%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o paper.ps paper.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1996.01.02:1633 %%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 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(man)o(y)e(cases.)21 b(Our)13 b(\\decision)h(pro)q(cedure")g(is)f (based)h(on)h(a)60 1347 y(decidable)f(and)j(length-decreasing)e (reduction)g(relation)h Fl(!)43 b(\032)29 b Fn(\(\006)10 b Fl([)g Fm(V)i Fn(\))1474 1329 y Fj(+)1513 1347 y Fl(\002)e Fn(\(\006)g Fl([)g Fm(V)h Fn(\))1727 1329 y Fj(+)1756 1347 y Fn(.)60 1407 y(Roughly)h(sp)q(eaking,)h(pattern)f Fm(\013)h Fn(reduces)e(to)i(another)f(pattern)h Fm(\013)1274 1389 y Fi(0)1298 1407 y Fn(\(whic)o(h)e(will)g(b)q(e)h(denoted)60 1467 y(b)o(y)k Fm(\013)e Fl(!)g Fm(\013)268 1449 y Fi(0)280 1467 y Fn(\))i(if)133 1567 y Fl(\017)24 b Fm(\013)213 1549 y Fi(0)239 1567 y Fn(can)15 b(b)q(e)f(obtained)h(from)e(the)h (pattern)h Fm(\013)g Fn(b)o(y)e(deleting)h(certain)g(v)m(ariables)g (from)f Fm(\013)p Fn(,)133 1668 y Fl(\017)24 b Fm(\013)14 b Fn(=)g Fm(\033)r Fn(\()p Fm(\013)359 1650 y Fi(0)370 1668 y Fn(\),)i(where)g Fm(\033)i Fn(is)e(a)g(\(linear\))g (substitution.)60 1768 y(Since)i Fl(!)h Fn(is)f(length-decreasing,)h (ev)o(ery)e(reduction)i(sequence)e(starting)j(from)e(a)h(pattern)60 1829 y Fm(\013)g Fn(m)o(ust)e(end)h(in)g(an)h(irreducible)d(pattern,)i (a)h(so-called)f(normal)g(form.)26 b(Moreo)o(v)o(er,)17 b Fl(!)h Fn(is)60 1889 y(language-preserving,)f(that)g(is,)e(if)h Fm(\013)f Fl(!)f Fm(\013)866 1871 y Fi(0)878 1889 y Fn(,)i(then)g Fm(L)1052 1896 y Fk(E)r(;)p Fj(\006)1117 1889 y Fn(\()p Fm(\013)p Fn(\))f(=)f Fm(L)1286 1896 y Fk(E)r(;)p Fj(\006)1351 1889 y Fn(\()p Fm(\013)1401 1871 y Fi(0)1413 1889 y Fn(\).)22 b(W)l(e)16 b(conjecture)60 1949 y(that)h(reduction)e(to)i(normal)e (form)h(yields)f(a)h(decision)g(pro)q(cedure:)60 2069 y Fg(Conjecture)23 b(1:)28 b Fn(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 2130 y(with)16 b Fm(car)q(d)p Fn(\(\006\))e Fl(\025)g Fn(3)i(if)f(and)h(only)g(if)f(their)g(normal)g(forms)g(with)h(resp)q (ect)f(to)i Fl(!)e Fn(are)h(iden)o(ti-)60 2190 y(cal,)f(up)i(to)f (renaming)g(of)g(v)m(ariables.)60 2310 y(W)l(e)21 b(will)e(pro)o(v)o(e) h(in)h(Section)f(4)i(that)f(the)g(conjecture)f(can)h(b)q(e)g (paraphrased)h(as)g(follo)o(ws:)60 2370 y(If)15 b Fm(\013;)8 b(\014)17 b Fl(2)d Fn(\(\006)c Fl([)h Fm(V)g Fn(\))419 2352 y Fj(+)464 2370 y Fn(are)16 b(patterns)h(and)f Fm(car)q(d)p Fn(\(\006\))e Fl(\025)g Fn(3,)i(then)g Fm(L)1265 2377 y Fk(E)r(;)p Fj(\006)1330 2370 y Fn(\()p Fm(\013)p Fn(\))e(=)g Fm(L)1498 2377 y Fk(E)r(;)p Fj(\006)1563 2370 y Fn(\()p Fm(\014)s Fn(\))h(if)g(and)60 2430 y(only)j(if)g(there)g(are)h (substitutions)g Fm(f)k Fn(:)17 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))1171 2412 y Fi(\003)1208 2430 y Fn(and)h Fm(g)h Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1751 2412 y Fi(\003)60 2491 y Fn(suc)o(h)e(that)h Fm(f)5 b Fn(\()p Fm(\013)p Fn(\))14 b(=)g Fm(\014)k Fn(and)f Fm(g)r Fn(\()p Fm(\014)s Fn(\))d(=)f Fm(\013)p Fn(.)133 2551 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 Fh(shortest)60 2611 y Fn(pattern)g(that)g(generates)g(a)g(giv)o(en)f (E-pattern)i(language.)21 b(This)13 b(is)f(b)q(ecause)h Fl(!)g Fn(is)g(a)g(length-)60 2671 y(decreasing)g(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 2731 y(is)k(a)h(minim)o(al)c(description)j(of)g (the)g(corresp)q(onding)i(language.)903 2856 y(2)p eop %%Page: 3 3 3 2 bop 133 203 a Fn(W)l(e)15 b(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)60 264 y(languages)20 b(can)g(in)f(fact)g(in)g(man)o(y)f(cases)h(b)q(e)g (decided)f(b)o(y)h(this)g(pro)q(cedure,)h(for)f(instance)60 324 y(whenev)o(er:)133 425 y Fl(\017)24 b Fn(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)182 486 y(patterns)i(\(this)f(is)g(alw) o(a)o(ys)g(true)g(if)f(\006)i(is)f(in\014nite\),)133 587 y Fl(\017)24 b Fm(car)q(d)p Fn(\(\006\))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)182 648 y(t)o(w)o(o,)133 749 y Fl(\017)24 b Fn(the)f(normal)g(form)f(of)i(one)g(pattern)f(has)i(indep)q(enden)o (t)d(v)m(ariable)h(segmen)o(ts)f(and)182 809 y Fm(car)q(d)p Fn(\(\006\))14 b Fl(\025)g Fn(4,)133 911 y Fl(\017)24 b Fn(the)16 b(normal)f(form)g(of)i(one)f(pattern)h(is)f(linear)f(and)i Fm(car)q(d)p Fn(\(\006\))d Fl(\025)g Fn(4,)133 1013 y Fl(\017)24 b Fn(the)16 b(normal)f(form)g(of)i(one)f(pattern)h(is)f(a)g (one-v)m(ariable)h(pattern)f(and)h Fm(car)q(d)p Fn(\(\006\))d Fl(\025)g Fn(3,)133 1115 y Fl(\017)24 b Fn(the)16 b(normal)f(form)g(of) h(one)g(pattern)h(do)q(es)g(not)f(con)o(tain)g(consecutiv)o(e)e(v)m (ariables)i(and)182 1175 y Fm(car)q(d)p Fn(\(\006\))e Fl(\025)g Fn(4.)60 1276 y(It)i(is)g(in)o(teresting)f(to)h(note)g(that)h (the)f(\014rst)g(t)o(w)o(o)g(sp)q(ecial)g(cases)h(can)f(already)g(b)q (e)g(pro)o(v)o(ed)g(b)o(y)60 1337 y(further)i(dev)o(eloping)f(a)h(pro)q (of)h(tec)o(hnique)e(used)h(in)g([JSSY93];)g(see)g(Section)f(3.)27 b(Ho)o(w)o(ev)o(er,)60 1397 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 1457 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 1517 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 1577 y(Note)j(that)g(in)g(the)g(conjecture)f(ab)q(o)o(v)o(e,)i(the)f (restriction)f Fm(car)q(d)p Fn(\(\006\))i Fl(\025)f Fn(3)h(is)e(essen)o (tial.)60 1638 y(This)d(is)g(witnessed)g(b)o(y)g(the)g(follo)o(wing)g (example.)60 1752 y Fg(Example)g(1.1)24 b Fn(Let)d(\006)h(=)g Fl(f)p Fn(0)p Fm(;)8 b Fn(1)p Fl(g)p Fn(,)22 b Fm(\013)g Fn(=)f Fm(x)p Fn(01)p Fm(y)r Fn(0)p Fm(z)r Fn(,)h(and)f Fm(\014)j Fn(=)e Fm(x)p Fn(0)p Fm(y)r Fn(10)p Fm(z)r Fn(.)35 b(Both)21 b Fm(\013)g Fn(and)h Fm(\014)60 1812 y Fn(are)d(in)g(normal)f(form)h(w.r.t.)f Fl(!)p Fn(;)i(see)f(Section)f (4.)31 b(Moreo)o(v)o(er,)18 b Fm(L)1296 1819 y Fk(E)r(;)p Fj(\006)1362 1812 y Fn(\()p Fm(\013)p Fn(\))h(=)g Fm(L)1540 1819 y Fk(E)r(;)p Fj(\006)1605 1812 y Fn(\()p Fm(\014)s Fn(\);)g(see)60 1872 y([JKS)164 1854 y Fj(+)193 1872 y Fn(94)q(].)i(Ho)o(w)o(ev)o(er,)14 b Fm(\013)i Fn(and)h Fm(\014)i Fn(are)d(not)h(iden)o(tical)d(up)j(to)f(renaming)f(of)i(v)m (ariables.)60 2039 y Fo(2)81 b(Basic)27 b(de\014nitions)60 2148 y Fn(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 2130 y Fi(\003)1488 2148 y Fn(,)h(whereas)f(\006)1740 2130 y Fj(+)60 2208 y Fn(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 Fm(\013)e Fl(2)g Fn(\006)1701 2190 y Fi(\003)1737 2208 y Fn(is)60 2268 y(denoted)i(b)o(y)g Fl(j)p Fm(\013)p Fl(j)p Fn(.)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 Fm(\025)p Fn(,)i(its)f(length)f(is)h(zero.)21 b(F)l(or)c(an)60 2329 y(arbitrary)f(set)g Fm(S)s Fn(,)g Fm(car)q(d)p Fn(\()p Fm(S)s Fn(\))h(denotes)f(its)g(cardinalit)o(y)l(.) 133 2389 y(In)i(the)f(sequel,)f(\006)i(and)h Fm(V)29 b Fn(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 2449 y Fh(terminals)12 b Fn(\(or)g Fh(c)n(onstants)p Fn(\))g(and)g(elemen)o (ts)c(of)k Fm(V)22 b Fn(are)12 b(called)e Fh(variables)p Fn(.)21 b(Usually)l(,)10 b(0)p Fm(;)e Fn(1)p Fm(;)g(a;)g(b;)g(c)60 2509 y Fn(will)k(denote)h(constan)o(ts,)g(whereas)h Fm(x;)8 b(y)r(;)g(z)r(;)g(x)884 2491 y Fi(0)894 2509 y Fm(;)g(y)942 2491 y Fi(0)953 2509 y Fm(;)g(z)1000 2491 y Fi(0)1011 2509 y Fm(;)g(x)1061 2516 y Fj(1)1080 2509 y Fm(;)g(:)g(:)g(:)k Fn(will)g(stand)i(for)f(v)m(ariables.)20 b(An)o(y)60 2569 y(w)o(ord)c Fm(\013)g Fn(o)o(v)o(er)f(the)h(union)g(\006)10 b Fl([)g Fm(V)27 b Fn(is)16 b(said)g(to)g(b)q(e)g(a)g Fh(p)n(attern)p Fn(.)22 b(The)15 b(n)o(um)o(b)q(er)f(of)i(o)q (ccurrences)60 2630 y(of)f(a)g(letter)f Fm(x)g Fl(2)g Fn(\006)8 b Fl([)g Fm(V)26 b Fn(in)15 b Fm(\013)g Fn(is)g(denoted)f(b)o (y)h Fl(j)p Fm(\013)p Fl(j)967 2637 y Fk(x)989 2630 y Fn(.)20 b(The)15 b(set)g(of)g(v)m(ariables)g(of)g Fm(V)26 b Fn(app)q(earing)60 2690 y(in)16 b Fm(\013)h Fn(will)e(b)q(e)h (denoted)g(b)o(y)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\).)903 2856 y(3)p eop %%Page: 4 4 4 3 bop 133 203 a Fn(Giv)o(en)19 b(\006)h(and)h Fm(V)11 b Fn(,)20 b(a)h Fh(substitution)g Fm(\033)h Fn(is)d(a)i(mapping)e(from) g Fm(V)31 b Fn(to)20 b(\(\006)14 b Fl([)g Fm(V)d Fn(\))1580 185 y Fi(\003)1600 203 y Fn(.)32 b(Ev)o(ery)60 264 y(substitution)17 b Fm(\033)i Fn(extends)d(uniquely)g(to)i(a)f Fh(morphism)f Fm(h)f Fn(:)g(\(\006)d Fl([)g Fm(V)f Fn(\))1326 246 y Fi(\003)1361 264 y Fl(!)k Fn(\(\006)d Fl([)g Fm(V)f Fn(\))1595 246 y Fi(\003)1615 264 y Fn(,)16 b(where)60 324 y Fm(h)p Fn(\()p Fm(a)p Fn(\))i(=)g Fm(a)h Fn(for)g(ev)o(ery)e Fm(a)h Fl(2)h Fn(\006)g(and)h Fm(h)p Fn(\()p Fm(x)p Fn(\))e(=)g Fm(\033)r Fn(\()p Fm(x)p Fn(\))g(for)h(ev)o(ery)f Fm(x)g Fl(2)g Fm(V)12 b Fn(.)29 b(Con)o(v)o(ersely)l(,)18 b(ev)o(ery)60 384 y(morphism)i Fm(h)25 b Fn(:)f(\(\006)15 b Fl([)h Fm(V)11 b Fn(\))561 366 y Fi(\003)606 384 y Fl(!)24 b Fn(\(\006)15 b Fl([)h Fm(V)11 b Fn(\))856 366 y Fi(\003)898 384 y Fn(with)23 b Fm(h)p Fn(\()p Fm(a)p Fn(\))h(=)h Fm(a)d Fn(for)g(ev)o(ery)f Fm(a)k Fl(2)f Fn(\006)f(can)g(b)q(e)60 444 y(view)o(ed)17 b(as)j(a)f(substitution.)29 b(In)19 b(the)f(sequel,)g(w)o(e)h(will)e(iden)o(tify)g(a)j(substitution)f(with) f(its)60 504 y(corresp)q(onding)h(morphism)d(that)j(k)o(eeps)f (terminals)e(\014xed.)28 b(Henceforth)17 b Fm(\033)o(;)8 b(\027;)g(f)s(;)g(g)r(;)g(h)18 b Fn(will)60 565 y(denote)j (substitutions.)37 b(A)21 b(bijectiv)o(e)d(substitution)k Fm(\033)i Fn(:)e Fm(V)33 b Fl(!)23 b Fm(V)32 b Fn(is)21 b(called)f(a)i Fh(variable)60 625 y(r)n(enaming)p Fn(.)f(Tw)o(o)15 b(patterns)g Fm(\013)f Fn(and)h Fm(\014)i 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926 y(ev)o(ery)10 b(pattern)i Fm(\013)g Fn(has)g(a)h(canonical)e(form)g Fm(\013)866 908 y Fi(0)878 926 y Fn(,)h(and)g(furthermore)e Fm(\013)i Fn(and)g Fm(\013)1423 908 y Fi(0)1447 926 y Fn(are)g(iden)o(tical)e(up)60 986 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 1046 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 1106 y(are)g(iden)o(tical)f(up)h(to)h(v)m(ariable)f (renaming.)133 1166 y(The)g(language)h Fm(L)469 1173 y Fk(E)r(;)p Fj(\006)534 1166 y Fn(\()p Fm(\013)p Fn(\))f(generated)g (b)o(y)g(an)g(E-pattern)g Fm(\013)f Fl(2)f Fn(\(\006)c Fl([)h Fm(V)g Fn(\))1457 1148 y Fj(+)1502 1166 y Fn(is)16 b(de\014ned)g(b)o(y)204 1276 y Fm(L)237 1283 y Fk(E)r(;)p Fj(\006)302 1276 y Fn(\()p Fm(\013)p Fn(\))e(=)g Fl(f)p Fm(w)h Fl(2)f Fn(\006)594 1256 y Fi(\003)630 1276 y Fl(j)i Fm(w)f Fn(=)f Fm(\033)r Fn(\()p Fm(\013)p Fn(\))i(for)h(some)e (substitution)h Fm(\033)g Fn(:)d Fm(V)25 b Fl(!)14 b Fn(\006)1568 1256 y Fi(\003)1588 1276 y Fl(g)p Fm(:)60 1386 y Fn(The)i(language)h Fm(L)396 1393 y Fk(N)t(E)r(;)p Fj(\006)493 1386 y Fn(\()p Fm(\013)p Fn(\))g(generated)f(b)o(y)g(an)h (NE-pattern)f Fm(\013)e Fl(2)g Fn(\(\006)d Fl([)h Fm(V)f Fn(\))1456 1368 y Fj(+)1502 1386 y Fn(is)16 b(de\014ned)g(as)183 1496 y Fm(L)216 1503 y Fk(N)t(E)r(;)p Fj(\006)313 1496 y Fn(\()p Fm(\013)p Fn(\))e(=)g Fl(f)p Fm(w)h Fl(2)f Fn(\006)605 1476 y Fi(\003)641 1496 y Fl(j)i Fm(w)f Fn(=)f Fm(\033)r Fn(\()p Fm(\013)p Fn(\))i(for)g(some)g(substitution)g Fm(\033)g Fn(:)d Fm(V)25 b Fl(!)14 b Fn(\006)1579 1476 y Fj(+)1608 1496 y Fl(g)p Fm(:)60 1606 y Fn(Giv)o(en)23 b(t)o(w)o(o)h(E-patterns)h Fm(\013)g Fn(and)g Fm(\014)s Fn(,)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 1667 y(substitution)d Fm(h)g Fn(with)g Fm(h)p Fn(\()p Fm(\014)s Fn(\))g(=)h Fm(\013)p Fn(,)g(then)f Fm(L)895 1674 y Fk(E)r(;)p Fj(\006)960 1667 y Fn(\()p Fm(\013)p Fn(\))h Fl(\022)g Fm(L)1145 1674 y Fk(E)r(;)p Fj(\006)1210 1667 y Fn(\()p Fm(\014)s Fn(\).)35 b(The)21 b(same)f(statemen)o(t)60 1727 y(holds)d(for)f(NE-patterns.)22 b(These)16 b(observ)m(ations)h(will)e(b)q(e)i(used)f(freely)f(in)h(the) g(sequel.)133 1787 y(Let)j Fm(\013)g Fn(b)q(e)g(a)h(pattern.)29 b Fm(\013)19 b Fn(has)h(a)f(unique)f(represen)o(tation)g Fm(\013)1282 1794 y Fj(0)1302 1787 y Fm(u)1330 1794 y Fj(1)1350 1787 y Fm(\013)1381 1794 y Fj(1)1400 1787 y Fm(u)1428 1794 y Fj(2)1456 1787 y Fm(:)8 b(:)g(:)g(\013)1553 1794 y Fk(m)p Fi(\000)p Fj(1)1631 1787 y Fm(u)1659 1794 y Fk(m)1692 1787 y Fm(\013)1723 1794 y Fk(m)1756 1787 y Fn(,)60 1847 y(where)22 b Fm(\013)238 1854 y Fj(0)257 1847 y Fm(;)8 b(\013)310 1854 y Fk(m)367 1847 y Fl(2)24 b Fm(V)464 1829 y Fi(\003)483 1847 y Fn(,)f Fm(\013)551 1854 y Fk(i)589 1847 y Fl(2)h Fm(V)686 1829 y Fj(+)737 1847 y Fn(for)e(1)j Fl(\024)e Fm(i)g(<)h(m)p Fn(,)f(and)g Fm(u)1239 1854 y Fk(i)1276 1847 y Fl(2)h Fn(\006)1368 1829 y Fj(+)1420 1847 y Fn(for)e(1)i Fl(\024)g Fm(i)f Fl(\024)h Fm(m)p Fn(,)60 1907 y Fm(m)17 b Fl(\025)h Fn(0.)28 b(W)l(e)18 b(call)g(the)g(sub)o(w)o(ords)h Fm(u)749 1914 y Fk(i)781 1907 y Fn(the)g Fh(terminal)h(se)n(gments)f Fn(of)g Fm(\013)p Fn(.)28 b(F)l(urthermore,)17 b(the)60 1968 y Fm(\013)91 1975 y Fk(i)125 1968 y Fn(are)i(called)g(the)g Fh(variable)i(se)n(gments)g Fn(of)f Fm(\013)p Fn(.)31 b(By)19 b(imp)q(osing)g(syn)o(tatic)f(restrictions,)h(w)o(e)60 2028 y(obtain)f(sp)q(ecial)f(patterns.)26 b(W)l(e)18 b(sa)o(y)f(that)h Fm(\013)g Fn(has)h Fh(indep)n(endent)h(variable)g(se) n(gments)f Fn(if)e(for)60 2088 y(all)h(1)g Fl(\024)g Fm(i)f(<)h(j)j Fl(\024)c Fm(m)i Fn(w)o(e)f(ha)o(v)o(e)g Fm(v)r(ar)q Fn(\()p Fm(\013)792 2095 y Fk(i)805 2088 y Fn(\))13 b Fl(\\)f Fm(v)r(ar)q Fn(\()p Fm(\013)1007 2095 y Fk(j)1025 2088 y Fn(\))18 b(=)f Fl(;)p Fn(.)29 b(P)o(attern)18 b Fm(\013)h Fn(is)f Fh(line)n(ar)25 b Fn(if)18 b(ev)o(ery)60 2148 y(v)m(ariable)i Fm(x)h Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))f(o)q(ccurs)h(exactly)e(once)h(in)g Fm(\013)p Fn(.)34 b(Ob)o(viously)l(,)20 b(ev)o(ery)f(linear)h(pattern) 60 2208 y(has)g(indep)q(enden)o(t)f(v)m(ariable)h(segmen)o(ts.)30 b(P)o(attern)19 b Fm(\013)i Fn(is)e(said)h(to)g(b)q(e)g Fh(terminal)h(fr)n(e)n(e)e Fn(if)h(no)60 2269 y(terminal)10 b(o)q(ccurs)j(in)f(it.)20 b(P)o(attern)12 b Fm(\013)h Fn(is)f(a)h Fh(one-variable)j(p)n(attern)c Fn(if)g(it)g(con)o(tains)g (o)q(ccurrences)60 2329 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 2389 y(A)i(pattern)g Fm(\013)g Fn(is)g Fh(unambiguous)h Fn(if,)f(for)g(ev)o(ery)e(w)o(ord)j Fm(w)g Fl(2)f Fm(L)1233 2396 y Fk(E)r(;)p Fj(\006)1299 2389 y Fn(\()p Fm(\013)p Fn(\),)g(there)g(is)f(a)i(unique)60 2449 y(substitution)e Fm(\033)h Fn(suc)o(h)f(that)g Fm(\033)r Fn(\()p Fm(\013)p Fn(\))f(=)g Fm(w)q Fn(.)28 b(It)19 b(is)f(decidable)g(whether)g(or)h(not)g(a)g(pattern)g(is)60 2509 y(unam)o(biguous;)c(see)g([MS94].)20 b(Moreo)o(v)o(er,)14 b(ev)o(ery)f(one-v)m(ariable)j(pattern)f(is)g(unam)o(biguous;)60 2570 y(see)h(also)h([MS94].)903 2856 y(4)p eop %%Page: 5 5 5 4 bop 60 203 a Fo(3)81 b(Preliminary)25 b(results)60 313 y Fn(On)12 b(the)h(one)f(hand,)i(Jiang)e Fh(et)j(al.)21 b Fn([JSSY93])12 b(ha)o(v)o(e)f(sho)o(wn)i(that)g(the)f(inclusion)g (problem)f(for)60 373 y(E-pattern)i(languages)h(is)e(in)g(general)g (undecidable.)19 b(On)13 b(the)f(other)h(hand,)g(this)f(problem)f(is)60 433 y(decidable)j(for)h(terminal)d(free)i(patterns;)i(see)e([JSSY93].) 20 b(Consequen)o(tly)l(,)14 b(the)h(E-language)60 493 y(equiv)m(alence)d(problem)h(for)i(terminal)d(free)h(patterns)i(is)f (decidable.)20 b(Despite)13 b(the)h(fact)h(that)60 554 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)60 614 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)60 674 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)60 734 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 Fh(et)i(al.)60 794 y Fn([JKS)164 776 y Fj(+)193 794 y Fn(94)q(,)f(JSSY93].)60 903 y Fg(Theorem)h(3.1)24 b Fn(Let)17 b Fm(\013;)8 b(\014)18 b Fl(2)d Fn(\(\006)d Fl([)g Fm(V)f Fn(\))798 884 y Fj(+)827 903 y Fn(.)24 b(Moreo)o(v)o(er,)16 b(let)g Fm(\013)g Fn(=)f Fm(\013)1296 910 y Fj(0)1316 903 y Fm(u)1344 910 y Fj(1)1363 903 y Fm(\013)1394 910 y Fj(1)1414 903 y Fm(u)1442 910 y Fj(2)1470 903 y Fm(:)8 b(:)g(:)f(\013)1566 910 y Fk(m)p Fi(\000)p Fj(1)1645 903 y Fm(u)1673 910 y Fk(m)1706 903 y Fm(\013)1737 910 y Fk(m)60 963 y Fn(and)22 b Fm(\014)k Fn(=)d Fm(\014)303 970 y Fj(0)322 963 y Fm(v)346 970 y Fj(1)365 963 y Fm(\014)393 970 y Fj(1)413 963 y Fm(v)437 970 y Fj(2)464 963 y Fm(:)8 b(:)g(:)g(\014)558 970 y Fk(n)p Fi(\000)p Fj(1)626 963 y Fm(v)650 970 y Fk(n)673 963 y Fm(\014)701 970 y Fk(n)746 963 y Fn(b)q(e)22 b(their)f(unique)f(represen)o(tations,)j(that)f(is)f (to)i(sa)o(y)l(,)60 1023 y Fm(\013)91 1030 y Fj(0)111 1023 y Fm(;)8 b(\014)161 1030 y Fj(0)180 1023 y Fm(;)g(\013)233 1030 y Fk(m)266 1023 y Fm(;)g(\014)316 1030 y Fk(n)357 1023 y Fl(2)19 b Fm(V)449 1005 y Fi(\003)468 1023 y Fn(,)h Fm(\013)533 1030 y Fk(i)565 1023 y Fl(2)f Fm(V)656 1005 y Fj(+)705 1023 y Fn(for)g(1)g Fl(\024)g Fm(i)f(<)h(m)p Fn(,)g Fm(\014)1079 1030 y Fk(j)1115 1023 y Fl(2)g Fm(V)1206 1005 y Fj(+)1255 1023 y Fn(for)g(1)g Fl(\024)f Fm(j)k(<)d(n)p Fn(,)g Fm(u)1621 1030 y Fk(i)1653 1023 y Fl(2)g Fn(\006)1740 1005 y Fj(+)60 1083 y Fn(for)h(1)h Fl(\024)f Fm(i)g Fl(\024)g Fm(m)p Fn(,)g Fm(v)439 1090 y Fk(j)477 1083 y Fl(2)g Fn(\006)565 1065 y Fj(+)615 1083 y Fn(for)g(1)h Fl(\024)f Fm(j)j Fl(\024)d Fm(n)p Fn(,)h Fm(m;)8 b(n)20 b Fl(\025)g Fn(0.)32 b(If)20 b Fm(L)1292 1090 y Fk(E)r(;)p Fj(\006)1357 1083 y Fn(\()p Fm(\013)p Fn(\))h(=)f Fm(L)1538 1090 y Fk(E)r(;)p Fj(\006)1603 1083 y Fn(\()p Fm(\014)s Fn(\))f(and)60 1143 y Fm(car)q(d)p Fn(\(\006\))g Fl(\025)f Fn(3,)i(then)f Fm(m)f Fn(=)h Fm(n)g Fn(and)h Fm(u)768 1150 y Fk(i)800 1143 y Fn(=)f Fm(v)881 1150 y Fk(i)913 1143 y Fn(for)h(1)f Fl(\024)f Fm(i)g Fl(\024)h Fm(m)p Fn(.)29 b(If)18 b Fm(car)q(d)p Fn(\(\006\))i Fl(\025)e Fn(4,)i(then)e(it)60 1203 y(further)e(follo)o (ws)g Fm(L)417 1210 y Fk(E)r(;)p Fj(\006)482 1203 y Fn(\()p Fm(\013)532 1210 y Fk(i)546 1203 y Fn(\))e(=)g Fm(L)664 1210 y Fk(E)r(;)p Fj(\006)729 1203 y Fn(\()p Fm(\014)776 1210 y Fk(i)790 1203 y Fn(\))i(for)h(0)d Fl(\024)f Fm(i)h Fl(\024)f Fm(m)p Fn(.)133 1312 y(Again,)h(the)f(assumption)g Fm(car)q(d)p Fn(\(\006\))h Fl(\025)f Fn(3)h(is)f(crucial;)g(see)g (Example)e(1.1.)21 b(It)13 b(is)g(unkno)o(wn)60 1372 y(whether)j(the)g(restriction)g Fm(car)q(d)p Fn(\(\006\))f Fl(\025)f Fn(4)i(can)h(b)q(e)g(w)o(eak)o(ened)e(to)i Fm(car)q(d)p Fn(\(\006\))e Fl(\025)f Fn(3)i(in)h(the)f(last)60 1432 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 1492 y(follo)o(wing)16 b(notion.)60 1600 y Fg(De\014nition)i(3.2)24 b Fn(Tw)o(o)19 b(patterns)f Fm(\013)h Fn(and)g Fm(\014)i Fn(are)e(called)e Fh(similar)h Fn(if)g(and)h(only)g(if)f(in)g(their)60 1660 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 1721 y Fm(\013)c Fn(=)g Fm(\013)188 1728 y Fj(0)208 1721 y Fm(u)236 1728 y Fj(1)255 1721 y Fm(\013)286 1728 y Fj(1)306 1721 y Fm(u)334 1728 y Fj(2)362 1721 y Fm(:)8 b(:)g(:)f(\013)458 1728 y Fk(m)p Fi(\000)p Fj(1)537 1721 y Fm(u)565 1728 y Fk(m)598 1721 y Fm(\013)629 1728 y Fk(m)678 1721 y Fn(and)17 b Fm(\014)f Fn(=)e Fm(\014)897 1728 y Fj(0)916 1721 y Fm(u)944 1728 y Fj(1)964 1721 y Fm(\014)992 1728 y Fj(1)1011 1721 y Fm(u)1039 1728 y Fj(2)1067 1721 y Fm(:)8 b(:)g(:)f(\014)1160 1728 y Fk(m)p Fi(\000)p Fj(1)1238 1721 y Fm(u)1266 1728 y Fk(m)1299 1721 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Fg(De\014nition)i(3.3)24 b Fn(Let)14 b Fm(V)25 b Fn(=)14 b Fl(f)p Fm(x)661 383 y Fj(1)680 376 y Fm(;)8 b(:)g(:)g(:)g(;)g(x)818 383 y Fk(n)841 376 y Fl(g)14 b Fn(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)60 437 y(with)h Fm(car)q(d)p Fn(\(\006\))f Fl(\025)f Fn(2.)22 b(F)l(or)14 b(ev)o(ery)g(pair)g(of)i(letters)d Fm(a;)8 b(b)14 b Fn(in)h(\006,)g Fm(a)e Fl(6)p Fn(=)h Fm(b)p Fn(,)g(and)h(an)h(in)o(teger)d Fm(k)j(>)e Fn(0,)60 497 y(w)o(e)i(de\014ne)g(a)g(substitution)h Fm(\034)606 504 y Fk(k)q(;a;b)695 497 y Fn(:)c Fm(V)25 b Fl(!)14 b(f)p Fm(a;)8 b(b)p Fl(g)958 479 y Fi(\003)993 497 y Fn(b)o(y)359 606 y Fm(\034)380 613 y Fk(k)q(;a;b)455 606 y Fn(\()p Fm(x)502 613 y Fk(i)515 606 y Fn(\))14 b(=)g Fm(ab)647 585 y Fk(k)q Fi(\003)p Fk(i)p Fj(+1)742 606 y Fm(aab)815 585 y Fk(k)q Fi(\003)p Fk(i)p Fj(+2)910 606 y Fm(a)8 b(:)g(:)g(:)g(ab)1057 585 y Fk(k)q Fi(\003)p Fj(\()p Fk(i)p Fj(+1\))1180 606 y Fm(a;)35 b Fn(1)14 b Fl(\024)g Fm(i)f Fl(\024)h Fm(n:)60 719 y Fg(Lemma)i(3.4)24 b Fn(Let)f(\006)h(b)q(e)f(an)h(alphab)q(et)g(and)g(let)e Fm(\013;)8 b(\014)28 b Fl(2)e Fn(\(\006)17 b Fl([)f Fm(V)11 b Fn(\))1390 701 y Fj(+)1443 719 y Fn(b)q(e)23 b(similar)e(pat-)60 779 y(terns)d(of)g(the)f(form)g Fm(\013)g Fn(=)f Fm(\013)576 786 y Fj(0)596 779 y Fm(u)624 786 y Fj(1)643 779 y Fm(\013)674 786 y Fj(1)694 779 y Fm(u)722 786 y Fj(2)750 779 y Fm(:)8 b(:)g(:)f(\013)846 786 y Fk(m)p Fi(\000)p Fj(1)925 779 y Fm(u)953 786 y Fk(m)986 779 y Fm(\013)1017 786 y Fk(m)1068 779 y Fn(and)18 b Fm(\014)h Fn(=)d Fm(\014)1293 786 y Fj(0)1313 779 y Fm(u)1341 786 y Fj(1)1360 779 y Fm(\014)1388 786 y Fj(1)1407 779 y Fm(u)1435 786 y Fj(2)1463 779 y Fm(:)8 b(:)g(:)g(\014)1557 786 y Fk(m)p Fi(\000)p Fj(1)1635 779 y Fm(u)1663 786 y Fk(m)1696 779 y Fm(\014)1724 786 y Fk(m)1756 779 y Fn(.)60 839 y(If)18 b(\006)i(con)o(tains)f(t)o(w)o(o) g(distinct)f(letters)g Fm(a)g Fn(and)i Fm(b)f Fn(whic)o(h)f(do)h(not)h (o)q(ccur)f(in)g Fm(\013)g Fn(and)h Fm(\014)s Fn(,)e(then)60 899 y Fm(\034)81 907 y Fi(j)p Fk(\014)r Fi(j)p Fk(;a;b)178 899 y Fn(\()p Fm(\013)p Fn(\))c Fl(2)g Fm(L)341 906 y Fk(E)r(;)p Fj(\006)406 899 y Fn(\()p Fm(\014)s Fn(\))e(if)f(and)i(only) f(if)g(there)g(exists)f(a)i(substitution)f Fm(h)i Fn(:)g Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))e Fl(!)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1751 881 y Fi(\003)60 959 y Fn(suc)o(h)i(that)h Fm(h)p Fn(\()p Fm(\014)s Fn(\))c(=)h Fm(\013)p Fn(.)60 1080 y Fg(Pro)r(of:)21 b Fn(The)15 b Fh(if)g Fn(part)h(is)g(trivially)d (true.)21 b(The)15 b Fh(only)i(if)f Fn(part)f(can)h(b)q(e)g(pro)o(v)o (ed)e(analogous)k(to)60 1140 y(Lemma)13 b(7.1)j(of)g([JSSY93])f(as)h (follo)o(ws.)21 b(Let)15 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))e(=)h Fl(f)p Fm(x)1159 1147 y Fj(1)1179 1140 y Fm(;)8 b(:)g(:)g(:)f(;)h(x)1316 1147 y Fk(n)1339 1140 y Fl(g)15 b Fn(and)h Fm(k)g Fn(=)e Fl(j)p Fm(\014)s Fl(j)p Fn(.)20 b(Since)60 1200 y Fm(\034)81 1207 y Fk(k)q(;a;b)156 1200 y Fn(\()p Fm(\013)p Fn(\))f Fl(2)f Fm(L)328 1207 y Fk(E)r(;)p Fj(\006)394 1200 y Fn(\()p Fm(\014)s Fn(\),)g(there)h(is)f (a)i Fm(\027)h Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)i Fn(\006)1060 1182 y Fi(\003)1099 1200 y Fn(suc)o(h)f(that)i Fm(\027)s Fn(\()p Fm(\014)s Fn(\))e(=)g Fm(\034)1511 1207 y Fk(k)q(;a;b)1586 1200 y Fn(\()p Fm(\013)p Fn(\).)30 b(F)l(or)60 1260 y(ev)o(ery)11 b Fm(x)i Fl(2)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\),)e Fm(\034)464 1267 y Fk(k)q(;a;b)539 1260 y Fn(\()p Fm(x)p Fn(\))g(consists)h(of)f Fm(k)i Fn(segmen)o(ts)d(of)h(the)g(form)f Fm(ab)1376 1242 y Fk(j)1394 1260 y Fm(a)p Fn(.)19 b(Th)o(us,)13 b(for)g(ev)o(ery)60 1321 y Fm(x)88 1328 y Fk(i)116 1321 y Fl(2)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\),)h(there)h(is)g(at)g(least)g(one)g(segmen)o(t)f Fm(ab)1008 1302 y Fk(j)1022 1307 y Ff(i)1037 1321 y Fm(a)p Fn(,)g Fm(k)e Fl(\003)e Fm(i)g Fn(+)f(1)15 b Fl(\024)e Fm(j)1353 1328 y Fk(i)1381 1321 y Fl(\024)h Fm(k)f Fl(\003)e Fn(\()p Fm(i)f Fn(+)h(1\),)16 b(suc)o(h)60 1381 y(that)i(none)g(of)h 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1862 y Fn(,)g(0)g Fm(<)g(i)f(<)h(m)g Fn(\(the)60 1922 y(cases)i Fm(i)e Fn(=)f(0)k(and)f Fm(i)d Fn(=)h Fm(m)i Fn(are)g(similar)d(and)k(simpler\).)i(Let)d Fm(c)f Fn(b)q(e)h(the)g (last)g(letter)f(of)h Fm(u)1662 1929 y Fk(i)1691 1922 y Fn(and)60 1983 y Fm(d)j Fn(the)f(\014rst)h(letter)e(of)i Fm(u)511 1990 y Fk(i)p Fj(+1)588 1983 y Fn(\()p Fm(c)f Fn(=)f Fm(d)i Fn(is)f(p)q(ossible\).)28 b(Let)19 b(the)f(last)g(letter) g(of)g Fm(u)1512 1990 y Fk(i)1545 1983 y Fn(b)q(e)g(the)g Fm(p)p Fn(th)60 2043 y(app)q(earance)d(of)g Fm(c)f Fn(and)h(the)g (\014rst)f(letter)f(of)i Fm(u)887 2050 y Fk(i)p Fj(+1)961 2043 y Fn(b)q(e)f(the)g Fm(q)r Fn(th)g(app)q(earance)h(of)g Fm(d)g Fn(in)f Fm(\014)s Fn(.)20 b(Since)60 2103 y Fm(\027)s Fn(\()p Fm(y)r Fn(\))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 Fm(\027)s Fn(\()p Fm(y)r Fn(\))h(is)g(in)h(b)q(et)o(w)o(een)e(the)60 2163 y Fm(p)p Fn(th)i(app)q(earance)g(of)g Fm(c)g Fn(and)g(the)f Fm(q)r Fn(th)g(app)q(earance)h(of)g Fm(d)g Fn(in)f Fm(\027)s Fn(\()p Fm(\014)s Fn(\).)20 b(This)c(observ)m(ation)g(pla)o(ys)60 2223 y(a)h(k)o(ey)e(r^)-24 b(ole)16 b(in)f(the)h(pro)q(of.)133 2284 y(W)l(e)22 b(de\014ne)h(a)f(substitution)h Fm(h)i Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1171 2266 y Fi(\003)1213 2284 y Fn(as)f(follo)o(ws.)40 b(F)l(or)23 b(eac)o(h)f Fm(y)k Fl(2)60 2344 y Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\),)14 b(let)611 2404 y Fm(ab)658 2383 y Fk(j)672 2388 y Ff(i)683 2395 y Fe(1)704 2404 y Fm(aab)777 2383 y Fk(j)791 2388 y Ff(i)802 2395 y Fe(2)823 2404 y Fm(a)8 b(:)g(:)g(:)g(ab)970 2383 y Fk(j)984 2388 y Ff(i)995 2392 y(r)1016 2404 y Fm(a;)35 b(r)15 b Fl(\025)f Fn(0)p Fm(;)60 2491 y Fn(b)q(e)23 b(the)g(w)o(ord)g(obtained)g(from)f Fm(\027)s Fn(\()p Fm(y)r Fn(\))g(b)o(y)h(deleting)f(all)g(the)h(incomplete)d(segmen)o(ts) h(and)60 2551 y(segmen)o(ts)e(that)i(are)f(not)h(anc)o(hor)g(segmen)o (ts.)33 b(Note)20 b(that)g(the)h(indices)e Fm(i)1462 2558 y Fj(1)1481 2551 y Fm(;)8 b(i)1520 2558 y Fj(2)1540 2551 y Fm(;)g(:)g(:)g(:)f(;)h(i)1666 2558 y Fk(r)1705 2551 y Fn(are)60 2611 y(not)14 b(necessarily)f(distinct.)19 b(Hereb)o(y)13 b(a)h(segmen)o(t)e Fm(ab)1015 2593 y Fk(j)1029 2598 y Ff(i)1040 2602 y(s)1060 2611 y Fm(a)i Fn(anc)o(hors)g(a)g(v)m (ariable)g Fm(x)1522 2618 y Fk(i)1534 2622 y Ff(s)1566 2611 y Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\),)60 2671 y(1)h Fl(\024)f Fm(i)169 2678 y Fk(s)201 2671 y Fl(\024)g Fm(n)p Fn(.)22 b(Then)17 b(w)o(e)f(de\014ne)g Fm(h)p Fn(\()p Fm(y)r Fn(\))e(=)h Fm(x)847 2678 y Fk(i)859 2683 y Fe(1)878 2671 y Fm(x)906 2678 y Fk(i)918 2683 y Fe(2)945 2671 y Fm(:)8 b(:)g(:)g(x)1039 2678 y Fk(i)1051 2682 y Ff(r)1070 2671 y Fn(.)22 b(Clearly)l(,)15 b Fm(h)p Fn(\()p Fm(\014)s Fn(\))f(=)g Fm(\013)j Fn(b)q(ecause)g(eac)o(h)60 2731 y(app)q(earance)g(of)g Fm(x)f Fn(in)f Fm(\013)i Fn(has)g(exactly)e(one)h(anc)o(hor)h(segmen)o(t)e(w.r.t.)g Fm(\014)s Fn(.)20 b Fd(2)903 2856 y Fn(6)p eop %%Page: 7 7 7 6 bop 133 203 a Fn(No)o(w)16 b(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.)60 310 y Fg(Theorem)j(3.5)24 b Fn(Let)13 b(\006)h(b)q(e)g(an)g(alphab)q (et)g(and)g(let)f Fm(\013;)8 b(\014)16 b Fl(2)e Fn(\(\006)6 b Fl([)g Fm(V)11 b Fn(\))1316 292 y Fj(+)1359 310 y Fn(b)q(e)j(similar) d(patterns.)60 370 y(If)17 b(\006)h(con)o(tains)f(t)o(w)o(o)h(distinct) e(constan)o(ts)i Fm(a)f Fn(and)i Fm(b)e Fn(not)h(o)q(ccurring)f(in)g Fm(\013)h Fn(and)g Fm(\014)s Fn(,)f(then)g(the)60 430 y(follo)o(wing)f(statemen)o(ts)f(are)h(equiv)m(alen)o(t:)120 526 y(1.)24 b Fm(L)215 533 y Fk(E)r(;)p Fj(\006)280 526 y Fn(\()p Fm(\013)p Fn(\))14 b Fl(\022)g Fm(L)449 533 y Fk(E)r(;)p Fj(\006)514 526 y Fn(\()p Fm(\014)s Fn(\).)120 626 y(2.)24 b Fm(\034)203 634 y Fi(j)p Fk(\014)r Fi(j)p Fk(;a;b)300 626 y Fn(\()p Fm(\013)p Fn(\))14 b Fl(2)g Fm(L)463 633 y Fk(E)r(;)p Fj(\006)528 626 y Fn(\()p Fm(\014)s Fn(\).)120 726 y(3.)24 b(There)16 b(exists)g(a)g(substitution)h Fm(h)d Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1203 707 y Fi(\003)1239 726 y Fn(suc)o(h)i(that)g Fm(h)p Fn(\()p Fm(\014)s Fn(\))d(=)h Fm(\013)p Fn(.)60 821 y Fg(Pro)r(of:)25 b Fn(The)18 b(implications)e (\(1\))h Fl(\))g Fn(\(2\))h(and)h(\(3\))e Fl(\))g Fn(\(1\))i(are)f (trivially)e(true.)27 b(\(2\))17 b Fl(\))g Fn(\(3\))60 882 y(has)g(b)q(een)f(pro)o(v)o(ed)g(in)g(Lemma)e(3.4.)21 b Fd(2)60 988 y Fg(Corollary)d(3.6)24 b Fn(F)l(or)15 b(t)o(w)o(o)g(similar)e(patterns)i(and)g(an)h(alphab)q(et)f(\006)h (with)e Fm(a;)8 b(b)13 b Fl(2)h Fn(\006,)h(where)60 1048 y Fm(a)h Fn(and)h Fm(b)f Fn(do)g(not)h(app)q(ear)g(in)f Fm(\013)h Fn(and)g Fm(\014)s Fn(,)e(it)h(is)g(decidable)f(whether)h Fm(L)1341 1055 y Fk(E)r(;)p Fj(\006)1406 1048 y Fn(\()p Fm(\013)p Fn(\))e Fl(\022)g Fm(L)1575 1055 y Fk(E)r(;)p Fj(\006)1640 1048 y Fn(\()p Fm(\014)s Fn(\).)60 1169 y Fg(Pro)r(of:)25 b Fn(By)17 b(Theorem)g(3.5,)h(it)f(su\016ces)h(to)g (sho)o(w)h(the)f(decidabilit)o(y)d(of)k(the)e(mem)o(b)q(ership)60 1229 y(problem.)31 b(It)20 b(is)g(not)h(v)o(ery)e(di\016cult)f(to)j (pro)o(v)o(e)e(that)i(the)f(mem)n(b)q(ership)e(problem)g(for)j(E-)60 1289 y(patterns)16 b(is)f(decidable)f(\(in)h(fact,)g(it)g(is)g (NP-complete;)e(see)i([Ang80a)q(,)g(JKS)1470 1271 y Fj(+)1500 1289 y Fn(94]\).)21 b(F)l(or)15 b(the)60 1349 y(sak)o(e)d(of)g (completeness,)e(w)o(e)i(include)e(the)i(pro)q(of)h(whic)o(h)e(app)q (eared)i(already)f(in)f([Ang80a)q(])h(for)60 1409 y(NE-patterns.)21 b(Let)14 b Fm(\013)g Fn(b)q(e)g(a)g(pattern)h(with)e Fm(k)j Fn(distinct)d(v)m(ariables,)h(and)h(let)e Fm(w)i Fn(b)q(e)f(a)g(w)o(ord)g(in)60 1470 y(\006)95 1451 y Fi(\003)131 1470 y Fn(of)j(length)f Fm(n)p Fn(.)21 b(There)16 b(are)h Fl(O)q Fn(\()p Fm(n)712 1451 y Fj(2)732 1470 y Fn(\))f(distinct)f(sub)o(w)o(ords)i(of)g Fm(w)q Fn(.)22 b(En)o(umerate)14 b(all)i Fm(k)r Fn(-tuples)60 1530 y(of)e(those)g(sub) o(w)o(ords)h(of)f Fm(w)h Fn({)f(there)f(are)h Fl(O)q Fn(\()p Fm(n)879 1512 y Fj(2)p Fk(k)918 1530 y Fn(\))g(of)g(them.)19 b(Eac)o(h)14 b(of)g(these)f Fm(k)r Fn(-tuples)h(de\014nes)60 1590 y(a)21 b(substitution)g Fm(\033)r Fn(.)34 b(Clearly)l(,)20 b(it)g(can)h(b)q(e)g(c)o(hec)o(k)o(ed)d(in)i Fl(O)q Fn(\()p Fm(n)p Fn(\))h(time)e(whether)h Fm(\033)r Fn(\()p Fm(\013)p Fn(\))h(=)g Fm(w)q Fn(.)60 1650 y(Hence)16 b(there)i(is)f(an)h (algorithm)f(whic)o(h)g(decides)g(mem)n(b)q(ership)e(of)j Fm(w)h Fn(in)f Fm(L)1465 1657 y Fk(E)r(;)p Fj(\006)1530 1650 y Fn(\()p Fm(\013)p Fn(\))g(in)f(time)60 1710 y Fl(O)q Fn(\()p Fm(n)149 1692 y Fj(2)p Fk(k)q Fj(+1)233 1710 y Fn(\).)36 b(Similarly)l(,)19 b(it)i(is)g(decidable)f(whether)h (or)g(not)h(there)f(exists)f(a)i(substitution)60 1771 y Fm(h)14 b Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))g Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))493 1752 y Fi(\003)529 1771 y Fn(suc)o(h)j(that)h Fm(h)p Fn(\()p Fm(\014)s Fn(\))c(=)g Fm(\013)p Fn(.)22 b(Let)16 b Fm(car)q(d)p Fn(\()p Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\)\))d(=)h Fm(k)k Fn(and)f(let)142 1874 y Fm(S)s(W)7 b Fn(\()p Fm(\013)p Fn(\))14 b(=)g Fl(f)p Fm(\015)j Fl(2)d Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))621 1853 y Fi(\003)656 1874 y Fl(j)i Fm(\015)k Fn(is)c(a)g(sub)o(w)o(ord)h(of)g(some)e(v)m(ariable)h(segmen) o(t)e(of)j Fm(\013)p Fl(g)p Fm(:)60 1977 y Fn(En)o(umerate)j(all)g Fm(k)r Fn(-tuples)h(of)h(elemen)o(ts)c(from)i Fm(S)s(W)7 b Fn(\()p Fm(\013)p Fn(\).)36 b(Eac)o(h)21 b(of)h(these)f(\014nitely)e (man)o(y)60 2037 y Fm(k)r Fn(-tuples)c(de\014nes)f(a)i(substitution)f Fm(h)f Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1147 2019 y Fi(\003)1181 2037 y Fn(in)h(the)f(ob)o(vious)h(w)o(a)o(y)l(.)20 b(Clearly)l(,)60 2097 y(it)c(can)g(b)q(e)h(c)o(hec)o(k)o(ed)c(in)j(linear)g(time)e(\(in) i Fl(j)p Fm(\013)p Fl(j)p Fn(\))g(whether)g(or)h(not)f Fm(h)p Fn(\()p Fm(\014)s Fn(\))d(=)h Fm(\013)p Fn(.)22 b Fd(2)60 2204 y Fg(Corollary)c(3.7)24 b Fn(If)16 b Fm(car)q(d)p Fn(\(\006\))f Fl(\025)g Fn(3,)i(then)f(the)h(equiv)m(alence)e(problem)g (for)i(E-pattern)h(lan-)60 2264 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 2324 y(that)f(do)f(not)h(o)q(ccur)g(in)e(the)h(patterns.) 60 2444 y Fg(Pro)r(of:)21 b Fn(This)c(follo)o(ws)f(from)f(Theorem)g (3.1)h(and)h(Corollary)g(3.6.)k Fd(2)133 2551 y Fn(Notice)f(that)h(the) f(inclusion)g(problem)f(\(and)i(hence)f(the)g(equiv)m(alence)f (problem\))g(for)60 2611 y(E-pattern)e(languages)g(is)f(particularly)g (decidable)f(if)g(\006)i(is)f(in\014nite.)133 2671 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 2731 y(lik)o(e)13 b(to)i(get)g(rid)f(of)h(the)g(anno)o(ying)g(condition)f (\\\006)i(has)f(to)g(con)o(tain)g(t)o(w)o(o)f(distinct)g(constan)o(ts) 903 2856 y(7)p eop %%Page: 8 8 8 7 bop 60 203 a Fm(a)17 b Fn(and)h Fm(b)f Fn(that)h(do)g(not)g(o)q (ccur)g(in)f Fm(\013)h Fn(and)g Fm(\014)s Fn(".)24 b(The)18 b(crucial)e(p)q(oin)o(t)i(is)f(that)h(this)f(condition)60 264 y(enabled)f(us)g(to)h(infer)f(the)g(second)g(implication)e(b)q(elo) o(w)i(\(cf.)f(Lemma)f(3.4\))278 374 y Fm(L)311 381 y Fk(E)r(;)p Fj(\006)376 374 y Fn(\()p Fm(\013)p Fn(\))h Fl(\022)e Fm(L)545 381 y Fk(E)r(;)p Fj(\006)610 374 y Fn(\()p Fm(\014)s Fn(\))g Fl(\))h Fm(\034)777 381 y Fi(j)p Fk(\014)r Fi(j)p Fk(;a;b)874 374 y Fn(\()p Fm(\013)p Fn(\))g Fl(2)g Fm(L)1037 381 y Fk(E)r(;)p Fj(\006)1102 374 y Fn(\()p Fm(\014)s Fn(\))f Fl(\))h(9)p Fm(h)g Fn(:)f Fm(h)p Fn(\()p Fm(\014)s Fn(\))g(=)h Fm(\013:)60 484 y Fn(Ho)o(w)o(ev)o(er,)e(if)h(there)f(are)i(no)g(extra)f(constan)o(ts,) h(then)g(it)e(is)i(not)g(clear)e(at)i(all)f(whic)o(h)g(enco)q(ding)60 544 y Fm(\034)81 552 y Fi(j)p Fk(\014)r Fi(j)p Fk(;l)144 557 y Fe(1)160 552 y Fk(;l)181 557 y Fe(2)200 544 y Fn(\()p Fm(\013)p Fn(\))21 b(should)h(b)q(e)g(tak)o(en,)f(i.e.,)g(whic)o(h)g (constan)o(ts)h Fm(l)1154 551 y Fj(1)1173 544 y Fm(;)8 b(l)1210 551 y Fj(2)1252 544 y Fl(2)22 b Fn(\006)p Fm(;)31 b(l)1402 551 y Fj(1)1444 544 y Fl(6)p Fn(=)22 b Fm(l)1519 551 y Fj(2)1560 544 y Fn(should)g(b)q(e)60 604 y(c)o(hosen)c(to)g(enco) q(de)g(the)g(v)m(ariables)g(in)f Fm(\013)p Fn(.)27 b(A)18 b(p)q(ossible)g(solution)g(w)o(ould)g(b)q(e)g(to)g(generalize)60 664 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 Fm(F)26 b Fl(\032)20 b Fm(L)1111 671 y Fk(E)r(;)p Fj(\006)1176 664 y Fn(\()p Fm(\013)p Fn(\))g(suc)o(h)f(that)h Fm(F)26 b Fl(\032)19 b Fm(L)1636 671 y Fk(E)r(;)p Fj(\006)1702 664 y Fn(\()p Fm(\014)s Fn(\))60 724 y(implies)f(the)j(existence)e(of)j(a)f (substitution)g Fm(h)g Fn(suc)o(h)g(that)g Fm(h)p Fn(\()p Fm(\014)s Fn(\))h(=)f Fm(\013)p Fn(.)36 b(A)21 b(v)o(ery)f(natural)60 785 y(candidate)c(w)o(ould)g(b)q(e)h(the)f(set)g(of)h(all)e(enco)q (dings,)i(i.e.,)d(the)i(set)526 895 y Fm(F)k Fn(=)14 b Fl(f)p Fm(\034)676 902 y Fi(j)p Fk(\014)r Fi(j)p Fk(;l)739 907 y Fe(1)754 902 y Fk(;l)775 907 y Fe(2)794 895 y Fn(\()p Fm(\013)p Fn(\))j Fl(j)f Fm(l)925 902 y Fj(1)944 895 y Fm(;)8 b(l)981 902 y Fj(2)1014 895 y Fl(2)14 b Fn(\006)p Fm(;)22 b(l)1147 902 y Fj(1)1180 895 y Fl(6)p Fn(=)14 b Fm(l)1247 902 y Fj(2)1266 895 y Fl(g)p Fm(:)60 1005 y Fn(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 Fm(car)q(d)p Fn(\(\006\))f(=)g(3)h (\(the)60 1065 y(situation)h(migh)o(t)f(b)q(e)h(di\013eren)o(t)g(for)g Fm(car)q(d)p Fn(\(\006\))f Fl(\025)e Fn(4\).)60 1179 y Fg(Example)j(3.8)24 b Fn(Let)d(\006)f(=)h Fl(f)p Fm(a;)8 b(b;)g(c)p Fl(g)p Fn(,)19 b Fm(\013)i Fn(=)g Fm(w)9 b(a)f(x)g(b)g(y)r (xz)22 b Fn(and)e Fm(\014)j Fn(=)31 b(\026)-34 b Fm(w)9 b(a)i Fn(\026)-27 b Fm(x)8 b(b)j Fn(\026)-27 b Fm(y)5 b Fn(\026)-27 b Fm(x)o Fn(.)33 b(There)20 b(is)60 1239 y(a)i(substitution)g Fm(f)29 b Fn(:)23 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))858 1221 y Fi(\003)898 1239 y Fn(suc)o(h)d(that)i Fm(f)5 b Fn(\()p Fm(\013)p Fn(\))24 b(=)f Fm(\014)s Fn(.)38 b(Hence)20 b Fm(L)1574 1246 y Fk(E)r(;)p Fj(\006)1640 1239 y Fn(\()p Fm(\014)s Fn(\))i Fl(\022)60 1299 y Fm(L)93 1306 y Fk(E)r(;)p Fj(\006)158 1299 y Fn(\()p Fm(\013)p Fn(\).)38 b(The)21 b(opp)q(osite)h(inclusion)f(do)q(es)h(not)g(hold)g (b)q(ecause)g Fm(cacbcb)g Fl(2)h Fm(L)1541 1306 y Fk(E)r(;)p Fj(\006)1606 1299 y Fn(\()p Fm(\013)p Fn(\))f(but)60 1359 y Fm(cacbcb)e Fl(62)h Fm(L)298 1366 y Fk(E)r(;)p Fj(\006)363 1359 y Fn(\()p Fm(\014)s Fn(\).)33 b(And)20 b(indeed,)g(there)g(is)g(no)h(substitution)f Fm(g)j Fn(:)e Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))e Fl(!)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1751 1341 y Fi(\003)60 1420 y Fn(suc)o(h)i(that)g Fm(g)r Fn(\()p Fm(\014)s Fn(\))h(=)h Fm(\013)p Fn(.)42 b(Despite)22 b(this)h(fact,)g(w)o(e)g(ha)o(v)o(e)f Fm(\034)1177 1427 y Fi(j)p Fk(\014)r Fi(j)p Fk(;l)1240 1432 y Fe(1)1256 1427 y Fk(;l)1277 1432 y Fe(2)1296 1420 y Fn(\()p Fm(\013)p Fn(\))j Fl(2)g Fm(L)1481 1427 y Fk(E)r(;)p Fj(\006)1546 1420 y Fn(\()p Fm(\014)s Fn(\))d(for)i(all)60 1480 y Fm(l)75 1487 y Fj(1)94 1480 y Fm(;)8 b(l)131 1487 y Fj(2)164 1480 y Fl(2)14 b Fn(\006)p Fm(;)22 b(l)297 1487 y Fj(1)330 1480 y Fl(6)p Fn(=)14 b Fm(l)397 1487 y Fj(2)416 1480 y Fn(.)21 b(The)c(easy)f(pro)q(of)h(is)f(left)g(as)h (an)f(exercise)f(to)i(the)f(reader.)133 1594 y(Note)j(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 1654 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 1714 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 1835 y Fg(Op)r(en)i(question)g(2:)j Fn(Do)q(es)c(the)f(equiv)m(alence:)424 1945 y Fm(L)457 1952 y Fk(E)r(;)p Fj(\006)522 1945 y Fn(\()p Fm(\013)p Fn(\))e(=)g Fm(L)690 1952 y Fk(E)r(;)p Fj(\006)755 1945 y Fn(\()p Fm(\014)s Fn(\))55 b Fl(,)g Fm(L)1017 1952 y Fk(E)r(;)p Fj(\006)1080 1943 y Fc(0)1094 1945 y Fn(\()p Fm(\013)p Fn(\))14 b(=)f Fm(L)1261 1952 y Fk(E)r(;)p Fj(\006)1324 1943 y Fc(0)1338 1945 y Fn(\()p Fm(\014)s Fn(\))60 2055 y(hold)j(for)h Fm(car)q(d)p Fn(\(\006\))d Fl(\025)g Fn(3,)i Fm(\013;)8 b(\014)16 b Fl(2)e Fn(\(\006)e Fl([)f Fm(V)g Fn(\))844 2037 y Fj(+)890 2055 y Fn(and)17 b(\006)1020 2037 y Fi(0)1045 2055 y Fn(=)d(\006)e Fl([)f(f)p Fm(a)p Fl(g)p Fn(,)k(where)h Fm(a)d Fl(62)h Fn(\006?)60 2175 y(Again,)j(the)h(condition)f Fm(car)q(d)p Fn(\(\006\))g Fl(\025)f Fn(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 2235 y(that)g(the)f Fh(only)i(if)e Fn(part)g(is)g(the)g(crucial)g(one)g(as)h(the)f Fh(if)g Fn(part)h(holds)f(trivially)l(.)133 2296 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 2356 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 2416 y(cial)g(cases.)60 2530 y Fg(Prop)r(osition)j(3.9)24 b Fn(Let)16 b(\006)h(b)q(e)f(an)h(alphab)q(et)g(and)f(let)g Fm(\013;)8 b(\014)16 b Fl(2)e Fn(\(\006)d Fl([)h Fm(V)f Fn(\))1411 2512 y Fj(+)1457 2530 y Fn(b)q(e)16 b(similar)e(pat-)60 2590 y(terns)k(of)g(the)f(form)g Fm(\013)g Fn(=)f Fm(\013)576 2597 y Fj(0)596 2590 y Fm(u)624 2597 y Fj(1)643 2590 y Fm(\013)674 2597 y Fj(1)694 2590 y Fm(u)722 2597 y Fj(2)750 2590 y Fm(:)8 b(:)g(:)f(\013)846 2597 y Fk(m)p Fi(\000)p Fj(1)925 2590 y Fm(u)953 2597 y Fk(m)986 2590 y Fm(\013)1017 2597 y Fk(m)1068 2590 y Fn(and)18 b Fm(\014)h Fn(=)d Fm(\014)1293 2597 y Fj(0)1313 2590 y Fm(u)1341 2597 y Fj(1)1360 2590 y Fm(\014)1388 2597 y Fj(1)1407 2590 y Fm(u)1435 2597 y Fj(2)1463 2590 y Fm(:)8 b(:)g(:)g(\014)1557 2597 y Fk(m)p Fi(\000)p Fj(1)1635 2590 y Fm(u)1663 2597 y Fk(m)1696 2590 y Fm(\014)1724 2597 y Fk(m)1756 2590 y Fn(.)60 2650 y(If)16 b Fm(car)q(d)p Fn(\(\006\))e Fl(\025)g Fm(m)c Fn(+)h(2,)17 b Fm(m)c Fl(\025)h Fn(0,)i(then)g(it)g(is)g (decidable)f(whether)h Fm(L)1307 2657 y Fk(E)r(;)p Fj(\006)1372 2650 y Fn(\()p Fm(\013)p Fn(\))e Fl(\022)g Fm(L)1541 2657 y Fk(E)r(;)p Fj(\006)1606 2650 y Fn(\()p Fm(\014)s Fn(\).)903 2856 y(8)p eop %%Page: 9 9 9 8 bop 60 203 a Fg(Pro)r(of:)21 b Fn(Let)16 b Fm(l)335 210 y Fj(1)369 203 y Fn(b)q(e)g(a)g(constan)o(t)g(in)f Fm(u)755 210 y Fj(1)775 203 y Fn(,)g Fm(l)819 210 y Fj(2)854 203 y Fn(b)q(e)h(a)g(constan)o(t)g(in)f Fm(u)1240 210 y Fj(2)1275 203 y Fn(\()p Fm(l)1309 210 y Fj(1)1342 203 y Fn(=)f Fm(l)1409 210 y Fj(2)1444 203 y Fn(is)h(p)q(ossible\))h(etc.) 60 264 y(Since)11 b Fm(car)q(d)p Fn(\()p Fl(f)p Fm(l)337 271 y Fj(1)356 264 y Fm(;)d(l)393 271 y Fj(2)412 264 y Fm(;)g(:)g(:)g(:)g(;)g(l)537 271 y Fk(m)570 264 y Fl(g)p Fn(\))13 b Fl(\024)h Fm(m)p Fn(,)e(there)f(are)h(t)o(w)o(o)g(distinct)f (constan)o(ts)h Fm(a)g Fn(and)g Fm(b)g Fn(in)f(the)h(set)60 324 y(\006)q Fl(n)q(f)p Fm(l)162 331 y Fj(1)180 324 y Fm(;)c(l)217 331 y Fj(2)236 324 y Fm(;)g(:)g(:)g(:)g(;)g(l)361 331 y Fk(m)394 324 y Fl(g)p Fn(.)19 b(It)11 b(follo)o(ws)g(as)g(in)g 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b(whether)h Fm(L)494 620 y Fk(E)r(;)p Fj(\006)559 613 y Fn(\()p Fm(\013)p Fn(\))e(=)g Fm(L)727 620 y Fk(E)r(;)p Fj(\006)792 613 y Fn(\()p Fm(\014)s Fn(\))i(pro)o(vided)f(that)i Fm(car)q(d)p Fn(\(\006\))d Fl(\025)g Fm(m)d Fn(+)g(2,)16 b Fm(m)d Fl(\025)h Fn(0.)60 734 y Fg(Pro)r(of:)k Fn(If)11 b Fm(\013)g Fn(and)h Fm(\014)h Fn(are)e(similar,)e(then)i Fm(L)831 741 y Fk(E)r(;)p Fj(\006)897 734 y Fn(\()p Fm(\013)p Fn(\))j(=)f Fm(L)1064 741 y Fk(E)r(;)p Fj(\006)1130 734 y Fn(\()p Fm(\014)s Fn(\))d(is)h(decidable)e(b)o(y)i(Prop)q(osition)60 794 y(3.9.)45 b(If)23 b Fm(m)k Fn(=)g(0)d(and)h Fm(\013)f Fn(and)h Fm(\014)h Fn(are)e(not)h(similar,)e(then)h Fm(\013)g Fn(is)g(terminal)e(free)h(and)i Fm(\014)60 854 y Fn(con)o(tains)f (terminals.)43 b(Th)o(us)24 b Fm(L)680 861 y Fk(E)r(;)p Fj(\006)745 854 y Fn(\()p Fm(\013)p Fn(\))j Fl(6)p Fn(=)g Fm(L)939 861 y Fk(E)r(;)p Fj(\006)1005 854 y Fn(\()p Fm(\014)s Fn(\))c(b)q(ecause)h(the)g(empt)o(y)d(w)o(ord)k(is)f(an)60 914 y(elemen)o(t)16 b(of)j Fm(L)334 921 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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 2190 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 2250 y(v)m(ariables;)i(see)f([Ang80a)q(].)30 b(F)l(or)20 b(E-patterns,)h(ho)o(w)o(ev)o(er,)d(this)h(is)h(not)f(true) h(at)f(all.)31 b(Jiang)60 2310 y Fh(et)17 b(al.)k Fn([JKS)292 2292 y Fj(+)322 2310 y Fn(94])15 b(write)f(that:)21 b(\\Tw)o(o)16 b(v)o(ery)e(di\013eren)o(t)g(lo)q(oking)i(E-patterns)g(ma)o(y)d (generate)60 2370 y(the)20 b(same)f(language.)34 b(F)l(or)21 b(instance,)f(if)g(a)g(terminal-free)e(pattern)i Fm(\013)h Fn(con)o(tains)f(exactly)60 2430 y(one)15 b(o)q(ccurrence)f(of)h(some)f (v)m(ariable,)g(then)g Fm(\013)h Fn(is)g(equiv)m(alen)o(t)e(to)i(the)g (pattern)g Fm(x)p Fn(".)20 b(The)15 b(fact)60 2491 y(that)f(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 2551 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 2611 y(a)f(v)m(ariable)f Fm(x)h Fn(in)f(an)h(E-pattern)g Fm(\013)j Fl(2)f Fn(\(\006)r Fl([)r Fm(V)d Fn(\))889 2593 y Fj(+)930 2611 y Fh(sup)n(er\015uous)p Fn(,)g(if)g(the)h (E-pattern)g Fm(\013)1562 2593 y Fi(0)1586 2611 y Fn(obtained)60 2671 y(from)18 b Fm(\013)h Fn(b)o(y)f(deleting)f(all)h(o)q(ccurrences)h (of)f Fm(x)h Fn(still)e(generates)i Fm(L)1270 2678 y Fk(E)r(;)p Fj(\006)1335 2671 y Fn(\()p Fm(\013)p Fn(\).)29 b(W)l(e)18 b(next)g(tac)o(kle)60 2731 y(the)h(equiv)m(alence)f(problem) g(of)i(t)o(w)o(o)g(E-patterns)g Fm(\013)g Fn(and)g Fm(\014)i Fn(b)o(y)e(eliminating)d(sup)q(er\015uous)903 2856 y(9)p eop %%Page: 10 10 10 9 bop 60 203 a Fn(v)m(ariables)20 b(in)f Fm(\013)h Fn(and)h Fm(\014)s Fn(.)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)60 264 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)60 324 y(not)g(kno)o(w)g(ho)o(w)f(to)h(decide)e(this)i(in)f(general,)g(w)o (e)g(b)o(y-pass)h(the)f(problem)f(b)o(y)h(successiv)o(ely)60 384 y(deleting)c(those)i(v)m(ariables)f(from)g Fm(\013)h Fn(for)f(whic)o(h)g(w)o(e)g(can)g(pro)o(v)o(e)g(that)h(they)f(are)g (sup)q(er\015uous.)60 495 y Fg(De\014nition)k(4.1)24 b Fn(Let)f Fm(\013)i Fl(2)g Fn(\(\006)16 b Fl([)g Fm(V)11 b Fn(\))803 477 y Fj(+)856 495 y Fn(b)q(e)23 b(a)g(pattern.)41 b(Let)23 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))i(=)g Fm(V)1542 502 y Fk(d)1578 495 y Fl([)16 b Fm(V)1655 502 y Fk(nd)1720 495 y Fn(b)q(e)60 555 y(an)24 b(arbitrary)g(partition)g (of)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))f(in)o(to)g(t)o(w)o(o)h (disjoin)o(t)f(sets)h Fm(V)1309 562 y Fk(nd)1375 555 y Fn(and)g Fm(V)1505 562 y Fk(d)1552 555 y Fl(6)p Fn(=)i Fl(;)p Fn(.)44 b(Let)60 615 y Fm(\013)91 597 y Fi(0)117 615 y Fl(2)14 b Fn(\(\006)p Fl([)p Fm(V)e Fn(\))310 597 y Fj(+)350 615 y Fn(b)q(e)f(the)g(pattern)g(obtained)g(from)f Fm(\013)h Fn(b)o(y)f(deleting)g(all)h(v)m(ariables)g(of)g Fm(V)1584 622 y Fk(d)1615 615 y Fn(from)f Fm(\013)p Fn(.)60 675 y(More)16 b(precisely)l(,)f Fm(\013)427 657 y Fi(0)454 675 y Fn(=)f Fm(d)p Fn(\()p Fm(\013)p Fn(\),)j(where)g Fm(d)e Fn(=)f Fl(f)p Fm(x)h Fl(7!)f Fm(\025)j Fl(j)g Fm(x)d Fl(2)h Fm(V)1191 682 y Fk(d)1212 675 y Fl(g)p Fn(.)23 b(Let)16 b Fm(V)1389 682 y Fk(nd)1446 675 y Fn(=)f Fl(f)p Fm(x)1552 682 y Fj(1)1571 675 y Fm(;)8 b(:)g(:)g(:)f(;)h(x)1708 682 y Fk(n)1732 675 y Fl(g)p Fn(.)60 736 y(Ev)o(ery)15 b(substitution)i(of)f(the)g(form)352 843 y Fm(\033)f Fn(=)f Fl(f)p Fm(x)500 850 y Fk(i)527 843 y Fl(7!)g Fm(\015)616 850 y Fk(i)630 843 y Fm(x)658 850 y Fk(i)672 843 y Fm(\015)700 822 y Fi(0)697 855 y Fk(i)728 843 y Fl(j)i Fm(x)786 850 y Fk(i)814 843 y Fl(2)e Fm(V)889 850 y Fk(nd)931 843 y Fm(;)35 b(\015)1005 850 y Fk(i)1020 843 y Fm(;)8 b(\015)1070 822 y Fi(0)1067 855 y Fk(i)1095 843 y Fl(2)14 b Fm(V)1181 822 y Fi(\003)1170 855 y Fk(d)1201 843 y Fm(;)36 b Fn(1)14 b Fl(\024)f Fm(i)h Fl(\024)f Fm(n)p Fl(g)60 950 y Fn(will)22 b(b)q(e)h(called)f Fh(line)n(ar)p Fn(.)41 b(No)o(w)23 b(for)g(all)f(linear)h(substitutions)g(satisfying)1489 916 y Fb(P)1533 930 y Fk(n)1533 962 y(i)p Fj(=1)1600 950 y Fl(j)p Fm(\015)1639 957 y Fk(i)1653 950 y Fm(\015)1681 932 y Fi(0)1678 962 y Fk(i)1693 950 y Fl(j)i Fn(=)60 1010 y Fl(j)p Fm(\013)p Fl(j)11 b(\000)g(j)p Fm(\013)225 992 y Fi(0)237 1010 y Fl(j)p Fn(,)16 b(c)o(hec)o(k)e(whether)i Fm(\033)r Fn(\()p Fm(\013)679 992 y Fi(0)691 1010 y Fn(\))e(=)g Fm(\013)p Fn(.)22 b(If)16 b(there)f(is)h(suc)o(h)h(a)f Fm(\033)r Fn(,)g(then)g(w)o(e)g(sa)o(y)g(that)h Fm(\013)g Fn(can)60 1082 y(b)q(e)h Fh(r)n(e)n(duc)n(e)n(d)f Fn(to)i Fm(\013)393 1064 y Fi(0)423 1082 y Fn(b)o(y)f(deleting)f(the)h(v)m (ariables)g Fm(V)997 1089 y Fk(d)1036 1082 y Fn(and)h(denote)f(this)g (b)o(y)g Fm(\013)1513 1053 y Fk(V)1534 1059 y Ff(d)1508 1082 y Fl(!)f Fm(\013)1606 1064 y Fi(0)1618 1082 y Fn(.)27 b(Since)60 1142 y Fm(V)88 1149 y Fk(d)125 1142 y Fn(can)16 b(b)q(e)h(inferred)e(from)g Fm(\013)i Fn(and)g Fm(\013)750 1124 y Fi(0)762 1142 y Fn(,)e(w)o(e)h(will)f(also)i(simply)d(write)i Fm(\013)e Fl(!)g Fm(\013)1474 1124 y Fi(0)1486 1142 y Fn(.)133 1253 y(In)22 b(the)f(ab)q(o)o(v)o(e)h(de\014nition,)g(it)g (su\016ces)f(to)i(c)o(hec)o(k)d(the)h(equalit)o(y)f Fm(\033)r Fn(\()p Fm(\013)1459 1235 y Fi(0)1471 1253 y Fn(\))j(=)h Fm(\013)e Fn(merely)60 1313 y(for)i(those)f(linear)g(substitutions)g Fm(\033)i Fn(whic)o(h)e(satisfy)1075 1280 y Fb(P)1119 1293 y Fk(n)1119 1326 y(i)p Fj(=1)1186 1313 y Fl(j)p Fm(\015)1225 1320 y Fk(i)1240 1313 y Fm(\015)1268 1295 y Fi(0)1265 1326 y Fk(i)1279 1313 y Fl(j)j Fn(=)f Fl(j)p Fm(\013)p Fl(j)16 b(\000)g(j)p Fm(\013)1557 1295 y Fi(0)1569 1313 y Fl(j)23 b Fn(b)q(ecause)60 1340 y Fb(P)104 1353 y Fk(n)104 1386 y(i)p Fj(=1)171 1373 y Fl(j)p Fm(\015)210 1380 y Fk(i)225 1373 y Fm(\015)253 1355 y Fi(0)250 1386 y Fk(i)264 1373 y Fl(j)e(6)p Fn(=)f Fl(j)p Fm(\013)p Fl(j)14 b(\000)g(j)p Fm(\013)528 1355 y Fi(0)539 1373 y Fl(j)20 b Fn(implies)e Fm(\033)r Fn(\()p Fm(\013)823 1355 y Fi(0)835 1373 y Fn(\))i Fl(6)p Fn(=)h Fm(\013)f Fn(already)l(.)33 b(W)l(e)20 b(illustrate)f(the)h(de\014nition)60 1434 y(b)o(y)15 b(a)i(small)d(example)g(whic)o(h)h(also)h(sho)o(ws)h (that)f(sometimes)d(it)i(is)h(necessary)g(to)g(eliminate)60 1494 y(more)f(than)i(one)f(v)m(ariable)g(in)g(one)g(reduction)g(step.) 60 1605 y Fg(Example)g(4.2)24 b Fn(Let)19 b Fm(\013)g Fn(=)g Fm(xz)r(y)r(z)r(y)r(z)r(z)r Fn(.)28 b(Then)19 b Fm(\013)g Fl(!)g Fm(\013)1086 1586 y Fi(0)1117 1605 y Fn(=)f Fm(x)h Fn(b)q(ecause)g(the)g(linear)g(substi-)60 1665 y(tution)f Fm(\033)f Fn(=)g Fl(f)p Fm(x)f Fl(7!)g Fm(xz)r(y)r(z)r(y)r(z)r(z)r Fl(g)g Fn(satis\014es)i Fm(\033)r Fn(\()p Fm(\013)925 1647 y Fi(0)936 1665 y Fn(\))e(=)g Fm(\013)p Fn(.)26 b(Note)18 b(that)g(neither)e Fm(\013)h Fl(!)f Fm(xy)r(y)j Fn(nor)60 1725 y Fm(\013)14 b Fl(!)g Fm(xz)r(z)r(z)r(z)r Fn(,)h(although)i Fm(xy)r(y)g Fn(and)g Fm(xz)r(z)r(z)r(z)h Fn(generate)e Fm(L)1091 1732 y Fk(E)r(;)p Fj(\006)1156 1725 y Fn(\()p Fm(\013)p Fn(\).)133 1836 y(Indeed,)g(it)h(is)g(decidable)f(whether)h(a)g(pattern)g Fm(\013)h Fn(is)f(reducible)e(or)j(not)f(b)q(ecause)h(there)60 1896 y(are)c(only)g(\014nitely)f(man)o(y)g(partitions)h(of)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))f(and)h(for)f(eac)o(h)g(partition)g (there)g(exist)f(only)60 1956 y(\014nitely)i(man)o(y)g(linear)g (substitutions)i(with)894 1923 y Fb(P)938 1936 y Fk(n)938 1968 y(i)p Fj(=1)1005 1956 y Fl(j)p Fm(\015)1044 1963 y Fk(i)1059 1956 y Fm(\015)1087 1938 y Fi(0)1084 1968 y Fk(i)1098 1956 y Fl(j)d Fn(=)g Fl(j)p Fm(\013)p Fl(j)d(\000)g(j)p Fm(\013)1343 1938 y Fi(0)1354 1956 y Fl(j)p Fn(.)60 2079 y Fg(Lemma)16 b(4.3)24 b Fn(If)18 b Fm(\013)461 2050 y Fk(V)482 2056 y Ff(d)456 2079 y Fl(!)g Fm(\013)555 2061 y Fi(0)567 2079 y Fn(,)h(then)g Fm(L)747 2086 y Fk(E)r(;)p Fj(\006)812 2079 y Fn(\()p Fm(\013)p Fn(\))g(=)g Fm(L)990 2086 y Fk(E)r(;)p Fj(\006)1055 2079 y Fn(\()p Fm(\013)1105 2061 y Fi(0)1117 2079 y Fn(\).)29 b(\(Hence)18 b(the)g(v)m(ariables)h Fm(V)1665 2086 y Fk(d)1705 2079 y Fn(are)60 2139 y(sup)q(er\015uous)f(in)e Fm(\013)p Fn(.\))60 2259 y Fg(Pro)r(of:)23 b Fm(d)p Fn(\()p Fm(\013)p Fn(\))16 b(=)f Fm(\013)429 2241 y Fi(0)458 2259 y Fn(implies)g Fm(L)658 2266 y Fk(E)r(;)p Fj(\006)723 2259 y Fn(\()p Fm(\013)773 2241 y Fi(0)785 2259 y Fn(\))h Fl(\022)f Fm(L)907 2266 y Fk(E)r(;)p Fj(\006)972 2259 y Fn(\()p Fm(\013)p Fn(\))i(and)h Fm(\033)r Fn(\()p Fm(\013)1234 2241 y Fi(0)1246 2259 y Fn(\))d(=)g Fm(\013)j Fn(implies)c Fm(L)1581 2266 y Fk(E)r(;)p Fj(\006)1647 2259 y Fn(\()p Fm(\013)p Fn(\))h Fl(\022)60 2320 y Fm(L)93 2327 y Fk(E)r(;)p Fj(\006)158 2320 y Fn(\()p Fm(\013)208 2302 y Fi(0)220 2320 y Fn(\).)21 b Fd(2)60 2430 y Fg(Lemma)16 b(4.4)24 b Fn(The)f(reduction)f(relation)h Fl(!)i(\032)h Fn(\(\006)16 b Fl([)g Fm(V)11 b Fn(\))1188 2412 y Fj(+)1233 2430 y Fl(\002)16 b Fn(\(\006)g Fl([)g Fm(V)11 b Fn(\))1465 2412 y Fj(+)1518 2430 y Fn(is)22 b(a)i(length-)60 2491 y(decreasing)16 b(partial)g(ordering.)60 2611 y Fg(Pro)r(of:)i Fn(In)11 b(order)g(to)h(pro)o(v)o(e)e(that)h Fl(!)g Fn(is)g(a)g (partial)g(ordering,)h(w)o(e)e(ha)o(v)o(e)g(to)i(sho)o(w)f (irre\015exivit)o(y)60 2671 y(and)20 b(transitivit)o(y)l(.)29 b(Irre\015exivit)o(y)16 b(of)k Fl(!)f Fn(follo)o(ws)h(from)e Fm(V)1141 2678 y Fk(d)1181 2671 y Fl(6)p Fn(=)h Fl(;)p Fn(.)31 b(W)l(e)19 b(sho)o(w)h(transitivit)o(y:)60 2731 y(Supp)q(ose)12 b Fm(\013)i Fl(!)g Fm(\013)386 2713 y Fi(0)409 2731 y Fn(and)d Fm(\013)529 2713 y Fi(0)555 2731 y Fl(!)j Fm(\013)650 2713 y Fi(00)671 2731 y Fn(,)e(i.e.,)e(there) g(are)h(deletion)f(substitutions)i Fm(d)1470 2713 y Fi(0)1482 2731 y Fm(;)c(d)1529 2713 y Fi(00)1561 2731 y Fn(and)k(linear)891 2856 y(10)p eop %%Page: 11 11 11 10 bop 60 203 a Fn(substitutions)15 b Fm(\033)379 185 y Fi(0)390 203 y Fm(;)8 b(\033)442 185 y Fi(00)476 203 y Fn(suc)o(h)14 b(that)h Fm(d)713 185 y Fi(0)725 203 y Fn(\()p Fm(\013)p Fn(\))f(=)g Fm(\013)891 185 y Fi(0)903 203 y Fn(,)g Fm(\033)961 185 y Fi(0)972 203 y Fn(\()p Fm(\013)1022 185 y Fi(0)1034 203 y Fn(\))g(=)f Fm(\013)i(d)1189 185 y Fi(00)1210 203 y Fn(\()p Fm(\013)1260 185 y Fi(0)1272 203 y Fn(\))f(=)g Fm(\013)1388 185 y Fi(00)1424 203 y Fn(and)g Fm(\033)1546 185 y Fi(0)1557 203 y Fn(\()p Fm(\013)1607 185 y Fi(00)1629 203 y Fn(\))g(=)f Fm(\013)1744 185 y Fi(0)1756 203 y Fn(.)60 264 y(De\014ne)k Fm(d)f Fn(=)g Fm(d)332 246 y Fi(00)366 264 y Fl(\016)c Fm(d)428 246 y Fi(0)457 264 y Fn(and)18 b Fm(\033)f Fn(=)f Fm(\033)682 246 y Fi(0)705 264 y Fl(\016)c Fm(\033)772 246 y Fi(00)793 264 y Fn(.)25 b(Ob)o(viously)l(,)16 b Fm(d)p Fn(\()p Fm(\013)p Fn(\))g(=)g Fm(\013)1267 246 y Fi(00)1306 264 y Fn(and)i Fm(\033)1432 246 y Fi(00)1453 264 y Fn(\()p Fm(\013)1503 246 y Fi(00)1524 264 y Fn(\))e(=)g Fm(\013)p Fn(.)25 b(It)17 b(is)60 324 y(not)i(to)q(o)h(di\016cult)d(to) j(pro)o(v)o(e)d(that)j Fm(\033)g Fn(is)e(linear,)h(hence)f(transitivit) o(y)f(of)i Fl(!)f Fn(follo)o(ws.)29 b(It)18 b(is)60 384 y(ob)o(vious)e(that)h Fm(\013)d Fl(!)g Fm(\013)482 366 y Fi(0)510 384 y Fn(implies)g Fl(j)p Fm(\013)p Fl(j)g Fm(>)f Fl(j)p Fm(\013)845 366 y Fi(0)857 384 y Fl(j)p Fn(,)j(th)o(us)g Fl(!)g Fn(is)g(length-decreasing.)21 b Fd(2)60 498 y Fg(De\014nition)d(4.5)24 b Fn(A)16 b(pattern)h Fm(\013)g Fn(is)f(said)h(to)g(b)q(e)g(in)f Fh(normal)i(form)e Fn(\(w.r.t.)22 b Fl(!)p Fn(\),)16 b(if)g(there)g(is)60 558 y(no)h(pattern)f Fm(\013)332 540 y Fi(0)360 558 y Fn(suc)o(h)g(that)h Fm(\013)d Fl(!)g Fm(\013)716 540 y Fi(0)728 558 y Fn(.)133 672 y(Since)i Fl(!)g Fn(is)g (length-decreasing,)g(the)h(successiv)o(e)e(reduction)h(of)g(a)h (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 Fm(\013)g Fn(ma)o(y)f(ha)o(v)o(e)g(t)o(w)o(o)60 793 y(distinct)12 b(normal)h(forms.)19 b(Just)14 b(consider)f Fm(\013)h Fn(=)g Fm(xy)r Fn(:)19 b Fm(\013)14 b Fl(!)g Fm(x)f Fn(and)g Fm(\013)i Fl(!)e Fm(y)r Fn(.)20 b(Our)13 b(next)g(goal)h(is)60 853 y(to)g(sho)o(w)h(that)f(the)g(normal)f(forms)g(of)i(a)f(pattern)g (are)g(iden)o(tical)e(up)j(to)f(v)m(ariable)g(renaming.)60 913 y(The)f(follo)o(wing)g(tec)o(hnical)f(lemm)o(a)f(essen)o(tially)g (enables)i(us)h(to)g(sho)o(w)f(that)h(if)f Fm(\013)h Fl(2)g Fn(\(\006)5 b Fl([)g Fm(V)12 b Fn(\))1741 895 y Fj(+)60 973 y Fn(is)j(a)g(pattern)g(in)g(normal)f(form)g(and)h Fm(h)f Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))h Fl(!)f Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1180 955 y Fi(\003)1215 973 y Fn(is)h(a)i(substitution)f(suc)o(h)g(that)60 1033 y Fm(\013)21 b Fn(is)g(a)g(\014xp)q(oin)o(t)g(of)h Fm(h)f Fn(\(i.e.,)f Fm(h)p Fn(\()p Fm(\013)p Fn(\))i(=)g Fm(\013)p Fn(\),)g(then)e Fm(h)h Fn(m)o(ust)f(b)q(e)h(the)g(iden)o(tit)o(y)e (function)i(on)60 1093 y Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\).)38 b(The)22 b(fact)g(that)g(the)g(normal)f(form)f(of)j(a)f (pattern)g(is)g(unique)f(up)h(to)g(v)m(ariable)60 1154 y(renaming)15 b(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 Fn(Let)17 b Fm(\013)e Fl(2)h Fn(\(\006)11 b Fl([)h Fm(V)f Fn(\))706 1250 y Fj(+)753 1268 y Fn(b)q(e)17 b(a)g(pattern)h(and)f(let)g Fm(h)e Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))h Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1642 1250 y Fi(\003)1678 1268 y Fn(b)q(e)j(a)60 1328 y(substitution)f(suc)o(h)f(that)g Fm(h)p Fn(\()p Fm(\013)p Fn(\))e(=)g Fm(\013)p Fn(.)22 b(Then)16 b(the)g(follo)o(wing)g(statemen)o(ts)f(hold.)120 1429 y(1.)24 b Fm(\013)18 b Fn(has)g(a)f(partition)h Fm(\013)e Fn(=)g Fm(\014)694 1436 y Fj(1)713 1429 y Fm(\014)741 1436 y Fj(2)768 1429 y Fm(:)8 b(:)g(:)g(\014)862 1436 y Fk(k)900 1429 y Fn(in)o(to)17 b(minimal)d(\014xp)q(oin)o(ts)j(of)h Fm(h)p Fn(,)f(i.e.,)f(for)i(eac)o(h)182 1490 y Fm(i)d Fl(2)g(f)p Fn(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(k)r Fl(g)p Fn(,)17 b Fm(\014)532 1497 y Fk(i)561 1490 y Fl(2)e Fn(\(\006)d Fl([)g Fm(V)f Fn(\))778 1472 y Fj(+)807 1490 y Fn(,)17 b(the)g(equalit)o(y)f Fm(h)p Fn(\()p Fm(\014)1185 1497 y Fk(i)1198 1490 y Fn(\))f(=)h Fm(\014)1314 1497 y Fk(i)1344 1490 y Fn(holds,)i(and)f(for)h(ev)o(ery)182 1550 y(prop)q(er)f(sub)o(w) o(ord)f Fm(\015)k Fn(of)c Fm(\014)658 1557 y Fk(i)688 1550 y Fn(w)o(e)g(ha)o(v)o(e)f Fm(h)p Fn(\()p Fm(\015)s Fn(\))f Fl(6)p Fn(=)g Fm(\015)s Fn(.)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 Fm(i)g Fl(2)h(f)p Fn(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(k)r Fl(g)p Fn(,)17 b(either)g Fm(\014)894 1760 y Fk(i)924 1753 y Fl(2)g Fn(\006)h(or)g Fm(\014)1116 1760 y Fk(i)1146 1753 y Fl(2)e Fm(V)1235 1735 y Fj(+)1264 1753 y Fn(.)26 b(Moreo)o(v)o(er,)17 b(if)g Fm(\014)1608 1760 y Fk(i)1638 1753 y Fl(2)g Fm(V)1727 1735 y Fj(+)1756 1753 y Fn(,)182 1813 y(then)f(it)g(con)o(tains)g(exactly)f(one)i(v)m (ariable)e Fm(x)h Fn(suc)o(h)g(that)210 1919 y(\()p Fl(\003)p Fn(\))101 b Fm(h)p Fn(\()p Fm(x)p Fn(\))13 b(=)h Fm(\015)558 1926 y Fj(1)578 1919 y Fm(x\015)631 1926 y Fj(2)651 1919 y Fm(;)57 b Fn(where)16 b Fm(\015)888 1926 y Fj(1)908 1919 y Fm(;)8 b(\015)955 1926 y Fj(2)989 1919 y Fl(2)14 b Fn(\()p Fm(v)r(ar)q Fn(\()p Fm(\014)1177 1926 y Fk(i)1190 1919 y Fn(\))d Fl(n)g(f)p Fm(x)p Fl(g)p Fn(\))1353 1901 y Fi(\003)1388 1919 y Fn(and)706 1979 y Fm(\014)734 1986 y Fk(i)761 1979 y Fn(=)j Fm(\016)835 1986 y Fj(1)854 1979 y Fm(\015)879 1986 y Fj(1)900 1979 y Fm(x\015)953 1986 y Fj(2)972 1979 y Fm(\016)994 1986 y Fj(2)1030 1979 y Fn(for)i(some)g Fm(\016)1249 1986 y Fj(1)1268 1979 y Fm(;)8 b(\016)1312 1986 y Fj(2)1345 1979 y Fl(2)14 b Fn(\()p Fm(v)r(ar)q Fn(\()p Fm(\014)1533 1986 y Fk(i)1546 1979 y Fn(\))d Fl(n)g(f)p Fm(x)p Fl(g)p Fn(\))1709 1961 y Fi(\003)1728 1979 y Fm(:)120 2109 y Fn(4.)24 b(If)18 b Fm(x)g Fn(is)g(a)h(v)m(ariable)f(suc)o(h)g(that)h Fm(h)p Fn(\()p Fm(x)p Fn(\))e(=)g Fm(\015)969 2116 y Fj(1)989 2109 y Fm(x\015)1042 2116 y Fj(2)1062 2109 y Fn(,)h(where)h Fm(\015)1263 2116 y Fj(1)1283 2109 y Fm(;)8 b(\015)1330 2116 y Fj(2)1367 2109 y Fl(2)17 b Fn(\()p Fm(v)r(ar)q Fn(\()p Fm(\014)1558 2116 y Fk(i)1571 2109 y Fn(\))c Fl(n)f(f)p Fm(x)p Fl(g)p Fn(\))1737 2091 y Fi(\003)1756 2109 y Fn(,)182 2169 y(and)17 b Fm(x)f Fn(o)q(ccurs)g(in)g(some)g Fm(\014)679 2176 y Fk(i)708 2169 y Fn(and)h Fm(\014)831 2176 y Fk(j)849 2169 y Fn(,)f Fm(i)d Fl(6)p Fn(=)h Fm(j)s Fn(,)i(then)g Fm(\014)1153 2176 y Fk(i)1180 2169 y Fn(=)e Fm(\014)1260 2176 y Fk(j)1278 2169 y Fn(.)60 2271 y Fg(Pro)r(of:)20 b Fn(\(1\))15 b(W)l(e)g(scan)g Fm(\013)g Fn(sym)o(b)q(ol)e(b)o(y)i(sym) o(b)q(ol)e(from)h(left)g(to)h(righ)o(t.)20 b(Since)14 b Fm(\013)h Fn(is)f(a)i(\014xp)q(oin)o(t,)60 2331 y(there)i(is)h(a)g (smallest)e(pre\014x)h(of)h Fm(\013)g Fn(whic)o(h)f(is)g(a)h(\014xp)q (oin)o(t)g(of)g Fm(h)p Fn(,)g(this)f(is)h Fm(\014)1434 2338 y Fj(1)1453 2331 y Fn(.)29 b(If)18 b(there)g(w)o(ere)60 2391 y(a)h(prop)q(er)g(sub)o(w)o(ord)g Fm(\015)i Fn(of)e Fm(\014)588 2398 y Fj(1)626 2391 y Fn(\(i.e.,)e Fm(\014)768 2398 y Fj(1)804 2391 y Fn(=)h Fm(\016)r(\015)s(\016)936 2373 y Fi(0)947 2391 y Fn(,)g(where)g Fm(\015)k Fn(and)d Fm(\016)r(\016)1314 2373 y Fi(0)1343 2391 y Fn(are)f(nonempt)o(y\))f (with)60 2452 y Fm(h)p Fn(\()p Fm(\015)s Fn(\))d(=)g Fm(\015)s Fn(,)h(then)i(it)e(w)o(ould)i(follo)o(w)478 2561 y Fm(\014)506 2568 y Fj(1)539 2561 y Fn(=)d Fm(h)p Fn(\()p Fm(\014)666 2568 y Fj(1)685 2561 y Fn(\))g(=)g Fm(h)p Fn(\()p Fm(\016)r Fn(\))p Fm(h)p Fn(\()p Fm(\015)s Fn(\))p Fm(h)p Fn(\()p Fm(\016)1025 2541 y Fi(0)1035 2561 y Fn(\))g(=)g Fm(h)p Fn(\()p Fm(\016)r Fn(\))p Fm(\015)s(h)p Fn(\()p Fm(\016)1309 2541 y Fi(0)1320 2561 y Fn(\))p Fm(:)60 2671 y Fn(By)i(induction,)g(w)o(e)g(obtain)h Fm(\014)620 2678 y Fj(1)654 2671 y Fn(=)e Fm(h)735 2653 y Fk(q)754 2671 y Fn(\()p Fm(\014)801 2678 y Fj(1)820 2671 y Fn(\))g(=)f Fm(h)934 2653 y Fk(q)954 2671 y Fn(\()p Fm(\016)r Fn(\))p Fm(\015)s(h)1072 2653 y Fk(q)1090 2671 y Fn(\()p Fm(\016)1133 2653 y Fi(0)1144 2671 y Fn(\))j(for)g(all)f Fm(q)g Fl(2)f Fa(I)-7 b(N)q Fn(.)22 b(In)17 b(particular,)60 2731 y Fm(h)88 2713 y Fk(q)107 2731 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y(there)19 b(is)h(a)g Fm(q)h Fl(2)f Fa(I)-7 b(N)20 b Fn(suc)o(h)f(that)i Fm(h)692 306 y Fk(q)q Fj(+1)756 324 y Fn(\()p Fm(\016)r Fn(\))e(=)g Fm(h)922 306 y Fk(q)941 324 y Fn(\()p Fm(\016)r Fn(\).)31 b(Hence)19 b Fm(h)1225 306 y Fk(q)1244 324 y Fn(\()p Fm(\016)r Fn(\))g(is)g(a)h(\014xp)q(oin)o(t)g(of)g Fm(h)p Fn(.)31 b(If)60 384 y Fm(h)88 366 y Fk(q)107 384 y Fn(\()p Fm(\016)r Fn(\))22 b(=)h Fm(\014)280 391 y Fj(1)300 384 y Fn(,)f(then)f(w)o(e)h(infer)e Fm(\014)675 391 y Fj(1)718 384 y Fn(=)i Fm(h)806 366 y Fk(q)825 384 y Fn(\()p Fm(\016)r Fn(\))p Fm(\015)s(h)943 366 y Fk(q)962 384 y Fn(\()p Fm(\016)1005 366 y Fi(0)1016 384 y Fn(\))h(=)g Fm(\014)1147 391 y Fj(1)1166 384 y Fm(\015)s(h)1222 366 y Fk(q)1241 384 y Fn(\()p Fm(\016)1284 366 y Fi(0)1295 384 y Fn(\))f(whic)o(h)f(is)g(imp)q(ossible.)60 444 y(Otherwise)16 b Fm(h)319 426 y Fk(q)339 444 y Fn(\()p Fm(\016)r Fn(\))g(is)h(a)h (smaller)d(pre\014x)h(of)i Fm(\013)f Fn(than)h Fm(\014)1063 451 y Fj(1)1082 444 y Fn(.)24 b(This,)17 b(ho)o(w)o(ev)o(er,)f(con)o (tradicts)h(our)60 504 y(minim)o(alit)o(y)c(assumption.)133 565 y(Let)f(us)f(resume)f(the)h(construction)g(of)h(the)f(partition)g (of)h Fm(\013)p Fn(.)20 b(Let)11 b Fm(\013)1344 546 y Fi(0)1367 565 y Fn(b)q(e)h(the)f(pattern)g(suc)o(h)60 625 y(that)17 b Fm(\013)f Fn(=)f Fm(\014)294 632 y Fj(1)313 625 y Fm(\013)344 607 y Fi(0)356 625 y Fn(.)24 b(Since)16 b Fm(h)p Fn(\()p Fm(\013)p Fn(\))g(=)f Fm(h)p Fn(\()p Fm(\014)763 632 y Fj(1)782 625 y Fn(\))p Fm(h)p Fn(\()p Fm(\013)879 607 y Fi(0)891 625 y Fn(\))g(=)g Fm(\014)1006 632 y Fj(1)1026 625 y Fm(h)p Fn(\()p Fm(\013)1104 607 y Fi(0)1116 625 y Fn(\))g(=)g Fm(\013)p Fn(,)i(the)g(equation)g Fm(h)p Fn(\()p Fm(\013)1628 607 y Fi(0)1640 625 y Fn(\))e(=)g Fm(\013)1758 607 y Fi(0)60 685 y Fn(m)o(ust)j(hold.)30 b(No)o(w)19 b(w)o(e)f(rep)q(eat)i(the)e(pro)q(cess)i(describ)q(ed)f(ab) q(o)o(v)o(e)g(with)g Fm(\013)1426 667 y Fi(0)1457 685 y Fn(and)h(obtain)f Fm(\014)1737 692 y Fj(2)1756 685 y Fn(.)60 745 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993 y Fj(1)516 986 y Fn(or)g(vice)e(v)o(ersa.)25 b(The)17 b(former)f(con)o(tradicts)h(the)g(minimali)o(t)o(y)d(of)k Fm(\015)1750 993 y Fj(1)60 1046 y Fn(and)f(the)g(latter)f(the)h(minim)o (alit)n(y)d(of)j Fm(\014)789 1053 y Fj(1)808 1046 y Fn(.)23 b(Th)o(us)17 b Fm(\014)997 1053 y Fj(1)1031 1046 y Fn(=)e Fm(\015)1109 1053 y Fj(1)1129 1046 y Fn(.)23 b(Rep)q(eated)16 b(application)h(of)g(this)60 1106 y(argumen)o(t)e(leads)h(to)h Fm(k)f Fn(=)e Fm(l)i Fn(and)h Fm(\014)707 1113 y Fk(i)735 1106 y Fn(=)d Fm(\015)812 1113 y Fk(i)842 1106 y Fn(for)j(ev)o(ery)d Fm(i)g Fl(2)g(f)p Fn(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(k)r Fl(g)p Fn(.)60 1166 y(\(3\))21 b(If)f Fm(\014)224 1173 y Fk(i)257 1166 y Fn(con)o(tains)h(a)g(letter)e Fm(b)h Fn(from)f(\006,)i(then)f Fm(\014)1005 1173 y Fk(i)1040 1166 y Fn(=)h Fm(b)f Fn(b)q(ecause)g Fm(h)p Fn(\()p Fm(b)p Fn(\))h(=)f Fm(b)h Fn(and)g Fm(\014)1659 1173 y Fk(i)1692 1166 y Fn(is)g(a)60 1227 y(minim)o(al)15 b(\014xp)q(oin)o(t.)25 b(Otherwise)17 b Fm(\014)714 1234 y 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847 y Fk(q)1523 866 y Fn(\()p Fm(x)p Fn(\))p Fm(h)p Fn(\()p Fm(w)q Fn(\).)20 b(In)60 926 y(other)c(w)o(ords,)h Fm(h)369 908 y Fk(q)388 926 y Fn(\()p Fm(x)p Fn(\))f(is)g(a)g(sub)o(w)o(ord)h(of)g Fm(h)836 908 y Fk(q)q Fj(+1)900 926 y Fn(\()p Fm(x)p Fn(\).)60 986 y(No)o(w)h(since)f Fm(\013)h Fn(is)f(a)h(\014xp)q(oin)o (t)g(of)g Fm(h)p Fn(,)g Fm(h)762 968 y Fk(q)781 986 y Fn(\()p Fm(x)p Fn(\))f(cannot)i(gro)o(w)f(b)q(ey)o(ond)g Fm(\013)p Fn(,)g(so)g(for)g(some)f Fm(q)h Fl(2)e Fa(I)-7 b(N)60 1046 y Fn(the)16 b(equalit)o(y)f Fm(h)358 1028 y Fk(q)377 1046 y Fn(\()p Fm(x)p Fn(\))e(=)h Fm(h)536 1028 y Fk(q)q Fj(+1)600 1046 y Fn(\()p Fm(x)p Fn(\))i(m)o(ust)f(hold.) 21 b Fd(2)60 1160 y Fg(Theorem)c(4.7)24 b Fn(Let)h Fm(\013)k Fl(2)g Fn(\(\006)17 b Fl([)g Fm(V)12 b Fn(\))792 1142 y Fj(+)846 1160 y Fn(b)q(e)25 b(a)h(pattern)f(in)g(normal)f(form)g(and) h(let)g Fm(h)j Fn(:)60 1220 y Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))15 b Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))428 1202 y Fi(\003)464 1220 y Fn(b)q(e)j(a)f(substitution)h(suc)o(h)e(that) i Fm(h)p Fn(\()p Fm(\013)p Fn(\))e(=)f Fm(\013)p Fn(.)24 b(Then)17 b Fm(h)g Fn(is)g(the)g(iden)o(tit)o(y)60 1281 y(function)f(on)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\).)60 1401 y Fg(Pro)r(of:)25 b Fn(According)17 b(to)i(Lemma)d(4.6,)i Fm(\013)h Fn(has)g(a)f(unique)g(partition)g Fm(\013)f Fn(=)g Fm(\014)1480 1408 y Fj(1)1499 1401 y Fm(\014)1527 1408 y Fj(2)1555 1401 y Fm(:)8 b(:)g(:)f(\014)1648 1408 y Fk(k)1687 1401 y Fn(in)o(to)60 1461 y(minim)o(al)15 b(\014xp)q(oin)o(ts)i(w.r.t.)f Fm(h)h Fn(suc)o(h)g(that)h(either)e Fm(\014)1012 1468 y Fk(i)1041 1461 y Fl(2)g Fn(\006)h(or)h Fm(\014)1231 1468 y Fk(i)1260 1461 y Fl(2)d Fm(V)1348 1443 y Fj(+)1394 1461 y Fn(where)i Fm(\014)1564 1468 y Fk(i)1595 1461 y Fn(con)o(tains)60 1521 y(exactly)k(one)i(v)m (ariable)g Fm(x)548 1528 y Fk(i)584 1521 y Fn(whic)o(h)f(o)q(ccurs)h (in)f Fm(h)p Fn(\()p Fm(x)1025 1528 y Fk(i)1039 1521 y Fn(\).)41 b(Let)23 b Fm(i)1224 1528 y Fj(1)1243 1521 y Fm(;)8 b(:)g(:)g(:)f(;)h(i)1369 1528 y Fk(p)1411 1521 y Fn(b)q(e)23 b(exactly)f(those)60 1582 y(indices)17 b(for)h(whic)o(h)g Fm(\014)468 1589 y Fk(i)480 1594 y Ff(j)514 1582 y Fl(2)f Fm(V)604 1563 y Fj(+)651 1582 y Fn(and)i(let)e Fm(x)848 1589 y Fk(i)860 1594 y Ff(j)896 1582 y Fn(b)q(e)h(the)g(corresp)q(onding)h(v)m(ariables)f(satisfying)60 1642 y(\()p Fl(\003)p Fn(\).)42 b(Let)24 b Fm(V)302 1649 y Fk(nd)370 1642 y Fn(=)h Fl(f)p Fm(x)486 1649 y Fk(i)498 1654 y Fe(1)517 1642 y Fm(;)8 b(:)g(:)g(:)g(;)g(x)655 1649 y Fk(i)667 1653 y Ff(p)686 1642 y Fl(g)24 b Fn(\(note)f(that)h Fm(x)1010 1649 y Fk(i)1022 1654 y Ff(j)1066 1642 y Fn(=)h Fm(x)1157 1649 y Fk(i)1169 1653 y Ff(q)1188 1642 y Fn(,)g Fm(i)1244 1649 y Fk(j)1288 1642 y Fl(6)p Fn(=)h Fm(i)1369 1649 y Fk(q)1387 1642 y Fn(,)f(is)e(p)q(ossible\))g(and)60 1702 y Fm(V)88 1709 y Fk(d)123 1702 y Fn(=)13 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))e Fl(n)g Fm(V)393 1709 y Fk(nd)435 1702 y Fn(.)21 b(Supp)q(ose)c(that)g Fm(V)795 1709 y Fk(d)830 1702 y Fl(6)p Fn(=)c Fl(;)j Fn(and)h(let)f Fm(\013)1119 1684 y Fi(0)1147 1702 y Fn(b)q(e)g(the)g(pattern)h(obtained)f(from)60 1762 y Fm(\013)c Fn(b)o(y)f(deleting)f(all)h(the)g(v)m(ariables)g(from) g Fm(V)823 1769 y Fk(d)843 1762 y Fn(,)h(i.e.,)e Fm(\013)988 1744 y Fi(0)1014 1762 y Fn(=)k Fm(d)p Fn(\()p Fm(\013)p Fn(\))e(where)f Fm(d)j Fn(=)g Fl(f)p Fm(x)g Fl(7!)f Fm(\025)f Fl(j)f Fm(x)j Fl(2)g Fm(V)1711 1769 y Fk(d)1732 1762 y Fl(g)p Fn(.)60 1822 y(Note)i(that)h Fm(v)r(ar)q Fn(\()p Fm(\013)409 1804 y Fi(0)420 1822 y Fn(\))d(=)f Fm(V)532 1829 y Fk(nd)574 1822 y Fn(.)22 b(De\014ne)16 b Fm(g)g Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\013)952 1804 y Fi(0)963 1822 y Fn(\))h Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1204 1804 y Fi(\003)1239 1822 y Fn(b)o(y)793 1932 y Fm(g)r Fn(\()p Fm(x)865 1939 y Fk(i)877 1944 y Ff(j)895 1932 y Fn(\))g(=)f Fm(\014)1007 1939 y Fk(i)1019 1944 y Ff(j)60 2042 y Fn(According)19 b(to)h(Lemma)d(4.6,)j Fm(g)i Fn(is)d(w)o (ell-de\014ned)f(b)q(ecause)i Fm(x)1210 2049 y Fk(i)1222 2054 y Ff(j)1259 2042 y Fn(=)f Fm(x)1344 2049 y Fk(i)1356 2053 y Ff(q)1395 2042 y Fn(implies)d Fm(\014)1591 2049 y Fk(i)1603 2054 y Ff(j)1640 2042 y Fn(=)k Fm(\014)1726 2049 y Fk(i)1738 2053 y Ff(q)1756 2042 y Fn(.)60 2103 y(F)l(urthermore,)12 b(w)o(e)i(claim)e Fm(g)r Fn(\()p Fm(\013)627 2084 y Fi(0)640 2103 y Fn(\))h(=)h Fm(\013)p Fn(.)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 2163 y Fm(g)r Fn(\()p Fm(d)p Fn(\()p Fm(\014)176 2170 y Fk(i)188 2175 y Ff(j)206 2163 y Fn(\)\))k(=)h Fm(\014)347 2170 y Fk(i)359 2175 y Ff(j)395 2163 y Fn(for)g(all)g(1)f Fl(\024)g Fm(j)j Fl(\024)d Fm(p)p Fn(.)29 b(W)l(e)19 b(ha)o(v)o(e)f Fm(d)p Fn(\()p Fm(\014)1081 2170 y Fk(i)1093 2175 y Ff(j)1111 2163 y Fn(\))g(=)g Fm(x)1232 2170 y Fk(i)1244 2175 y Ff(j)1281 2163 y Fn(b)q(ecause)h Fm(x)1492 2170 y Fk(i)1504 2175 y Ff(j)1541 2163 y Fn(is)g(the)f(only)60 2223 y(v)m(ariable)e(from)g Fm(V)387 2230 y Fk(nd)445 2223 y Fn(in)h Fm(\014)531 2230 y Fk(i)543 2235 y Ff(j)560 2223 y Fn(.)23 b(Hence)15 b Fm(g)r Fn(\()p Fm(d)p Fn(\()p Fm(\014)858 2230 y Fk(i)870 2235 y Ff(j)888 2223 y Fn(\)\))f(=)h Fm(g)r Fn(\()p Fm(x)1065 2230 y Fk(i)1077 2235 y Ff(j)1095 2223 y Fn(\))f(=)h Fm(\014)1209 2230 y Fk(i)1221 2235 y Ff(j)1255 2223 y Fn(and)i(the)f(claim)f(is)h(pro)o(v)o(ed.)60 2283 y(Since)h Fm(\014)217 2290 y Fk(i)229 2295 y Ff(j)265 2283 y Fn(has)h(a)g (represen)o(tation)f Fm(\016)737 2290 y Fj(1)757 2283 y Fm(\015)782 2290 y Fj(1)802 2283 y Fm(x)830 2290 y Fk(i)842 2295 y Ff(j)860 2283 y Fm(\015)885 2290 y Fj(2)905 2283 y Fm(\016)927 2290 y Fj(2)946 2283 y Fn(,)h(where)f Fm(\015)1145 2290 y Fj(1)1165 2283 y Fm(;)8 b(\015)1212 2290 y Fj(2)1232 2283 y Fm(;)g(\016)1276 2290 y Fj(1)1295 2283 y Fm(;)g(\016)1339 2290 y Fj(2)1375 2283 y Fl(2)16 b Fm(V)1464 2265 y Fi(\003)1452 2295 y Fk(d)1483 2283 y Fn(,)i Fm(g)i Fn(is)e(a)g(linear)60 2360 y(substitution.)31 b(Th)o(us)19 b(it)g(follo)o(ws)g Fm(\013)757 2331 y Fk(V)778 2337 y Ff(d)751 2360 y Fl(!)g Fm(\013)851 2342 y Fi(0)863 2360 y Fn(.)30 b(This,)20 b(ho)o(w)o(ev)o(er,)e(con)o(tradicts)h(the)g (fact)h(that)60 2420 y Fm(\013)f Fn(is)f(in)g(normal)f(form.)27 b(So)19 b Fm(V)626 2427 y Fk(d)664 2420 y Fn(=)f Fl(;)p Fn(.)27 b(It)18 b(further)g(follo)o(ws)h(that)f Fm(\014)1305 2427 y Fk(i)1317 2432 y Ff(j)1353 2420 y Fn(=)f Fm(x)1436 2427 y Fk(i)1448 2432 y Ff(j)1485 2420 y Fn(and)i(therefore)60 2481 y Fm(h)p Fn(\()p Fm(x)135 2488 y Fk(i)147 2493 y Ff(j)165 2481 y Fn(\))14 b(=)g Fm(x)278 2488 y Fk(i)290 2493 y Ff(j)324 2481 y Fn(for)j(1)d Fl(\024)f Fm(j)k Fl(\024)d Fm(p)p Fn(.)21 b(Hence)15 b Fm(h)i Fn(is)f(the)g(iden)o(tit)o (y)e(on)j Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))c(=)h Fl(f)p Fm(x)1471 2488 y Fk(i)1483 2493 y Fe(1)1502 2481 y Fm(;)8 b(:)g(:)g(:)f(;)h(x)1639 2488 y Fk(i)1651 2492 y Ff(p)1671 2481 y Fl(g)p Fn(.)21 b Fd(2)133 2595 y Fn(The)16 b(simple)e(example)f Fm(\013)h Fn(=)g Fm(xy)r Fn(,)h Fm(h)p Fn(\()p Fm(x)p Fn(\))e(=)h Fm(xy)j Fn(and)g Fm(h)p Fn(\()p Fm(y)r Fn(\))c(=)h Fm(\025)i Fn(sho)o(ws)h(that)f(in)f(Theorem)60 2655 y(4.7)i(the)e (normal)h(form)f(requiremen)o(t)e(on)j Fm(\013)h Fn(cannot)g(b)q(e)f (dropp)q(ed)h(b)q(ecause)f Fm(h)p Fn(\()p Fm(\013)p Fn(\))f(=)e Fm(\013)k Fn(but)60 2715 y Fm(h)f Fn(is)g(not)h(the)f(iden)o(tit)o(y)e (on)j Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\).)891 2856 y(13)p eop %%Page: 14 14 14 13 bop 60 203 a Fg(Corollary)18 b(4.8)24 b Fn(If)16 b(a)i(pattern)f Fm(\013)f Fl(2)f Fn(\(\006)d Fl([)g Fm(V)f Fn(\))934 185 y Fj(+)981 203 y Fn(reduces)16 b(to)i(t)o(w)o(o)f (distinct)f(normal)g(forms)60 264 y Fm(\013)91 246 y Fi(0)122 264 y Fn(and)k Fm(\013)251 246 y Fi(00)272 264 y Fn(,)f(then)g(there)f(is)h(a)h(v)m(ariable)e(renaming)g Fm(h)g Fn(:)g Fm(v)r(ar)q Fn(\()p Fm(\013)1247 246 y Fi(0)1259 264 y Fn(\))g Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)1489 246 y Fi(00)1510 264 y Fn(\))1529 246 y Fi(\003)1568 264 y Fn(suc)o(h)h(that)60 324 y Fm(h)p Fn(\()p Fm(\013)138 306 y Fi(0)150 324 y Fn(\))14 b(=)g Fm(\013)266 306 y Fi(00)287 324 y Fn(.)60 444 y Fg(Pro)r(of:)21 b Fn(Consider)c(t)o(w)o (o)f(reductions)g(of)g Fm(\013)h Fn(to)g(normal)e(forms)g Fm(\013)1258 426 y Fi(0)1286 444 y Fn(and)i Fm(\013)1412 426 y Fi(00)1434 444 y Fn(:)79 553 y Fm(\013)110 533 y Fi(0)136 553 y Fn(=)d Fm(\013)219 533 y Fi(0)219 566 y Fk(n)256 553 y Fl( )g Fm(\013)351 533 y Fi(0)351 566 y Fk(n)p Fi(\000)p Fj(1)433 553 y Fl( )g Fm(:)8 b(:)g(:)13 b Fl( )h Fm(\013)663 533 y Fi(0)663 566 y Fj(1)696 553 y Fl( )g Fm(\013)791 533 y Fi(0)791 566 y Fj(0)824 553 y Fn(=)g Fm(\013)g Fn(=)g Fm(\013)1004 533 y Fi(00)1004 566 y Fj(0)1039 553 y Fl(!)g Fm(\013)1134 533 y Fi(00)1134 566 y Fj(1)1170 553 y Fl(!)f Fm(:)8 b(:)g(:)13 b Fl(!)h Fm(\013)1399 533 y Fi(00)1399 566 y Fk(m)p Fi(\000)p Fj(1)1491 553 y Fl(!)g Fm(\013)1586 533 y Fi(00)1586 566 y Fk(m)1633 553 y Fn(=)g Fm(\013)1716 533 y Fi(00)1737 553 y Fm(:)60 663 y Fn(Since)i Fl(!)g Fn(is)h(transitiv)o(e,)e(w)o(e)h (obtain)h Fm(\013)790 644 y Fi(0)817 663 y Fl( )d Fm(\013)h Fl(!)g Fm(\013)1023 644 y Fi(00)1044 663 y Fn(.)23 b(By)16 b(de\014nition)g(of)h(reduction,)f(there)60 723 y(are)j(substitutions)g Fm(d)462 705 y Fi(0)474 723 y Fm(;)8 b(d)521 705 y Fi(00)561 723 y Fn(and)19 b Fm(\033)688 705 y Fi(0)699 723 y Fm(;)8 b(\033)751 705 y Fi(00)791 723 y Fn(suc)o(h)18 b(that)h Fm(d)1036 705 y Fi(0)1048 723 y Fn(\()p Fm(\013)p Fn(\))g(=)f Fm(\013)1223 705 y Fi(0)1235 723 y Fn(,)g Fm(d)1292 705 y Fi(00)1314 723 y Fn(\()p Fm(\013)p Fn(\))g(=)g Fm(\013)1488 705 y Fi(00)1510 723 y Fn(,)h Fm(\033)1573 705 y Fi(0)1584 723 y Fn(\()p Fm(\013)1634 705 y Fi(0)1646 723 y Fn(\))f(=)g Fm(\013)60 783 y Fn(and)i Fm(\033)188 765 y Fi(00)209 783 y Fn(\()p Fm(\013)259 765 y Fi(00)280 783 y Fn(\))g(=)f Fm(\013)p Fn(.)32 b(De\014ne)20 b Fm(f)25 b Fn(:)19 b Fm(v)r(ar)q Fn(\()p Fm(\013)815 765 y Fi(0)826 783 y Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\013)1059 765 y Fi(00)1080 783 y Fn(\))1099 765 y Fi(\003)1138 783 y Fn(b)o(y)f Fm(f)25 b Fn(=)19 b Fm(d)1340 765 y Fi(00)1375 783 y Fl(\016)14 b Fm(\033)1444 765 y Fi(0)1455 783 y Fn(.)31 b(Analogously)l(,)60 843 y(de\014ne)17 b Fm(g)h Fn(:)e Fm(v)r(ar)q Fn(\()p Fm(\013)398 825 y Fi(00)419 843 y Fn(\))g Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)645 825 y Fi(0)656 843 y Fn(\))675 825 y Fi(\003)712 843 y Fn(b)o(y)h Fm(g)i Fn(=)d Fm(d)902 825 y Fi(0)926 843 y Fl(\016)11 b Fm(\033)992 825 y Fi(00)1013 843 y Fn(.)25 b(Then)18 b Fm(f)5 b Fn(\()p Fm(\013)1260 825 y Fi(0)1272 843 y Fn(\))16 b(=)g Fm(\013)1392 825 y Fi(00)1432 843 y Fn(and)i Fm(g)r Fn(\()p Fm(\013)1603 825 y Fi(00)1624 843 y Fn(\))e(=)g Fm(\013)1744 825 y Fi(0)1756 843 y Fn(.)60 903 y(Hence)d Fm(f)5 b Fn(\()p Fm(g)r Fn(\()p Fm(\013)326 885 y Fi(00)348 903 y Fn(\)\))14 b(=)g Fm(\013)483 885 y Fi(00)519 903 y Fn(and)i Fm(g)r Fn(\()p Fm(f)5 b Fn(\()p Fm(\013)736 885 y Fi(0)748 903 y Fn(\)\))14 b(=)g Fm(\013)883 885 y Fi(0)895 903 y Fn(.)20 b(By)14 b(Theorem)g(4.7,)h Fm(f)e Fl(\016)8 b Fm(g)17 b Fn(is)d(the)h(iden)o (tit)o(y)e(on)60 964 y Fm(v)r(ar)q Fn(\()p Fm(\013)185 945 y Fi(00)206 964 y Fn(\))k(and)i Fm(g)14 b Fl(\016)e Fm(f)23 b Fn(is)17 b(the)h(iden)o(tit)o(y)d(on)j Fm(v)r(ar)q Fn(\()p Fm(\013)971 945 y Fi(0)983 964 y Fn(\).)25 b(Th)o(us)18 b Fm(f)23 b Fn(and)18 b Fm(g)i Fn(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 Fn(Th)o(us)14 b(the)f(normal)f(form)h(of)g(pattern)h Fm(\013)g Fn(w.r.t.)e Fl(!)h Fn(is)g(unique)g(\(up)h(to)f(v)m(ariable)g (renamings\);)60 1197 y(from)k(no)o(w)i(on)f(it)g(will)f(b)q(e)h (denoted)g(b)o(y)g Fm(\013)p Fl(#)p Fn(.)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 Fn(Let)c Fm(\013;)8 b(\014)22 b Fl(2)f Fn(\(\006)13 b Fl([)h Fm(V)e Fn(\))880 1352 y Fj(+)929 1371 y Fn(b)q(e)20 b(patterns)g(and)h(let)e Fm(car)q(d)p Fn(\(\006\))i Fl(\025)e Fn(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 Fm(\013)p Fl(#)16 b Fn(and)h Fm(\014)s Fl(#)f Fn(are)g(iden)o(tical)e(up)j(to)g(v)m(ariable)e(renaming.)120 1633 y(2.)24 b(There)g(are)h(substitutions)g Fm(f)749 1615 y Fi(0)789 1633 y Fn(:)i Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\))h Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\))1271 1615 y Fi(\003)1316 1633 y Fn(and)d Fm(g)1444 1615 y Fi(0)1483 1633 y Fn(:)j Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\))g Fl(!)182 1693 y Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\))350 1675 y Fi(\003)386 1693 y Fn(suc)o(h)16 b(that)h Fm(f)631 1675 y Fi(0)643 1693 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))d(=)g Fm(\014)s Fl(#)h Fn(and)i Fm(g)994 1675 y Fi(0)1006 1693 y Fn(\()p Fm(\014)s Fl(#)o Fn(\))d(=)g Fm(\013)p Fl(#)p Fn(.)120 1795 y(3.)24 b(There)19 b(are)h (substitutions)g Fm(f)k Fn(:)19 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))g Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))1162 1777 y Fi(\003)1200 1795 y Fn(and)h Fm(g)i Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\014)s Fn(\))f Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fn(\))1751 1777 y Fi(\003)182 1855 y Fn(suc)o(h)d(that)h Fm(f)5 b Fn(\()p Fm(\013)p Fn(\))14 b(=)g Fm(\014)k Fn(and)f Fm(g)r Fn(\()p Fm(\014)s Fn(\))c(=)h Fm(\013)p Fn(.)60 1956 y Fg(Pro)r(of:)36 b Fn(The)23 b(implication)e(\(1\))26 b Fl(\))g Fn(\(2\))e(holds)g(trivially)l(.)41 b(The)23 b(con)o(v)o(erse)g(implication)60 2016 y(\(2\))j Fl(\))g Fn(\(1\))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 Fl(,)e Fn(\(3\))h(follo)o(ws)g(from)e(the)h (equalities)f Fm(d)p Fn(\()p Fm(\013)p Fn(\))j(=)f Fm(\013)p Fl(#)p Fn(,)g Fm(\033)r Fn(\()p Fm(\013)p Fl(#)p Fn(\))g(=)g Fm(\013)p Fn(,)60 2137 y Fm(d)85 2119 y Fi(0)97 2137 y Fn(\()p Fm(\014)s Fn(\))h(=)h Fm(\014)s Fl(#)e Fn(and)h Fm(\033)456 2119 y Fi(0)467 2137 y Fn(\()p Fm(\014)s Fl(#)o Fn(\))h(=)g Fm(\014)s Fn(.)37 b(The)22 b(existence)d(of)j(a)g (substitution)g Fm(f)1457 2119 y Fi(0)1492 2137 y Fn(:)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\))g Fl(!)60 2197 y Fm(v)r(ar)q Fn(\()p 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Fk(E)r(;)p Fj(\006)1067 2491 y Fn(\()p Fm(\014)s Fn(\))f(if)h(and)h(only)f(if)f Fm(\013)p Fl(#)i Fn(and)g Fm(\014)s Fl(#)e Fn(are)60 2551 y(iden)o(tical)i(up)h(to)h(v)m(ariable)f(renaming.)36 b(The)21 b Fh(if)h Fn(direction)e(is)i(ob)o(vious)f(and)h(the)f Fh(only)i(if)60 2611 y Fn(direction)c(can)g(b)q(e)h(seen)f(as)i(follo)o (ws.)31 b(If)19 b Fm(\013)p Fl(#)h Fn(and)g Fm(\014)s Fl(#)f Fn(are)h(not)g(iden)o(tical)e(up)h(to)h(v)m(ariable)60 2671 y(renaming)f(but)h(similar,)e(then)i(it)f(follo)o(ws)h(from)f (Prop)q(osition)i(4.9)f(in)f(conjunction)h(with)60 2731 y(Theorem)15 b(3.5)h(and)g(Prop)q(osition)h(3.9,)f(resp)q(ectiv)o(ely)l (,)d(that)j(either)f Fm(L)1343 2738 y Fk(E)r(;)p Fj(\006)1409 2731 y Fn(\()p Fm(\013)p Fn(\))f Fl(6\022)f Fm(L)1577 2738 y Fk(E)r(;)p Fj(\006)1643 2731 y Fn(\()p Fm(\014)s Fn(\))i(or)891 2856 y(14)p eop %%Page: 15 15 15 14 bop 60 203 a Fm(L)93 210 y Fk(E)r(;)p Fj(\006)158 203 y Fn(\()p Fm(\014)s Fn(\))13 b Fl(6\022)h Fm(L)326 210 y Fk(E)r(;)p Fj(\006)391 203 y Fn(\()p Fm(\013)p Fn(\),)i(hence)e Fm(L)657 210 y Fk(E)r(;)p Fj(\006)723 203 y Fn(\()p Fm(\013)p Fn(\))g Fl(6)p Fn(=)f Fm(L)890 210 y Fk(E)r(;)p Fj(\006)956 203 y Fn(\()p Fm(\014)s Fn(\).)20 b(Finally)l(,)14 b(if)h Fm(\013)p Fl(#)g Fn(and)h Fm(\014)s Fl(#)f Fn(are)g(not)h(sim-)60 264 y(ilar)h(\(hence)f(not)i (iden)o(tical)d(up)j(to)f(v)m(ariable)g(renaming\),)f(then)h Fm(\013)h Fn(and)g Fm(\014)h Fn(are)e(not)h(similar)60 324 y(and)h(th)o(us)g Fm(L)300 331 y Fk(E)r(;)p Fj(\006)365 324 y Fn(\()p Fm(\013)p Fn(\))f Fl(6)p Fn(=)g Fm(L)541 331 y Fk(E)r(;)p Fj(\006)606 324 y Fn(\()p Fm(\014)s Fn(\))g(\(cf.)28 b(Theorem)18 b(3.1)h(and)g(the)f(pro)q(of)i(of)f (Corollary)g(3.10,)60 384 y(resp)q(ectiv)o(ely\).)133 444 y(W)l(e)12 b(ha)o(v)o(e)e(not)j(in)o(v)o(estigated)d(y)o(et)h (whether)g(reduction)g(to)h(normal)f(form)f(is)i(less)f(complex)60 504 y(than)17 b(solving)f(the)g(mem)o(b)q(ership)d(problem)i(whic)o(h)g (is)h(NP-complete.)60 670 y Fo(5)81 b(Equiv)l(alence)24 b(tests)j(based)f(on)i(normal)e(forms)60 779 y Fn(W)l(e)16 b(next)g(sho)o(w)h(the)f(b)q(ene\014ts)g(of)h(our)f(normal)g(form)f (approac)o(h.)60 884 y Fg(Prop)r(osition)j(5.1)24 b Fn(Let)13 b Fm(\013;)8 b(\014)17 b Fl(2)d Fn(\(\006)6 b Fl([)g Fm(V)11 b Fn(\))845 866 y Fj(+)888 884 y Fn(b)q(e)i(patterns)h(and)h (let)d Fm(car)q(d)p Fn(\(\006\))j Fl(\025)e Fn(4.)21 b(Supp)q(ose)60 944 y Fm(\013)p Fl(#)e Fn(has)h(indep)q(enden)o(t)f(v)m (ariable)g(segmen)o(ts.)29 b(Then)19 b Fm(L)1090 951 y Fk(E)r(;)p Fj(\006)1155 944 y Fn(\()p Fm(\013)p Fn(\))g(=)g Fm(L)1333 951 y Fk(E)r(;)p Fj(\006)1398 944 y Fn(\()p Fm(\014)s Fn(\))g(if)g(and)h(only)f(if)60 1005 y Fm(\013)p Fl(#)e Fn(and)f Fm(\014)s Fl(#)g Fn(are)g(iden)o(tical)f(up)h(to)h(v)m (ariable)f(renaming.)60 1125 y Fg(Pro)r(of:)41 b Fn(W)l(e)26 b(sho)o(w)h Fm(L)511 1132 y Fk(E)r(;)p Fj(\006)576 1125 y Fn(\()p Fm(\013)p Fn(\))32 b(=)e Fm(L)778 1132 y Fk(E)r(;)p Fj(\006)844 1125 y Fn(\()p Fm(\014)s Fn(\))25 b(if)h(and)h(only)f(if)g 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1366 y(Theorem)d(3.1,)i Fm(\013)p Fl(#)f Fn(and)h Fm(\014)s Fl(#)e Fn(are)h(similar,)e(so)j (let)e Fm(\013)p Fl(#)j Fn(=)e Fm(\013)1196 1373 y Fj(0)1216 1366 y Fm(u)1244 1373 y Fj(1)1264 1366 y Fm(\013)1295 1373 y Fj(1)1314 1366 y Fm(u)1342 1373 y Fj(2)1370 1366 y Fm(:)8 b(:)g(:)g(\013)1467 1373 y Fk(m)p Fi(\000)p Fj(1)1545 1366 y Fm(u)1573 1373 y Fk(m)1606 1366 y Fm(\013)1637 1373 y Fk(m)1691 1366 y Fn(and)60 1426 y Fm(\014)s Fl(#)13 b Fn(=)h Fm(\014)209 1433 y Fj(0)228 1426 y Fm(u)256 1433 y Fj(1)276 1426 y Fm(\014)304 1433 y Fj(1)323 1426 y Fm(u)351 1433 y Fj(2)379 1426 y Fm(:)8 b(:)g(:)f(\014)472 1433 y Fk(m)p Fi(\000)p Fj(1)550 1426 y Fm(u)578 1433 y Fk(m)611 1426 y Fm(\014)639 1433 y Fk(m)672 1426 y Fn(.)21 b(W)l(e)16 b(sho)o(w:)120 1521 y(1.)24 b(F)l(or)16 b(an)o(y)g Fm(\013)392 1528 y Fk(i)407 1521 y Fn(,)f(there)h(is)g(an)h Fm(f)702 1528 y Fk(i)730 1521 y Fn(:)c Fm(v)r(ar)q Fn(\()p Fm(\013)882 1528 y Fk(i)896 1521 y Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\014)1114 1528 y 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Fn(can)g(b)q(e)g(com)o(bined)e(in)o(to) h(a)i(global)f(substitution)g Fm(f)18 b Fn(suc)o(h)182 1840 y(that)f Fm(f)5 b Fn(\()p Fm(\013)p Fl(#)p Fn(\))14 b(=)g Fm(\014)s Fl(#)o Fn(.)120 1940 y(4.)24 b(The)13 b(lo)q(cal)g(substitutions)h Fm(g)702 1947 y Fk(i)730 1940 y Fn(can)f(b)q(e)h(com)o(bined)d(in)o(to)i(a)h(global)f (substitution)h Fm(g)h Fn(suc)o(h)182 2000 y(that)i Fm(g)r Fn(\()p Fm(\014)s Fl(#)o Fn(\))d(=)g Fm(\013)p Fl(#)p Fn(.)60 2095 y(In)19 b(order)h(to)g(pro)o(v)o(e)e(\(1\))i(and)g(\(2\),) h(w)o(e)e(consider)g(only)g(the)g(case)h(0)g Fm(<)f(i)g(<)g(m)h Fn(\(the)f(cases)60 2155 y Fm(i)14 b Fn(=)g(0)j(and)g Fm(i)d Fn(=)h Fm(m)h Fn(are)h(similar)d(and)j(simpler\).)j(Let)d Fm(c)f Fn(b)q(e)h(the)f(last)h(letter)f(of)g Fm(u)1552 2162 y Fk(i)1583 2155 y Fn(and)h Fm(d)g Fn(b)q(e)60 2215 y(the)f(\014rst)h(letter)e(of)h Fm(u)458 2222 y Fk(i)p Fj(+1)517 2215 y Fn(.)21 b(W)l(e)16 b(c)o(ho)q(ose)h Fm(a;)8 b(b)13 b Fl(2)h Fn(\006)d Fl(n)g(f)p Fm(c;)d(d)p Fl(g)17 b Fn(suc)o(h)f(that)g Fm(a)e Fl(6)p Fn(=)g Fm(b)p Fn(.)120 2310 y(1.)24 b(Consider)17 b Fm(\034)407 2318 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)522 2310 y Fn(\()p Fm(\014)s Fl(#)o Fn(\).)22 b Fm(\034)672 2318 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)787 2310 y Fn(\()p Fm(\014)s Fl(#)o Fn(\))17 b(can)f(b)q(e)h(written)f(as)h Fm(w)1319 2317 y Fj(0)1338 2310 y Fm(u)1366 2317 y Fj(1)1386 2310 y Fm(w)1421 2317 y Fj(1)1448 2310 y Fm(:)8 b(:)g(:)g(w)1549 2317 y Fk(m)p Fi(\000)p Fj(1)1627 2310 y Fm(u)1655 2317 y Fk(m)1688 2310 y Fm(w)1723 2317 y Fk(m)1756 2310 y Fn(,)182 2370 y(where)23 b Fm(w)365 2377 y Fk(i)405 2370 y Fn(=)j Fm(\034)490 2378 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)605 2370 y Fn(\()p Fm(\014)652 2377 y Fk(i)666 2370 y Fn(\).)42 b(Since)23 b Fm(L)909 2377 y Fk(E)r(;)p Fj(\006)974 2370 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))j(=)g Fm(L)1191 2377 y Fk(E)r(;)p Fj(\006)1256 2370 y Fn(\()p Fm(\014)s Fl(#)o Fn(\),)f(there)e(is)g(a)g Fm(\033)i Fn(suc)o(h)182 2430 y(that)c Fm(\033)r Fn(\()p Fm(\013)p Fl(#)p Fn(\))g(=)g Fm(w)531 2437 y Fj(0)550 2430 y Fm(u)578 2437 y Fj(1)598 2430 y Fm(w)633 2437 y Fj(1)660 2430 y Fm(:)8 b(:)g(:)g(w)761 2437 y Fk(m)p Fi(\000)p Fj(1)839 2430 y Fm(u)867 2437 y Fk(m)900 2430 y Fm(w)935 2437 y Fk(m)968 2430 y Fn(.)34 b(Note)20 b(that)h(if)f Fm(\013)1328 2437 y Fk(i)1363 2430 y Fn(is)g(b)q(et)o(w)o(een)g(the)g Fm(k)r Fn(th)182 2491 y(app)q(earance)d(of)g Fm(c)g Fn(and)g(the)f Fm(l)q Fn(th)g(app)q(earance)i(of)f Fm(d)g Fn(in)f Fm(\013)p Fl(#)p Fn(,)g(then)h Fm(w)1434 2498 y Fk(i)1464 2491 y Fn(is)g(b)q(et)o(w)o(een)e(the)182 2551 y Fm(k)r Fn(th)20 b(app)q(earance)i(of)e Fm(c)g Fn(and)h(the)f Fm(l)q Fn(th)g(app)q (earance)i(of)e Fm(d)h Fn(in)f Fm(\034)1353 2559 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1468 2551 y Fn(\()p Fm(\014)s Fl(#)p Fn(\).)33 b(Since)19 b Fm(\033)182 2611 y Fn(cannot)h(in)o(tro)q (duce)f(the)g(constan)o(ts)h Fm(c)g Fn(and)g Fm(d)p Fn(,)g(it)f(follo)o (ws)g Fm(\033)r Fn(\()p Fm(\013)1364 2618 y Fk(i)1378 2611 y Fn(\))g(=)g Fm(w)1508 2618 y Fk(i)1522 2611 y Fn(.)31 b(As)19 b(in)g(the)182 2671 y(pro)q(of)f(of)f(Lemma)d(3.4,)j(w) o(e)f(obtain)h(a)g(substitution)g Fm(f)1194 2678 y Fk(i)1222 2671 y Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\013)1375 2678 y Fk(i)1388 2671 y Fn(\))h Fl(!)f Fm(v)r(ar)q Fn(\()p Fm(\014)1608 2678 y Fk(i)1621 2671 y Fn(\))1640 2653 y Fi(\003)1676 2671 y Fn(suc)o(h)182 2731 y(that)j Fm(f)312 2738 y Fk(i)326 2731 y Fn(\()p Fm(\013)376 2738 y Fk(i)390 2731 y Fn(\))d(=)f Fm(\014)502 2738 y Fk(i)516 2731 y Fn(.)891 2856 y(15)p eop %%Page: 16 16 16 15 bop 120 203 a Fn(2.)24 b(Let)16 b Fm(\034)21 b Fn(b)q(e)16 b(the)f(substitution)h(whic)o(h)f(enco)q(des)h(only)f(v)m (ariables)g(from)g Fm(v)r(ar)q Fn(\()p Fm(\013)1597 210 y Fk(i)1610 203 y Fn(\).)21 b(That)182 264 y(is,)d Fm(\034)6 b Fn(\()p Fm(x)p Fn(\))16 b(=)h Fm(\034)432 271 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)546 264 y Fn(\()p Fm(x)p Fn(\))h(if)g Fm(x)e Fl(2)h Fm(v)r(ar)q Fn(\()p Fm(\013)896 271 y Fk(i)910 264 y Fn(\))h(and)h Fm(\034)6 b Fn(\()p Fm(x)p Fn(\))16 b(=)h Fm(\025)h Fn(otherwise.)27 b(Since)17 b Fm(\013)p Fl(#)h Fn(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 Fm(\034)6 b Fn(\()p Fm(\013)p Fl(#)p Fn(\))14 b(=)g Fm(u)765 441 y Fj(0)784 434 y Fm(u)812 441 y Fj(1)840 434 y Fm(:)8 b(:)g(:)g(u)934 441 y Fk(i)948 434 y Fm(\034)969 442 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)1083 434 y Fn(\()p Fm(\013)1133 441 y Fk(i)1147 434 y Fn(\))p Fm(u)1194 441 y Fk(i)p Fj(+1)1261 434 y Fm(:)g(:)g(:)g(u)1355 441 y Fk(m)1388 434 y Fm(:)182 544 y Fn(Since)19 b Fm(L)346 551 y Fk(E)r(;)p Fj(\006)412 544 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))i(=)g Fm(L)619 551 y Fk(E)r(;)p Fj(\006)684 544 y Fn(\()p Fm(\014)s Fl(#)o Fn(\),)g(there)f(is)g(a)h(substitution)g Fm(\033)g Fn(suc)o(h)g(that)f Fm(\033)r Fn(\()p Fm(\014)s Fl(#)o Fn(\))h(=)182 604 y Fm(u)210 611 y Fj(0)229 604 y Fm(u)257 611 y Fj(1)285 604 y Fm(:)8 b(:)g(:)g(u)379 611 y Fk(i)393 604 y Fm(\034)414 612 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)528 604 y Fn(\()p Fm(\013)578 611 y Fk(i)592 604 y Fn(\))p Fm(u)639 611 y Fk(i)p Fj(+1)706 604 y Fm(:)g(:)g(:)g(u)800 611 y Fk(m)833 604 y Fn(.)32 b(Since)18 b Fm(\033)k Fn(cannot)e(in)o(tro)q(duce)f(the)h(constan)o (ts)g Fm(c)182 664 y Fn(and)f Fm(d)p Fn(,)f(it)g(follo)o(ws)g Fm(\033)r Fn(\()p Fm(\014)626 671 y Fk(i)639 664 y Fn(\))f(=)g Fm(\034)751 672 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)866 664 y Fn(\()p Fm(\013)916 671 y Fk(i)930 664 y Fn(\))h(and)h Fm(\033)r Fn(\()p Fm(\014)1141 671 y Fk(j)1158 664 y Fn(\))e(=)g Fm(\025)i Fn(for)f(ev)o(ery)f Fm(j)j Fl(6)p Fn(=)d Fm(i)p Fn(.)26 b(As)18 b(in)182 724 y(the)g(pro)q(of)h(of)g (Lemma)d(3.4,)j(w)o(e)e(obtain)i(a)g(substitution)f Fm(g)1292 731 y Fk(i)1324 724 y Fn(:)f Fm(v)r(ar)q Fn(\()p Fm(\014)1477 731 y Fk(i)1490 724 y Fn(\))g Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)1718 731 y Fk(i)1731 724 y Fn(\))1750 706 y Fi(\003)182 785 y Fn(suc)o(h)g(that)g Fm(g)422 792 y Fk(i)436 785 y Fn(\()p Fm(\014)483 792 y Fk(i)497 785 y Fn(\))e(=)g Fm(\013)615 792 y Fk(i)629 785 y Fn(.)23 b(Moreo)o(v)o(er)15 b Fm(g)903 792 y Fk(i)918 785 y Fn(\()p Fm(x)p Fn(\))f(=)h Fm(\025)j Fn(whenev)o(er)d Fm(x)g Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\014)1524 792 y Fk(i)1537 785 y Fn(\))c Fl(\\)h Fm(v)r(ar)q Fn(\()p Fm(\014)1734 792 y Fk(j)1751 785 y Fn(\))182 845 y(for)k(some)g Fm(j)g Fl(6)p Fn(=)e Fm(i)p Fn(.)120 946 y(3.)24 b(Let)e Fm(x)h Fl(2)h Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)p Fn(\).)38 b(Since)21 b Fm(\013)p Fl(#)h Fn(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 Fm(i)h Fn(suc)o(h)g(that)g Fm(x)i Fl(2)h Fm(v)r(ar)q Fn(\()p Fm(\013)1097 1014 y Fk(i)1110 1007 y Fn(\).)42 b(De\014ne)23 b Fm(f)5 b Fn(\()p Fm(x)p Fn(\))25 b(=)h Fm(f)1551 1014 y Fk(i)1565 1007 y Fn(\()p Fm(x)p Fn(\).)41 b(The)182 1067 y(substitution)11 b Fm(f)16 b Fn(obtained)11 b(in)g(this)g(manner)e(is)i(w)o (ell-de\014ned)f(and)h(moreo)o(v)o(er)e Fm(f)c Fn(\()p Fm(\013)p Fl(#)p Fn(\))14 b(=)182 1127 y Fm(\014)s Fl(#)o Fn(.)120 1229 y(4.)24 b(De\014ne)481 1308 y Fm(g)r Fn(\()p Fm(x)p Fn(\))14 b(=)637 1235 y Fb(\()692 1277 y Fm(g)715 1284 y Fk(i)729 1277 y Fn(\()p Fm(x)p Fn(\))57 b(if)16 b Fm(x)e Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\014)1108 1284 y Fk(i)1121 1277 y Fn(\))d Fl(n)1187 1244 y Fb(S)1221 1287 y Fk(j)r Fi(6)p Fj(=)p Fk(i)1287 1277 y Fm(v)r(ar)q Fn(\()p Fm(\014)1409 1284 y Fk(j)1426 1277 y Fn(\))692 1337 y Fm(\025)116 b Fn(otherwise)182 1426 y(The)16 b(substitution)h Fm(g)h Fn(is)e(w)o(ell-de\014ned)f(and)i(moreo)o(v)o(er)d Fm(g)r Fn(\()p Fm(\014)s Fl(#)o Fn(\))g(=)g Fm(\013)p Fl(#)p Fn(.)60 1528 y Fd(2)60 1642 y Fg(Corollary)k(5.2)24 b Fn(Let)f Fm(\013;)8 b(\014)28 b Fl(2)e Fn(\(\006)16 b Fl([)f Fm(V)d Fn(\))844 1624 y Fj(+)896 1642 y Fn(b)q(e)24 b(patterns)f(and)h(let)e Fm(car)q(d)p Fn(\(\006\))k Fl(\025)f Fn(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 Fm(\013)p Fl(#)16 b Fn(has)h(indep)q(enden)o(t)f(v)m(ariable)g(segmen)o(ts.)120 1905 y(2.)24 b Fm(\013)17 b Fn(has)g(indep)q(enden)o(t)e(v)m(ariable)h (segmen)o(ts.)120 2007 y(3.)24 b Fm(\013)p Fl(#)16 b Fn(is)g(linear.)120 2109 y(4.)24 b Fm(\013)17 b Fn(is)f(linear.)60 2210 y Fg(Pro)r(of:)120 2312 y Fn(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 Fm(\013)g Fn(has)h(indep)q(enden)o(t)f(v)m(ariable)g(segmen)o(ts,)e (then)i(so)h(do)q(es)g Fm(\013)p Fl(#)p Fn(.)120 2516 y(3.)24 b(If)16 b Fm(\013)p Fl(#)g Fn(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 Fm(\013)g Fn(is)g(linear,)f(then)h Fm(\013)p Fl(#)h Fn(is)f(linear.)60 2719 y Fd(2)891 2856 y Fn(16)p eop %%Page: 17 17 17 16 bop 133 203 a Fn(It)11 b(has)g(already)g(b)q(een)g(sho)o(wn)g(b)o (y)f(Jiang)i Fh(et)h(al.)20 b Fn([JKS)1096 185 y Fj(+)1125 203 y Fn(94)q(])10 b(that)h(the)g(inclusion)f Fm(L)1608 210 y Fk(E)r(;)p Fj(\006)1673 203 y Fn(\()p Fm(\013)p Fn(\))15 b Fl(\022)60 264 y Fm(L)93 271 y Fk(E)r(;)p Fj(\006)158 264 y Fn(\()p Fm(\014)s Fn(\))20 b(is)f(decidable)g(if)h Fm(\014)i Fn(is)e(a)g(one-v)m(ariable)g(pattern)h(and)f Fm(car)q(d)p Fn(\(\006\))h Fl(\025)f Fn(2)g(\(the)g(same)60 324 y(result)h(for)h(NE-patterns)g(is)g(con)o(tained)f(in)h ([Ang80a]\).)37 b(Th)o(us)22 b(our)h(next)e(result)g(is)h(not)60 384 y(surprising.)f(Its)16 b(pro)q(of,)h(ho)o(w)o(ev)o(er,)e(is)g(m)o (uc)o(h)f(more)h(complicated)f(than)j(exp)q(ected)e(b)q(ecause)60 444 y(only)h(one)g(normal)g(form)f(is)h(assumed)g(to)g(b)q(e)h(a)f (one-v)m(ariable)h(pattern.)60 558 y Fg(Prop)r(osition)h(5.3)24 b Fn(Let)18 b Fm(\013;)8 b(\014)21 b Fl(2)d Fn(\(\006)13 b Fl([)f Fm(V)g Fn(\))872 540 y Fj(+)920 558 y Fn(b)q(e)18 b(patterns)h(and)h(let)d Fm(car)q(d)p Fn(\(\006\))i Fl(\025)e Fn(3.)29 b(If)18 b Fm(\013)p Fl(#)60 619 y Fn(is)g(a)h(one-v)m(ariable) g(pattern,)g(then)f Fm(L)765 626 y Fk(E)r(;)p Fj(\006)831 619 y Fn(\()p Fm(\013)p Fn(\))g(=)f Fm(L)1006 626 y Fk(E)r(;)p Fj(\006)1072 619 y Fn(\()p Fm(\014)s Fn(\))h(if)g(and)h(only)f(if)g Fm(\013)p Fl(#)h Fn(and)g Fm(\014)s Fl(#)f Fn(are)60 679 y(iden)o(tical)c(up)j(to)g(v)m(ariable)e(renaming.)60 799 y Fg(Pro)r(of:)21 b Fn(Again,)16 b(w)o(e)g(sho)o(w)h Fm(L)615 806 y Fk(E)r(;)p Fj(\006)680 799 y Fn(\()p Fm(\013)p Fn(\))d(=)g Fm(L)848 806 y Fk(E)r(;)p Fj(\006)913 799 y Fn(\()p Fm(\014)s Fn(\))i(if)g(and)h(only)f(if)f(there)h(are)h (substitutions)60 859 y Fm(f)28 b Fn(:)23 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)p Fn(\))g Fl(!)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)o Fn(\))581 841 y Fi(\003)622 859 y Fn(and)h Fm(g)i Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)o Fn(\))h Fl(!)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\))1239 841 y Fi(\003)1281 859 y Fn(suc)o(h)e(that)h Fm(f)5 b Fn(\()p Fm(\013)p Fl(#)q Fn(\))23 b(=)g Fm(\014)s Fl(#)60 919 y Fn(and)d Fm(g)r Fn(\()p Fm(\014)s Fl(#)o Fn(\))e(=)h Fm(\013)p Fl(#)p Fn(.)30 b(In)18 b(order)h(to)h(pro)o(v)o(e)e(the)h Fh(only)h(if)f Fn(case,)g(let)g Fm(L)1312 926 y Fk(E)r(;)p Fj(\006)1377 919 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))g(=)f Fm(L)1579 926 y Fk(E)r(;)p Fj(\006)1644 919 y Fn(\()p Fm(\013)p Fn(\))h(=)60 980 y Fm(L)93 987 y Fk(E)r(;)p Fj(\006)158 980 y Fn(\()p Fm(\014)s Fn(\))g(=)g Fm(L)336 987 y Fk(E)r(;)p Fj(\006)401 980 y Fn(\()p Fm(\014)s Fl(#)o Fn(\).)31 b(According)19 b(to)h(Theorem)e(3.1,)i Fm(\013)p Fl(#)g Fn(and)g Fm(\014)s Fl(#)f Fn(are)g(similar,)f(so)i (let)60 1040 y Fm(\013)p Fl(#)h Fn(=)g Fm(\013)227 1047 y Fj(0)246 1040 y Fm(u)274 1047 y Fj(1)294 1040 y Fm(\013)325 1047 y Fj(1)344 1040 y Fm(u)372 1047 y Fj(2)400 1040 y Fm(:)8 b(:)g(:)g(\013)497 1047 y Fk(m)p Fi(\000)p Fj(1)575 1040 y Fm(u)603 1047 y Fk(m)636 1040 y Fm(\013)667 1047 y Fk(m)721 1040 y Fn(and)21 b Fm(\014)s Fl(#)f Fn(=)g Fm(\014)982 1047 y Fj(0)1001 1040 y Fm(u)1029 1047 y Fj(1)1049 1040 y Fm(\014)1077 1047 y Fj(1)1096 1040 y Fm(u)1124 1047 y Fj(2)1152 1040 y Fm(:)8 b(:)g(:)f(\014)1245 1047 y Fk(m)p Fi(\000)p Fj(1)1323 1040 y Fm(u)1351 1047 y Fk(m)1384 1040 y Fm(\014)1412 1047 y Fk(m)1445 1040 y Fn(.)33 b(Let)21 b Fm(x)f Fn(b)q(e)g(the)60 1100 y(\(only\))d(v)m (ariable)h(o)q(ccurring)f(in)h Fm(\013)p Fl(#)p Fn(,)f(and)i(let)e Fm(x)950 1107 y Fj(1)969 1100 y Fm(;)8 b(x)1019 1107 y Fj(2)1038 1100 y Fm(;)g(:)g(:)g(:)g(;)g(x)1176 1107 y Fk(k)1214 1100 y Fn(b)q(e)18 b(the)f(v)m(ariables)h(o)q(ccurring)60 1160 y(in)g Fm(\014)s Fl(#)p Fn(.)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 Fm(\013)1301 1167 y Fj(0)1339 1160 y Fn(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 Fm(x)p Fn(.)25 b(Let)18 b Fm(c)g Fn(b)q(e)f(the)h(\014rst)g (letter)e(of)i Fm(u)1487 1227 y Fj(1)1507 1220 y Fn(.)25 b(W)l(e)18 b(c)o(ho)q(ose)60 1281 y Fm(a;)8 b(b)13 b Fl(2)h Fn(\006)5 b Fl(n)g(f)p Fm(c)p Fl(g)14 b Fn(suc)o(h)f(that)h Fm(a)g Fl(6)p Fn(=)f Fm(b)p Fn(.)20 b(Consider)14 b Fm(\034)922 1288 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1038 1281 y Fn(\()p Fm(\014)s Fl(#)o Fn(\).)20 b Fm(\034)1186 1288 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1302 1281 y Fn(\()p Fm(\014)s 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b(Clearly)l(,)18 b Fm(\033)r Fn(\()p Fm(\013)1675 1408 y Fj(0)1695 1401 y Fn(\))g(=)60 1461 y Fm(w)95 1468 y Fj(0)115 1461 y Fn(.)24 b(It)16 b(is)h(not)h(di\016cult)e(to)h(pro)o(v)o(e)f(that)i Fm(\033)r Fn(\()p Fm(x)p Fn(\))d(=)g Fm(\034)1008 1469 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1123 1461 y Fn(\()p Fm(x)1170 1468 y Fk(i)1182 1473 y Fe(1)1202 1461 y Fn(\))p Fm(\034)1242 1469 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1357 1461 y Fn(\()p Fm(x)1404 1468 y Fk(i)1416 1473 y Fe(2)1435 1461 y Fn(\))8 b Fm(:)g(:)g(:)g(\034)1549 1469 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1665 1461 y Fn(\()p Fm(x)1712 1468 y Fk(i)1724 1474 y Ff(l)1737 1461 y Fn(\),)60 1521 y(where)24 b Fm(i)226 1528 y Fj(1)245 1521 y Fm(;)8 b(i)284 1528 y Fj(2)303 1521 y Fm(;)g(:)g(:)g(:)g(i)408 1528 y Fk(l)447 1521 y Fl(2)27 b(f)p Fn(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(k)r Fl(g)p Fn(,)25 b(b)q(ecause)f Fm(\013)976 1528 y Fj(0)1020 1521 y Fn(con)o(tains)g(only)g Fm(x)p Fn('s.)44 b(Supp)q(ose)25 b Fm(\033)r Fn(\()p Fm(\013)1732 1528 y Fj(1)1751 1521 y Fn(\))60 1582 y(con)o(tains)d(a)g(pre\014x)f Fm(w)i Fn(of)f Fm(u)592 1589 y Fj(2)633 1582 y Fn(\()p Fm(w)i Fn(=)f Fm(u)800 1589 y Fj(2)841 1582 y Fn(is)e(p)q(ossible\).)37 b(Then)22 b Fm(w)h Fn(m)o(ust)d(b)q(e)i(of)g(the)f(form)60 1642 y Fm(\034)81 1650 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)197 1642 y Fn(\()p Fm(x)244 1649 y Fk(j)258 1654 y Fe(1)277 1642 y Fn(\))p Fm(\034)317 1650 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)432 1642 y Fn(\()p Fm(x)479 1649 y Fk(j)493 1654 y Fe(2)513 1642 y Fn(\))8 b Fm(:)g(:)g(:)g(\034)627 1650 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)742 1642 y Fn(\()p Fm(x)789 1649 y Fk(j)803 1659 y Ff(l)813 1652 y Fc(0)829 1642 y Fn(\),)17 b(where)g Fm(j)1041 1649 y Fj(1)1061 1642 y Fm(;)8 b(j)1103 1649 y Fj(2)1122 1642 y Fm(;)g(:)g(:)g(:)g(j)1230 1649 y Fk(l)1241 1640 y Fc(0)1270 1642 y Fl(2)16 b(f)p Fn(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(k)r Fl(g)p Fn(.)24 b(This)18 b(con-)60 1702 y(tradicts)24 b(the)f(fact)h(that)g Fl(j)p Fm(\034)590 1710 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)706 1702 y Fn(\()p Fm(x)753 1709 y Fk(j)771 1702 y Fn(\))p Fl(j)i Fm(>)h Fl(j)p Fm(u)937 1709 y Fj(2)956 1702 y Fl(j)d Fn(for)g(ev)o(ery)e Fm(x)1240 1709 y Fk(j)1285 1702 y Fl(2)27 b Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\).)44 b(Rep)q(eated)60 1762 y(application)18 b(of)g(this)g(argumen)o(t)e(yields)h Fm(\033)r Fn(\()p Fm(\013)906 1769 y Fk(i)920 1762 y Fn(\))f(=)h Fm(w)1045 1769 y Fk(i)1075 1762 y Fn(=)g Fm(\034)1151 1770 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1266 1762 y Fn(\()p Fm(\014)1313 1769 y Fk(i)1327 1762 y Fn(\).)26 b(As)18 b(in)f(Lemma)f(3.4,)60 1822 y(w)o(e)21 b(obtain)i(a)f(substitution)g Fm(f)29 b Fn(:)22 b Fl(f)p Fm(x)p Fl(g)h(!)g Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)o Fn(\))1049 1804 y Fi(\003)1090 1822 y Fn(suc)o(h)f(that)g Fm(f)5 b Fn(\()p Fm(\013)p Fl(#)q Fn(\))23 b(=)g Fm(\014)s Fl(#)o Fn(,)g(namely)60 1883 y Fm(f)5 b Fn(\()p Fm(x)p Fn(\))17 b(=)g Fm(x)255 1890 y Fk(i)267 1895 y Fe(1)286 1883 y Fm(x)314 1890 y Fk(i)326 1895 y Fe(2)353 1883 y Fm(:)8 b(:)g(:)g(x)447 1890 y Fk(i)459 1896 y Ff(l)472 1883 y Fn(.)27 b(Moreo)o(v)o(er,)17 b(if)g Fl(j)p Fm(\013)834 1890 y Fk(i)848 1883 y Fl(j)862 1890 y Fk(x)901 1883 y Fn(=)f Fm(r)q Fn(,)j(then)e Fm(\014)1151 1890 y Fk(i)1182 1883 y Fn(=)g Fm(f)5 b Fn(\()p Fm(x)p Fn(\))j Fm(:)g(:)g(:)f(f)e Fn(\()p Fm(x)p Fn(\),)18 b(where)g Fm(f)5 b Fn(\()p Fm(x)p Fn(\))60 1943 y(o)q(ccurs)19 b Fm(r)g Fn(times)e(in)h Fm(\014)474 1950 y Fk(i)488 1943 y Fn(.)27 b(Th)o(us)19 b(ev)o(ery)e(v)m(ariable)h Fm(x)999 1950 y Fk(j)1034 1943 y Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\))g(o)q(ccurs)h(at)g(least)f(as)h(often)60 2003 y(in)i Fm(\014)s Fl(#)g Fn(as)h Fm(x)f Fn(in)g Fm(\013)p Fl(#)p Fn(.)37 b(No)o(w)21 b(consider)g Fm(\034)814 2011 y Fi(j)p Fk(\014)r Fi(#j)p Fk(;a;b)929 2003 y Fn(\()p Fm(\013)p Fl(#)p Fn(\).)37 b(Again,)22 b(there)f(is)g(a)h(substitution) g Fm(\027)60 2063 y Fn(suc)o(h)e(that)h Fm(\027)s Fn(\()p Fm(\014)s Fl(#)p Fn(\))g(=)g Fm(\034)506 2071 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)620 2063 y Fn(\()p Fm(\013)p Fl(#)q Fn(\))f(and)h(furthermore)e Fm(\027)s Fn(\()p Fm(\014)1182 2070 y Fj(0)1202 2063 y Fn(\))i(=)g Fm(\013)1332 2070 y Fj(0)1352 2063 y Fn(.)34 b(Let)21 b Fm(x)1520 2070 y Fk(j)1558 2063 y Fn(b)q(e)g(a)g(v)m(ari-)60 2123 y(able)16 b(of)h Fm(v)r(ar)q Fn(\()p Fm(\014)341 2130 y Fj(0)360 2123 y Fn(\))f(suc)o(h)h(that)g Fm(\027)s Fn(\()p Fm(x)686 2130 y Fk(j)704 2123 y Fn(\))f(con)o(tains)h(an)g(anc) o(hor)g(segmen)o(t)e(of)i Fm(x)p Fn(.)22 b(It)16 b(follo)o(ws)h(that)60 2183 y Fl(j)p Fm(\013)p Fl(#j)144 2190 y Fk(x)180 2183 y Fn(=)d Fl(j)p Fm(\014)s Fl(#)o(j)315 2190 y Fk(x)335 2195 y Ff(j)353 2183 y Fn(.)21 b(Since)15 b Fm(\014)s Fl(#)h Fn(is)g(in)g(normal)f(form,)g(w)o(e)h(infer)f Fm(j)i Fn(=)d Fm(k)i Fn(=)d(1,)k(that)f(is)g(to)h(sa)o(y)l(,)f Fm(\014)s Fl(#)60 2244 y Fn(con)o(tains)h(only)g(the)f(v)m(ariable)h Fm(x)656 2251 y Fj(1)675 2244 y Fn(.)24 b(Ob)o(viously)l(,)15 b Fm(\013)p Fl(#)i Fn(and)h Fm(\014)s Fl(#)e Fn(are)h(iden)o(tical)e (up)i(to)g(v)m(ariable)60 2304 y(renaming.)j Fd(2)133 2418 y Fn(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 2478 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 2538 y(a)j(result)e(similar)g(to)h(Prop)q(osition)i(5.3.)60 2653 y Fg(Prop)r(osition)g(5.4)24 b Fn(Let)19 b Fm(\013;)8 b(\014)21 b Fl(2)e Fn(\(\006)13 b Fl([)h Fm(V)d Fn(\))875 2634 y Fj(+)924 2653 y Fn(b)q(e)19 b(patterns)g(and)h(let)e Fm(car)q(d)p Fn(\(\006\))i Fl(\025)e Fn(4.)30 b(Sup-)60 2713 y(p)q(ose)20 b Fm(\013)p Fl(#)f Fn(=)f Fm(\013)335 2720 y Fj(0)355 2713 y Fm(u)383 2720 y Fj(1)402 2713 y Fm(\013)433 2720 y Fj(1)453 2713 y Fm(u)481 2720 y Fj(2)509 2713 y Fm(:)8 b(:)g(:)f(\013)605 2720 y Fk(m)p Fi(\000)p Fj(1)684 2713 y Fm(u)712 2720 y Fk(m)745 2713 y Fm(\013)776 2720 y Fk(m)809 2713 y Fn(,)19 b(where)g Fl(j)p Fm(\013)1031 2720 y Fj(0)1050 2713 y Fl(j)g(\024)f Fn(1,)i Fl(j)p Fm(\013)1243 2720 y Fk(m)1276 2713 y Fl(j)e(\024)g Fn(1,)i(and)g Fl(j)p Fm(\013)1566 2720 y Fk(i)1580 2713 y Fl(j)e Fn(=)g(1)i(for)891 2856 y(17)p eop %%Page: 18 18 18 17 bop 60 203 a Fn(0)21 b Fm(<)g(i)g(<)f(m)p Fn(.)34 b(In)20 b(other)g(w)o(ords,)i Fm(\013)p Fl(#)e Fn(do)q(es)i(not)e(con)o (tain)h(consecutiv)o(e)d(v)m(ariables.)34 b(Then)60 264 y Fm(L)93 271 y Fk(E)r(;)p Fj(\006)158 264 y Fn(\()p Fm(\013)p Fn(\))17 b(=)f Fm(L)331 271 y Fk(E)r(;)p Fj(\006)396 264 y Fn(\()p Fm(\014)s Fn(\))h(if)h(and)g(only)f(if)h Fm(\013)p Fl(#)g Fn(and)g Fm(\014)s Fl(#)f Fn(are)h(iden)o(tical)e(up)h (to)i(v)m(ariable)e(renam-)60 324 y(ing.)60 444 y Fg(Pro)r(of:)25 b Fn(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 Fm(L)1315 451 y Fk(E)r(;)p Fj(\006)1380 444 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))f(=)f Fm(L)1580 451 y Fk(E)r(;)p Fj(\006)1645 444 y Fn(\()p Fm(\013)p Fn(\))h(=)60 504 y Fm(L)93 511 y Fk(E)r(;)p Fj(\006)158 504 y Fn(\()p Fm(\014)s Fn(\))h(=)h Fm(L)337 511 y Fk(E)r(;)p Fj(\006)403 504 y Fn(\()p Fm(\014)s Fl(#)o Fn(\).)32 b(By)19 b(Theorem)f(3.1,)j Fm(\014)s Fl(#)e Fn(=)h Fm(\014)1089 511 y Fj(0)1108 504 y Fm(u)1136 511 y Fj(1)1156 504 y Fm(\014)1184 511 y Fj(1)1203 504 y Fm(u)1231 511 y Fj(2)1259 504 y Fm(:)8 b(:)g(:)f(\014)1352 511 y Fk(m)p Fi(\000)p Fj(1)1430 504 y Fm(u)1458 511 y Fk(m)1491 504 y Fm(\014)1519 511 y Fk(m)1552 504 y Fn(.)32 b(Let)20 b Fm(x)g Fl(2)60 565 y Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)p Fn(\).)31 b(There)19 b(is)h(an)g(index)f Fm(i)p Fn(,)g(0)h Fl(\024)g Fm(i)f Fl(\024)g Fm(m)p Fn(,)h(suc)o(h)g(that)g Fm(\013)1255 572 y Fk(i)1288 565 y Fn(=)g Fm(x)p Fn(.)31 b(W.l.o.g.)18 b(w)o(e)i(ma)o(y)60 625 y(assume)12 b(that)g(0)j Fm(<)e(i)h(<)g(m)d Fn(\(again,)j(the)e(cases)g Fm(i)i Fn(=)g(0)e(and)h Fm(i)h Fn(=)f Fm(m)f Fn(are)h(similar)d(and)j(simpler\).)60 685 y(Let)18 b Fm(c)g Fn(b)q(e)h(the)e(last)i(letter)e(of)h Fm(u)654 692 y Fk(i)686 685 y Fn(and)h Fm(d)f Fn(b)q(e)g(the)g(\014rst) h(letter)e(of)h Fm(u)1300 692 y Fk(i)p Fj(+1)1359 685 y Fn(.)27 b(Again,)18 b(w)o(e)g(c)o(ho)q(ose)60 745 y Fm(a;)8 b(b)13 b Fl(2)h Fn(\006)d Fl(n)g(f)p Fm(c;)d(d)p Fl(g)17 b Fn(suc)o(h)f(that)g Fm(a)e Fl(6)p Fn(=)g Fm(b)p Fn(.)20 b(De\014ne)570 886 y Fm(\034)6 b Fn(\()p Fm(z)r Fn(\))14 b(=)726 813 y Fb(\()780 856 y Fm(\034)801 863 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)915 856 y Fn(\()p Fm(z)r Fn(\))58 b(if)15 b Fm(x)f Fn(=)g Fm(z)r(;)780 916 y(\025)212 b Fn(otherwise.)60 1027 y Fm(\034)6 b Fn(\()p Fm(\013)p Fl(#)p Fn(\))17 b(can)h(b)q(e)g(written)f(as)h Fm(w)625 1034 y Fj(0)644 1027 y Fm(u)672 1034 y Fj(1)692 1027 y Fm(w)727 1034 y Fj(1)755 1027 y Fm(:)8 b(:)g(:)f(w)855 1034 y Fk(m)p Fi(\000)p Fj(1)933 1027 y Fm(u)961 1034 y Fk(m)994 1027 y Fm(w)1029 1034 y Fk(m)1063 1027 y Fn(,)17 b(where)g Fm(w)1271 1034 y Fk(j)1305 1027 y Fn(=)f Fm(\034)1380 1035 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)1494 1027 y Fn(\()p Fm(x)p Fn(\))h(if)g Fm(\013)1654 1034 y Fk(j)1688 1027 y Fn(=)f Fm(x)60 1088 y Fn(and)f Fm(w)188 1095 y Fk(j)220 1088 y Fn(=)f Fm(\025)h Fn(otherwise.)21 b(Since)14 b Fm(L)710 1095 y Fk(E)r(;)p Fj(\006)775 1088 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))g(=)g Fm(L)968 1095 y Fk(E)r(;)p Fj(\006)1033 1088 y Fn(\()p Fm(\014)s Fl(#)o Fn(\),)h(there)f(is)g(a)h (substitution)g Fm(\027)j Fn(suc)o(h)60 1148 y(that)f Fm(\027)s Fn(\()p Fm(\014)s Fl(#)p Fn(\))d(=)h Fm(w)389 1155 y Fj(0)409 1148 y Fm(u)437 1155 y Fj(1)456 1148 y Fm(w)491 1155 y Fj(1)519 1148 y Fm(:)8 b(:)g(:)g(w)620 1155 y Fk(m)p Fi(\000)p Fj(1)698 1148 y Fm(u)726 1155 y Fk(m)759 1148 y Fm(w)794 1155 y Fk(m)827 1148 y Fn(.)23 b(Clearly)l(,)16 b Fm(\027)s Fn(\()p Fm(\014)1118 1155 y Fk(i)1131 1148 y Fn(\))f(=)g Fm(\034)6 b Fn(\()p Fm(\013)1295 1155 y Fk(i)1309 1148 y Fn(\))15 b(=)f Fm(\034)6 b Fn(\()p Fm(x)p Fn(\))15 b(=)f Fm(\034)1576 1156 y Fi(j)p Fk(\014)r Fi(#)o(j)p Fk(;a;b)1691 1148 y Fn(\()p Fm(x)p Fn(\).)60 1208 y(Let)19 b Fm(y)h Fn(b)q(e)f(the)g(v)m(ariable)f(in)h Fm(v)r(ar)q Fn(\()p Fm(\014)717 1215 y Fk(i)730 1208 y Fn(\))g(suc)o(h)f(that)h Fm(\027)s Fn(\()p Fm(y)r Fn(\))g(con)o (tains)g(the)f(anc)o(hor)i(segmen)o(t)d(of)60 1268 y Fm(x)p Fn(.)22 b(It)16 b(is)h(clear)f(that)h Fm(y)h Fn(o)q(ccurs)f (exactly)e(once)i(in)f Fm(\014)1008 1275 y Fk(i)1022 1268 y Fn(;)g(in)g(the)h(sequel)e(w)o(e)h(call)g Fm(y)i Fn(the)f(anc)o(hor)60 1328 y(v)m(ariable)f(of)g Fm(x)p Fn(.)21 b(F)l(urthermore,)14 b(it)i(follo)o(ws)g Fl(j)p Fm(\014)s Fl(#)o(j)949 1335 y Fk(y)984 1328 y Fl(\024)d(j)p Fm(\013)p Fl(#j)1120 1335 y Fk(x)1142 1328 y Fn(.)133 1389 y(No)o(w)25 b(consider)f Fm(\034)472 1396 y Fi(j)p Fk(\013)p Fi(#j)p Fk(;a;b)588 1389 y Fn(\()p Fm(\014)s Fl(#)o Fn(\))k(=)h Fm(v)800 1396 y Fj(0)819 1389 y Fm(u)847 1396 y Fj(1)866 1389 y Fm(v)890 1396 y Fj(1)918 1389 y Fm(:)8 b(:)g(:)g(v)1008 1396 y Fk(m)p Fi(\000)p Fj(1)1086 1389 y Fm(u)1114 1396 y Fk(m)1147 1389 y Fm(v)1171 1396 y Fk(m)1204 1389 y Fn(,)26 b(where)e Fm(v)1417 1396 y Fk(k)1467 1389 y Fn(=)k Fm(\034)1554 1396 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1670 1389 y Fn(\()p Fm(\014)1717 1396 y Fk(k)1737 1389 y Fn(\).)60 1449 y(Since)g Fm(L)233 1456 y Fk(E)r(;)p Fj(\006)298 1449 y Fn(\()p Fm(\013)p Fl(#)p Fn(\))36 b(=)f Fm(L)534 1456 y Fk(E)r(;)p Fj(\006)599 1449 y Fn(\()p Fm(\014)s Fl(#)o Fn(\),)d(there)c(is)h(a)g(substitution) g Fm(\033)h Fn(suc)o(h)f(that)g Fm(\033)r Fn(\()p Fm(\013)p Fl(#)p Fn(\))35 b(=)60 1509 y Fm(v)84 1516 y Fj(0)103 1509 y Fm(u)131 1516 y Fj(1)151 1509 y Fm(v)175 1516 y Fj(1)202 1509 y Fm(:)8 b(:)g(:)g(v)292 1516 y Fk(m)p Fi(\000)p Fj(1)370 1509 y Fm(u)398 1516 y Fk(m)431 1509 y Fm(v)455 1516 y Fk(m)488 1509 y Fn(.)32 b(Clearly)l(,)19 b Fm(\033)r Fn(\()p Fm(\013)797 1516 y Fk(i)810 1509 y Fn(\))h(=)g Fm(\033)r Fn(\()p Fm(x)p Fn(\))f(=)h Fm(v)1104 1516 y Fk(i)1137 1509 y Fn(=)g Fm(\034)1216 1517 y Fi(j)p Fk(\013)p Fi(#)o(j)p Fk(;a;b)1332 1509 y Fn(\()p Fm(\014)1379 1516 y Fk(i)1392 1509 y Fn(\).)32 b(Hence)19 b Fl(j)p Fm(\013)p Fl(#j)1690 1516 y Fk(x)1731 1509 y Fl(\024)60 1569 y(j)p Fm(\014)s Fl(#)o(j)143 1576 y Fk(y)164 1569 y Fn(.)h(Both)12 b(inequalities)f(yield)h Fl(j)p Fm(\013)p Fl(#j)767 1576 y Fk(x)802 1569 y Fn(=)i Fl(j)p Fm(\014)s Fl(#)o(j)937 1576 y Fk(y)958 1569 y Fn(.)20 b(As)12 b(a)i(further)e (consequence,)g(w)o(e)h(observ)o(e)60 1629 y(that)g(if)f Fm(y)i Fn(o)q(ccurs)e(in)g Fm(\014)468 1636 y Fk(j)486 1629 y Fn(,)h(0)h Fl(\024)g Fm(j)i Fl(\024)e Fm(m)p Fn(,)e(then)h Fm(\014)898 1636 y Fk(j)929 1629 y Fn(=)h Fm(\015)s(\014)1037 1636 y Fk(i)1051 1629 y Fm(\016)r Fn(.)19 b(Let)13 b(us)f(compare)f (the)i(equal)e(w)o(ords)60 1689 y Fm(\034)6 b Fn(\()p Fm(\013)p Fl(#)p Fn(\))17 b(and)h Fm(\027)s Fn(\()p Fm(\014)s Fl(#)p Fn(\))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 Fm(\027)s Fn(\()p Fm(z)r Fn(\))d(=)h Fm(\025)i Fn(for)60 1750 y(all)e Fm(z)g Fl(2)f Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\))d Fl(n)f Fm(v)r(ar)q Fn(\()p Fm(\014)552 1757 y Fk(i)565 1750 y Fn(\).)22 b(Secondly)l(,)15 b(the)h(same)g(length)g(argumen)o(t)g 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Fn(\()p Fm(\014)981 2039 y Fk(j)998 2032 y Fn(\))i(and)h Fl(j)p Fm(\014)1170 2039 y Fk(j)1188 2032 y Fl(j)1202 2039 y Fk(y)1236 2032 y Fn(=)d(1,)i(and)133 2134 y Fl(\017)24 b Fm(\013)213 2141 y Fk(j)245 2134 y Fl(6)p Fn(=)14 b Fm(x)p Fn(,)h(0)g Fl(\024)e Fm(j)k Fl(\024)d Fm(m)p Fn(,)h(implies)f Fm(y)h Fl(62)f Fm(v)r(ar)q Fn(\()p Fm(\014)981 2141 y Fk(j)998 2134 y Fn(\).)60 2235 y(Just)i(compare)e(the)h(equal)g(w)o(ords)h Fm(\034)6 b Fn(\()p Fm(\013)p Fl(#)p Fn(\))16 b(and)g Fm(\027)s Fn(\()p Fm(\014)s Fl(#)o Fn(\))g(from)e(left)h(to)g(righ)o (t.)21 b(Supp)q(ose)c(there)60 2296 y(is)g(another)h(index)f Fm(k)h Fl(2)e(f)p Fn(0)p Fm(;)8 b Fn(1)p Fm(;)g(:)g(:)g(:)g(;)g(m)p Fl(g)p Fn(,)17 b Fm(k)h Fl(6)p Fn(=)e Fm(i)p Fn(,)h(suc)o(h)g(that)h Fm(\013)1213 2303 y Fk(k)1250 2296 y Fn(=)e Fm(x)p Fn(.)24 b(It)17 b(follo)o(ws)h(from)e(the)60 2356 y(ab)q(o)o(v)o(e)f(that)g Fm(\014)329 2363 y Fk(k)363 2356 y Fn(=)f Fm(\015)s(\014)471 2363 y Fk(i)485 2356 y Fm(\016)r Fn(.)20 b(Since)13 b Fm(i)i 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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 Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\))60 2657 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 Fm(y)h Fl(2)e Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)o Fn(\))h(is)g(the)h(anc)o(hor)g(v)m(ariable)f(of)60 2717 y Fm(x)d Fl(2)g Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)o Fn(\),)i(then)g Fm(y)i Fn(o)q(ccurs)e(in)g Fm(\014)735 2724 y Fk(j)769 2717 y Fn(if)g(and)h(only)f(if)g Fm(\013)1092 2724 y Fk(j)1124 2717 y Fn(=)e Fm(x)p Fn(.)891 2856 y(18)p eop %%Page: 19 19 19 18 bop 133 203 a Fn(With)12 b(these)f(preparatory)h(considerations,) h(w)o(e)e(are)h(no)o(w)g(able)f(to)h(de\014ne)g(substitutions)60 264 y Fm(f)21 b Fn(:)15 b Fm(v)r(ar)q Fn(\()p Fm(\013)p Fl(#)p Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)n Fn(\))551 246 y Fi(\003)588 264 y Fn(and)i Fm(g)g Fn(:)d Fm(v)r(ar)q Fn(\()p Fm(\014)s Fl(#)o Fn(\))g Fl(!)h Fm(v)r(ar)q Fn(\()p 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