only)60 2244 y(the)f(v)m(ariable)g Fo(x)355 2251 y Fl(1)374 2244 y Fp(.)22 b(Ob)o(viously)l(,)14 b Fo(\013)p Fn(#)j Fp(and)g Fo(\014)s Fn(#)e Fp(are)h(iden)o(tical)f (up)h(to)h(v)m(ariable)f(renaming.)k Fd(2)133 2358 y Fp(It)12 b(remains)f(to)h(b)q(e)h(in)o(v)o(estigated)e(whether)h(Prop)q (osition)h(5.3)g(extends)f(to)g(unam)o(biguous)60 2418 y(patterns)j(\(recall)e(that)i(ev)o(ery)e(one-v)m(ariable)h(pattern)h (is)f(unam)o(biguous\).)21 b(W)l(e)14 b(next)g(deriv)o(e)60 2478 y(a)j(result)e(similar)g(to)h(Prop)q(osition)i(5.3.)60 2592 y Fg(Prop)r(osition)g(5.4)24 b Fp(Let)19 b Fo(\013;)8 b(\014)21 b Fn(2)e Fp(\(\006)13 b Fn([)h Fo(V)d Fp(\))875 2574 y Fl(+)924 2592 y Fp(b)q(e)19 b(patterns)g(and)h(let)e Fo(car)q(d)p Fp(\(\006\))i Fn(\025)e Fp(4.)30 b(Sup-)60 2653 y(p)q(ose)20 b Fo(\013)p Fn(#)f Fp(=)f Fo(\013)335 2660 y Fl(0)355 2653 y Fo(u)383 2660 y Fl(1)402 2653 y Fo(\013)433 2660 y Fl(1)453 2653 y Fo(u)481 2660 y Fl(2)509 2653 y Fo(:)8 b(:)g(:)f(\013)605 2660 y Fm(m)p Fh(\000)p Fl(1)684 2653 y Fo(u)712 2660 y Fm(m)745 2653 y Fo(\013)776 2660 y Fm(m)809 2653 y Fp(,)19 b(where)g Fn(j)p Fo(\013)1031 2660 y Fl(0)1050 2653 y Fn(j)g(\024)f Fp(1,)i Fn(j)p Fo(\013)1243 2660 y Fm(m)1276 2653 y Fn(j)e(\024)g Fp(1,)i(and)g Fn(j)p Fo(\013)1566 2660 y Fm(i)1580 2653 y Fn(j)e Fp(=)g(1)i(for)60 2713 y(0)h Fo(<)g(i)g(<)f(m)p Fp(.)34 b(In)20 b(other)g(w)o(ords,)i Fo(\013)p Fn(#)e Fp(do)q(es)i(not)e(con)o(tain)h(consecutiv)o(e)d(v)m(ariables.)34 b(Then)891 2856 y(17)p eop %%Page: 18 18 18 17 bop 60 203 a Fo(L)93 210 y Fm(E)r(;)p Fl(\006)158 203 y Fp(\()p Fo(\013)p Fp(\))17 b(=)f Fo(L)331 210 y Fm(E)r(;)p Fl(\006)396 203 y Fp(\()p Fo(\014)s Fp(\))h(if)h(and)g(only) f(if)h Fo(\013)p Fn(#)g Fp(and)g Fo(\014)s Fn(#)f Fp(are)h(iden)o (tical)e(up)h(to)i(v)m(ariable)e(renam-)60 264 y(ing.)60 384 y Fg(Pro)r(of:)25 b Fp(W)l(e)18 b(pro)q(ceed)h(as)g(in)f(Prop)q (ositions)i(5.1)e(and)h(5.3.)28 b(Let)19 b Fo(L)1315 391 y Fm(E)r(;)p Fl(\006)1380 384 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))f(=)f Fo(L)1580 391 y Fm(E)r(;)p Fl(\006)1645 384 y Fp(\()p Fo(\013)p Fp(\))h(=)60 444 y Fo(L)93 451 y Fm(E)r(;)p Fl(\006)158 444 y Fp(\()p Fo(\014)s Fp(\))h(=)h Fo(L)337 451 y Fm(E)r(;)p Fl(\006)403 444 y Fp(\()p Fo(\014)s Fn(#)o Fp(\).)32 b(By)19 b(Theorem)f(3.1,)j Fo(\014)s Fn(#)e Fp(=)h Fo(\014)1089 451 y Fl(0)1108 444 y Fo(u)1136 451 y Fl(1)1156 444 y Fo(\014)1184 451 y Fl(1)1203 444 y Fo(u)1231 451 y Fl(2)1259 444 y Fo(:)8 b(:)g(:)f(\014)1352 451 y Fm(m)p Fh(\000)p Fl(1)1430 444 y Fo(u)1458 451 y Fm(m)1491 444 y Fo(\014)1519 451 y Fm(m)1552 444 y Fp(.)32 b(Let)20 b Fo(x)g Fn(2)60 504 y Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)p Fp(\).)31 b(There)19 b(is)h(an)g(index)f Fo(i)p Fp(,)g(0)h Fn(\024)g Fo(i)f Fn(\024)g Fo(m)p Fp(,)h(suc)o(h)g (that)g Fo(\013)1255 511 y Fm(i)1288 504 y Fp(=)g Fo(x)p Fp(.)31 b(W.l.o.g.)18 b(w)o(e)i(ma)o(y)60 565 y(assume)12 b(that)g(0)j Fo(<)e(i)h(<)g(m)d Fp(\(again,)j(the)e(cases)g Fo(i)i Fp(=)g(0)e(and)h Fo(i)h Fp(=)f Fo(m)f Fp(are)h(similar)d(and)j (simpler\).)60 625 y(Let)18 b Fo(c)g Fp(b)q(e)h(the)e(last)i(letter)e (of)h Fo(u)654 632 y Fm(i)686 625 y Fp(and)h Fo(d)f Fp(b)q(e)g(the)g (\014rst)h(letter)e(of)h Fo(u)1300 632 y Fm(i)p Fl(+1)1359 625 y Fp(.)27 b(Again,)18 b(w)o(e)g(c)o(ho)q(ose)60 685 y Fo(a;)8 b(b)13 b Fn(2)h Fp(\006)d Fn(n)g(f)p Fo(c;)d(d)p Fn(g)17 b Fp(suc)o(h)f(that)g Fo(a)e Fn(6)p Fp(=)g Fo(b)p Fp(.)20 b(De\014ne)570 815 y Fo(\034)6 b Fp(\()p Fo(z)r Fp(\))14 b(=)726 742 y Fb(\()780 784 y Fo(\034)801 792 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)915 784 y Fp(\()p Fo(z)r Fp(\))58 b(if)15 b Fo(x)f Fp(=)g Fo(z)r(;)780 844 y(\025)212 b Fp(otherwise.)60 944 y Fo(\034)6 b Fp(\()p Fo(\013)p Fn(#)p Fp(\))17 b(can)h(b)q(e)g(written)f(as)h Fo(w)625 951 y Fl(0)644 944 y Fo(u)672 951 y Fl(1)692 944 y Fo(w)727 951 y Fl(1)755 944 y Fo(:)8 b(:)g(:)f(w)855 951 y Fm(m)p Fh(\000)p Fl(1)933 944 y Fo(u)961 951 y Fm(m)994 944 y Fo(w)1029 951 y Fm(m)1063 944 y Fp(,)17 b(where)g Fo(w)1271 951 y Fm(j)1305 944 y Fp(=)f Fo(\034)1380 952 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)1494 944 y Fp(\()p Fo(x)p Fp(\))h(if)g Fo(\013)1654 951 y Fm(j)1688 944 y Fp(=)f Fo(x)60 1005 y Fp(and)i Fo(w)191 1012 y Fm(j)226 1005 y Fp(=)e Fo(\025)j Fp(otherwise.)25 b(Since)17 b Fo(L)729 1012 y Fm(E)r(;)p Fl(\006)795 1005 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))f(=)h Fo(L)993 1012 y Fm(E)r(;)p Fl(\006)1058 1005 y Fp(\()p Fo(\014)s Fn(#)o Fp(\),)h(there)f(is)h(a)g (morphism)d Fo(\027)21 b Fp(suc)o(h)60 1065 y(that)c Fo(\027)s Fp(\()p Fo(\014)s Fn(#)p Fp(\))d(=)h Fo(w)389 1072 y Fl(0)409 1065 y Fo(u)437 1072 y Fl(1)456 1065 y Fo(w)491 1072 y Fl(1)519 1065 y Fo(:)8 b(:)g(:)g(w)620 1072 y Fm(m)p Fh(\000)p Fl(1)698 1065 y Fo(u)726 1072 y Fm(m)759 1065 y Fo(w)794 1072 y Fm(m)827 1065 y Fp(.)23 b(Clearly)l(,)16 b Fo(\027)s Fp(\()p Fo(\014)1118 1072 y Fm(i)1131 1065 y Fp(\))f(=)g Fo(\034)6 b Fp(\()p Fo(\013)1295 1072 y Fm(i)1309 1065 y Fp(\))15 b(=)f Fo(\034)6 b Fp(\()p Fo(x)p Fp(\))15 b(=)f Fo(\034)1576 1073 y Fh(j)p Fm(\014)r Fh(#)o(j)p Fm(;a;b)1691 1065 y Fp(\()p Fo(x)p Fp(\).)60 1125 y(Let)19 b Fo(y)h Fp(b)q(e)f(the)g(v)m(ariable)f(in)h Fo(v)r(ar)q Fp(\()p Fo(\014)717 1132 y Fm(i)730 1125 y Fp(\))g(suc)o(h)f(that)h Fo(\027)s Fp(\()p Fo(y)r Fp(\))g(con)o (tains)g(the)f(anc)o(hor)i(segmen)o(t)d(of)60 1185 y Fo(x)p Fp(.)22 b(It)16 b(is)h(clear)f(that)h Fo(y)h Fp(o)q(ccurs)f (exactly)e(once)i(in)f Fo(\014)1008 1192 y Fm(i)1022 1185 y Fp(;)g(in)g(the)h(sequel)e(w)o(e)h(call)g Fo(y)i Fp(the)f(anc)o(hor)60 1245 y(v)m(ariable)f(of)g Fo(x)p Fp(.)21 b(F)l(urthermore,)14 b(it)i(follo)o(ws)g Fn(j)p Fo(\014)s Fn(#)o(j)949 1252 y Fm(y)984 1245 y Fn(\024)d(j)p Fo(\013)p Fn(#j)1120 1252 y Fm(x)1142 1245 y Fp(.)133 1306 y(No)o(w)25 b(consider)f Fo(\034)472 1313 y Fh(j)p Fm(\013)p Fh(#j)p Fm(;a;b)588 1306 y Fp(\()p Fo(\014)s Fn(#)o Fp(\))k(=)h Fo(v)800 1313 y Fl(0)819 1306 y Fo(u)847 1313 y Fl(1)866 1306 y Fo(v)890 1313 y Fl(1)918 1306 y Fo(:)8 b(:)g(:)g(v)1008 1313 y Fm(m)p Fh(\000)p Fl(1)1086 1306 y Fo(u)1114 1313 y Fm(m)1147 1306 y Fo(v)1171 1313 y Fm(m)1204 1306 y Fp(,)26 b(where)e Fo(v)1417 1313 y Fm(k)1467 1306 y Fp(=)k Fo(\034)1554 1313 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1670 1306 y Fp(\()p Fo(\014)1717 1313 y Fm(k)1737 1306 y Fp(\).)60 1366 y(Since)j Fo(L)236 1373 y Fm(E)r(;)p Fl(\006)301 1366 y Fp(\()p Fo(\013)p Fn(#)p Fp(\))41 b(=)f Fo(L)547 1373 y Fm(E)r(;)p Fl(\006)612 1366 y Fp(\()p Fo(\014)s Fn(#)p Fp(\),)35 b(there)c(is)h(a)g(morphism)e Fo(\033)j Fp(suc)o(h)f(that)g Fo(\033)r Fp(\()p Fo(\013)p Fn(#)p Fp(\))40 b(=)60 1426 y Fo(v)84 1433 y Fl(0)103 1426 y Fo(u)131 1433 y Fl(1)151 1426 y Fo(v)175 1433 y Fl(1)202 1426 y Fo(:)8 b(:)g(:)g(v)292 1433 y Fm(m)p Fh(\000)p Fl(1)370 1426 y Fo(u)398 1433 y Fm(m)431 1426 y Fo(v)455 1433 y Fm(m)488 1426 y Fp(.)32 b(Clearly)l(,)19 b Fo(\033)r Fp(\()p Fo(\013)797 1433 y Fm(i)810 1426 y Fp(\))h(=)g Fo(\033)r Fp(\()p Fo(x)p Fp(\))f(=)h Fo(v)1104 1433 y Fm(i)1137 1426 y Fp(=)g Fo(\034)1216 1434 y Fh(j)p Fm(\013)p Fh(#)o(j)p Fm(;a;b)1332 1426 y Fp(\()p Fo(\014)1379 1433 y Fm(i)1392 1426 y Fp(\).)32 b(Hence)19 b Fn(j)p Fo(\013)p Fn(#j)1690 1433 y Fm(x)1731 1426 y Fn(\024)60 1486 y(j)p Fo(\014)s Fn(#)o(j)143 1493 y Fm(y)164 1486 y Fp(.)h(Both)12 b(inequalities)f(yield)h Fn(j)p Fo(\013)p Fn(#j)767 1493 y Fm(x)802 1486 y Fp(=)i Fn(j)p Fo(\014)s Fn(#)o(j)937 1493 y Fm(y)958 1486 y Fp(.)20 b(As)12 b(a)i(further)e (consequence,)g(w)o(e)h(observ)o(e)60 1546 y(that)g(if)f Fo(y)i Fp(o)q(ccurs)e(in)g Fo(\014)468 1553 y Fm(j)486 1546 y Fp(,)h(0)h Fn(\024)g Fo(j)i Fn(\024)e Fo(m)p Fp(,)e(then)h Fo(\014)898 1553 y Fm(j)929 1546 y Fp(=)h Fo(\015)s(\014)1037 1553 y Fm(i)1051 1546 y Fo(\016)r Fp(.)19 b(Let)13 b(us)f(compare)f (the)i(equal)e(w)o(ords)60 1606 y Fo(\034)6 b Fp(\()p Fo(\013)p Fn(#)p Fp(\))17 b(and)h Fo(\027)s Fp(\()p Fo(\014)s Fn(#)p Fp(\))f(again.)25 b(Firstly)l(,)16 b(b)o(y)h(a)h(length)f (argumen)o(t,)f(w)o(e)h(see)g(that)h Fo(\027)s Fp(\()p Fo(z)r Fp(\))d(=)h Fo(\025)i Fp(for)60 1667 y(all)e Fo(z)g Fn(2)f Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)n Fp(\))d Fn(n)f Fo(v)r(ar)q Fp(\()p Fo(\014)552 1674 y Fm(i)565 1667 y Fp(\).)22 b(Secondly)l(,)15 b(the)h(same)g(length)g(argumen)o(t)g (has)h(the)f(follo)o(wing)60 1727 y(consequence:)26 b(if)18 b Fo(y)433 1709 y Fh(0)463 1727 y Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\014)637 1734 y Fm(i)650 1727 y Fp(\),)g Fo(y)728 1709 y Fh(0)758 1727 y Fn(6)p Fp(=)g Fo(y)i Fp(and)e Fo(\027)s Fp(\()p Fo(y)1029 1709 y Fh(0)1041 1727 y Fp(\))f Fn(6)p Fp(=)h Fo(\025)p Fp(,)h(then)f Fn(j)p Fo(\014)s Fn(#)o(j)1394 1734 y Fm(y)1412 1725 y Fc(0)1444 1727 y Fp(=)g Fn(j)p Fo(\014)1543 1734 y Fm(i)1556 1727 y Fn(j)1570 1734 y Fm(y)1588 1725 y Fc(0)1615 1727 y Fn(\003)13 b(j)p Fo(\014)s Fn(#)o(j)1736 1734 y Fm(y)1756 1727 y Fp(.)60 1787 y(In)k(other)f(w)o(ords,)i Fo(y)430 1769 y Fh(0)458 1787 y Fp(o)q(ccurs)f(solely)f(within)g(the)h(con)o(text)f Fo(\014)1183 1794 y Fm(i)1213 1787 y Fp(in)h Fo(\014)s Fn(#)o Fp(.)23 b(No)o(w)17 b(it)f(is)h(relativ)o(ely)60 1847 y(easy)f(to)h(sho)o(w)g(that)133 1939 y Fn(\017)24 b Fo(\013)213 1946 y Fm(j)245 1939 y Fp(=)14 b Fo(x)p Fp(,)h(0)g Fn(\024)e Fo(j)k Fn(\024)d Fo(m)p Fp(,)h(implies)f Fo(y)h Fn(2)f Fo(v)r(ar)q Fp(\()p Fo(\014)981 1946 y Fm(j)998 1939 y Fp(\))i(and)h Fn(j)p Fo(\014)1170 1946 y Fm(j)1188 1939 y Fn(j)1202 1946 y Fm(y)1236 1939 y Fp(=)d(1,)i(and)133 2038 y Fn(\017)24 b Fo(\013)213 2045 y Fm(j)245 2038 y Fn(6)p Fp(=)14 b Fo(x)p Fp(,)h(0)g Fn(\024)e Fo(j)k Fn(\024)d Fo(m)p Fp(,)h(implies)f Fo(y)h Fn(62)f Fo(v)r(ar)q Fp(\()p Fo(\014)981 2045 y Fm(j)998 2038 y Fp(\).)60 2130 y(Just)i(compare)e(the)h(equal)g(w)o(ords)h Fo(\034)6 b Fp(\()p Fo(\013)p Fn(#)p Fp(\))16 b(and)g Fo(\027)s Fp(\()p Fo(\014)s Fn(#)o Fp(\))g(from)e(left)h(to)g(righ)o (t.)21 b(Supp)q(ose)c(there)60 2190 y(is)g(another)h(index)f Fo(k)h Fn(2)e(f)p Fp(0)p Fo(;)8 b Fp(1)p Fo(;)g(:)g(:)g(:)g(;)g(m)p Fn(g)p Fp(,)17 b Fo(k)h Fn(6)p 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b(ev)o(ery)60 2430 y Fo(x)14 b Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)o Fp(\))f(has)g(an)h(anc)o(hor)f(v)m(ariable)f Fo(y)k Fn(2)e Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)n Fp(\).)20 b(This)13 b(anc)o(hor)g(v)m(ariable)g(is)f(uniquely)60 2491 y(determined)d(b)o(y)i(the)g(ab)q(o)o(v)o(e)h(construction.)19 b(Moreo)o(v)o(er,)11 b(di\013eren)o(t)g(v)m(ariables)g(from)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)o Fp(\))60 2551 y(ha)o(v)o(e)17 b(di\013eren)o(t)f(anc)o(hor)i(v)m(ariables.)25 b(Finally)l(,)16 b(if)h Fo(y)h Fn(2)e Fo(v)r(ar)q Fp(\()p Fo(\014)s Fn(#)o Fp(\))h(is)g(the)h(anc)o(hor)g(v)m(ariable)f(of)60 2611 y Fo(x)d Fn(2)g Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)o Fp(\),)i(then)g Fo(y)i Fp(o)q(ccurs)e(in)g Fo(\014)735 2618 y Fm(j)769 2611 y Fp(if)g(and)h(only)f(if)g Fo(\013)1092 2618 y Fm(j)1124 2611 y Fp(=)e Fo(x)p Fp(.)133 2671 y(With)i(these)g (preparatory)h(considerations,)f(w)o(e)f(are)i(no)o(w)f(able)g(to)g (de\014ne)g(morphisms)60 2731 y Fo(f)21 b Fp(:)15 b Fo(v)r(ar)q Fp(\()p Fo(\013)p 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b(w)o(ell-de\014ned)e(and)j(ob)o(viously)60 324 y Fo(f)5 b Fp(\()p Fo(\013)p Fn(#)q Fp(\))13 b(=)h Fo(\014)s Fn(#)o Fp(.)22 b(On)16 b(the)g(other)g(hand,)h(de\014ne)f Fo(g)i Fp(b)o(y)250 453 y Fo(g)r Fp(\()p Fo(y)r Fp(\))c(=)404 380 y Fb(\()459 422 y Fo(x)58 b Fp(if)15 b Fo(y)j Fp(is)e(the)g(anc)o (hor)h(v)m(ariable)f(of)g(some)g Fo(x)d Fn(2)h Fo(v)r(ar)q Fp(\()p Fo(\013)p Fn(#)p Fp(\))p Fo(;)459 482 y(\025)42 b Fp(otherwise)o Fo(:)60 581 y(g)21 b Fp(is)d(w)o(ell-de\014ned,)f(to)q (o,)j(and)f(the)f(equalit)o(y)f Fo(g)r Fp(\()p Fo(\014)s Fn(#)o Fp(\))h(=)g Fo(\013)p Fn(#)g Fp(holds)h(true.)28 b(This)18 b(pro)o(v)o(es)g(the)60 642 y(prop)q(osition.)k Fd(2)133 742 y Fp(Note)f(that)i(if)e(a)h(pattern)g(in)f(normal)g(form)g (is)g(linear,)h(then)g(its)f(canonical)h(form)e(is)60 802 y Fo(x)88 809 y Fl(0)107 802 y Fo(u)135 809 y Fl(1)155 802 y Fo(x)183 809 y Fl(1)202 802 y Fo(u)230 809 y Fl(2)258 802 y Fo(:)8 b(:)g(:)g(x)352 809 y Fm(m)p Fh(\000)p Fl(1)430 802 y Fo(u)458 809 y Fm(m)491 802 y Fo(x)519 809 y Fm(m)552 802 y Fp(.)23 b(Th)o(us)17 b(Prop)q(osition)h(5.4)f(also)h(generalizes) e(Corollary)h(5.2,)g(\(3\))60 862 y(and)g(\(4\).)60 1027 y Fq(6)81 b(Related)27 b(W)-7 b(ork)60 1136 y Fp(In)18 b(this)h(pap)q(er,)g(w)o(e)f(did)g(not)h(in)o(v)o(estigate)e(the)i (cases)f Fo(car)q(d)p Fp(\(\006\))h Fn(\024)e Fp(2.)29 b(In)18 b([3],)g(it)g(is)g(sho)o(wn)60 1196 y(that)g(the)f(equiv)m (alence)f(of)i(E-patterns)g(is)f(decidable)f(for)i Fo(car)q(d)p Fp(\(\006\))f(=)e(1)j(and)g(a)g(necessary)60 1257 y(condition)g(for)g (the)f(equiv)m(alence)f(of)j(E-patterns)f(in)g(case)g Fo(car)q(d)p Fp(\(\006\))f(=)f(2)i(is)g(giv)o(en.)25 b(More-)60 1317 y(o)o(v)o(er,)13 b(D\023)-24 b(an)o(yi)14 b(and)h(F)q(\177)-26 b(ul\177)i(op)15 b([3])f(claim)e(that)j(for)g Fo(car)q(d)p Fp(\(\006\))f Fn(\025)g Fp(2)g(the)g(inclusion)g(and)h (hence)e(the)60 1377 y(equiv)m(alence)g(problem)f(for)j(similar)e (E-patterns)i(is)f(decidable)f(b)o(y)h(testing)h(mem)n(b)q(ership)d(of) 60 1437 y(a)17 b(certain)e(w)o(ord;)i(see)f(b)q(elo)o(w)g(for)h(the)f (exact)g(claim.)j(Ho)o(w)o(ev)o(er,)14 b(Example)h(3.8)h(refutes)g(the) 60 1497 y(claim.)28 b(T)l(o)19 b(see)g(this,)g(let)f(us)i(\014rst)f (recall)f(the)h(de\014nition)g(of)g(the)g(morphism)e Fo(\034)1565 1504 y Fm(k)q(;a;b)1659 1497 y Fp(in)i([3)o(])60 1558 y(\(whic)o(h)d(is)g(a)g(mo)q(di\014cation)g(of)g(De\014nition)g (3.3\).)60 1658 y Fg(De\014nition)i(6.1)24 b Fp(Let)16 b Fo(\013)e Fp(=)g Fo(\013)633 1665 y Fl(0)652 1658 y Fo(u)680 1665 y Fl(1)700 1658 y Fo(\013)731 1665 y Fl(1)751 1658 y Fo(u)779 1665 y Fl(2)806 1658 y Fo(:)8 b(:)g(:)g(\013)903 1665 y Fm(m)p Fh(\000)p Fl(1)981 1658 y Fo(u)1009 1665 y Fm(m)1042 1658 y Fo(\013)1073 1665 y Fm(m)1121 1658 y Fn(2)14 b Fp(\(\006)d Fn([)f Fo(V)i Fp(\))1335 1640 y Fl(+)1364 1658 y Fp(,)k Fo(V)25 b Fp(=)14 b Fn(f)p Fo(x)1552 1665 y Fl(1)1571 1658 y Fo(;)8 b(:)g(:)g(:)f(;)h(x)1708 1665 y Fm(n)1732 1658 y Fn(g)p Fp(,)60 1718 y(and)17 b Fo(car)q(d)p Fp(\(\006\))e Fn(\025)g Fp(2.)22 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y(Computer)i(Scienc)n(e) h Fg(54)p Fp(,)d(pages)h(194{206,)h(1994.)60 2082 y([11])24 b(A.)18 b(Salomaa.)28 b(Return)18 b(to)h(patterns.)29 b Fi(Bul)r(letin)22 b(of)d(the)h(Eur)n(op)n(e)n(an)f(Asso)n(ciation)h (for)160 2142 y(The)n(or)n(etic)n(al)d(Computer)g(Scienc)n(e)j Fg(55)p Fp(,)c(pages)h(144{155,)h(1995.)891 2856 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF