%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: csl.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSCommandLine: dvips -o csl.ps csl.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1997.03.24:0924 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 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/rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 39158280 55380996 1000 300 300 (csl.dvi) @start /Fa 48 122 df12 DI< 90380FC3F09038382E089038E0781C3801C0701518390380E000A5003FB512F8390701C0 70A6390E0380E0A6391C0701C0A239FF1FC7F81E1A7F9920>I<12381278A212381208A2 1210A2122012401280050B7E830B>44 DI<121CA2123C121812 00A81270A212F0126006107D8F0B>58 D<144014C01301801302A21304A2EB0870A21310 A2132080134013C0EBFFF838010038801202A25AA2000C7F001C131EB4EBFFC01A1A7F99 1D>65 D<380FFFF03801C01C140E140F1407EA0380140FA3141E143C380700F8EBFFE0EB 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y(\(A)o(C\).)13 b(W)m(e)f(call)h(this)g(the)h Fm(termination)g(hier)n(ar)n(chy)p Fy(.)e(One)i(motiv)n(atio)o(n)d(for)h(the)i(prop)q(erties)183 2378 y(stronger)f(than)f(termination)f(is)h(that)h(they)f(ob)q(ey)h(b)q (etter)h(decomp)q(osition)d(theorems.)h(F)m(or)183 2428 y(instance,)18 b Fn(!)q Fy(-termination)d(and)j(simple)e(termination)g (satisfy)h(direct)h(sum)f(mo)q(dularit)o(y)p eop %%Page: 2 2 2 1 bop 340 194 a Fy(\([13]\),)18 b(and)h(total)f(termination)f(allo)o (ws)h(distribution)h(elimination)d(without)i(linearit)o(y)340 244 y(conditions)e(\([21]\).)f(T)m(ermination)f(itself)i(do)q(es)h(not) f(ha)o(v)o(e)h(these)g(prop)q(erties.)h(The)e(prop-)340 293 y(erties)g(w)o(eak)o(er)e(than)g(termination)e(are)j(motiv)n(ated)d (b)o(y)i(frequen)o(tly)g(o)q(ccurring)h(shap)q(es)g(of)340 343 y(in\014nite)h(reductions.)h(An)g(extra)f(implication)d(SN)k Fo(\))e Fy(WN)h(\(w)o(eak)h(normalization\))c(can)340 393 y(b)q(e)k(added)f(as)f(an)h(indep)q(enden)o(t)h(branc)o(h)f(in)f 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y(Undecidabilit)o(y)e(of)g(con\015uence)i(is)f(w)o(ell-kno)o(wn)f (\([10)o(]\),)g(for)g(the)i(other)f(prop)q(erties)i(in)340 1047 y(the)e(con\015uence)i(hierarc)o(h)o(y)d(it)g(is)h(easy)g(to)f (see)i(to)q(o.)d(Also)h(undecidabilit)o(y)g(of)g(most)f(of)h(the)340 1097 y(prop)q(erties)18 b(related)f(to)f(termination)e(is)i(kno)o(wn)g (\([8)o(,)g(18)o(,)g(1,)f(16,)g(23]\),)g(sometimes)g(ev)o(en)340 1146 y(for)e(single)f(rules)h(\([2)o(,)f(16,)g(14)o(]\).)g(The)h (undecidabilit)o(y)f(of)f Fn(!)q Fy(-termination)g(is)i(a)f(new)h (result;)340 1196 y(this)h(pap)q(er)h(includes)f(a)g(sk)o(etc)o(h)h(of) e(the)i(pro)q(of.)403 1246 y(In)9 b(this)h(pap)q(er)h(w)o(e)f(do)f(not) h(only)f(pro)o(vide)g(a)h(general)g(framew)o(ork)e(for)h(pro)o(ving)g (this)h(kind)340 1296 y(of)h(undecidabilit)o(y)m(,)f(for)h(all)f (implications)f(in)i(the)h(hierarc)o(hies)h(except)g(one|PT)f Fo(\))f Fn(!)q Fy(T|)340 1346 y(w)o(e)17 b(pro)o(v)o(e)g(the)h (stronger)f(result)h(of)e Fm(r)n(elative)h(unde)n(cidability)t Fy(:)f(for)g(suc)o(h)i(an)e(implication)340 1395 y Fn(X)22 b Fo(\))17 b Fn(Y)27 b Fy(w)o(e)18 b(pro)o(v)o(e)g(that)g(the)g(prop)q (ert)o(y)h Fn(X)i Fy(is)d(undecidable)g(for)f(TRSs)h(satisfying)e Fn(Y)10 b Fy(.)340 1445 y(As)17 b(a)f(consequence,)i(relativ)o(e)f (undecidabilit)o(y)e(of)h Fn(X)j Fo(\))c Fn(Z)k Fy(immediately)13 b(follo)o(ws)i(from)340 1495 y(v)n(alidit)o(y)d(of)h(the)i(implication) c Fn(Y)21 b Fo(\))11 b Fn(Z)17 b Fy(and)c(relativ)o(e)h(undecidabilit)o (y)f(of)g Fn(X)i Fo(\))c Fn(Y)f Fy(.)403 1545 y(All)e(of)h(our)g(pro)q (ofs)h(are)f(giv)o(en)g(b)o(y)h(means)e(of)h(P)o(ost's)g(Corresp)q (ondence)j(Problem)d(\(PCP\))340 1595 y(in)j(the)h(follo)o(wing)d(w)o (a)o(y:)h(for)h(all)f(of)g(the)i(implications)d Fn(X)15 b Fo(\))c Fn(Y)22 b Fy(and)12 b(all)f(instances)i(of)f(PCP)340 1644 y(w)o(e)17 b(construct)h(a)d(TRS)h(that)g(alw)o(a)o(ys)g (satis\014es)h Fn(Y)9 b Fy(,)16 b(and)g(either)h(satis\014es)g Fn(X)j Fy(if)15 b(and)h(only)340 1694 y(if)i(the)h(PCP)g(instance)g (admits)e(a)h(solution,)g(or)g(satis\014es)h Fn(X)k Fy(if)17 b(and)h(only)g(if)g(the)h(PCP)340 1744 y(instance)d(admits)d(no)h (solution.)g(Since)h(PCP)g(is)g(kno)o(wn)f(to)h(b)q(e)g(undecidable)g (\([19)o(]\),)f(this)340 1794 y(pro)o(v)o(es)h(relativ)o(e)e (undecidabilit)o(y)g(of)h(the)g(implication)d Fn(X)k Fo(\))c Fn(Y)e Fy(.)403 1844 y(The)g(main)f(part)h(of)g(the)h(pap)q(er) f(consists)i(of)d(constructions)j(of)e(suc)o(h)h(TRSs)f(parametrized) 340 1894 y(b)o(y)14 b(PCP)h(instances)g(and)f(corresp)q(onding)h(pro)q (ofs)f(of)f(the)i(ab)q(o)o(v)o(e)e(men)o(tioned)g(prop)q(erties.)340 1943 y(In)f(the)g(next)g(section)g(this)f(is)h(done)f(for)g(the)h (con\015uence)i(hierarc)o(h)o(y)d(and)h(in)f(Sect.)h(3)f(for)g(the)340 1993 y(termination)i(hierarc)o(h)o(y)m(.)h(These)i(t)o(w)o(o)e (sections)i(can)e(b)q(e)i(read)f(indep)q(enden)o(tly)m(.)f(The)h (de\014-)340 2043 y(nitions)f(of)f(the)i(v)n(arious)e(prop)q(erties)j (are)e(giv)o(en)g(in)f(the)i(resp)q(ectiv)o(e)h(sections;)f(for)e (further)340 2093 y(preliminaries)g(on)g(term)h(rewriting)f(w)o(e)h (refer)h(to)f([3)o(,)g(11)o(].)403 2143 y(F)m(or)c(all)g(implications)f (in)h(the)i(hierarc)o(hies)g(there)g(are)g(w)o(ell-kno)o(wn)e(examples) g(sho)o(wing)340 2192 y(the)i(in)o(v)n(alidit)o(y)c(of)i(the)i(con)o(v) o(erse)g(of)e(the)i(in)o(v)o(olv)o(ed)d(implication.)f(All)i(of)g(our)h (constructions)340 2242 y(are)17 b(designed)f(b)o(y)g(plugging)f(in)g (the)i(essen)o(tials)f(of)g(the)g(actual)g(example)f(in)g(some)g(basic) 340 2292 y(TRS)f(related)g(to)g(PCP)m(.)403 2342 y(W)m(e)h(conclude)i (this)f(in)o(tro)q(duction)f(b)o(y)h(giving)e(the)j(form)o(ulation)c (of)i(PCP)h(as)g(w)o(e)g(use)340 2392 y(it:)p eop %%Page: 3 3 3 2 bop 253 194 a Fy(giv)o(en)15 b(a)h(\014nite)f(alphab)q(et)h Fn(\000)21 b Fy(and)15 b(a)g(\014nite)h(set)g Fn(P)k Fo(\032)15 b Fn(\000)1146 179 y Fx(+)1183 194 y Fo(\002)c Fn(\000)1258 179 y Fx(+)1285 194 y Fy(,)k(is)g(there)i(some)253 244 y(natural)j(n)o(um)o(b)q(er)g Fn(n)j(>)g Fy(0)d(and)h(\()p Fn(\013)838 250 y Fk(i)851 244 y Fn(;)7 b(\014)893 250 y Fk(i)907 244 y Fy(\))23 b Fo(2)f Fn(P)k Fy(for)20 b Fn(i)j Fy(=)g(1)p Fn(;)7 b(:)g(:)g(:)12 b(;)7 b(n)20 b Fy(suc)o(h)h(that)253 293 y Fn(\013)280 299 y Fx(1)298 293 y Fn(\013)325 299 y Fx(2)350 293 y Fo(\001)7 b(\001)g(\001)f Fn(\013)433 299 y Fk(n)467 293 y Fy(=)11 b Fn(\014)533 299 y Fx(1)552 293 y Fn(\014)575 299 y Fx(2)601 293 y Fo(\001)c(\001)g(\001)f Fn(\014)680 299 y Fk(n)703 293 y Fy(?)183 376 y(The)19 b(set)g Fn(P)24 b Fy(is)19 b(called)f(an)h Fm(instanc)n(e)j Fy(of)c(PCP)m(,)g(the)h(string)g Fn(\013)1185 382 y Fx(1)1203 376 y Fn(\013)1230 382 y Fx(2)1255 376 y Fo(\001)7 b(\001)g(\001)f Fn(\013)1338 382 y Fk(n)1379 376 y Fy(=)20 b Fn(\014)1454 382 y Fx(1)1473 376 y Fn(\014)1496 382 y Fx(2)1522 376 y Fo(\001)7 b(\001)g(\001)f Fn(\014)1601 382 y Fk(n)183 426 y Fy(a)20 b Fm(solution)j Fy(for)e Fn(P)6 b Fy(.)19 b(Without)h(loss)g(of)g(generalit)o(y)g(w)o(e)g (require)i Fn(P)k Fy(to)20 b(b)q(e)h(non-empt)o(y)m(.)183 476 y(Matiy)o(asevic)o(h)10 b(and)h(Senizergues)h([15)o(])e(recen)o (tly)i(sho)o(w)o(ed)f(that)g(PCP)g(is)g(undecidable)g(ev)o(en)183 526 y(when)h(restricted)i(to)e(instances)i(consisting)e(of)f(sev)o(en)j (pairs.)d(W)m(e)h(assume)g(that)g Fn(\000)17 b Fy(is)12 b(\014xed)183 576 y(throughout)19 b(the)h(pap)q(er.)g(\(One)g(ma)o(y)e (assume)h(that)h Fn(\000)26 b Fy(=)21 b Fo(f)p Fy(0)p Fn(;)7 b Fy(1)p Fo(g)p Fy(.\))18 b(In)h(our)h(TRSs)f(w)o(e)183 625 y(need)h(for)e(ev)o(ery)i Fn(a)g Fo(2)f Fn(\000)24 b Fy(a)19 b(unary)g(sym)o(b)q(ol)e Fn(a)i Fy(\(and)g(sometimes)e(also)h (unary)h(sym)o(b)q(ols)183 675 y(\026)-21 b Fn(a)p Fy(,)23 b(_)-17 b Fn(a)19 b Fy(and)h(\177)-22 b Fn(a)p Fy(\).)19 b(F)m(or)g(an)o(y)f(string)i Fn(\013)g Fy(=)g Fn(a)839 681 y Fx(1)858 675 y Fn(a)880 681 y Fx(2)905 675 y Fo(\001)7 b(\001)g(\001)f Fn(a)983 681 y Fk(n)1025 675 y Fo(2)20 b Fn(\000)1105 660 y Fl(\003)1143 675 y Fy(and)f(an)o(y)g(term)f Fn(t)h Fy(w)o(e)h(de\014ne)183 725 y Fn(\013)p Fy(\()p Fn(t)p Fy(\))11 b(=)h Fn(a)334 731 y Fx(1)353 725 y Fy(\()p Fn(a)391 731 y Fx(2)409 725 y Fy(\()p Fo(\001)7 b(\001)g(\001)f Fy(\()p Fn(a)519 731 y Fk(n)541 725 y Fy(\()p Fn(t)p Fy(\)\))h Fo(\001)g(\001)g(\001)f Fy(\)\).)183 860 y Fp(2)56 b(The)18 b(Con\015uence)h(Hierarc)n(h)n(y)183 959 y Fy(In)14 b(this)h(section)g(w)o(e)g(sho)o(w)f(relativ)o(e)h (undecidabilit)o(y)e(of)h(all)f(implications)f(in)i(the)h(con\015u-)183 1009 y(ence)h(hierarc)o(h)o(y)f(as)g(presen)o(ted)i(in)d(the)h(in)o (tro)q(duction.)f(Actually)g(w)o(e)h(sho)o(w)g(the)h(stronger)183 1059 y(result)e(that)g(relativ)o(e)g(undecidabilit)o(y)f(holds)g(for)h (linear)f(TRSs.)245 1109 y(Let)23 b(us)g(\014rst)g(recall)f(the)h (de\014nitions)g(of)f(the)h(six)f(prop)q(erties)i(prop)q(erties)g(in)e (the)183 1158 y(con\015uence)e(hierarc)o(h)o(y)m(.)d(A)h(TRS)g Fo(R)g Fy(is)g(called)g Fm(c)n(on\015uent)24 b Fy(\(or)18 b(Ch)o(urc)o(h-Rosser,)h(CR\))e(if)183 1208 y Fo( )225 1193 y Fl(\003)225 1220 y(R)266 1208 y Fo(\001)11 b(!)331 1193 y Fl(\003)331 1220 y(R)383 1208 y Fo(\022)22 b(!)479 1193 y Fl(\003)479 1220 y(R)521 1208 y Fo(\001)11 b( )586 1193 y Fl(\003)586 1220 y(R)616 1208 y Fy(,)20 b(or,)f(equiv)n(alen)o (tly)m(,)f(ev)o(ery)j(t)o(w)o(o)f(con)o(v)o(ertible)g(terms)g(ha)o(v)o (e)g(a)183 1258 y(common)10 b(reduct.)15 b(A)e(TRS)g Fo(R)h Fy(is)f(called)g Fm(lo)n(c)n(al)r(ly)h(c)n(on\015uent)19 b Fy(\(or)13 b(w)o(eakly)g(Ch)o(urc)o(h-Rosser,)183 1308 y(W)o(CR\))20 b(if)f Fo( )408 1314 y Fl(R)450 1308 y Fo(\001)11 b(!)515 1314 y Fl(R)568 1308 y Fo(\022)23 b(!)665 1293 y Fl(\003)665 1319 y(R)707 1308 y Fo(\001)11 b( )772 1293 y Fl(\003)772 1319 y(R)802 1308 y Fy(.)21 b(A)f(TRS)h Fo(R)g Fy(is)f(called)h Fm(str)n(ongly)f(c)n(on\015uent)26 b Fy(\(or)183 1358 y(strongly)17 b(Ch)o(urc)o(h-Rosser,)h(SCR\))f(if)f Fo( )840 1364 y Fl(R)882 1358 y Fo(\001)11 b(!)947 1364 y Fl(R)994 1358 y Fo(\022)17 b(!)1085 1343 y Fx(=)1085 1369 y Fl(R)1127 1358 y Fo(\001)11 b( )1192 1343 y Fl(\003)1192 1369 y(R)1222 1358 y Fy(.)17 b(A)g(TRS)g Fo(R)h Fy(is)f(said)g(to)183 1407 y(ha)o(v)o(e)c(the)h Fm(normal)g(form)g(pr)n(op)n(erty)i Fy(\(NF\))e(if)e(ev)o(ery)j(term)d(con)o(v)o(ertible)i(to)f(a)g(normal) e(form)183 1457 y(reduces)21 b(to)e(that)h(normal)d(form,)g(or,)i (equiv)n(alen)o(tly)m(,)e(ev)o(ery)k(term)d(that)i(has)g(a)f(normal)183 1507 y(form)11 b(is)j(con\015uen)o(t.)g(A)g(TRS)f Fo(R)h Fy(is)f(said)g(to)h(ha)o(v)o(e)f Fm(unique)j(normal)e(forms)i Fy(\(UN\))e(if)f(di\013er-)183 1557 y(en)o(t)j(normal)d(forms)h(are)i (not)g(con)o(v)o(ertible.)f(A)h(TRS)f Fo(R)g Fy(is)h(said)f(to)g(ha)o (v)o(e)h Fm(unique)h(normal)183 1607 y(forms)g(with)h(r)n(esp)n(e)n(ct) f(to)h(r)n(e)n(duction)j Fy(\(UN)853 1592 y Fl(!)889 1607 y Fy(\))c(if)g(ev)o(ery)h(term)f(has)g(at)h(most)e(one)i(normal) 183 1657 y(form.)12 b(The)k(ab)q(o)o(v)o(e)e(de\014nition)h(of)f (strong)h(con\015uence)i(originates)d(from)g(Huet)h([9])f(and)h(is)183 1706 y(di\013eren)o(t)k(from)d(the)j(one)f(in)g(Dersho)o(witz)g(and)g (Jouannaud)g([3)o(].)f(They)i(call)e(a)h(TRS)f Fo(R)183 1756 y Fy(strongly)d(con\015uen)o(t)i(if)e Fo( )605 1762 y Fl(R)647 1756 y Fo(\001)d(!)712 1762 y Fl(R)755 1756 y Fo(\022)j(!)843 1741 y Fx(=)843 1768 y Fl(R)884 1756 y Fo(\001)d( )949 1741 y Fx(=)949 1768 y Fl(R)979 1756 y Fy(.)k(Klop)f([11)o(])h(calls)f(the)i(latter)f(prop)q(ert)o(y)183 1806 y(sub)q(comm)o(utativit)o(y)10 b(\(W)o(CR)644 1791 y Fl(\024)p Fx(1)688 1806 y Fy(\).)245 1856 y(Belo)o(w)k(w)o(e)h(use)g (PCP)g(to)g(sho)o(w)f(that)h(for)f(eac)o(h)h(of)f(the)h(\014v)o(e)f (implications)e Fn(X)k Fo(\))c Fn(Y)24 b Fy(in)183 1906 y(the)15 b(con\015uence)h(hierarc)o(h)o(y)e(the)h(prop)q(ert)o(y)g Fn(X)j Fy(is)d(undecidable)f(for)g(TRSs)h(satisfying)e(the)183 1955 y(prop)q(ert)o(y)18 b Fn(Y)9 b Fy(.)16 b(A)h(k)o(ey)g(observ)n (ation)g(is)g(that)g(an)g(arbitrary)g(PCP)g(instance)h Fn(P)k Fy(admits)16 b(a)183 2005 y(solution)d(if)g(and)g(only)h(if)f Fn(A)e Fo(!)672 1990 y Fl(\003)672 2019 y(R)701 2023 y Fj(0)716 2019 y Fx(\()p Fk(P)t Fx(\))782 2005 y Fn(B)16 b Fy(for)e(the)g(TRS)253 2148 y Fo(R)288 2154 y Fx(0)307 2148 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)427 2063 y Fi(8)427 2101 y(<)427 2175 y(:)562 2098 y Fn(A)g Fo(!)h Fn(f)t Fy(\()p Fn(\013)p Fy(\()p Fn(c)p Fy(\))p Fn(;)7 b(\014)r Fy(\()p Fn(c)p Fy(\)\))36 b(for)14 b(all)e(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)472 2148 y(f)t Fy(\()p Fn(x;)c(y)q Fy(\))12 b Fo(!)g Fn(f)t Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(\014)r Fy(\()p Fn(y)q Fy(\)\))27 b(for)14 b(all)e(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)470 2198 y(f)t Fy(\()p Fn(x;)c(x)p Fy(\))k Fo(!)h Fn(B)183 2279 y Fy(This)j(observ)n(ation)g(is)g(quite)g (simple:)e Fn(B)18 b Fy(can)d(b)q(e)h(reac)o(hed)g(if)e(and)h(only)g (if)f(a)h(term)f(of)h(the)183 2329 y(shap)q(e)d Fn(f)t Fy(\()p Fn(t;)7 b(t)p Fy(\))k(can)g(b)q(e)h(reac)o(hed,)g(and)f(this)g (can)g(b)q(e)h(reac)o(hed)g(from)e Fn(A)h Fy(if)f(and)h(only)f(if)g(a)h (PCP)p eop %%Page: 4 4 4 3 bop 340 194 a Fy(solution)12 b(for)h Fn(P)18 b Fy(exists.)13 b(T)m(o)f(arriv)o(e)h(at)g(results)g(for)g(linear)f(TRSs)h(and)g(for)f (some)g(tec)o(hnical)340 244 y(con)o(v)o(enience)k(this)d(basic)i (system)e(is)h(replaced)h(b)o(y)411 474 y Fo(R)446 480 y Fx(1)465 474 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)585 289 y Fi(8)585 326 y(>)585 339 y(>)585 351 y(>)585 364 y(>)585 376 y(>)585 389 y(>)585 401 y(>)585 413 y(>)585 426 y(<)585 501 y(>)585 513 y(>)585 526 y(>)585 538 y(>)585 550 y(>)585 563 y(>)585 575 y(>)585 588 y(>)585 600 y(:)822 324 y Fn(A)h Fo(!)f Fn(f)t Fy(\()p Fn(\013)p Fy(\()p Fn(c)p Fy(\))p Fn(;)c(\014)r Fy(\()p Fn(c)p Fy(\)\))37 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)732 374 y(f)t Fy(\()p Fn(x;)c(y)q Fy(\))13 b Fo(!)e Fn(f)t Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)c(\014)r Fy(\()p Fn(y)q Fy(\)\))28 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)732 423 y(f)t Fy(\()p Fn(x;)c(y)q Fy(\))13 b Fo(!)e Fn(g)q Fy(\()p Fn(x;)c(y)q Fy(\))732 473 y Fn(f)t Fy(\()p Fn(x;)g(y)q Fy(\))13 b Fo(!)e Fn(A)736 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861 y Fy(\))17 b Fo(2)g Fn(P)23 b Fy(for)17 b Fn(i)g Fy(=)h(1)p Fn(;)7 b(:)g(:)g(:)12 b(;)7 b(n)p Fy(.)16 b(W)m(e)h(ha)o(v)o(e)g(the)h(follo)o(wing)d (reduction)j(in)340 911 y Fo(R)375 917 y Fx(1)394 911 y Fy(\()p Fn(P)6 b Fy(\):)411 996 y Fn(A)12 b Fo(!)f Fn(f)t Fy(\()p Fn(\013)574 1002 y Fk(n)597 996 y Fy(\()p Fn(c)p Fy(\))p Fn(;)c(\014)689 1002 y Fk(n)712 996 y Fy(\()p Fn(c)p Fy(\)\))k Fo(!)831 979 y Fl(\003)862 996 y Fn(f)t Fy(\()p Fn(\013)929 1002 y Fx(1)955 996 y Fo(\001)c(\001)g (\001)e Fn(\013)1037 1002 y Fk(n)1059 996 y Fy(\()p Fn(c)p Fy(\))p Fn(;)i(\014)1151 1002 y Fx(1)1177 996 y Fo(\001)g(\001)g(\001)e Fn(\014)1255 1002 y Fk(n)1278 996 y Fy(\()p Fn(c)p Fy(\)\))12 b(=)g Fn(f)t Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)7 b(\015)r Fy(\()p Fn(c)p Fy(\)\))454 1063 y Fo(!)k Fn(g)q Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)c(\015)r Fy(\()p Fn(c)p Fy(\)\))13 b Fo(!)780 1046 y Fx(+)819 1063 y Fn(g)q Fy(\()p Fn(c;)7 b(c)p Fy(\))k Fo(!)g Fn(B)r(:)340 1148 y Fy(Con)o(v)o(ersely)m(,)20 b(supp)q(ose)h(that)f Fn(A)h Fo(!)917 1133 y Fl(\003)917 1162 y(R)946 1166 y Fj(1)962 1162 y Fx(\()p Fk(P)t Fx(\))1037 1148 y Fn(B)r Fy(.)f(Bey)o(ond)g(the)h(last)e Fn(A)h Fy(o)q(ccurring)h(in)e(this)340 1198 y(reduction)c(sequence)h(it)e(is)f(of)h(the)g(form)411 1283 y Fn(A)e Fo(!)f Fn(f)t Fy(\()p Fn(\013)574 1289 y Fk(n)597 1283 y Fy(\()p Fn(c)p Fy(\))p Fn(;)c(\014)689 1289 y Fk(n)712 1283 y Fy(\()p Fn(c)p Fy(\)\))k Fo(!)831 1266 y Fl(\003)862 1283 y Fn(f)t Fy(\()p Fn(\013)929 1289 y Fx(1)955 1283 y Fo(\001)c(\001)g(\001)e Fn(\013)1037 1289 y Fk(n)1059 1283 y Fy(\()p Fn(c)p Fy(\))p Fn(;)i(\014)1151 1289 y Fx(1)1177 1283 y Fo(\001)g(\001)g(\001)e Fn(\014)1255 1289 y Fk(n)1278 1283 y Fy(\()p Fn(c)p Fy(\)\))454 1346 y Fo(!)11 b Fn(g)q Fy(\()p Fn(\013)571 1352 y Fx(1)596 1346 y Fo(\001)c(\001)g(\001)f Fn(\013)679 1352 y Fk(n)701 1346 y Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\014)793 1352 y Fx(1)818 1346 y Fo(\001)g(\001)g(\001)f Fn(\014)897 1352 y Fk(n)920 1346 y Fy(\()p Fn(c)p Fy(\)\))12 b Fo(!)1040 1328 y Fl(\003)1070 1346 y Fn(g)q Fy(\()p Fn(c;)7 b(c)p Fy(\))507 1373 y Fi(|)p 526 1373 299 5 v 299 w({z)p 863 1373 V 299 w(})1189 1346 y Fo(!)k Fn(B)r(:)340 1457 y Fy(for)j(some)g Fn(n)e Fo(\025)g Fy(1)i(with)g(\()p Fn(\013)763 1463 y Fk(i)776 1457 y Fn(;)7 b(\014)818 1463 y Fk(i)832 1457 y Fy(\))12 b Fo(2)g Fn(P)19 b Fy(for)14 b Fn(i)f Fy(=)f(1)p Fn(;)7 b(:)g(:)g(:)12 b(;)7 b(n)p Fy(.)13 b(In)h(the)h(underbraced)h(part)e(only)340 1506 y(rewrite)i(rules)g(of) e(the)h(form)e Fn(g)q Fy(\()p Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(a)p Fy(\()p Fn(y)q Fy(\)\))14 b Fo(!)e Fn(g)q Fy(\()p Fn(x;)7 b(y)q Fy(\))16 b(are)f(used.)g(Hence)h Fn(\013)1556 1512 y Fx(1)1581 1506 y Fo(\001)7 b(\001)g(\001)f Fn(\013)1664 1512 y Fk(n)1686 1506 y Fy(\()p Fn(c)p Fy(\))13 b(=)340 1556 y Fn(\014)363 1562 y Fx(1)389 1556 y Fo(\001)7 b(\001)g(\001)f Fn(\014)468 1562 y Fk(n)491 1556 y Fy(\()p Fn(c)p Fy(\),)13 b(giving)g(a)g(solution)g(for)h Fn(P)6 b Fy(.)764 b Fo(u)-28 b(t)403 1648 y Fy(Belo)o(w)13 b(w)o(e)i(mak)o(e)d (frequen)o(t)j(use)f(of)g(the)g(follo)o(wing)d(result)k(of)e(Huet)i([9) o(].)340 1739 y Fh(Theorem)7 b(2.)21 b Fm(Every)15 b(line)n(ar)f(str)n (ongly)h(close)n(d)f(TRS)i(is)e(str)n(ongly)h(c)n(on\015uent.)174 b Fo(u)-28 b(t)403 1830 y Fy(Here)11 b(a)f(TRS)f Fo(R)h Fy(is)g(called)g(strongly)g(closed)h(if)e(b)q(oth)h Fn(s)i Fo(!)1295 1815 y Fx(=)1295 1842 y Fl(R)1337 1830 y Fo(\001)f( )1402 1815 y Fl(\003)1402 1842 y(R)1443 1830 y Fn(t)f Fy(and)g Fn(t)i Fo(!)1614 1815 y Fx(=)1614 1842 y Fl(R)1655 1830 y Fo(\001)f( )1720 1815 y Fl(\003)1720 1842 y(R)1762 1830 y Fn(s)340 1880 y Fy(for)j(ev)o(ery)h(critical)e(pair)h Fo(h)p Fn(s;)7 b(t)p Fo(i)14 b Fy(of)f Fo(R)p Fy(.)403 1930 y(No)o(w)c(the)i(approac)o(h)f(for)g(pro)o(ving)f(relativ)o(e)h (undecidabilit)o(y)f(in)h(the)g(con\015uence)i(hierar-)340 1980 y(c)o(h)o(y)g(is)g(as)f(follo)o(ws:)f(for)h(ev)o(ery)i(implicatio) o(n)c Fn(X)15 b Fo(\))c Fn(Y)21 b Fy(w)o(e)12 b(construct)h(a)f(minor)e (extension)i(of)340 2029 y Fo(R)375 2035 y Fx(1)394 2029 y Fy(\()p Fn(P)6 b Fy(\))12 b(that)g(alw)o(a)o(ys)e(satis\014es)j Fn(Y)d Fy(,)h(and)h(satis\014es)g Fn(X)k Fy(if)11 b(and)h(only)f(if)g Fn(P)17 b Fy(admits)10 b(a)i(solution.)340 2079 y(Only)i(for)g(the)g (implication)d(UN)j Fo(\))f Fy(UN)988 2064 y Fl(!)1038 2079 y Fy(the)h(approac)o(h)g(is)g(sligh)o(tly)e(di\013eren)o(t.)340 2204 y Fh(2.1)48 b(NF)16 b Fg(\))g Fh(UN)340 2287 y(Prop)q(osition)5 b(3.)20 b Fm(The)d(TRS)h Fo(R)871 2293 y Fx(1)889 2287 y Fy(\()p Fn(P)6 b Fy(\))17 b Fm(has)g(unique)h(normal)f(forms)f(for)h (every)f(PCP)h(in-)340 2337 y(stanc)n(e)f Fn(P)6 b Fm(.)p eop %%Page: 5 5 5 4 bop 183 194 a Fm(Pr)n(o)n(of.)20 b Fy(Consider)11 b(the)g(TRS)f Fo(R)680 179 y Fl(0)680 204 y Fx(1)698 194 y Fy(\()p Fn(P)c Fy(\))12 b(=)g Fo(R)854 200 y Fx(1)872 194 y Fy(\()p Fn(P)6 b Fy(\))r Fo([)r(f)p Fn(A)12 b Fo(!)f Fn(B)r(;)c(f)t Fy(\()p Fn(x;)g(y)q Fy(\))13 b Fo(!)e Fn(B)r(;)c(g)q Fy(\()p Fn(x;)g(y)q Fy(\))12 b Fo(!)f Fn(B)r Fo(g)p 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Fx(1)490 621 y Fy(\()p Fn(P)6 b Fy(\))15 b Fm(is)f(c)n(on\015uent.)199 670 y(3.)20 b(The)15 b(PCP)g(instanc)n(e) g Fn(P)21 b Fm(admits)14 b(a)h(solution.)183 759 y(Pr)n(o)n(of.)20 b Fy(Since)14 b(con\015uence)i(implies)c(the)i(normal)e(form)g(prop)q (ert)o(y)m(,)i(according)g(to)g(Prop.)g(1)183 809 y(it)h(su\016ces)i (to)f(sho)o(w)g(that)g(\(i\))f Fn(A)g Fo(!)766 793 y Fl(\003)766 822 y(R)795 826 y Fj(1)811 822 y Fx(\()p Fk(P)t Fx(\))879 809 y Fn(B)j Fy(whenev)o(er)g Fo(R)1148 815 y Fx(1)1167 809 y Fy(\()p Fn(P)6 b Fy(\))15 b(has)h(the)h(normal)c (form)183 864 y(prop)q(ert)o(y)19 b(and)f(\(ii\))f Fo(R)550 870 y Fx(1)569 864 y Fy(\()p Fn(P)6 b Fy(\))18 b(is)g(con\015uen)o(t)h (whenev)o(er)g Fn(A)g Fo(!)1159 849 y Fl(\003)1159 878 y(R)1188 882 y Fj(1)1204 878 y Fx(\()p Fk(P)t Fx(\))1276 864 y Fn(B)r Fy(.)f(F)m(or)g(\(i\))g(w)o(e)g(note)183 920 y(that)d Fn(A)f Fo( )f Fn(g)q Fy(\()p Fn(c;)7 b(c)p Fy(\))13 b Fo(!)h Fn(B)j Fy(in)e Fo(R)684 926 y Fx(1)703 920 y Fy(\()p Fn(P)6 b Fy(\))15 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1884 y Fn(f)t Fy(\()p Fn(x;)c(y)q(;)g(z)r(;)g(i)p Fy(\()p Fn(w)q Fy(\)\))12 b Fo(!)f Fn(f)t Fy(\()p Fn(\013)1038 1890 y Fk(i)1052 1884 y Fy(\()p Fn(x)p Fy(\))p Fn(;)c(\014)1150 1890 y Fk(i)1164 1884 y Fy(\()p Fn(y)q Fy(\))p Fn(;)g(i)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(w)q Fy(\))26 b(for)14 b(all)e Fn(i)g Fo(2)f(f)p Fy(1)p Fn(;)c(:)g(:)g(:)12 b(;)7 b(n)p Fo(g)662 1934 y Fn(f)t Fy(\()p Fn(x;)g(y)q(;)g(i)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(c)p Fy(\))13 b Fo(!)e Fn(g)q Fy(\()p Fn(x;)c(y)q(;)g(i)p Fy(\()p Fn(z)r Fy(\)\))221 b(for)14 b(all)e Fn(i)g Fo(2)f(f)p Fy(1)p Fn(;)c(:)g(:)g(:)12 b(;)7 b(n)p Fo(g)702 1984 y Fn(g)q Fy(\()p Fn(x;)g(y)q(;)g(i)p Fy(\()p Fn(z)r Fy(\)\))13 b Fo(!)e Fn(g)q Fy(\()p Fn(x;)c(y)q(;)g(z)r Fy(\))267 b(for)14 b(all)e Fn(i)g Fo(2)f(f)p Fy(1)p Fn(;)c(:)g(:)g(:)12 b(;)7 b(n)p Fo(g)628 2034 y Fn(g)q Fy(\()p Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)g(a)p Fy(\()p Fn(y)q Fy(\))p Fn(;)g(c)p Fy(\)\))12 b Fo(!)f Fn(g)q Fy(\()p Fn(x;)c(y)q(;)g(c)p Fy(\))270 b(for)14 b(all)e Fn(a)g Fo(2)f 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Fy(for)g(all)f(rewrite)i(steps)h Fn(t)e Fo(!)1699 644 y Fl(R)1746 638 y Fn(u)p Fy(.)340 688 y(F)m(or)f(compatibilit)o(y)d(with)j(a)g(monotonic)e(order)j(it)f (su\016ces)h(to)f(c)o(hec)o(k)h(that)f Fn(l)q(\033)h(>)e(r)q(\033)j Fy(for)340 738 y(all)e(rules)i Fn(l)g Fo(!)e Fn(r)h Fy(in)g Fo(R)g Fy(and)g(all)f(ground)g(substitutions)i Fn(\033)q Fy(.)f(An)g Fo(F)t Fy(-algebra)f(consists)i(of)340 788 y(a)f(set)h Fn(A)f Fy(and)f(for)h(ev)o(ery)g Fn(f)22 b Fo(2)16 b(F)21 b Fy(a)16 b(function)h Fn(f)1116 794 y Fk(A)1148 788 y Fy(:)d Fn(A)1205 773 y Fk(n)1244 788 y Fo(!)i Fn(A)p Fy(,)h(where)h Fn(n)e Fy(is)h(the)h(arit)o(y)e(of)340 838 y Fn(f)t Fy(.)i(A)g Fm(monotone)k Fo(F)t Fy(-algebra)17 b(\()p Fn(A;)7 b(>)p Fy(\))18 b(is)g(an)f Fo(F)t Fy(-algebra)g Fn(A)h Fy(for)g(whic)o(h)f(the)i(underlying)340 887 y(set)c(is)f(pro)o (vided)g(with)g(an)f(order)i Fn(>)f Fy(suc)o(h)h(that)f(ev)o(ery)h (algebra)f(op)q(eration)f(is)h(monotonic)340 937 y(in)j(all)e(of)h(its) 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y Fx(1)543 1137 y Fn(;)7 b(:)g(:)g(:)12 b(;)7 b(a)665 1143 y Fk(n)687 1137 y Fy(\))16 b Fo(\025)h Fn(a)790 1143 y Fk(i)804 1137 y Fy(.)f(Ev)o(ery)h(monotone) e Fo(F)t Fy(-algebra)h(\()p Fn(A;)7 b(>)p Fy(\))17 b(induces)h(an)f (order)340 1186 y Fn(>)372 1192 y Fk(A)414 1186 y Fy(on)d(the)g(set)h (of)f(terms)f Fo(T)e Fy(\()p Fo(F)t Fn(;)c Fo(X)f Fy(\))13 b(as)h(follo)o(ws:)e Fn(t)g(>)1197 1192 y Fk(A)1236 1186 y Fn(u)i Fy(if)f(and)h(only)f(if)h([)p Fn(\013)p Fy(]\()p Fn(t)p Fy(\))c Fn(>)j Fy([)p Fn(\013)p Fy(]\()p Fn(u)p Fy(\))340 1236 y(for)i(all)f(assignmen)o(ts)g Fn(\013)5 b Fy(:)13 b Fo(X)19 b(!)13 b Fn(A)p Fy(.)i(Here)h([)p Fn(\013)p Fy(])e(denotes)i(the)f(homomorphic)d(extension)k(of)340 1286 y Fn(\013)p Fy(,)c(i.e.,)g([)p Fn(\013)p Fy(]\()p Fn(x)p Fy(\))e(=)i Fn(\013)p Fy(\()p Fn(x)p Fy(\))h(and)f([)p Fn(\013)p Fy(]\()p Fn(f)t Fy(\()p Fn(t)927 1292 y Fx(1)946 1286 y Fn(;)7 b(:)g(:)g(:)k(;)c(t)1060 1292 y Fk(n)1083 1286 y Fy(\)\))k(=)h Fn(f)1190 1292 y Fk(A)1218 1286 y Fy(\([)p 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Fn(c;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))589 1895 y Fo(!)k Fn(F)6 b Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))p Fn(:)183 1967 y Fy(Con)o(v)o(ersely)m(,)k(supp)q(ose)j(that)f Fo(R)p Fy(\()p Fn(P)6 b Fy(\))12 b(admits)f(an)h(in\014nite)g(reduction.)h(It) f(is)h(not)f(di\016cult)g(to)183 2017 y(see)g(that)f(there)h(exists)g (an)f(in\014nite)g(reduction)h(in)e(whic)o(h)h(all)f(steps)j(tak)o(e)e (place)g(at)g(the)h(ro)q(ot)183 2067 y(p)q(osition)j(and)g(b)q(oth)h (kinds)f(of)g(rewrite)i(rules)f(are)g(used)g(in\014nitely)f(often.)g (\(This)h(can)g(b)q(e)183 2116 y(sho)o(wn)d(formally)c(using)k(t)o(yp)q (e)h(eliminatio)o(n)c([21)o(].\))i(An)o(y)h(suc)o(h)h(reduction)g(m)o (ust)d(con)o(tain)i(a)183 2166 y(subsequence)j(of)d(the)i(form)253 2236 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(t)p Fy(\)\))k Fo(!)g Fn(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)h(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)g(a)p Fy(\()p Fn(t)p Fy(\)\))j Fo(!)887 2219 y Fx(+)926 2236 y Fn(F)c Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(t)p Fy(\)\)\))525 2264 y Fi(|)p 544 2264 275 5 v 275 w({z)p 857 2264 V 275 w(})183 2332 y Fy(where)13 b(in)g(the)g(underbraced)h(part)f(only)f (rewrite)i(rules)f(of)f(the)h(form)e Fn(F)6 b Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))12 b Fo(!)183 2382 y Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r Fy(\))12 b(are)i(used.)g(Hence)h Fn(a)p Fy(\()p Fn(t)p Fy(\))c(=)h Fn(\013)813 2388 y Fx(1)838 2382 y Fo(\001)7 b(\001)g(\001)f Fn(\013)921 2388 y Fk(n)943 2382 y Fy(\()p Fn(c)p Fy(\))11 b(=)h Fn(\014)1071 2388 y Fx(1)1097 2382 y Fo(\001)7 b(\001)g(\001)f Fn(\014)1176 2388 y Fk(n)1199 2382 y Fy(\()p Fn(c)p Fy(\))13 b(for)g(some)g Fn(n)e Fo(\025)h Fy(1)h(with)183 2432 y Fn(\013)210 2438 y Fk(i)223 2432 y Fn(;)7 b(\014)265 2438 y Fk(i)279 2432 y Fy(\))k Fo(2)h Fn(P)19 b Fy(for)13 b Fn(i)f Fy(=)g(1)p Fn(;)7 b(:)g(:)g(:)12 b(;)7 b(n)p Fy(,)12 b(giving)h(a)g(solution)g(for)h Fn(P)6 b Fy(.)477 b Fo(u)-28 b(t)p eop %%Page: 10 10 10 9 bop 403 194 a Fy(This)14 b(pro)o(v)o(es)h(undecidabilit)o(y)f(of)g (termination.)f(Since)i(the)g(constructed)i(in\014nite)e(re-)340 244 y(duction)f(is)f(alw)o(a)o(ys)g(cyclic,)g(this)g(also)g(pro)o(v)o (es)h(undecidabilit)o(y)f(of)g(b)q(oth)g(lo)q(opingness)g(and)340 293 y(cyclicit)o(y)m(.)e(The)h(adv)n(an)o(tage)f(of)g Fo(S)s Fy(\()p Fn(P)6 b Fy(\))12 b(o)o(v)o(er)f Fo(R)p Fy(\()p Fn(P)6 b Fy(\))12 b(is)g(that)f(it)h(is)f(length-preserving,)h (whic)o(h)340 343 y(means)f(that)h Fo(j)p Fn(l)q(\033)q Fo(j)f Fy(=)h Fo(j)p Fn(r)q(\033)q Fo(j)f Fy(for)h(all)f(rules)h Fn(l)h Fo(!)e Fn(r)i Fy(in)e Fo(S)s Fy(\()p Fn(P)6 b Fy(\))12 b(and)g(all)e(ground)i(substitutions)h Fn(\033)q Fy(.)340 393 y(Here)h Fo(j)p Fn(t)p Fo(j)c Fy(denotes)k(the)e(n)o(um)o (b)q(er)f(of)h(function)f(sym)o(b)q(ols)g(in)g Fn(t)p Fy(.)g(Since)i(for)e(length-preserving)340 443 y(TRSs)h(termination)f (and)g(simple)g(termination)f(coincide,)i(this)g(pro)o(v)o(es)h(that)f (b)q(oth)g(simple)340 493 y(termination)h(and)i(self-em)o(b)q (eddingness)g(are)g(undecidable.)g(The)g(main)e(result)j(of)e([23)o(])g (is)340 542 y(that)k Fo(S)s Fy(\()p Fn(P)6 b Fy(\))18 b(is)g(totally)e(terminating)g(if)h(and)h(only)f(if)g Fn(P)23 b Fy(admits)16 b(no)i(solution,)e(pro)o(ving)340 592 y(undecidabilit)o(y)d(of)g(total)h(termination.)403 642 y(No)o(w)g(the)h(approac)o(h)g(for)f(pro)o(ving)g(relativ)o(e)h (undecidabilit)o(y)e(in)i(the)g(termination)e(hi-)340 692 y(erarc)o(h)o(y)j(is)e(as)h(follo)o(ws:)e(for)h(ev)o(ery)i (implicatio)o(n)c Fn(X)17 b Fo(\))12 b Fn(Y)24 b Fy(w)o(e)15 b(construct)h(a)e(mo)q(di\014cation)340 742 y(of)j Fo(R)p Fy(\()p Fn(P)6 b Fy(\))17 b(or)g Fo(S)s Fy(\()p Fn(P)6 b Fy(\))17 b(that)g(alw)o(a)o(ys)f(satis\014es)i Fn(Y)9 b Fy(,)17 b(and)g(satis\014es)h Fn(X)j Fy(if)16 b(and)h(only)f(if)g Fn(P)23 b Fy(ad-)340 791 y(mits)10 b(no)i(solution.)e(A)i(n)o(um)o(b)q (er)e(of)h(times)g(this)g(mo)q(di\014cation)f(is)h(triggered)h(b)o(y)g (w)o(ell-kno)o(wn)340 841 y(examples.)340 966 y Fh(3.1)48 b(NL)16 b Fg(\))g Fh(A)o(C)340 1049 y Fy(Let)411 1146 y Fo(S)436 1152 y Fx(1)455 1146 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)575 1088 y Fi(\032)684 1121 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(z)r Fy(\)\))k Fo(!)g Fn(g)q Fy(\()p Fn(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)h(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(a)p Fy(\()p Fn(z)r Fy(\)\)\))26 b(for)13 b(all)g Fn(a)e Fo(2)g Fn(\000)612 1171 y(F)6 b Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))12 b Fo(!)f Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r Fy(\))238 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))k Fo(2)h Fn(P)340 1263 y Fh(Prop)q(osition)5 b(15.)20 b Fm(The)15 b(TRS)g Fo(S)880 1269 y Fx(1)899 1263 y Fy(\()p Fn(P)6 b Fy(\))15 b Fm(is)f(acyclic)h(for)f(every)h(PCP) g(instanc)n(e)g Fn(P)6 b Fm(.)340 1354 y(Pr)n(o)n(of.)20 b Fy(F)m(or)e(a)g(pro)q(of)f(b)o(y)h(con)o(tradiction,)f(assume)h(a)f (cyclic)i Fo(S)1360 1360 y Fx(1)1378 1354 y Fy(\()p Fn(P)6 b Fy(\)-reduction)19 b Fn(t)f Fo(!)1721 1339 y Fx(+)1766 1354 y Fn(t)340 1404 y Fy(exists.)e(Applying)f(rules)h(of)f(the)h(form) e Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(z)r Fy(\)\))13 b Fo(!)h Fn(g)q Fy(\()p Fn(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)h(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(a)p Fy(\()p Fn(z)r Fy(\)\)\))16 b(strictly)340 1454 y(increases)e(the)f(n)o (um)o(b)q(er)e(of)g Fn(g)i Fy(sym)o(b)q(ols,)d(while)i(the)g(other)h (kind)e(of)h(rules)g(do)q(es)h(not)f(c)o(hange)340 1503 y(the)j(n)o(um)o(b)q(er)f(of)g Fn(g)i Fy(sym)o(b)q(ols.)c(Hence)k(in)e (a)g(reduction)h Fn(t)e Fo(!)1281 1488 y Fx(+)1320 1503 y Fn(t)i Fy(only)e(rules)i(of)f(the)h(second)340 1553 y(kind)g(are)h(applied.)e(But)i(these)h(rules)f(constitute)g(a)f (terminating)f(system,)h(yielding)f(the)340 1603 y(desired)h(con)o (tradiction.)1022 b Fo(u)-28 b(t)340 1694 y Fh(Prop)q(osition)5 b(16.)20 b Fm(The)14 b(TRS)h Fo(S)879 1700 y Fx(1)898 1694 y Fy(\()p Fn(P)6 b Fy(\))14 b Fm(is)f(lo)n(oping)i(if)e(and)i (only)g(if)e Fn(P)20 b Fm(admits)14 b(a)g(solution.)340 1786 y(Pr)n(o)n(of.)20 b Fy(If)14 b Fn(\015)g Fo(2)e Fn(\000)619 1771 y Fx(+)659 1786 y Fy(is)i(a)g(solution)f(for)g Fn(P)19 b Fy(then)c(w)o(e)f(ha)o(v)o(e)g(the)g Fo(S)1350 1792 y Fx(1)1369 1786 y Fy(\()p Fn(P)6 b Fy(\)-lo)q(op)411 1869 y Fn(F)g Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))13 b Fo(!)788 1852 y Fx(+)827 1869 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))k Fo(!)g Fn(g)q Fy(\()p Fn(F)6 b Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\)\))p Fn(:)340 1952 y Fy(Con)o(v)o(ersely)m(,)20 b(if)g Fo(S)635 1958 y Fx(1)653 1952 y Fy(\()p Fn(P)6 b Fy(\))20 b(is)g(lo)q(oping)f(then)i(it)f(admits)f(an)h(in\014nite)g (reduction.)g(Erasing)340 2001 y(all)g(o)q(ccurrences)k(of)c Fn(g)i Fy(in)e(an)o(y)g(in\014nite)h Fo(S)1044 2007 y Fx(1)1062 2001 y Fy(\()p Fn(P)6 b Fy(\)-reduction)21 b(yields)g(an)f(in\014nite)h Fo(R)p Fy(\()p Fn(P)6 b Fy(\)-)340 2051 y(reduction.)15 b(According)f(to)g(Prop.)f(14,)g Fn(P)19 b Fy(admits)13 b(a)g(solution.)416 b Fo(u)-28 b(t)340 2176 y Fh(3.2)48 b(SN)16 b Fg(\))g Fh(NL)340 2259 y Fy(The)f(system)411 2342 y Fn(h)p Fy(\()p Fn(c)p Fy(\))d Fo(!)f Fn(g)q Fy(\()p Fn(c)p Fy(\))p Fn(;)30 b(h)p Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))12 b Fo(!)f Fn(g)q Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\))p Fn(;)30 b(f)t Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))13 b Fo(!)e Fn(f)t Fy(\()p Fn(h)p Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\)\))p eop %%Page: 11 11 11 10 bop 183 194 a Fy(is)12 b(kno)o(wn)f(to)h(b)q(e)h(non-lo)q(oping)e (and)h(non-terminating)e(\([24)o(]\).)h(It)i(can)f(b)q(e)h(com)o(bined) e(with)183 244 y(the)j(basic)g Fo(R)p Fy(\()p Fn(P)6 b Fy(\))14 b(construction)h(as)f(follo)o(ws.)e(Let)253 401 y Fo(S)278 407 y Fx(2)297 401 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)417 291 y Fi(8)417 329 y(>)417 341 y(>)417 354 y(<)417 428 y(>)417 441 y(>)417 453 y(:)476 326 y Fn(h)p Fy(\()p Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(z)r Fy(\)\)\))k Fo(!)g Fn(g)q Fy(\()p Fn(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)h(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(a)p Fy(\()p Fn(z)r Fy(\)\)\))26 b(for)13 b(all)g Fn(a)e Fo(2)g Fn(\000)460 376 y(F)6 b Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))12 b Fo(!)f Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r Fy(\))238 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))k Fo(2)h Fn(P)612 426 y(h)p Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))g Fo(!)f Fn(g)q Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\))611 476 y Fn(f)t Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))i Fo(!)e Fn(f)t Fy(\()p Fn(h)p Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\)\))183 567 y Fh(Prop)q(ositi)o(on)5 b(17.)21 b Fm(The)14 b(TRS)i Fo(S)723 573 y Fx(2)741 567 y Fy(\()p Fn(P)6 b Fy(\))15 b Fm(is)f(non-lo)n(oping)i(for)f(every)f(PCP)h(instanc)n(e)h Fn(P)6 b Fm(.)183 659 y(Pr)n(o)n(of.)20 b Fy(F)m(or)13 b(arbitrary)h(terms)g Fn(t)f Fy(de\014ne)i(inductiv)o(ely:)438 739 y Fn(\036)p Fy(\()p Fn(x)p Fy(\))c(=)h(0)348 b Fn( )q Fy(\()p Fn(x)p Fy(\))13 b(=)e(0)259 b(for)13 b Fn(x)e Fo(2)h(X)444 789 y Fn(\036)p Fy(\()p Fn(c)p Fy(\))f(=)h(0)354 b Fn( )q Fy(\()p Fn(c)p Fy(\))13 b(=)e(0)392 839 y Fn(\036)p Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\)\))h(=)g(0)303 b Fn( )q Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\)\))13 b(=)e(1)e(+)h Fn( )q Fy(\()p Fn(t)p Fy(\))133 b(for)13 b Fn(a)f Fo(2)f Fn(\000)389 888 y(\036)p Fy(\()p Fn(f)t Fy(\()p Fn(t)p Fy(\)\))i(=)f(0)300 b Fn( )q Fy(\()p Fn(f)t Fy(\()p Fn(t)p Fy(\)\))14 b(=)d(1)e(+)h Fn( )q Fy(\()p Fn(t)p Fy(\))393 938 y Fn(\036)p Fy(\()p Fn(g)q Fy(\()p Fn(t)p Fy(\)\))i(=)g(1)d(+)h Fn(\036)p Fy(\()p Fn(t)p Fy(\))181 b Fn( )q Fy(\()p Fn(g)q Fy(\()p Fn(t)p Fy(\)\))13 b(=)e Fn( )q Fy(\()p Fn(t)p Fy(\))390 988 y Fn(\036)p Fy(\()p Fn(h)p Fy(\()p Fn(t)p Fy(\)\))h(=)g(1)d(+)h Fn(\036)p Fy(\()p Fn(t)p Fy(\))178 b Fn( )q Fy(\()p Fn(h)p Fy(\()p Fn(t)p Fy(\)\))13 b(=)e Fn( )q Fy(\()p Fn(t)p Fy(\))259 1038 y Fn(\036)p Fy(\()p Fn(F)6 b Fy(\()p Fn(t)364 1044 y Fx(1)382 1038 y Fn(;)h(t)416 1044 y Fx(2)434 1038 y Fn(;)g(t)468 1044 y Fx(3)486 1038 y Fy(\)\))12 b(=)g(0)170 b Fn( )q Fy(\()p Fn(F)6 b Fy(\()p Fn(t)873 1044 y Fx(1)892 1038 y Fn(;)h(t)926 1044 y Fx(2)944 1038 y Fn(;)g(t)978 1044 y Fx(3)996 1038 y Fy(\)\))12 b(=)f(1)e(+)h(max)n Fo(f)p Fn( )q Fy(\()p Fn(t)1312 1044 y Fx(1)1331 1038 y Fy(\))p Fn(;)d( )q Fy(\()p Fn(t)1425 1044 y Fx(2)1444 1038 y Fy(\))p Fn(;)g( )q Fy(\()p Fn(t)1538 1044 y Fx(3)1557 1038 y Fy(\))p Fo(g)183 1121 y Fy(F)m(or)16 b(ev)o(ery)i Fo(S)398 1127 y Fx(2)417 1121 y Fy(\()p Fn(P)6 b Fy(\)-reduction)17 b(step)h Fn(t)e Fo(!)h Fn(u)f Fy(w)o(e)h(ha)o(v)o(e)g Fn(\036)p Fy(\()p Fn(t)p Fy(\))g(=)g Fn(\036)p Fy(\()p Fn(u)p Fy(\))f(and)h Fn( )q Fy(\()p Fn(t)p Fy(\))h Fo(\025)e Fn( )q Fy(\()p Fn(u)p Fy(\).)183 1171 y(Assume)f Fo(S)363 1177 y Fx(2)382 1171 y Fy(\()p Fn(P)6 b Fy(\))15 b(admits)e(a)i(lo)q(op.)f(Cho)q(ose)i(a)f(lo)q(op)f Fn(t)g Fo(!)1084 1156 y Fx(+)1125 1171 y Fn(C)s Fy([)p Fn(t\033)q Fy(])g(for)h(whic)o(h)g(the)h(nesting)183 1221 y(of)f Fn(F)22 b Fy(sym)o(b)q(ols)15 b(in)h Fn(t)g Fy(is)h(minim)o(al)o(.)c(F)m(rom)i Fn( )q Fy(\()p Fn(t)p Fy(\))h Fo(\025)g Fn( )q Fy(\()p Fn(C)s Fy([)p Fn(t\033)q Fy(]\))g(w)o(e)h(conclude)g(that)f Fn(C)j Fy(only)183 1271 y(consists)f(of)f Fn(g)h Fy(and)g Fn(h)f Fy(sym)o(b)q(ols;)e(from) h Fn(\036)p Fy(\()p Fn(t)p Fy(\))i(=)f Fn(\036)p Fy(\()p Fn(C)s Fy([)p Fn(t\033)q Fy(]\))f(w)o(e)i(conclude)g(that)g Fn(C)i Fy(is)d(the)183 1320 y(trivial)9 b(con)o(text.)j(Hence)g Fn(t)g Fo(!)649 1305 y Fx(+)687 1320 y Fn(t\033)q Fy(.)f(W)m(rite)g Fn(t)g Fy(=)h Fn(D)q Fy([)p Fn(F)6 b Fy(\()p Fn(t)1044 1326 y Fx(1)1063 1320 y Fn(;)h(t)1097 1326 y Fx(2)1115 1320 y Fn(;)g(t)1149 1326 y Fx(3)1167 1320 y Fy(\)])j(for)h Fn(D)i Fy(consisting)e(only)f(of)183 1370 y(unary)h(sym)o(b)q(ols.)e (Hence)k Fn(D)q Fy([)p Fn(F)6 b Fy(\()p Fn(t)698 1376 y Fx(1)716 1370 y Fn(;)h(t)750 1376 y Fx(2)768 1370 y Fn(;)g(t)802 1376 y Fx(3)820 1370 y Fy(\)])12 b Fo(!)902 1355 y Fx(+)940 1370 y Fn(D)q Fy([)p Fn(F)6 b Fy(\()p Fn(t)1051 1376 y Fx(1)1069 1370 y Fn(\033)o(;)h(t)1126 1376 y Fx(2)1144 1370 y Fn(\033)o(;)g(t)1201 1376 y Fx(3)1219 1370 y Fn(\033)q Fy(\)].)k(Due)g(to)h(minim)o(al)o(it)o(y)183 1420 y(not)g(all)g(steps)i(tak)o(e)f(place)h(inside)e Fn(t)741 1426 y Fx(1)760 1420 y Fy(,)g Fn(t)799 1426 y Fx(2)818 1420 y Fy(,)g Fn(t)857 1426 y Fx(3)876 1420 y Fy(.)g(Moreo)o(v)o(er,)h(at)g(least)g(one)g(of)g(the)g(steps)i(m)o (ust)183 1470 y(a\013ect)c(the)f(con)o(text)h Fn(D)h Fy(for)e(otherwise)h(w)o(e)f(w)o(ould)g(ha)o(v)o(e)f Fn(F)d Fy(\()p Fn(t)1119 1476 y Fx(1)1138 1470 y Fn(;)h(t)1172 1476 y Fx(2)1190 1470 y Fn(;)g(t)1224 1476 y Fx(3)1242 1470 y Fy(\))k Fo(!)1311 1455 y Fx(+)1350 1470 y Fn(F)6 b Fy(\()p Fn(t)1414 1476 y Fx(1)1432 1470 y Fn(\033)o(;)h(t)1489 1476 y Fx(2)1507 1470 y Fn(\033)o(;)g(t)1564 1476 y Fx(3)1582 1470 y Fn(\033)q Fy(\))183 1520 y(with)i(only)g(applications)g(of)g (the)h(rules)g Fn(F)c Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))12 b Fo(!)f Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r Fy(\))i(at)h(ro)q(ot)f(p)q(ositions,)183 1570 y(but)h(then)h(the)g (size)h(of)d(the)i(maximal)c(topmost)i(part)h(of)g Fn(t)1071 1576 y Fx(1)1100 1570 y Fy(consisting)h(en)o(tirely)f(of)g(sym)o(b)q (ols)183 1619 y(in)17 b Fn(\000)22 b Fy(m)o(ust)17 b(exceed)i(that)e (of)g Fn(t)685 1625 y Fx(1)704 1619 y Fn(\033)q Fy(,)g(whic)o(h)g(is)g (clearly)g(imp)q(ossible.)f(Hence)j(w)o(e)e(obtain)g(a)183 1669 y(non-empt)o(y)g(reduction)j Fn(D)q Fy([)p Fn(c)p Fy(])g Fo(!)724 1654 y Fx(+)771 1669 y Fn(D)q Fy([)p Fn(c)p Fy(])f(in)g(the)g(TRS)g(consisting)g(of)g(the)g(three)i(rules) 183 1719 y Fn(h)p Fy(\()p Fn(c)p Fy(\))d Fo(!)f Fn(g)q Fy(\()p Fn(c)p Fy(\),)h Fn(h)p Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))g Fo(!)f Fn(g)q Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\),)h(and)f Fn(f)t Fy(\()p Fn(g)q Fy(\()p Fn(x)p Fy(\)\))j Fo(!)d Fn(f)t Fy(\()p Fn(h)p Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\)\).)h(Considering)g(size)183 1769 y(and)12 b(observing)g(that)h(the)g(\014rst)g(t)o(w)o(o)f(rules)h (are)f(terminating)f(yields)h(a)g(con)o(tradiction.)41 b Fo(u)-28 b(t)183 1860 y Fh(Prop)q(ositi)o(on)5 b(18.)21 b Fm(The)e(TRS)i Fo(S)733 1866 y Fx(2)751 1860 y Fy(\()p Fn(P)6 b Fy(\))20 b Fm(is)f(terminating)h(if)f(and)i(only)f(if)f Fn(P)25 b Fm(admits)20 b(no)183 1910 y(solution.)183 2001 y(Pr)n(o)n(of.)g Fy(Let)15 b Fn(\015)h Fo(2)c Fn(\000)497 1986 y Fx(+)539 2001 y Fy(b)q(e)j(a)g(solution)f(for)g Fn(P)6 b Fy(.)14 b(W)m(rite)g Fn(t)f Fy(=)g Fn(F)6 b Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\).)16 b(F)m(or)f(ev)o(ery)183 2051 y Fn(i)c(>)h Fy(0)i(w)o(e)g(ha)o(v)o(e)g (the)g Fo(S)540 2057 y Fx(2)559 2051 y Fy(\()p Fn(P)6 b Fy(\)-reduction)253 2136 y Fn(f)t Fy(\()p Fn(h)317 2119 y Fk(i)332 2136 y Fy(\()p Fn(t)p Fy(\)\))12 b(=)g Fn(f)t Fy(\()p Fn(h)515 2119 y Fk(i)529 2136 y Fy(\()p Fn(F)6 b Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\)\)\))14 b Fo(!)955 2119 y Fx(+)994 2136 y Fn(f)t Fy(\()p Fn(h)1058 2119 y Fk(i)1072 2136 y Fy(\()p Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\)\)\))407 2204 y Fo(!)k Fn(f)t Fy(\()p Fn(h)524 2187 y Fk(i)p Fl(\000)p Fx(1)581 2204 y Fy(\()p Fn(g)q Fy(\()p Fn(t)p Fy(\)\)\))i Fo(!)752 2187 y Fl(\003)782 2204 y Fn(f)t Fy(\()p Fn(g)q Fy(\()p Fn(h)883 2187 y Fk(i)p Fl(\000)p Fx(1)940 2204 y Fy(\()p Fn(t)p Fy(\)\)\))f Fo(!)f Fn(f)t Fy(\()p Fn(h)1148 2187 y Fk(i)p Fx(+1)1205 2204 y Fy(\()p Fn(t)p Fy(\)\))p Fn(;)183 2289 y Fy(easily)16 b(extending)h(to)f(an)g(in\014nite)h(reduction.)g(Con)o(v)o(ersely)m(,) f(assume)g Fo(S)1341 2295 y Fx(2)1360 2289 y Fy(\()p Fn(P)6 b Fy(\))16 b(admits)f(an)183 2339 y(in\014nite)d(reduction.)h (Erasing)f(all)f(o)q(ccurrences)16 b(of)c Fn(f)t Fy(,)g Fn(g)q Fy(,)h(and)f Fn(h)g Fy(yields)g(an)h(in\014nite)f Fo(R)p Fy(\()p Fn(P)6 b Fy(\)-)183 2389 y(reduction.)14 b(According)g(to)g(Prop.)g(14,)e Fn(P)20 b Fy(admits)12 b(a)i(solution.)415 b Fo(u)-28 b(t)p eop %%Page: 12 12 12 11 bop 340 194 a Fh(3.3)48 b(NSE)16 b Fg(\))g Fh(SN)340 277 y Fy(Let)411 399 y Fo(S)436 405 y Fx(3)455 399 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)575 314 y Fi(8)575 351 y(<)575 426 y(:)690 349 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(z)r Fy(\)\))k Fo(!)g Fn(G)p Fy(\()p Fn(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)c(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(a)p Fy(\()p Fn(z)r Fy(\)\))25 b(for)14 b(all)e Fn(a)g Fo(2)f Fn(\000)617 399 y(G)p Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)c(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))13 b Fo(!)e Fn(G)p Fy(\()p Fn(x;)c(y)q(;)g(z)r Fy(\))184 b(for)14 b(all)e(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)744 449 y(G)p Fy(\()p Fn(c;)c(c;)g(z)r Fy(\))k Fo(!)g Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(z)r Fy(\))340 540 y Fh(Prop)q(osition)e(19.)20 b Fm(The)15 b(TRS)g Fo(S)880 546 y Fx(3)899 540 y Fy(\()p Fn(P)6 b Fy(\))15 b Fm(is)f(terminating)h(for)f(every)h(PCP)g(instanc)n (e)g Fn(P)6 b Fm(.)340 632 y(Pr)n(o)n(of.)20 b Fy(W)m(e)13 b(apply)f(seman)o(tic)g(lab)q(elling)f(as)h(describ)q(ed)j(in)d([22)o (].)g(As)h(mo)q(del)e Fo(M)i Fy(w)o(e)f(c)o(ho)q(ose)340 681 y(t)o(w)o(o)k(elemen)o(ts)h(0,)f(1,)g(with)g(in)o(terpretations)h Fn(F)1099 687 y Fl(M)1140 681 y Fy(\()p Fn(x;)7 b(y)q(;)g(z)r Fy(\))16 b(=)g Fn(G)1373 687 y Fl(M)1414 681 y Fy(\()p Fn(x;)7 b(y)q(;)g(z)r Fy(\))16 b(=)g Fn(a)1636 687 y Fl(M)1677 681 y Fy(\()p Fn(x)p Fy(\))g(=)340 731 y Fn(c)358 737 y Fl(M)417 731 y Fy(=)j(1)e(and)h Fn(h)615 737 y Fl(M)656 731 y Fy(\()p Fn(x)p Fy(\))g(=)g(0)f(for)h(all)e Fn(x;)7 b(y)q(;)g(z)20 b Fo(2)d(f)p Fy(0)p Fn(;)7 b Fy(1)p Fo(g)17 b Fy(and)g Fn(a)h Fo(2)g Fn(\000)6 b Fy(.)16 b(Since)i(all)f(left)g(and)340 781 y(righ)o(t-hand)e(sides)i(of)e(the)i (rules)f(are)g(equal)g(to)f(1)h(in)f(this)h(in)o(terpretation,)g Fo(M)f Fy(is)h(indeed)340 831 y(a)f(mo)q(del)e(for)i Fo(S)590 837 y Fx(3)609 831 y Fy(\()p Fn(P)6 b Fy(\).)14 b(Using)g(this)h(mo)q(del)f(w)o(e)h(lab)q(el)f(the)h(sym)o(b)q(ol)e Fn(F)21 b Fy(b)o(y)14 b(the)i(v)n(alue)e(of)g(its)340 881 y(\014rst)h(argumen)o(t,)d(yielding)h(the)i(lab)q(elled)e(system) 411 1011 y Fo(S)439 994 y Fl(0)436 1021 y Fx(3)455 1011 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)575 926 y Fi(8)575 963 y(<)575 1038 y(:)677 961 y Fn(F)704 967 y Fx(1)722 961 y Fy(\()p Fn(c;)c(c;)g(a)p Fy(\()p Fn(z)r Fy(\)\))12 b Fo(!)f Fn(G)p Fy(\()p Fn(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)c(a)p Fy(\()p Fn(z)r Fy(\))p Fn(;)g(a)p Fy(\()p Fn(z)r Fy(\)\))25 b(for)14 b(all)e Fn(a)g Fo(2)f Fn(\000)617 1011 y(G)p Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)c(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))13 b Fo(!)e Fn(G)p Fy(\()p Fn(x;)c(y)q(;)g(z)r Fy(\))184 b(for)14 b(all)e(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)744 1061 y(G)p Fy(\()p Fn(c;)c(c;)g(z)r Fy(\))k Fo(!)g Fn(F)995 1067 y Fx(0)1013 1061 y Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)c(c;)g(z)r Fy(\))340 1144 y(The)19 b(main)c(result)k(of)e(seman)o (tic)g(lab)q(elling)f(states)j(that)f Fo(S)1296 1150 y Fx(3)1314 1144 y Fy(\()p Fn(P)6 b Fy(\))18 b(is)g(terminating)e(if)g (and)340 1194 y(only)h(if)f Fo(S)504 1179 y Fl(0)501 1204 y Fx(3)519 1194 y Fy(\()p Fn(P)6 b Fy(\))17 b(is)g(terminating.)e (The)i(latter)g(holds)g(b)o(y)g(recursiv)o(e)h(path)f(order:)g(c)o(ho)q (ose)340 1244 y Fn(F)367 1250 y Fx(1)397 1244 y Fn(>)12 b(G)f(>)h(F)556 1250 y Fx(0)586 1244 y Fn(>)g(h)p Fy(.)1088 b Fo(u)-28 b(t)340 1335 y Fh(Prop)q(osition)5 b(20.)20 b Fm(The)e(TRS)g Fo(S)886 1341 y Fx(3)905 1335 y Fy(\()p Fn(P)6 b Fy(\))17 b Fm(is)g(self-emb)n(e)n(dding)g(if)g(and)i(only)f (if)f Fn(P)23 b Fm(admits)17 b(a)340 1385 y(solution.)340 1476 y(Pr)n(o)n(of.)j Fy(Let)15 b Fn(\015)f Fo(2)d Fn(\000)651 1461 y Fx(+)692 1476 y Fy(b)q(e)k(a)e(solution)g(for)h Fn(P)6 b Fy(.)13 b(Then)h(w)o(e)g(ha)o(v)o(e)g(the)g Fo(S)1423 1482 y Fx(3)1442 1476 y Fy(\()p Fn(P)6 b Fy(\)-reduction)411 1559 y Fn(F)g Fy(\()p Fn(c;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))k Fo(!)g Fn(G)p Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)c(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))14 b Fo(!)1065 1542 y Fx(+)1103 1559 y Fn(G)p Fy(\()p Fn(c;)7 b(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))12 b Fo(!)f Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))p Fn(:)340 1642 y Fy(Since)19 b Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\))18 b Fo(!)781 1648 y Fl(E)r Fx(m)o(b)866 1642 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\)\),)17 b Fo(S)1132 1648 y Fx(3)1150 1642 y Fy(\()p Fn(P)6 b Fy(\))18 b(is)g(self-em)o(b)q(edding.)e(Con)o(v)o(ersely)m(,)340 1692 y(assume)e Fo(S)509 1698 y Fx(3)527 1692 y Fy(\()p Fn(P)6 b Fy(\))14 b(is)f(self-em)o(b)q(edding.)f(Let)i Fn( )q Fy(\()p Fn(t)p Fy(\))h(denote)f(the)g(maxima)o(l)c(nesting)k(of) f Fn(F)19 b Fy(and)340 1742 y Fn(G)14 b Fy(sym)o(b)q(ols)e(in)i(a)f (term)h Fn(t)p Fy(:)464 1822 y Fn( )q Fy(\()p Fn(x)p Fy(\))e(=)g(0)173 b(for)14 b Fn(x)d Fo(2)g(X)419 1872 y Fn( )q Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\)\))h(=)g Fn( )q Fy(\()p Fn(t)p Fy(\))119 b(for)14 b Fn(a)d Fo(2)g Fn(\000)470 1922 y( )q Fy(\()p Fn(c)p Fy(\))h(=)g(0)173 b Fn( )q Fy(\()p Fn(F)6 b Fy(\()p Fn(t)906 1928 y Fx(1)925 1922 y Fn(;)h(t)959 1928 y Fx(2)977 1922 y Fn(;)g(t)1011 1928 y Fx(3)1029 1922 y Fy(\)\))12 b(=)g(1)d(+)h(max)n Fo(f)p Fn( )q Fy(\()p Fn(t)1346 1928 y Fx(1)1365 1922 y Fy(\))p Fn(;)d( )q Fy(\()p Fn(t)1459 1928 y Fx(2)1478 1922 y Fy(\))p Fn(;)g( )q Fy(\()p Fn(t)1572 1928 y Fx(3)1591 1922 y Fy(\))p Fo(g)417 1972 y Fn( )q Fy(\()p Fn(h)p Fy(\()p Fn(t)p Fy(\)\))12 b(=)g Fn( )q Fy(\()p Fn(t)p Fy(\))119 b Fn( )q Fy(\()p Fn(G)p Fy(\()p Fn(t)906 1978 y Fx(1)925 1972 y Fn(;)7 b(t)959 1978 y Fx(2)977 1972 y Fn(;)g(t)1011 1978 y Fx(3)1029 1972 y Fy(\)\))12 b(=)g(1)d(+)h(max)n Fo(f)p Fn( )q Fy(\()p Fn(t)1346 1978 y Fx(1)1365 1972 y Fy(\))p Fn(;)d( )q Fy(\()p Fn(t)1459 1978 y Fx(2)1478 1972 y Fy(\))p Fn(;)g( )q Fy(\()p Fn(t)1572 1978 y Fx(3)1591 1972 y Fy(\))p Fo(g)340 2060 y Fy(Ob)o(viously)m(,)17 b Fn(t)h Fo(!)623 2067 y Fl(S)643 2071 y Fj(3)659 2067 y Fx(\()p Fk(P)t Fx(\))731 2060 y Fn(u)g Fy(implies)e Fn( )q Fy(\()p Fn(t)p Fy(\))j(=)g Fn( )q Fy(\()p Fn(u)p Fy(\).)f(Let)g Fn(t)h Fo(!)1331 2042 y Fx(+)1331 2074 y Fl(S)1351 2078 y Fj(3)1367 2074 y Fx(\()p Fk(P)t Fx(\))1439 2060 y Fn(u)f Fo(!)1523 2045 y Fl(\003)1523 2071 y(E)r Fx(m)o(b)1608 2060 y Fn(t)g Fy(b)q(e)h(suc)o(h)340 2116 y(that)f Fn( )q Fy(\()p Fn(t)p Fy(\))h(is)e(minimal)o(.)d(W)m(e)k(ma)o (y)e(assume)h(that)h(the)g(topmost)f(sym)o(b)q(ol)f(of)h Fn(t)h Fy(is)f(either)340 2166 y(\(i\))i Fn(F)k Fy(or)c(\(ii\))e Fn(G)p Fy(.)h(Since)h Fn( )q Fy(\()p Fn(t)p Fy(\))g(=)h Fn( )q Fy(\()p Fn(u)p Fy(\))e(and)h Fn(u)f Fo(!)1176 2151 y Fl(\003)1176 2178 y(E)r Fx(m)o(b)1262 2166 y Fn(t)g Fy(w)o(e)h(conclude)g(that)f(the)h(ro)q(ot)340 2216 y(sym)o(b)q(ols)14 b(of)h Fn(t)g Fy(and)g Fn(u)g Fy(coincide.)g(Moreo)o(v)o(er,)h(b)o(y)f (our)g(minim)o(ali)o(t)o(y)d(assumption,)i(there)i(is)340 2266 y(at)e(least)g(one)g(reduction)g(step)h(in)e Fn(t)f Fo(!)953 2248 y Fx(+)953 2280 y Fl(S)973 2284 y Fj(3)989 2280 y Fx(\()p Fk(P)t Fx(\))1054 2266 y Fn(u)h Fy(at)h(the)g(ro)q(ot)g (p)q(osition.)e(First)i(w)o(e)g(consider)340 2322 y(case)19 b(\(i\).)d(The)i Fo(S)618 2328 y Fx(3)637 2322 y Fy(\()p Fn(P)6 b Fy(\)-reduction)17 b(from)f Fn(t)h Fy(to)g Fn(u)h Fy(m)o(ust)e(start)i(as)f Fn(t)g Fy(=)h Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(t)1654 2328 y Fx(1)1671 2322 y Fy(\)\))18 b Fo(!)1763 2307 y Fl(\003)340 2372 y Fn(F)6 b Fy(\()p Fn(c;)h(c;)g(a)p Fy(\()p Fn(t)516 2378 y Fx(2)534 2372 y Fy(\)\))12 b Fo(!)f Fn(G)p Fy(\()p Fn(a)p Fy(\()p Fn(t)733 2378 y Fx(2)751 2372 y Fy(\))p Fn(;)c(a)p Fy(\()p Fn(t)839 2378 y Fx(2)858 2372 y Fy(\))p Fn(;)g(a)p Fy(\()p Fn(t)946 2378 y Fx(2)964 2372 y Fy(\)\),)14 b(where)h Fn(t)1157 2378 y Fx(1)1187 2372 y Fo(!)1229 2357 y Fl(\003)1260 2372 y Fn(t)1275 2378 y Fx(2)1293 2372 y Fy(,)f(for)g(otherwise)h (there)g(w)o(ould)340 2422 y(b)q(e)c(no)e(reduction)h(step)h(at)e(a)h (ro)q(ot)f(p)q(osition.)g(Since)h(the)g(ro)q(ot)g(sym)o(b)q(ols)e(of)h Fn(t)h Fy(and)f Fn(u)h Fy(coincide,)p eop %%Page: 13 13 13 12 bop 183 194 a Fy(it)16 b(further)i(follo)o(ws)e(that)h Fn(G)p Fy(\()p Fn(a)p Fy(\()p Fn(t)705 200 y Fx(2)724 194 y Fy(\))p Fn(;)7 b(a)p Fy(\()p Fn(t)812 200 y Fx(2)830 194 y Fy(\))p Fn(;)g(a)p Fy(\()p Fn(t)918 200 y Fx(2)936 194 y Fy(\)\))18 b Fo(!)1028 179 y Fx(+)1072 194 y Fn(G)p Fy(\()p Fn(c;)7 b(c;)g(t)1210 200 y Fx(3)1227 194 y Fy(\))17 b Fo(!)g Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(t)1513 200 y Fx(3)1530 194 y Fy(\))17 b Fo(!)1605 179 y Fl(\003)183 244 y Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(t)377 250 y Fx(4)394 244 y Fy(\))12 b(=)f Fn(u)p Fy(.)i(Consequen)o(tly)m(,)f Fn(a)p Fy(\()p Fn(t)831 250 y Fx(2)850 244 y Fy(\))h(is)f(a)h(solution)f(for)g Fn(P)6 b Fy(.)12 b(Next)i(w)o(e)f(consider)h(case)183 293 y(\(ii\).)d(W)m(rite)h Fn(t)f Fy(=)h Fn(G)p Fy(\()p Fn(t)510 299 y Fx(1)529 293 y Fn(;)7 b(t)563 299 y Fx(2)581 293 y Fn(;)g(t)615 299 y Fx(3)633 293 y Fy(\))12 b(and)h Fn(u)e Fy(=)h Fn(G)p Fy(\()p Fn(u)893 299 y Fx(1)911 293 y Fn(;)7 b(u)954 299 y Fx(2)972 293 y Fn(;)g(u)1015 299 y Fx(3)1033 293 y Fy(\).)12 b(Note)g(that)h(the)g(rule)g Fn(G)p Fy(\()p Fn(c;)7 b(c;)g(z)r Fy(\))j Fo(!)183 343 y Fn(F)c Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(z)r Fy(\))13 b(is)i(not)f(applicable)g(at)h(ro)q(ot)f(p)q(ositions)h(in)f (the)h Fo(S)1189 349 y Fx(3)1208 343 y Fy(\()p Fn(P)6 b Fy(\)-reduction)15 b(\()p Fo(\003)p Fy(\))f(from)183 393 y Fn(t)e Fy(to)g Fn(u)g Fy(b)q(ecause)j(for)d(no)g(term)g Fn(t)679 378 y Fl(0)703 393 y Fy(there)i(is)e(an)g Fo(S)929 399 y Fx(3)948 393 y Fy(\()p Fn(P)6 b Fy(\)-reduction)13 b(from)d Fn(F)c Fy(\()p Fn(h)p Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(t)1500 378 y Fl(0)1510 393 y Fy(\))13 b(to)f Fn(u)p Fy(.)183 443 y(Hence)i(only)d(rules)j(of)e(the)h(form)d Fn(G)p Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)d(\014)r Fy(\()p Fn(y)q Fy(\))p Fn(;)g(z)r Fy(\))12 b Fo(!)f Fn(G)p Fy(\()p Fn(x;)c(y)q(;)g(z)r Fy(\))13 b(are)f(applicable)g(at)h(ro)q(ot) 183 493 y(p)q(ositions)i(in)h(\()p Fo(\003)p Fy(\))g(and)g(th)o(us)h (the)f(ro)q(ot)g(sym)o(b)q(ol)f(of)g(ev)o(ery)i(term)e(in)h(\()p Fo(\003)p Fy(\))g(is)g Fn(G)p Fy(.)f(F)m(or)h(ev)o(ery)183 542 y(reduction)j(step)g Fn(G)p Fy(\()p Fn(t)528 527 y Fl(0)528 553 y Fx(1)546 542 y Fn(;)7 b(t)580 527 y Fl(0)580 553 y Fx(2)599 542 y Fn(;)g(t)633 527 y Fl(0)633 553 y Fx(3)651 542 y Fy(\))19 b Fo(!)f Fn(G)p Fy(\()p Fn(u)819 527 y Fl(0)819 553 y Fx(1)837 542 y Fn(;)7 b(u)880 527 y Fl(0)880 553 y Fx(2)898 542 y Fn(;)g(u)941 527 y Fl(0)941 553 y Fx(3)959 542 y Fy(\))19 b(in)f(\()p Fo(\003)p Fy(\))g(w)o(e)g(ha)o(v)o(e)h(either)g(\(1\))f Fn(t)1492 527 y Fl(0)1492 553 y Fx(1)1530 542 y Fy(=)h Fn(u)1605 527 y Fl(0)1605 553 y Fx(1)183 592 y Fy(if)d(the)h(reduction) h(to)q(ok)f(place)g(in)g Fn(t)760 577 y Fl(0)760 602 y Fx(2)795 592 y Fy(or)g Fn(t)864 577 y Fl(0)864 602 y Fx(3)883 592 y Fy(,)f(\(2\))h Fn(t)996 577 y Fl(0)996 602 y Fx(1)1032 592 y Fo(!)1074 599 y Fl(S)1094 603 y Fj(3)1110 599 y Fx(\()p Fk(P)t Fx(\))1180 592 y Fn(u)1204 577 y Fl(0)1204 602 y Fx(1)1239 592 y Fy(if)g(the)g(reduction)h(to)q (ok)183 642 y(place)f(in)g Fn(t)359 627 y Fl(0)359 652 y Fx(1)377 642 y Fy(,)g(or)g(\(3\))g Fn(t)545 627 y Fl(0)545 652 y Fx(1)581 642 y Fn(.)g(u)643 627 y Fl(0)643 652 y Fx(1)678 642 y Fy(\(i.e.,)f Fn(u)800 627 y Fl(0)800 652 y Fx(1)835 642 y Fy(is)h(a)g(prop)q(er)h(subterm)f(of)g Fn(t)1285 627 y Fl(0)1285 652 y Fx(1)1303 642 y Fy(\))h(if)e(the)i (reduction)183 692 y(to)q(ok)11 b(place)h(at)g(the)g(ro)q(ot.)g(By)g (assumption)e(alternativ)o(e)i(\(3\))g(o)q(ccurs)h(at)f(least)g(once.)g (Using)183 742 y(the)k(w)o(ell-kno)o(wn)e(facts)i(that)g Fn(.)10 b Fo(\001)g(!)760 749 y Fl(S)780 753 y Fj(3)796 749 y Fx(\()p Fk(P)t Fx(\))865 742 y Fo(\022)k(!)953 749 y Fl(S)973 753 y Fj(3)990 749 y Fx(\()p Fk(P)t Fx(\))1054 742 y Fo(\001)c Fn(.)15 b Fy(and)g Fn(.)c Fo(\001)f Fn(.)k Fo(\022)h Fn(.)g Fy(it)g(follo)o(ws)f(that)183 791 y Fn(t)198 797 y Fx(1)235 791 y Fo(!)277 776 y Fl(\003)277 805 y(S)297 809 y Fj(3)313 805 y Fx(\()p Fk(P)t Fx(\))385 791 y Fo(\001)19 b Fn(.)f(u)479 797 y Fx(1)497 791 y Fy(.)g(Because)i Fn( )q Fy(\()p Fn(t)p Fy(\))g(=)f Fn( )q Fy(\()p Fn(u)p Fy(\),)f(there)h(are)g(no)f(reduction)h(steps)g(at)g(ro) q(ot)183 841 y(p)q(ositions)12 b(in)h Fn(u)e Fo(!)481 826 y Fl(\003)481 853 y(E)r Fx(m)o(b)559 841 y Fn(t)p Fy(.)i(Hence)h Fn(u)745 847 y Fx(1)775 841 y Fo(!)817 826 y Fl(\003)817 853 y(E)r Fx(m)o(b)895 841 y Fn(t)910 847 y Fx(1)929 841 y Fy(.)f(Com)o(bining)d(this)j(with)f Fn(t)1352 847 y Fx(1)1383 841 y Fo(!)1425 826 y Fl(\003)1425 855 y(S)1445 859 y Fj(3)1461 855 y Fx(\()p Fk(P)t Fx(\))1526 841 y Fo(\001)f Fn(.)g(u)1605 847 y Fx(1)183 902 y Fy(yields)g Fn(t)312 908 y Fx(1)342 902 y Fo(!)384 884 y Fx(+)384 916 y Fl(S)404 920 y Fj(3)420 916 y Fx(\()p Fk(P)t Fx(\))485 902 y Fo(\001)g(!)550 887 y Fl(\003)550 914 y(E)r Fx(m)o(b)628 902 y Fn(t)643 908 y Fx(1)662 902 y Fy(.)g(Ho)o(w)o(ev)o(er,)g Fn( )q Fy(\()p Fn(t)920 908 y Fx(1)939 902 y Fy(\))h Fn(<)f( )q Fy(\()p Fn(t)p Fy(\),)h(con)o(tradicting)f(the)h(minim)o(al) o(it)o(y)183 958 y(of)h Fn( )q Fy(\()p Fn(t)p Fy(\).)h(W)m(e)g (conclude)g(that)g(case)h(\(ii\))e(is)h(imp)q(ossible.)532 b Fo(u)-28 b(t)183 1083 y Fh(3.4)47 b(ST)16 b Fg(\))g Fh(NSE)183 1166 y Fy(Let)e Fo(S)282 1172 y Fx(4)301 1166 y Fy(\()p Fn(P)6 b Fy(\))13 b(b)q(e)i(the)f(TRS)253 1215 y Fi(\032)305 1249 y Fn(F)6 b Fy(\()p Fn(x;)g Fy(\026)-20 b Fn(a)o Fy(\()p Fn(y)q Fy(\))p Fn(;)7 b(x;)g Fy(\026)-21 b Fn(a)o Fy(\()p Fn(y)q Fy(\)\))13 b Fo(!)e Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(a)p Fy(\()p Fn(x)p Fy(\)\))p Fn(;)h(h)p Fy(\()p Fn(y)q Fy(\))p Fn(;)g(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)g(y)q Fy(\))41 b(for)13 b(all)g Fn(a)f Fo(2)f Fn(\000)290 1298 y(F)6 b Fy(\()p Fn(\013)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(y)q(;)g(\014)r Fy(\()p Fn(z)r Fy(\))p Fn(;)g(w)q Fy(\))k Fo(!)g Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\))p Fn(;)h(h)p Fy(\()t(\026)-25 b Fn(\013)o Fy(\()p Fn(y)q Fy(\)\))p Fn(;)7 b(z)r(;)1037 1288 y Fy(\026)1031 1298 y Fn(\014)s Fy(\()p Fn(w)q Fy(\)\))26 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))12 b Fo(2)f Fn(P)183 1390 y Fh(Prop)q(ositi)o(on)5 b(21.)21 b Fm(The)12 b(TRS)g Fo(S)717 1396 y Fx(4)736 1390 y Fy(\()p Fn(P)6 b Fy(\))11 b Fm(is)h (non-self-emb)n(e)n(dding)h(for)e(every)h(PCP)g(instanc)n(e)183 1440 y Fn(P)6 b Fm(.)183 1531 y(Pr)n(o)n(of.)20 b Fy(F)m(or)9 b(a)g(term)h Fn(t)p Fy(,)f(let)g Fo(k)p Fn(t)p Fo(k)h Fy(denote)g(the)h(n)o(um)o(b)q(er)e(of)g Fn(F)d Fy(,)i Fn(a)p Fy(,)h(and)i(\026)-22 b Fn(a)10 b Fy(sym)o(b)q(ols)e(in)h Fn(t)p Fy(.)g(Clearly)183 1581 y Fo(k)p Fn(t)p Fo(k)17 b Fy(=)g Fo(k)p Fn(u)p Fo(k)g Fy(for)g(ev)o(ery)i(reduction)f(step)g Fn(t)g Fo(!)924 1588 y Fl(S)944 1592 y Fj(4)960 1588 y Fx(\()p Fk(P)t Fx(\))1031 1581 y Fn(u)p Fy(.)f(F)m(or)g(a)g(pro)q(of) g(b)o(y)g(con)o(tradiction,)183 1636 y(assume)f(a)g(self-em)o(b)q (edding)f(reduction)i Fn(t)e Fo(!)910 1618 y Fx(+)910 1650 y Fl(S)930 1654 y Fj(4)946 1650 y Fx(\()p Fk(P)t Fx(\))1016 1636 y Fn(u)g Fo(!)1097 1621 y Fl(\003)1097 1648 y(E)r Fx(m)o(b)1179 1636 y Fn(t)i Fy(\()p Fo(\003)p Fy(\))f(exists.)g(Since)h Fo(k)p Fn(t)p Fo(k)e Fy(=)183 1692 y Fo(k)p Fn(u)p Fo(k)p Fy(,)i(in)g Fn(u)i Fo(!)415 1677 y Fl(\003)415 1704 y(E)r Fx(m)o(b)500 1692 y Fn(t)f Fy(only)f(the)i(rule)f Fn(h)p Fy(\()p Fn(x)p Fy(\))h Fo(!)f Fn(x)g Fy(is)g(applied.)f(Just)i(as)f(in)g(the)g(pro)q(of)g(of) 183 1742 y(Prop.)13 b(20)h(w)o(e)g(ma)o(y)e(assume)i(that)g(there)h(is) f(at)f(least)i(one)f(reduction)g(step)h(in)f Fn(t)d Fo(!)1498 1725 y Fx(+)1498 1756 y Fl(S)1518 1760 y Fj(4)1535 1756 y Fx(\()p Fk(P)t Fx(\))1600 1742 y Fn(u)183 1799 y Fy(at)i(the)i(ro)q (ot)f(p)q(osition.)f(Hence)i(w)o(e)f(ma)o(y)e(write)i(\()p Fo(\003)p Fy(\))g(as)253 1882 y Fn(t)e Fy(=)g Fn(F)6 b Fy(\()p Fn(t)388 1888 y Fx(1)406 1882 y Fn(;)h(t)440 1888 y Fx(2)458 1882 y Fn(;)g(t)492 1888 y Fx(3)510 1882 y Fn(;)g(t)544 1888 y Fx(4)562 1882 y Fy(\))12 b Fo(!)632 1865 y Fl(\003)662 1882 y Fn(F)6 b Fy(\()p Fn(u)735 1888 y Fx(1)753 1882 y Fn(;)h(u)796 1888 y Fx(2)814 1882 y Fn(;)g(u)857 1888 y Fx(3)874 1882 y Fn(;)g(u)917 1888 y Fx(4)935 1882 y Fy(\))324 1910 y Fi(|)p 343 1910 277 5 v 277 w({z)p 658 1910 V 277 w(})963 1882 y Fo(!)k Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(v)1125 1888 y Fx(1)1143 1882 y Fy(\))p Fn(;)h(h)p Fy(\()p Fn(v)1238 1888 y Fx(2)1257 1882 y Fy(\))p Fn(;)g(v)1312 1888 y Fx(3)1330 1882 y Fn(;)g(v)1369 1888 y Fx(4)1387 1882 y Fy(\))12 b Fo(!)1457 1865 y Fl(\003)1487 1882 y Fn(u)f Fo(!)1564 1865 y Fl(\003)1595 1882 y Fn(t)1016 1910 y Fi(|)p 1035 1910 260 5 v 260 w({z)p 1333 1910 V 260 w(})183 1991 y Fy(where)i(in)f(the)h (underbraced)h(parts)f(no)f(steps)i(tak)o(e)f(place)f(at)h(ro)q(ot)f(p) q(ositions.)g(\(Note)g(that)183 2041 y(no)g(term)h(of)f(the)i(form)d Fn(F)6 b Fy(\()p Fn(h)p Fy(\()p Fn(v)661 2047 y Fx(1)680 2041 y Fy(\))p Fn(;)h(h)p Fy(\()p Fn(v)775 2047 y Fx(2)793 2041 y Fy(\))p Fn(;)g(v)848 2047 y Fx(3)867 2041 y Fn(;)g(v)906 2047 y Fx(4)924 2041 y Fy(\))13 b(is)g(a)g(redex.\))g(W)m(e)g(obtain)f Fo(k)p Fn(t)1401 2047 y Fx(1)1420 2041 y Fo(k)f Fy(=)h Fo(k)p Fn(u)1541 2047 y Fx(1)1559 2041 y Fo(k)f(6)p Fy(=)183 2090 y Fo(k)p Fn(h)p Fy(\()p Fn(v)264 2096 y Fx(1)282 2090 y Fy(\))p Fo(k)h Fy(=)f Fo(k)p Fn(t)410 2096 y Fx(1)429 2090 y Fo(k)p Fy(,)i(whic)o(h)h(is)f(a)h(con)o(tradiction.)675 b Fo(u)-28 b(t)183 2182 y Fh(Prop)q(ositi)o(on)5 b(22.)21 b Fm(The)13 b(TRS)h Fo(S)720 2188 y Fx(4)738 2182 y Fy(\()p Fn(P)6 b Fy(\))13 b Fm(is)h(simply)f(terminating)f(if)h(and)h(only)g (if)f Fn(P)19 b Fm(admits)183 2232 y(no)c(solution.)183 2323 y(Pr)n(o)n(of.)20 b Fy(According)c(to)h(Prop.)f(14)f(it)h(is)g (su\016cien)o(t)h(to)f(sho)o(w)h(that)f Fo(S)1280 2329 y Fx(4)1299 2323 y Fy(\()p Fn(P)6 b Fy(\))16 b(is)g(simply)e(ter-)183 2373 y(minating)e(if)j(and)g(only)f(if)h Fo(S)s Fy(\()p Fn(P)6 b Fy(\))15 b(is)g(terminating.)e(Supp)q(ose)k Fo(S)s Fy(\()p Fn(P)6 b Fy(\))15 b(is)g(non-terminating.)183 2422 y(Since)j Fo(!)337 2429 y Fl(S)r Fx(\()p Fk(P)t Fx(\))432 2422 y Fo(\022)h(!)525 2405 y Fx(+)525 2437 y Fl(S)545 2441 y Fj(4)561 2437 y Fx(\()p Fk(P)t Fx(\))p Fl([E)r Fx(m)o(b)702 2422 y Fy(,)f(also)f Fo(S)844 2428 y Fx(4)863 2422 y Fy(\()p Fn(P)6 b Fy(\))12 b Fo([)g(E)t Fy(m)o(b)k(is)i(non-terminating)e(and)i(hence)p eop %%Page: 14 14 14 13 bop 340 194 a Fo(S)365 200 y Fx(4)384 194 y Fy(\()p Fn(P)6 b Fy(\))13 b(is)f(not)h(simply)e(terminating.)g(Con)o(v)o (ersely)m(,)h(assume)h Fo(S)s Fy(\()p Fn(P)6 b Fy(\))12 b(is)h(terminating.)e(Since)340 244 y Fo(S)s Fy(\()p Fn(P)6 b Fy(\))13 b(is)f(length-preserving)h(it)f(is)g(simply)e (terminating)h(and)h(th)o(us)g(admits)f(a)h(compatible)340 293 y(simple)h(monotone)f(algebra)h(\()p Fn(A;)7 b(>)p Fy(\).)13 b(By)h(de\014ning)g Fn(h)1191 299 y Fk(A)1218 293 y Fy(\()p Fn(x)p Fy(\))d(=)h Fn(x)i Fy(for)f Fn(x)e Fo(2)g Fn(A)j Fy(this)g(b)q(ecomes)340 343 y(a)i(simple)e(monotone)h (algebra)g(compatible)g(with)g Fo(S)1185 349 y Fx(4)1204 343 y Fy(\()p Fn(P)6 b Fy(\),)15 b(hence)j Fo(S)1439 349 y Fx(4)1457 343 y Fy(\()p Fn(P)6 b Fy(\))16 b(is)g(simply)e(ter-) 340 393 y(minating.)1238 b Fo(u)-28 b(t)340 517 y Fh(3.5)48 b(TT)16 b Fg(\))g Fh(ST)340 600 y Fy(Let)411 748 y Fo(S)436 754 y Fx(5)455 748 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)575 638 y Fi(8)575 675 y(>)575 688 y(>)575 700 y(<)575 775 y(>)575 787 y(>)575 800 y(:)632 673 y Fn(F)6 b Fy(\()p Fn(x;)h Fy(\026)-21 b Fn(a)o Fy(\()p Fn(y)q Fy(\))p Fn(;)7 b(x;)g Fy(\026)-21 b Fn(a)p Fy(\()p Fn(y)q Fy(\)\))13 b Fo(!)e Fn(F)6 b Fy(\()f(_)-17 b Fn(a)o Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q(;)h 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Fy(\))18 b(=)f(1)g(for)h(all)340 1204 y Fn(x;)7 b(y)q(;)g(z)r(;)g(w)13 b Fo(2)g(f)p Fy(0)p Fn(;)7 b Fy(1)p Fo(g)13 b Fy(and)h Fn(a)f Fo(2)g Fn(\000)6 b Fy(.)13 b(Since)i(all)f(left)g(and)h(righ)o (t-hand)f(sides)h(of)f(the)i(rules)f(are)340 1254 y(equal)k(to)f(0)g (in)g(this)h(in)o(terpretation,)f Fo(M)h Fy(is)f(indeed)h(a)f(mo)q(del) f(for)h Fo(S)1479 1260 y Fx(5)1498 1254 y Fy(\()p Fn(P)6 b Fy(\).)18 b(Using)g(this)340 1304 y(mo)q(del)12 b(w)o(e)i(lab)q(el)f (the)h(sym)o(b)q(ol)d Fn(F)19 b Fy(as)13 b(follo)o(ws:)f(it)h(is)g(lab) q(elled)g(b)o(y)g(1)g(if)g(the)h(in)o(terpretations)340 1354 y(of)i(the)h(\014rst)h(and)e(the)h(third)g(argumen)o(t)e(are)i (equal,)f(and)g(b)o(y)h(0)f(otherwise.)h(This)f(yields)340 1404 y(the)f(lab)q(elled)e(system)411 1584 y Fo(S)439 1567 y Fl(0)436 1594 y Fx(5)455 1584 y Fy(\()p Fn(P)6 b Fy(\))11 b(=)575 1449 y Fi(8)575 1487 y(>)575 1499 y(>)575 1511 y(>)575 1524 y(>)575 1536 y(<)575 1611 y(>)575 1624 y(>)575 1636 y(>)575 1648 y(>)575 1661 y(:)632 1484 y Fn(F)659 1490 y Fx(1)678 1484 y Fy(\()p Fn(x;)c Fy(\026)-21 b Fn(a)o Fy(\()p Fn(y)q Fy(\))p Fn(;)7 b(x;)g Fy(\026)-21 b Fn(a)p Fy(\()p Fn(y)q Fy(\)\))13 b Fo(!)e Fn(F)1057 1490 y Fx(0)1075 1484 y Fy(\()5 b(_)-17 b Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q(;)g Fy(\177)-21 b Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q Fy(\))41 b(for)13 b(all)g Fn(a)e Fo(2)g Fn(\000)632 1534 y(F)659 1540 y Fx(1)678 1534 y Fy(\()5 b(_)-17 b Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q(;)12 b Fy(_)-17 b Fn(a)o Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q Fy(\))13 b Fo(!)e Fn(F)1057 1540 y Fx(1)1075 1534 y Fy(\()p Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)c(y)q(;)g(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)g(y)q Fy(\))41 b(for)13 b(all)g Fn(a)e Fo(2)g Fn(\000)632 1584 y(F)659 1590 y Fx(1)678 1584 y Fy(\()q(\177)-22 b Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q(;)g Fy(\177)-21 b Fn(a)o Fy(\()p Fn(x)p Fy(\))p Fn(;)7 b(y)q Fy(\))13 b Fo(!)e Fn(F)1057 1590 y Fx(1)1075 1584 y Fy(\()p Fn(a)p Fy(\()p Fn(x)p Fy(\))p Fn(;)c(y)q(;)g(a)p Fy(\()p 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b(is)g(terminating)f(if)h(and)g(only)g(if)g Fo(S)1426 1752 y Fl(0)1423 1777 y Fx(5)1442 1767 y Fy(\()p Fn(P)6 b Fy(\))11 b(is)h(terminating.)340 1817 y(The)k(latter)f(holds)f(b)o(y) h(lexicographic)f(path)h(order:)g(c)o(ho)q(ose)g Fn(F)1338 1823 y Fx(1)1370 1817 y Fn(>)e(F)1442 1823 y Fx(0)1473 1817 y Fn(>)19 b Fy(_)-17 b Fn(a)13 b(>)g Fy(\177)-21 b Fn(a)13 b(>)g(a)g(>)h Fy(\026)-22 b Fn(a)340 1866 y Fy(for)14 b(all)f Fn(a)e Fo(2)g Fn(\000)19 b Fy(and)14 b(compare)f(the)i(argumen)o(ts)e(of)g Fn(F)1174 1872 y Fx(1)1206 1866 y Fy(from)f(left)i(to)g(righ)o(t.)227 b Fo(u)-28 b(t)340 1958 y Fh(Prop)q(osition)5 b(24.)20 b Fm(The)14 b(TRS)g Fo(S)878 1964 y Fx(5)896 1958 y Fy(\()p Fn(P)6 b Fy(\))14 b Fm(is)f(total)r(ly)g(terminating)g(if)g(and)i(only) e(if)g Fn(P)19 b Fm(admits)340 2008 y(no)d(solution.)340 2099 y(Pr)n(o)n(of.)k Fy(Assume)13 b Fn(P)18 b Fy(admits)11 b(no)i(solution.)e(According)i(to)g([23)o(])f(the)h(TRS)f Fo(S)s Fy(\()p Fn(P)6 b Fy(\))13 b(is)g(totally)340 2149 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Fn(;)7 b(u)p Fy(\))k Fn(>)h(F)6 b Fy(\()f(_)-17 b Fn(a)o Fy(\()p Fn(t)p Fy(\))p Fn(;)7 b(u;)k Fy(_)-16 b Fn(a)o Fy(\()p Fn(t)p Fy(\))p Fn(;)7 b(u)p Fy(\))k Fn(>)h(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)h(u;)g(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)g(u)p Fy(\))p Fn(:)183 1244 y Fy(Hence)19 b(for)e(all)g(ground)g(terms)h Fn(t)p Fy(,)f Fn(u)h Fy(and)f(all)g Fn(a)h Fo(2)f Fn(\000)23 b Fy(w)o(e)18 b(obtain)f Fn(F)6 b Fy(\()f(_)-17 b Fn(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)7 b(u;)f Fy(\177)-20 b Fn(a)o Fy(\()p Fn(t)p Fy(\))p Fn(;)7 b(u)p Fy(\))17 b Fn(>)183 1294 y(F)6 b Fy(\()p Fn(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)h(u;)g(a)p Fy(\()p Fn(t)p Fy(\))p Fn(;)g(u)p Fy(\).)j(Using)j(this)g(result)h(and) f(the)h(compatibilit)o(y)c(of)i Fo(S)1299 1300 y Fx(5)1318 1294 y Fy(\()p Fn(P)6 b Fy(\))13 b(and)f Fn(>)i Fy(yields)253 1379 y Fn(F)6 b Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(c)p Fy(\))12 b Fn(>)g Fo(\001)7 b(\001)g(\001)j Fn(>)h(F)6 b Fy(\()p Fn(c;)i 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Fn(;)h(y)q(;)g(\014)r Fy(\()p Fn(z)r Fy(\))p Fn(;)g(w)q(;)g(u)p Fy(\))j Fo(!)h Fn(F)6 b Fy(\()p Fn(x;)k Fy(\026)-24 b Fn(\013)o Fy(\()p Fn(y)q Fy(\))p Fn(;)7 b(z)r(;)1013 2107 y Fy(\026)1007 2118 y Fn(\014)r Fy(\()p Fn(w)q Fy(\))p Fn(;)g(u)p Fy(\))121 b(for)13 b(all)g(\()p Fn(\013;)7 b(\014)r Fy(\))k Fo(2)h Fn(P)183 2210 y Fh(Prop)q(ositi)o (on)5 b(25)15 b([6)q(].)21 b Fm(The)16 b(TRS)g Fo(S)791 2216 y Fx(6)810 2210 y Fy(\()p Fn(P)6 b Fy(\))16 b Fm(is)f(total)r(ly)h (terminating)f(for)h(every)g(PCP)g(in-)183 2259 y(stanc)n(e)f Fn(P)6 b Fm(.)p eop %%Page: 16 16 16 15 bop 340 194 a Fm(Pr)n(o)n(of.)20 b Fy(W)m(e)14 b(use)g(the)g(Kn)o(uth-Bendix)h(order)f(\([12)o(]\))g(where)g Fn(h)g Fy(is)f(assigned)h(w)o(eigh)o(t)g(0,)e(and)340 244 y(ev)o(ery)17 b(other)g(function)f(sym)o(b)q(ol)e(w)o(eigh)o(t)i (1.)f(F)m(or)h(the)h(precedence)i(w)o(e)e(c)o(ho)q(ose)f Fn(h)g(>)f(F)21 b(>)340 293 y(G)16 b(>)h(a)f(>)h Fy(\026)-22 b Fn(a)16 b(>)g(c)h Fy(for)f(all)g Fn(a)g Fo(2)f Fn(\000)6 b Fy(.)16 b(Note)h(that)f Fn(h)h Fy(satis\014es)h(the)f(constrain)o(t)g (the)g(Kn)o(uth-)340 343 y(Bendix)f(order)g(requires,)g(namely)d(that)i (ev)o(ery)h(function)f(sym)o(b)q(ol)e(of)i(w)o(eigh)o(t)f(0)h(is)g (unary)340 393 y(and)e(greatest)i(in)e(precedence.)i(W)m(e)e(tak)o(e)h (lexicographic)e(status)i(for)f(eac)o(h)h(function)f(sym-)340 443 y(b)q(ol,)f Fn(F)18 b Fy(left-to-righ)o(t)11 b(and)h Fn(G)g Fy(righ)o(t-to-left.)f(The)i(induced)g(Kn)o(uth-Bendix)g(order)g (orien)o(ts)340 493 y(eac)o(h)j(rule)g(from)e(left)i(to)f(righ)o(t.)g (By)g(a)h(result)g(of)f(F)m(erreira)h([4,)f(Theorem)g(4.47])f(the)i (TRS)340 542 y Fo(S)365 548 y Fx(6)384 542 y Fy(\()p Fn(P)6 b Fy(\))14 b(is)f(not)h(only)f(terminating,)f(but)i(ev)o(en)h (totally)d(terminating.)318 b Fo(u)-28 b(t)340 624 y Fh(Prop)q(osition)5 b(26)15 b([6].)22 b Fm(The)15 b(TRS)h Fo(S)948 630 y Fx(6)967 624 y Fy(\()p Fn(P)6 b Fy(\))15 b Fm(is)g Fn(!)q Fm(-terminating)g(if)g(and)h(only)g(if)f Fn(P)20 b Fm(admits)340 673 y(no)c(solution.)340 755 y(Pr)n(o)n(of.)k Fy(Let)15 b Fn(\015)g Fo(2)c Fn(\000)652 740 y Fx(+)693 755 y Fy(b)q(e)k(a)f(solution)f(for)h Fn(P)6 b Fy(.)13 b(Then)i(w)o(e)f(ha)o(v)o(e)g(for)g(all)f(ground)h (terms)f Fn(t)i Fy(an)340 805 y Fo(S)365 811 y Fx(6)384 805 y Fy(\()p Fn(P)6 b Fy(\)-reduction)411 882 y Fn(h)p Fy(\()p Fn(G)p Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)h(c;)g(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(c;)g(t)p Fy(\)\))k Fo(!)858 865 y Fx(+)897 882 y Fn(F)6 b Fy(\()p Fn(c;)i Fy(\026)-22 b Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)7 b(c;)g Fy(\026)-21 b Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)7 b(c;)g(t)p Fy(\))816 948 y Fo(!)858 931 y Fx(+)897 948 y Fn(G)p Fy(\()p Fn(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(c;)g(\015)r Fy(\()p Fn(c)p Fy(\))p Fn(;)g(c;)g(h)p Fy(\()p Fn(h)p Fy(\()p Fn(t)p Fy(\)\)\))p Fn(:)340 1026 y Fy(F)m(or)12 b(a)g(pro)q(of)g(b)o(y)h(con)o(tradiction,)e(assume)h (that)g Fo(S)1127 1032 y Fx(6)1146 1026 y Fy(\()p Fn(P)6 b Fy(\))12 b(is)g Fn(!)q Fy(-terminating,)f(with)h(compat-)340 1075 y(ible)i(w)o(ell-founded)f(monotone)f(algebra)i(\()p Fn(A;)7 b(>)p Fy(\))14 b(where)h Fn(A)c Fy(=)h Fc(N)p Fy(.)g(De\014ne)411 1150 y Fn(f)431 1156 y Fk(A)458 1150 y Fy(\()p Fn(t)p Fy(\))g(=)g Fn(G)594 1156 y Fk(A)621 1150 y Fy(\([)p Fn(\015)r Fy(\()p Fn(c)p Fy(\)])p Fn(;)7 b(c)771 1156 y Fk(A)797 1150 y Fn(;)g Fy([)p Fn(\015)r Fy(\()p Fn(c)p Fy(\)])p Fn(;)g(c)950 1156 y Fk(A)976 1150 y Fn(;)g(t)p Fy(\))p Fn(:)340 1225 y Fy(Then)20 b(the)f(ab)q(o)o(v)o(e)f(reduction)i(pro)o(v)o(es)f Fn(!)q Fy(-termination)e(of)h(the)h(single)f(rule)h Fn(h)p Fy(\()p Fn(f)t Fy(\()p Fn(x)p Fy(\)\))i Fo(!)340 1275 y Fn(f)t Fy(\()p Fn(h)p Fy(\()p Fn(h)p Fy(\()p Fn(x)p Fy(\)\)\),)15 b(con)o(tradicting)f([21)o(,)f(Prop.)h(11].)e(Hence)k Fo(S)1231 1281 y Fx(6)1249 1275 y Fy(\()p Fn(P)6 b Fy(\))14 b(is)g(not)f Fn(!)q Fy(-terminating.)403 1325 y(This)h(lea)o(v)o(es)h (to)g(pro)o(v)o(e)g(that)g Fo(S)899 1331 y Fx(6)917 1325 y Fy(\()p Fn(P)6 b Fy(\))15 b(is)g Fn(!)q Fy(-terminating)e(if)h Fn(P)20 b Fy(has)15 b(no)g(solution.)e(Here)340 1375 y(w)o(e)g(sk)o(etc)o(h)h(the)f(pro)q(of,)f(for)h(more)e(details)i(w)o (e)g(refer)h(to)e([6].)f(F)m(or)i(a)f(ground)h(term)f Fn(t)p Fy(,)g(let)h Fo(k)p Fn(t)p Fo(k)340 1425 y Fy(denote)18 b(the)f(n)o(um)o(b)q(er)f(of)g(barred)i(and)e(un)o(barred)i(letters)g (in)e Fn(t)g Fy(not)h(b)q(elo)o(w)f(an)h Fn(F)6 b Fy(,)16 b Fn(G)p Fy(,)f(or)340 1474 y Fn(h)h Fy(sym)o(b)q(ol.)d(An)i(analysis)g (of)g(reduction)h(patterns)g(sho)o(ws)g(that)f(an)o(y)g(reduction)h (starting)340 1524 y(from)h(a)h(term)g(of)g(the)h(shap)q(e)g Fn(h)859 1509 y Fk(k)879 1524 y Fy(\()p Fn(G)p Fy(\()p Fn(p;)7 b(q)q(;)g(r)o(;)g(s;)g(t)p Fy(\)\))17 b(or)h Fn(h)1241 1509 y Fk(k)1262 1524 y Fy(\()p Fn(F)6 b Fy(\()p Fn(p;)h(q)q(;)g(r)o(;)g(s;)g(t)p Fy(\)\))16 b(with)i(at)g(least)340 1574 y(3)6 b Fo(\001)g Fy(min)n Fo(fk)p Fn(p)p Fo(k)g Fy(+)g Fo(k)p Fn(q)q Fo(k)p Fn(;)h Fo(k)p Fn(r)q Fo(k)f Fy(+)g Fo(k)p Fn(s)p Fo(kg)12 b Fy(steps)h(at)g(the)g(topmost)e Fn(F)18 b Fy(or)12 b Fn(G)g Fy(sym)o(b)q(ol,)e(giv)o(es)i(rise)h(to)g (a)340 1624 y(solution)g(for)g Fn(P)6 b Fy(.)12 b(Hence)j(assuming)d Fn(P)19 b Fy(has)13 b(no)g(solution)g(implies)e(an)i(e\013ectiv)o(e)i (b)q(ound)f(on)340 1674 y(reduction)i(lengths.)g(No)o(w)f(the)g(idea)h (is)f(to)g(enco)q(de)h(this)g(b)q(ound)f(in)o(to)g(an)g(in)o (terpretation)340 1723 y([)p Fo(\001)p Fy(])g(in)g(the)i(p)q(ositiv)o (e)f(in)o(tegers.)g(There)h(are,)f(ho)o(w)o(ev)o(er,)g(a)g(few)g(tec)o (hnical)g(problems)f(to)h(b)q(e)340 1773 y(solv)o(ed.)c(The)g (reduction)g(length)g(for)f(the)i(in)o(terpretation)f(m)o(ust)f(b)q(e)h (determined)g(from)e(the)340 1823 y(in)o(terpretations)17 b(of)f(the)h(argumen)o(ts)e(of)h Fn(F)21 b Fy(and)16 b Fn(G)p Fy(,)g(rather)h(than)f(from)e(the)j(argumen)o(ts)340 1873 y(themselv)o(es.)d(So)g(one)g(m)o(ust)f(ha)o(v)o(e)h(access)i(to)e (the)h(structure)h(of)d(the)i(argumen)o(ts)e(through)340 1923 y(their)i(in)o(terpretations.)f(This)g(is)f(done)i(b)o(y)e(means)g (of)h(a)f(function)h Fn(\031)h Fy(suc)o(h)f(that)411 1998 y(for)g(all)e(strings)j Fn(\020)i Fy(of)c(barred)i(and)f(un)o (barred)h(letters)g(and)f(for)f(all)g(ground)h(terms)411 2047 y Fn(t)k Fy(and)f Fn(t)543 2032 y Fl(0)555 2047 y Fy(,)g(the)h(equalit)o(y)f Fn(\031)q Fy(\([)p Fn(t)p Fy(]\))g(=)h Fn(\020)s Fy(\()p Fn(t)1036 2032 y Fl(0)1048 2047 y Fy(\))g(is)g(equiv)n(alen)o(t)f(to)g(the)h(existence)i(of)d(a) 411 2097 y(ground)d(term)f Fn(t)666 2082 y Fl(00)701 2097 y Fy(suc)o(h)i(that)f Fn(t)d Fy(=)h Fn(\020)s Fy(\()p Fn(t)1007 2082 y Fl(00)1029 2097 y Fy(\))i(and)f Fn(\031)q Fy(\([)p Fn(t)1207 2082 y Fl(00)1228 2097 y Fy(]\))e(=)h Fn(t)1326 2082 y Fl(0)1338 2097 y Fy(.)340 2172 y(Next)h(the)g (reduction)g(length)g(map)e(ob)o(viously)g(decreases)k(along)c(a)h (rule)h(application,)d(but)340 2222 y(it)k(is)f(not)h(monotonic.)d (Therefore)16 b(a)d(summand)e(is)j(inserted)h(that)f(o)o(v)o(errides)g (its)g(gro)o(wth,)340 2272 y(yielding)h(a)h(function)g(that)g(is)g 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y(instances.)e(This)f(can)h(b)q(e)g(repaired)g(b)o(y)f(adding)g (appropriate)g(rewrite)i(rules)f(of)f(the)h(shap)q(e)183 375 y Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r(;)g(:)g(:)g(:)t Fy(\))k Fo(!)g Fn(c)p Fy(,)g Fn(G)p Fy(\()p Fn(x;)c(y)q(;)g(z)r(;)g(:)g (:)g(:)e Fy(\))11 b Fo(!)g Fn(c)p Fy(,)g Fn(f)t Fy(\()p Fn(c)p Fy(\))i Fo(!)e Fn(c)p Fy(,)g Fn(g)q Fy(\()p Fn(c)p Fy(\))h Fo(!)f Fn(c)p Fy(,)g(and)h Fn(h)p Fy(\()p Fn(c)p Fy(\))f Fo(!)g Fn(c)h Fy(without)183 425 y(a\013ecting)i(an)o(y)g(of)f (our)h(prop)q(ositions.)g(This)g(has)g(the)h(additional)d(b)q(ene\014t) k(of)d(making)f(the)183 475 y(TRSs)20 b Fo(S)328 481 y Fx(1)347 475 y Fy(\()p Fn(P)6 b Fy(\))19 b(and)h Fo(S)543 481 y Fx(2)562 475 y Fy(\()p Fn(P)6 b Fy(\))20 b(w)o(eakly)g (normalizing.)d(\(Note)j(that)h Fo(S)1279 481 y Fx(3)1298 475 y Fy(\()p Fn(P)6 b Fy(\))19 b(to)h Fo(S)1464 481 y Fx(6)1483 475 y Fy(\()p Fn(P)6 b Fy(\))20 b(are)183 524 y(alw)o(a)o(ys)15 b(terminating.\))g(Consequen)o(tly)i(all)f(of)g (our)g(results)i(hold)e(for)h(con\015uen)o(t)g(w)o(eakly)183 574 y(normalizing)11 b(TRSs.)245 624 y(In)i(particular,)g(the)h(union)e (of)h Fo(R)p Fy(\()p Fn(P)6 b Fy(\))13 b(and)g(the)h(rule)g Fn(F)6 b Fy(\()p Fn(x;)h(y)q(;)g(z)r Fy(\))k Fo(!)g Fn(c)i Fy(is)g(easily)g(seen)i(to)183 674 y(b)q(e)d(w)o(eakly)f(normalizing)e (while)j(it)f(is)h(terminating)e(if)h(and)h(only)f(if)g Fn(P)17 b Fy(admits)11 b(no)g(solution)183 724 y(as)h(in)f(Prop.)h(14.) f(This)h(pro)o(v)o(es)h(relativ)o(e)f(undecidabilit)o(y)f(of)g(the)i (implication)c(SN)j Fo(\))f Fy(WN.)183 857 y Fp(4)56 b(Conclusions)183 955 y Fy(F)m(or)16 b(most)g(of)g(the)h(implications)d (in)i(the)i(con\015uence)g(and)f(termination)e(hierarc)o(hies)i(w)o(e) 183 1005 y(pro)o(v)o(ed)e(relativ)o(e)g(undecidabilit)o(y;)f(only)g (for)h(p)q(olynomial)d(termination)h(the)j(question)f(of)183 1055 y(\(relativ)o(e\))f(undecidabilit)o(y)e(is)i(still)f(op)q(en.)245 1104 y(One)18 b(can)f(w)o(onder)g(whether)h(similar)c(results)k(hold)f (for)f(TRSs)h(consisting)g(of)f(single)183 1154 y(rules.)e (Undecidabilit)o(y)f(of)g(termination)g(\([2)o(]\),)g(and)h (non-self-em)o(b)q(eddingness)g(and)g(sim-)183 1204 y(ple)g (termination)e(\([16]\))i(of)f(single)h(rules)h(w)o(as)f(already)g(kno) o(wn.)g(Recen)o(tly)g(for)g(the)h(lo)o(w)o(er)183 1254 y(\014v)o(e)f(implications)e(in)i(the)h(termination)d(hierarc)o(h)o(y)j (w)o(e)g(pro)o(v)o(ed)f(relativ)o(e)h(undecidabilit)o(y)183 1304 y(for)e(single)h(\(ev)o(en)g(orthogonal\))f(rules;)h(the)h(result) f(app)q(ears)h(in)f([7)o(].)245 1354 y(Some)f(of)g(the)i(results)h (also)d(hold)h(elemen)o(t)o(wise.)f(F)m(or)h(instance,)g(giv)o(en)g(a)g (lo)q(cally)e(con-)183 1403 y(\015uen)o(t)k(TRS)f(and)g(a)h(\014xed)g (term,)e(it)i(is)f(undecidable)h(whether)h(that)f(term)f(is)g (con\015uen)o(t.)183 1453 y(Here)j(w)o(e)g(can)f(c)o(ho)q(ose)h(the)g (TRS)e Fo(R)773 1459 y Fx(4)792 1453 y Fy(\()p Fn(P)6 b Fy(\))17 b(as)g(in)g(Section)h(2.4)e(and)h Fn(g)q Fy(\()p Fn(c;)7 b(c)p Fy(\))17 b(as)g(the)h(\014xed)183 1503 y(term:)f(then)i(the)g(claimed)e(result)j(follo)o(ws)d(from)f(the)j (results)h(of)e(Section)h(2.4)f(and)g(the)183 1553 y(observ)n(ation)11 b(that)h(if)f Fn(g)q Fy(\()p Fn(c;)c(c)p Fy(\))12 b(is)g(con\015uen)o (t)g(then)h Fn(A)f Fo(!)1039 1560 y Fl(R)1068 1564 y Fj(4)1083 1560 y Fx(\()p Fk(P)t Fx(\))1148 1553 y Fn(C)i Fy(and)e(hence)h Fn(A)f Fo(!)1469 1560 y Fl(R)1498 1564 y Fj(4)1513 1560 y Fx(\()p Fk(P)t Fx(\))1578 1553 y Fn(B)r Fy(,)183 1603 y(whic)o(h)f(w)o(as)h(already)g(kno)o(wn)f(to)h(b)q(e)h (equiv)n(alen)o(t)e(to)g(the)i(existence)h(of)d(a)h(PCP)g(solution)f (for)183 1652 y Fn(P)6 b Fy(.)12 b(In)i(the)h(termination)d(hierarc)o (h)o(y)i(elemen)o(t)o(wise)f(results)i(are)f(not)g(obtained)g(that)g (easy;)183 1702 y(w)o(e)g(did)f(not)h(examine)f(this.)183 1835 y Fp(References)202 1929 y Fv(1.)20 b(A.-C.)11 b(Caron,)h Fa(Linear)h(Bounded)h(Automata)f(and)g(Rewrite)f(Systems:)h (In\015uence)g(of)f(Initial)252 1975 y(Con\014guration)g(on)e(Decision) i(Prop)q(erties)p Fv(,)f(Pro)q(ceedings)g(of)f(the)f(15th)h(Collo)q (quium)j(on)d(T)m(rees)252 2021 y(in)j(Algebra)h(and)f(Programming,)g (Lecture)g(Notes)f(in)h(Computer)f(Science)i Ft(493)e Fv(\(1991\))h(74{)252 2066 y(89.)202 2112 y(2.)20 b(M.)9 b(Dauc)o(het,)h Fa(Sim)o(ulation)i(of)d(T)m(uring)h(Mac)o(hines)g(b)o (y)g(a)f(Regular)h(Rewrite)g(Rule)p Fv(,)g(Theoretical)252 2158 y(Computer)k(Science)g Ft(103)f Fv(\(1992\))h(109{120.)202 2203 y(3.)20 b(N.)10 b(Dersho)o(witz)h(and)g(J.-P)m(.)d(Jouannaud,)k Fa(Rewrite)e(Systems)p Fv(,)h(in:)f(Handb)q(o)q(ok)i(of)e(Theoretical) 252 2249 y(Computer)k(Science,)g(V)m(ol.)f(B)g(\(ed.)g(J.)f(v)n(an)h (Leeu)o(w)o(en\),)g(North-Holland)i(\(1990\))f(243{320.)202 2295 y(4.)20 b(M.C.F.)13 b(F)m(erreira,)g Fa(T)m(ermination)j(of)d(T)m (erm)g(Rewriting:)h(W)m(ell-foundedne)q(ss,)i(T)m(otalit)o(y)m(,)e(and) 252 2340 y(T)m(ransformations)p Fv(,)h(PhD)e(thesis,)h(Univ)o(ersit)o (y)h(of)d(Utrec)o(h)o(t,)h(1995.)202 2386 y(5.)20 b(M.C.F.)9 b(F)m(erreira)h(and)h(H.)d(Zan)o(tema,)j Fa(T)m(otal)f(T)m(ermination)h (of)e(T)m(erm)g(Rewriting)p Fv(,)i(Applicabl)q(e)252 2432 y(Algebra)k(in)e(Engineering,)j(Comm)o(unication)f(and)f (Computing)g Ft(7)f Fv(\(1996\))h(133{162.)p eop %%Page: 18 18 18 17 bop 360 194 a Fv(6.)20 b(A.)9 b(Geser,)g Fa(Omega-termination)j (is)d(Undecidable)j(for)e(T)m(otally)g(T)m(erminating)h(T)m(erm)e (Rewrit-)410 239 y(ing)14 b(Systems)p Fv(,)g(Journal)g(of)f(Sym)o(b)q (olic)i(Computation)g Ft(23)e Fv(\(1997\).)g(T)m(o)f(app)q(ear.)360 285 y(7.)20 b(A.)14 b(Geser,)g(A.)f(Middeldorp)q(,)j(E.)e(Ohlebusc)o(h) i(and)f(H.)f(Zan)o(tema,)g Fa(Relativ)o(e)j(undecidabil)q(it)o(y)410 331 y(in)12 b(the)g(termination)i(hierarc)o(h)o(y)f(of)e(single)i (rewrite)f(rules)p Fv(,)g(Pro)q(ceedings)h(of)f(the)f(Collo)q(quium)410 376 y(on)g(T)m(rees)g(in)g(Algebra)h(and)f(Programming,)i(Lecture)e (Notes)f(in)i(Computer)f(Science)h(\(1997\).)410 422 y(T)m(o)h(app)q(ear.)360 468 y(8.)20 b(G.)15 b(Huet)f(and)i(D.)e (Lankford,)i Fa(On)e(the)h(Uniform)g(Halting)i(Problem)f(for)e(T)m(erm) g(Rewriting)410 513 y(Systems)p Fv(,)g(rep)q(ort)f(283,)g(INRIA)f (\(1978\).)360 559 y(9.)20 b(G.)f(Huet,)f Fa(Con\015uen)o(t)i (Reductions:)h(Abstract)f(Prop)q(erties)g(ans)g(Applications)i(to)d(T)m (erm)410 605 y(Rewriting)c(Systems)p Fv(,)e(Journal)i(of)d(the)h(A)o (CM)g Ft(27)g Fv(\(1980\))g(797{821.)340 650 y(10.)21 b(G.)c(Huet)h(and)g(D.C.)e(Opp)q(en,)i Fa(Equations)i(and)e(Rewrite)g (Rules:)g(A)f(Surv)o(ey)p Fv(,)h(in:)g(F)m(ormal)410 696 y(Language)11 b(Theory:)f(P)o(ersp)q(ectiv)o(es)i(and)f(Op)q(en)f (Problems)h(\(ed.)f(R.)g(Bo)q(ok\),)g(Academic)h(Press)410 742 y(\(1980\))j(349{405.)340 787 y(11.)21 b(J.W.)12 b(Klop,)i Fa(T)m(erm)e(Rewriting)j(Systems)p Fv(,)e(in:)h(Handb)q(o)q (ok)g(of)f(Logic)h(in)f(Computer)h(Science)410 833 y(\(eds.)j(S.)g (Abramski,)h(D.)f(Gabba)o(y)i(and)e(T.)g(Maibaum\),)h(V)m(olume)g(2,)f (Oxford)g(Univ)o(ersit)o(y)410 879 y(Press)d(\(1992\))f(1{116.)340 924 y(12.)21 b(D.E.)16 b(Kn)o(uth)g(and)h(P)m(.B.)e(Bendix,)j Fa(Simple)g(W)m(ord)e(Problems)i(in)f(Univ)o(ersal)h(Algebras)p Fv(,)f(in:)410 970 y(Computational)23 b(Problems)f(in)f(Abstract)f (Algebra)i(\(ed.)e(J.)f(Leec)o(h\),)i(P)o(ergamon)g(Press)410 1016 y(\(1970\))14 b(263{297.)340 1061 y(13.)21 b(M.)11 b(Kurihara)i(and)e(A.)g(Oh)o(uc)o(hi,)h Fa(Mo)q(dularit)o(y)h(of)e (Simple)i(T)m(ermination)g(of)e(T)m(erm)f(Rewriting)410 1107 y(Systems)p Fv(,)k(Journal)g(of)f(IPS)g(Japan)g Ft(31)g Fv(\(1990\))h(633{642.)340 1153 y(14.)21 b(P)m(.)10 b(Lescanne,)i Fa(On)e(T)m(ermination)j(of)d(One)h(Rule)h(Rewrite)f (Systems)p Fv(,)g(Theoretical)i(Computer)410 1198 y(Science)i Ft(132)e Fv(\(1994\))g(395{401.)340 1244 y(15.)21 b(Y.)16 b(Matiy)o(asevic)o(h)j(and)f(G.)e(Senizergues,)i Fa(Decision)h (Problems)f(for)f(Semi-Th)o(ue)g(Systems)410 1290 y(with)e(a)e(F)m(ew)h (Rules)p Fv(.)g(Pro)q(ceedings)i(of)e(the)g(11th)g(IEEE)g(Ann)o(ual)h (Symp)q(osium)h(on)f(Logic)f(in)410 1335 y(Computer)g(Science,)g(New)e (Brunswic)o(k,)i(pp.)f(523{531,)h(1996.)340 1381 y(16.)21 b(A.)11 b(Middeldorp)k(and)d(B.)g(Gramlic)o(h,)h Fa(Simple)g(T)m (ermination)h(is)e(Di\016cult)p Fv(,)h(Applicable)i(Alge-)410 1427 y(bra)f(in)f(Engineering,)j(Comm)o(unication)f(and)f(Computing)g Ft(6)f Fv(\(1995\))h(115{128.)340 1472 y(17.)21 b(A.)15 b(Middeldorp)k(and)d(H.)f(Zan)o(tema,)h Fa(Simple)i(T)m(ermination)f (of)f(Rewrite)g(Systems)p Fv(,)g(Theo-)410 1518 y(retical)f(Computer)e (Science)h Ft(175)g Fv(\(1997\))f(127{158.)340 1564 y(18.)21 b(D.A.)10 b(Plaisted,)i Fa(The)f(Undecidabil)q(it)o(y)j(of)c(Self-Em)o (b)q(edding)k(for)c(T)m(erm)g(Rewriting)j(Systems)p Fv(,)410 1609 y(Information)h(Pro)q(cessing)h(Letters)e Ft(20)g Fv(\(1985\))h(61{64.)340 1655 y(19.)21 b(E.)12 b(P)o(ost,)f Fa(A)h(V)m(arian)o(t)h(of)e(a)h(Recursiv)o(ely)j(Unsolv)n(able)f (Problem)p Fv(,)f(Bulletin)h(of)e(the)g(American)410 1701 y(Mathematical)j(So)q(ciet)o(y)g Ft(52)e Fv(\(1946\))g(264{268.) 340 1746 y(20.)21 b(A.)11 b(Rubio,)i Fa(Extension)g(Orderings)p Fv(,)g(Pro)q(ceedings)g(of)e(the)h(22nd)g(In)o(ternational)i(Collo)q (quium)410 1792 y(on)h(Automata,)f(Languages)i(and)f(Programming,)h (Szeged,)f(Lecture)f(Notes)h(in)g(Computer)410 1838 y(Science)g Ft(944)e Fv(\(1995\))g(511{522.)340 1883 y(21.)21 b(H.)9 b(Zan)o(tema,)h Fa(T)m(ermination)h(of)e(T)m(erm)g(Rewriting:)i(In)o (terpretation)g(and)f(T)o(yp)q(e)g(Elimination)q Fv(,)410 1929 y(Journal)k(of)f(Sym)o(b)q(olic)j(Computation)e Ft(17)f Fv(\(1994\))h(23{50.)340 1975 y(22.)21 b(H.)13 b(Zan)o(tema,)g Fa(T)m(ermination)i(of)e(T)m(erm)g(Rewriting)i(b)o(y)e (Seman)o(tic)i(Lab)q(elling)p Fv(,)h(F)m(undamen)o(ta)410 2020 y(Informaticae)e Ft(24)f Fv(\(1995\))h(89{105.)340 2066 y(23.)21 b(H.)10 b(Zan)o(tema,)g Fa(T)m(otal)h(T)m(ermination)h (of)e(T)m(erm)f(Rewriting)j(is)f(Undecidable)p Fv(,)i(Journal)e(of)f (Sym-)410 2111 y(b)q(olic)15 b(Computation)g Ft(20)e Fv(\(1995\))g(43{60.)340 2157 y(24.)21 b(H.)8 b(Zan)o(tema)i(and)g(A.)e (Geser,)g Fa(Non-Lo)q(oping)j(Rewriting)p Fv(,)f(Utrec)o(h)o(t)f(Univ)o (ersit)o(y)m(,)h(Departmen)o(t)410 2203 y(of)j(Computer)g(Science,)h (rep)q(ort)g(UU-CS-1996-03)f(\(1996\).)340 2424 y(This)h(article)g(w)o (as)f(pro)q(cessed)i(using)f(the)f(L)968 2415 y Fw(A)985 2424 y Fv(T)1007 2432 y(E)1028 2424 y(X)f(macro)h(pac)o(k)n(age)h(with) g(LLNCS)e(st)o(yle)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF