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b(Relation)523 886 y FA(Enno)27 b(Ohlebusc)n(h)523 1060 y Fz(Univ)n(ersit)n(y)e(of)h(Bielefeld,)i(T)-6 b(ec)n(hnisc)n(he) 26 b(F)-6 b(akult\177)-38 b(at)523 1152 y(P)-6 b(.O.)26 b(Bo)n(x)g(100131,)i(33501)g(Bielefeld,)f(German)n(y)523 1243 y(enno@T)-6 b(ec)n(hF)g(ak.Uni-Bielefeld.DE)523 1489 y Fy(Abstract.)42 b Fz(A)33 b(v)n(ery)g(p)r(o)n(w)n(erful)h(metho) r(d)e(for)i(pro)n(ving)f(the)g(Ch)n(urc)n(h-Rosser)g(prop)r(ert)n(y)g (for)523 1580 y(abstract)27 b(rewriting)h(systems)e(has)h(b)r(een)f (dev)n(elop)r(ed)h(b)n(y)e(v)l(an)h(Oostrom.)h(In)f(this)h(pap)r(er,)g (his)523 1672 y(tec)n(hnique)35 b(is)h(extended)e(in)i(t)n(w)n(o)g(w)n (a)n(ys)g(to)g(abstract)g(rewriting)h(mo)r(dulo)e(an)h(equiv)l(alence) 523 1763 y(relation.)27 b(It)e(is)h(sho)n(wn)g(that)f(kno)n(wn)g(Ch)n (urc)n(h-Rosser)g(theorems)g(can)h(b)r(e)g(view)n(ed)f(as)h(sp)r(ecial) 523 1854 y(cases)32 b(of)g(the)f(new)h(criteria.)g(Moreo)n(v)n(er,)h (applications)f(of)g(the)f(new)g(criteria)i(yield)e(sev)n(eral)523 1946 y(new)26 b(results.)523 2217 y Fx(1)112 b(In)m(tro)s(duction)523 2407 y FA(An)27 b(abstract)f(reduction)1323 2377 y Fw(1)1386 2407 y FA(system)h(\(ARS\))h(is)e(just)i(a)e(set)h(of)f(ob)5 b(jects)27 b(and)f(a)g(sequence)523 2507 y(of)38 b(binary)g(relations)f (on)h(it.)h(ARSs)g(are)e(called)h(general)f(replacemen)n(t)h(systems)g (in)523 2606 y([Ros73)n(,Sta75)o(].)g(Abstract)f(rewriting)f(comprises) g(sev)n(eral)f(kinds)i(of)g(rewriting)f(lik)n(e)523 2706 y(term-,)25 b(string-,)f(graph-,)f(and)i(conditional)f(rewriting.)g(Th) n(us)g(a)h(rep)r(etition)g(of)g(similar)523 2806 y(de\014nitions)34 b(and)f(concepts)g(is)h(a)n(v)n(oided)e(b)n(y)h(stating)g(them)h(once)f (and)h(for)f(all)g(on)g(an)523 2905 y(abstract)21 b(lev)n(el)h(\(for)g (ARSs,)h(that)f(is\).)h(In)f(essence,)g(a)g(rewriting)f(relation)g(mo)r (dels)h(non-)523 3005 y(deterministic)27 b(computations.)g(A)h(k)n(ey)e (prop)r(ert)n(y)g(in)i(the)g(theory)e(of)h(rewriting)f(is)i(the)523 3104 y(Ch)n(urc)n(h-Rosser)e(prop)r(ert)n(y)h(whic)n(h)h(guaran)n(tees) f(that)h(the)h(results)f(of)g(suc)n(h)h(computa-)523 3204 y(tions)i(are)f(unique.)h(Ch)n(urc)n(h-Rosser)d(theorems)i(for)h (abstract)f(rewriting)g(w)n(ere)g(giv)n(en)523 3304 y(for)20 b(instance)g(b)n(y)g(Hindley)h([Hin64)o(],)g(Rosen)f([Ros73)n(],)h (Staples)f([Sta75)o(],)g(Huet)h([Hue80],)523 3403 y(Geser)e([Ges90)o (],)h(and)f(v)-5 b(an)20 b(Oostrom)e([Oos94a)n(].)h(The)h(reader)e(is)i (referred)f(to)g(the)h(surv)n(ey)523 3503 y(of)25 b(Klop)e([Klo92)o(])h (for)g(details.)h(In)f(this)h(pap)r(er,)g(w)n(e)f(study)g(ARSs)h(whic)n (h)g(are)e(addition-)523 3603 y(ally)30 b(equipp)r(ed)g(with)h(an)f (equiv)-5 b(alence)29 b(relation)g Fv(\030)h FA(on)g(the)g(set)g(of)g (ob)5 b(jects.)30 b(Suc)n(h)g(an)523 3702 y(ARS)e(is)e(called)h(Ch)n (urc)n(h-Rosser)e(if)i(ev)n(ery)f(t)n(w)n(o)g(ob)5 b(jects)26 b(whic)n(h)h(are)f(con)n(v)n(ertible)g(ha)n(v)n(e)523 3802 y(reducts)c(that)h(are)f(equiv)-5 b(alen)n(t.)22 b(It)h(is)g(w)n(ell-kno)n(wn)e(that)i(for)f(abstract)g(reduction)g(mo)r (d-)523 3901 y(ulo)k Fv(\030)g FA(the)g(Ch)n(urc)n(h-Rosser)d(prop)r (ert)n(y)i(do)r(es)h(not)g(coincide)g(with)g(con\015uence.)g(This)g(is) 523 4001 y(in)34 b(sharp)f(con)n(trast)g(to)h(pure)g(abstract)f (rewriting.)g(Sethi's)h([Set74)o(])g(Ch)n(urc)n(h-Rosser)523 4101 y(theorem)20 b(for)g(b)r(ounded)h(ARSs)f(has)g(subsequen)n(tly)g (b)r(een)h(impro)n(v)n(ed)e(b)n(y)h(Huet)h([Hue80].)523 4200 y(F)-7 b(urther)33 b(Ch)n(urc)n(h-Rosser)d(theorems)i(for)h (abstract)e(rewriting)h(mo)r(dulo)h Fv(\030)g FA(w)n(ere)f(ob-)523 4300 y(tained)27 b(in)g(the)h(con)n(text)e(of)h(completion)g(mo)r(dulo) f(equations;)g(see)h(e.g.)g([JM84)n(,JK86)o(].)523 4400 y(In)k(this)h(pap)r(er,)e(w)n(e)h(will)g(giv)n(e)g(sev)n(eral)e(new)i (su\016cien)n(t)g(conditions)g(for)g(the)g(Ch)n(urc)n(h-)523 4499 y(Rosser)21 b(prop)r(ert)n(y)g(for)h(abstract)f(rewriting)h(mo)r (dulo)g Fv(\030)g FA(and)g(review)g(kno)n(wn)g(metho)r(ds.)523 4599 y(This)28 b(is)g(essen)n(tially)f(done)h(b)n(y)g(generalizing)e (the)i(p)r(o)n(w)n(erful)g(metho)r(d)g(devised)g(b)n(y)g(v)-5 b(an)523 4698 y(Oostrom)26 b([Oos94a)m(].)p 523 4768 473 4 v 540 4822 a Fu(1)600 4853 y Fz(The)21 b(terms)f(reduction)h(and) f(rewriting)i(are)f(used)g(synon)n(ymously)e(throughout)h(the)g(pap)r (er.)p eop %%Page: 2 2 2 1 bop 523 448 a Fx(2)112 b(Preliminaries)523 625 y Ft(De\014nition)31 b(1.)41 b FA(An)23 b Fs(abstr)l(act)i(r)l(e)l (duction)g(system)k Fv(A)23 b FA(=)g(\()p Fr(A;)14 b Fv(h!)2573 637 y Fq(\013)2621 625 y Fv(i)2653 637 y Fq(\013)p Fp(2)p Fq(I)2779 625 y Fr(;)g Fv(\030)p FA(\))23 b(is)f(a)h(struc-)523 725 y(ture)31 b(consisting)g(of)g(a)f(set)i(of)f(ob)5 b(jects)31 b Fr(A)p FA(,)g(a)g(sequence)g(of)g(relations)f Fv(!)2839 737 y Fq(\013)2918 725 y FA(on)g Fr(A)p FA(,)i(and)523 825 y(an)39 b(equiv)-5 b(alence)39 b(relation)f Fv(\030)h FA(on)g Fr(A)p FA(.)g(A)h(relation)e Fv(!)2272 837 y Fq(\013)2359 825 y FA(is)h(said)f(to)h(b)r(e)h(a)f Fs(r)l(e)l(duction) 523 924 y FA(relation)c Fs(lab)l(ele)l(d)46 b FA(b)n(y)36 b Fr(\013)p FA(.)g(The)g(reduction)g(relation)f(of)h Fv(A)g FA(is)g(the)h(union)f(of)f(its)i(con-)523 1024 y(stituen)n(t)d(reduction)e(relations:)g Fv(!)1659 1036 y Fp(A)1717 1024 y FA(=)1813 962 y Fo(S)1882 1049 y Fq(\013)p Fp(2)p Fq(I)2040 1024 y Fv(!)2123 1036 y Fq(\013)2171 1024 y FA(.)h(When)g(the)h(ARS)f(is)g(clear)f(from)523 1123 y(the)42 b(con)n(text,)f(it)g(will)h(b)r(e)f(suppressed.)g(Tw)n(o) f(ARSs)i Fv(A)k FA(=)g(\()p Fr(A;)14 b Fv(h!)2772 1135 y Fq(\013)2820 1123 y Fv(i)2852 1135 y Fq(\013)p Fp(2)p Fq(I)2978 1123 y Fr(;)g Fv(\030)p FA(\))41 b(and)523 1223 y Fv(B)25 b FA(=)e(\()p Fr(A;)14 b Fv(h!)937 1235 y Fq(\014)982 1223 y Fv(i)1014 1235 y Fq(\014)s Fp(2)p Fq(J)1146 1223 y Fr(;)g Fv(\030)p FA(\))28 b(are)e Fs(r)l(e)l(duction)k (e)l(quivalent)36 b FA(if)28 b Fv(!)2359 1235 y Fp(A)2431 1223 y FA(=)14 b Fv(!)2593 1235 y Fp(B)2641 1223 y FA(.)523 1378 y(The)40 b(sym)n(b)r(ols)g Fv( )p FA(,)68 b Fv(!)1293 1348 y Fw(=)1376 1378 y FA(,)g Fv(!)1550 1348 y Fw(+)1633 1378 y FA(,)40 b(and)68 b Fv(!)1981 1348 y Fp(\003)2087 1378 y FA(denote)40 b(the)h(in)n(v)n(erse,)d(the)j(re\015exiv)n(e)523 1478 y(closure,)31 b(the)h(transitiv)n(e)f(closure,)g(and)h(the)g (re\015exiv)n(e)f(transitiv)n(e)g(closure)g(of)h Fv(!)p FA(,)g(re-)523 1577 y(sp)r(ectiv)n(ely)-7 b(.)28 b(W)-7 b(e)29 b(use)f Fv(!)1294 1589 y Fq(\013)1366 1577 y Fv(\001)c(!)1496 1589 y Fq(\014)1569 1577 y FA(to)29 b(denote)f(the)h(comp)r(osition)e (of)i Fv(!)2727 1589 y Fq(\013)2802 1577 y FA(and)g Fv(!)3048 1589 y Fq(\014)3092 1577 y FA(.)g(The)523 1677 y(relation)63 b Fv(!)947 1647 y Fp(\003)1038 1677 y Fv(\001)1127 1647 y Fp(\003)1160 1677 y Fv( )i FA(\()28 b Fv(!)1451 1647 y Fp(\003)1517 1677 y Fv(\001)38 b(\030)f(\001)1745 1647 y Fp(\003)1779 1677 y Fv( )27 b FA(\))37 b(is)g(called)f Fs(joinability)j FA(\()p Fs(mo)l(dulo)g Fv(\030)p FA(\))e(and)523 1776 y(denoted)h(b)n(y)h Fv(#)f FA(\()p Fv(#)1128 1788 y Fp(\030)1183 1776 y FA(\).)h(W)-7 b(e)39 b(further)f(de\014ne)h Fv(j)-14 b FA(=)c Fv(j)40 b FA(=)h Fv(!)g([)g( )g([)h(\030)p FA(,)c Fv(\031)i FA(=)h Fv(j)-14 b FA(=)c Fv(j)3053 1735 y Fp(\003)3092 1776 y FA(,)38 b(and)551 1876 y Fv(!)634 1888 y Fp(\030)741 1876 y FA(=)73 b Fv(\030)23 b(\001)g(!)g(\001)g (\030)p FA(.)k(The)h(relation)e Fv(\031)i FA(is)f(called)g Fs(c)l(onversion)p FA(.)648 1976 y(If)d Fr(a)f Fv(!)g Fr(b)p FA(,)i(then)g(w)n(e)f(sp)r(eak)g(of)g(a)g Fs(r)l(e)l(duction)j (step)e FA(from)f Fr(a)g FA(to)g Fr(b)p FA(.)h(An)g(elemen)n(t)f Fr(a)f Fv(2)h Fr(A)523 2075 y FA(is)31 b(in)g Fs(normal)i(form)f FA(if)g(there)e(is)h(no)g(elemen)n(t)g Fr(b)d Fv(2)h Fr(A)i FA(with)g Fr(a)e Fv(!)f Fr(b)p FA(;)j(it)g Fs(has)j(a)f(normal) 523 2175 y(form)j FA(if)g Fr(a)28 b Fv(!)968 2145 y Fp(\003)1033 2175 y Fr(b)35 b FA(for)g(some)g(normal)f(form)h Fr(b)p FA(.)g(The)h(ARS)g Fv(A)f FA(is)g(called)g Fs(normalizing)523 2275 y FA(\(w)n(eakly)24 b(normalizing{WN\))g(if)i(ev)n(ery)e(term)h (has)g(a)f(normal)g(form.)h Fv(A)h FA(is)f Fs(terminating)523 2374 y FA(\(strongly)e(normalizing{SN\))g(if)i(there)f(is)g(no)g (in\014nite)h(reduction)f(sequence)f(w.r.t.)i Fv(!)p FA(.)523 2474 y Fv(A)31 b FA(is)g Fs(terminating)i(mo)l(dulo)g Fv(\030)d FA(\(SN)p Fv(\030)p FA(\))i(if)f(there)f(is)h(no)g (in\014nite)g(reduction)f(sequence)523 2573 y(w.r.t.)55 b Fv(!)855 2585 y Fp(\030)939 2573 y FA(.)648 2673 y(The)32 b(lab)r(el)g(of)g(a)g(\014nite)g(reduction)g(sequence)f(is)h(the)h (string)e(of)h(the)h(lab)r(els)f(of)g(its)523 2773 y(constituen)n(t)g (reduction)g(steps.)h(The)f(Greek)g(letters)g Fr(\033)n(;)14 b(\034)5 b(;)14 b(\026;)g(\027)38 b FA(etc.)33 b(will)f(b)r(e)h(used)f (to)523 2872 y(denote)c(strings.)e(The)i(concatenation)e(of)i(t)n(w)n (o)f(strings)f Fr(\033)31 b FA(and)d Fr(\034)37 b FA(is)28 b(denoted)f(b)n(y)h Fr(\033)s(\034)9 b FA(.)648 2972 y(The)21 b(rest)g(of)h(this)g(section)f(is)g(copied)h(from)f([Oos94a)m (].)h(The)g(reader)e(should)h(consult)523 3072 y([Oos94a)m(])28 b(or)f([Oos94b)n(])h(for)f(more)g(details)g(and)g(in)n(tuitiv)n(e)h (explanations.)523 3215 y Ft(De\014nition)j(2.)41 b FA(A)28 b(\()p Fs(gener)l(al)9 b FA(\))30 b Fs(multiset)35 b FA(is)28 b(a)g(collection)g(in)h(whic)n(h)f(elemen)n(ts)g(are)g(al-)523 3315 y(lo)n(w)n(ed)d(to)h(o)r(ccur)f(more)h(than)g(once)f(or)h(ev)n(en) f(inf)6 b(initely)28 b(of)6 b(ten.)27 b(A)f Fs(\014nite)32 b FA(m)n(ultiset)26 b(has)523 3415 y(\014nitely)c(man)n(y)e(dif)6 b(feren)n(t)22 b(elemen)n(ts)f(whic)n(h)g(o)r(ccur)g(f)6 b(initely)22 b(of)6 b(ten.)22 b(A)g Fs(set)29 b FA(is)21 b(a)f(m)n(ultiset)523 3514 y(in)28 b(whic)n(h)f(elemen)n(ts)h(o)r(ccur) f(either)g(not)h(at)f(all)h(or)e(inf)6 b(initely)29 b(of)6 b(ten.)523 3658 y(T)-7 b(o)20 b(distinguish)h(b)r(et)n(w)n(een)g(set)g (comprehension)e(and)i(f)6 b(inite)22 b(m)n(ultiset)f(comprehension,) 523 3758 y(braces)29 b(will)i(b)r(e)g(used)g(to)f(denote)h(the)g (former)f(and)g(square)g(brac)n(k)n(ets)e(to)j(denote)g(the)523 3857 y(latter.)26 b(F)-7 b(or)26 b(example)g([)p Fr(\013;)14 b(\014)t FA(])28 b(denotes)e(the)h(f)6 b(inite)28 b(m)n(ultiset)f(with) g(exactly)f(one)g(o)r(ccur-)523 3957 y(rence)j(of)g(b)r(oth)g Fr(\013)h FA(and)f Fr(\014)t FA(,)h(whereas)e Fv(f)p Fr(\013)p Fv(g)h FA(denotes)f(the)i(set)f(m)n(ultiset)h(with)f(inf)6 b(initely)523 4056 y(man)n(y)27 b(o)r(ccurrences)f(of)i Fr(\013)p FA(.)648 4156 y(The)k(follo)n(wing)e(\(in\)equalities)j (illustrate)e(the)i(dif)6 b(ferences)32 b(b)r(et)n(w)n(een)g(f)6 b(inite)33 b(and)523 4256 y(set)e(m)n(ultisets,)g(as)f(w)n(ell)h(as)f (sum)h(and)g(union:)g([)p Fr(\013)p FA(])21 b Fv(])g FA([)p Fr(\013)p FA(])29 b(=)f([)p Fr(\013;)14 b(\013)p FA(])29 b Fv(6)p FA(=)f([)p Fr(\013)p FA(])h(=)f([)p Fr(\013)p FA(])21 b Fv([)g FA([)p Fr(\013)p FA(],)523 4355 y Fv(f)p Fr(\013)p Fv(g)14 b(])g(f)p Fr(\013)p Fv(g)23 b FA(=)f Fv(f)p Fr(\013;)14 b(\013)p Fv(g)23 b FA(=)g Fv(f)p Fr(\013)p Fv(g)f FA(=)h Fv(f)p Fr(\013)p Fv(g)13 b([)i(f)p Fr(\013)p Fv(g)p Fr(;)f FA([)p Fr(\013;)g(\013)p FA(])g Fv(\000)g FA([)p Fr(\013)p FA(])23 b(=)g([)p Fr(\013)p FA(])p Fr(;)14 b Fv(f)p Fr(\013)p Fv(g)g(\000)g FA([)p Fr(\013;)g(\013)p FA(])24 b(=)e Fv(f)p Fr(\013)p Fv(g)p FA(,)523 4455 y(and)27 b([)p Fr(\013;)14 b(\013)p FA(])20 b Fv(\000)e(f)p Fr(\013)p Fv(g)k FA(=)h Fv(;)p Fr(:)648 4555 y FA(The)j(m)n(ultiset)h([)p Fr(\033)s FA(])h(of)e(lab)r(els)h(of) g(a)f(string)g Fr(\033)k FA(is)d(the)g(sum)g(of)f(all)h(lab)r(el)f(o)r (ccurrences)523 4654 y(in)i(it,)f(so)g(in)h(particular)e(w)n(e)h(ha)n (v)n(e)f([)p Fr(\033)s(\034)9 b FA(])24 b(=)f([)p Fr(\033)s FA(])18 b Fv(])g FA([)p Fr(\034)9 b FA(].)29 b(F)-7 b(or)26 b(example,)h(if)h(w)n(e)f(ha)n(v)n(e)f(digits)523 4754 y(as)h(lab)r(els,)g([132343])21 b(=)h([1)p Fr(;)14 b FA(3)p Fr(;)g FA(2)p Fr(;)g FA(3)p Fr(;)g FA(4)p Fr(;)g FA(3].)648 4853 y(In)27 b(the)h(sequel,)g Fv(\036)f FA(denotes)g(a)g (strict)h(partial)f(order)f(on)h(the)h(set)g(of)f(lab)r(els)h Fr(I)7 b FA(.)p eop %%Page: 3 3 3 2 bop 523 448 a Ft(De\014nition)31 b(3.)41 b FA(\(1\))31 b(The)g Fs(down-set)39 b Fn(g)q Fr(\013)31 b FA(of)g Fr(\013)h FA(is)f(de\014ned)h(b)n(y)e Fn(g)q Fr(\013)f FA(=)g Fv(f)p Fr(\014)35 b Fv(j)c Fr(\014)i Fv(\036)c Fr(\013)p Fv(g)p Fr(:)523 548 y FA(This)h(is)f(extended)h(to)g(m)n (ultisets)g(and)g(strings)e(b)n(y)i(def)6 b(ining)30 b Fn(g)q Fr(M)35 b FA(=)2762 486 y Fo(S)2831 573 y Fq(\013)p Fp(2)p Fq(M)3007 548 y Fn(g)p Fr(\013)30 b FA(and)523 648 y Fn(g)p Fr(\033)d FA(=)22 b Fn(g)p FA([)p Fr(\033)s FA(])p Fr(:)29 b FA(F)-7 b(or)27 b(example,)g Fn(g)p FA(2)c(=)f Fn(g)p FA([0)p Fr(;)14 b FA(2])22 b(=)h Fn(g)p FA(212)f(=)g Fv(f)p FA(0)p Fr(;)14 b FA(1)p Fv(g)p Fr(:)648 747 y FA(\(2\))32 b(The)h(\()p Fs(standar)l(d)9 b FA(\))34 b Fs(multiset)g(extension)k FA(\(denoted)33 b(b)n(y)g Fv(\036)2611 759 y Fq(mul)2734 747 y FA(\))g(of)g(the)g(partial)523 847 y(order)26 b Fv(\036)i FA(is)f(def)6 b(ined)29 b(b)n(y)590 1013 y Fr(M)j Fv(\036)768 1025 y Fq(mul)914 1013 y Fr(N)119 b FA(if)28 b Fv(9)g Fr(X)r(;)14 b(Y)5 b(;)14 b(Z)28 b FA(:)23 b Fr(M)32 b FA(=)23 b Fr(Z)h Fv(])18 b Fr(X)r(;)c(N)32 b FA(=)23 b Fr(Z)h Fv(])18 b Fr(Y)5 b(;)14 b(X)29 b Fv(\022)23 b Fn(g)p Fr(Y)47 b FA(and)27 b Fr(Y)42 b Fv(6)p FA(=)22 b Fv(;)p FA(.)523 1179 y(F)-7 b(urthermore,)54 b Fv(\026)1120 1191 y Fq(mul)1299 1179 y FA(will)28 b(b)r(e)g(used)f(to)h(denote)f (the)h(ref)6 b(lexiv)n(e)27 b(closure)g(of)55 b Fv(\036)3069 1191 y Fq(mul)3220 1179 y FA(.)523 1362 y Ft(De\014nition)31 b(4.)41 b Fs(The)22 b(\(lexic)l(o)l(gr)l(aphic)h(maximum\))e(me)l(asur) l(e)k FA(grades)16 b(strings)i(b)n(y)g(f)6 b(inite)523 1461 y(m)n(ultisets)28 b(and)g(is)f(denoted)h(b)n(y)f Fv(j)19 b(\001)g(j)p FA(.)28 b(It)g(is)f(def)6 b(ined)29 b(inductiv)n(ely)f(b)n(y)f Fv(j)p Fr(")p Fv(j)d FA(=)e Fv(;)p FA(,)28 b(where)f Fr(")523 1561 y FA(denotes)g(the)h(empt)n(y)g (string,)f(and)g Fv(j)p Fr(\013\033)s Fv(j)d FA(=)f([)p Fr(\013)p FA(])c Fv(])g FA(\()p Fv(j)p Fr(\033)s Fv(j)g(\000)f Fn(g)p Fr(\013)p FA(\))p Fr(:)523 1744 y FA(The)30 b(measure)e(of)h(a)g (rewrite)g(sequence)g Fr(a)1862 1756 y Fw(0)1925 1744 y Fv(!)2008 1756 y Fq(\013)2051 1764 y Fm(0)2114 1744 y Fr(a)2158 1756 y Fw(1)2221 1744 y Fv(!)2304 1756 y Fq(\013)2347 1764 y Fm(1)2410 1744 y Fr(:)14 b(:)g(:)g(a)2565 1756 y Fq(m)2654 1744 y Fv(!)2737 1756 y Fq(\013)2780 1764 y Fl(m)2866 1744 y Fr(a)2910 1756 y Fq(m)p Fw(+1)3086 1744 y FA(is)30 b(the)523 1843 y(measure)d(of)g(its)h(lab)r(els,)f (i.e.,)h Fv(j)p Fr(\013)1546 1855 y Fw(0)1584 1843 y Fr(\013)1637 1855 y Fw(1)1688 1843 y Fr(:)14 b(:)g(:)f(\013)1851 1855 y Fq(m)1915 1843 y Fv(j)p FA(.)648 1943 y(In)n(tuitiv)n(ely)-7 b(,)42 b(the)i(lexicographic)c(maxim)n(um)j(measure)f(tak)n(es)g(the)h (m)n(ultiset)g(of)523 2043 y(elemen)n(ts)c(whic)n(h)h(are)e(maximal)h (\(in)h(the)g Fv(\037)f FA(ordering\))f(with)i(resp)r(ect)f(to)g(the)h (ele-)523 2142 y(men)n(ts)e(to)f(their)h(lef)6 b(t)38 b(in)g(the)g(string.)f(Op)r(erationally)-7 b(,)36 b(one)i(can)f(think)h (of)f(f)6 b(iltering)523 2242 y(out)33 b(the)g(noise)f(b)r(efore)g(pro) r(ceeding)g(to)g(the)h(righ)n(t.)f(F)-7 b(or)32 b(instance,)h(taking)f (the)h(mea-)523 2342 y(sures)28 b(of)g(the)h(strings)f(of)g(digits)h (132343)d(and)i(211)f(yields)h Fv(j)p FA(132343)p Fv(j)22 b FA(=)j([1)p Fr(;)14 b FA(3)p Fr(;)g FA(3)p Fr(;)g FA(4])26 b(and)523 2441 y Fv(j)p FA(211)p Fv(j)h FA(=)g([2].)j(T)-7 b(aking)30 b(the)h(measure)f(of)g(the)h(rewrite)f(sequence)g Fr(a)e Fv(!)2738 2453 y Fw(2)2803 2441 y Fr(b)f Fv(!)2949 2453 y Fw(1)3015 2441 y Fr(c)j FA(yields)523 2541 y Fv(j)p Fr(a)23 b Fv(!)696 2553 y Fw(2)756 2541 y Fr(b)g Fv(!)898 2553 y Fw(1)958 2541 y Fr(c)p Fv(j)g FA(=)g Fv(j)p FA(21)p Fv(j)f FA(=)h([2].)523 2807 y Fx(3)112 b(Prop)s(erties)36 b(of)i(ARSs)523 3006 y FA(The)20 b(follo)n(wing)e(de\014nitions)i (describ)r(e)f(basic)g(prop)r(erties)f(of)i(ARSs.)f(Supp)r(ose)h(that)g Fv(`)-28 b(a)19 b FA(is)523 3106 y(a)25 b(symmetric)f(relation)g(on)h Fr(A)g FA(with)h Fv(`)-28 b(a)1754 3068 y Fp(\003)1815 3106 y FA(=)23 b Fv(\030)p FA(.)h(Note)h(that)h Fv(\030)2456 3075 y Fp(\003)2493 3106 y FA(=)d Fv(\030)p FA(.)i(Throughout)f(this) 523 3205 y(section,)29 b Fv(`)-28 b(a)28 b FA(indeed)h(coincides)g (with)g Fv(\030)p FA(.)g(A)g(\(partial\))f(analysis)g(of)h(the)g (relationships)523 3305 y(b)r(et)n(w)n(een)f(the)g(last)f(six)g(prop)r (erties)g(can)g(b)r(e)h(found)g(in)g(the)g(next)g(section.)523 3471 y Ft(De\014nition)j(5.)41 b FA(Let)27 b Fv(A)d FA(=)e(\()p Fr(A;)14 b Fv(h!)1656 3483 y Fq(\013)1704 3471 y Fv(i)1736 3483 y Fq(\013)p Fp(2)p Fq(I)1863 3471 y Fr(;)g Fv(\030)p FA(\))27 b(b)r(e)h(an)f(ARS.)558 3637 y(1.)41 b(W)-7 b(e)25 b(write)f Fk(3)p FA(\()p Fv(!)1195 3607 y Fp(\003)1255 3637 y Fv(\001)g(\030)p FA(\))g(and)g(sa)n(y)f(that)i(the)g Fs(diamond)k(pr)l(op)l(erty)c FA(holds)f(for)g Fv(!)3115 3607 y Fp(\003)3176 3637 y Fv(\001)f(\030)664 3737 y FA(if)79 b Fv(\030)23 b(\001)943 3707 y Fp(\003)977 3737 y Fv( )k(\001)d(!)1217 3707 y Fp(\003)1278 3737 y Fv(\001)f(\030)73 b(\022)51 b(#)1620 3749 y Fp(\030)1675 3737 y FA(.)558 3836 y(2.)41 b Fv(A)28 b FA(is)g Fs(Chur)l(ch-R)l(osser)i(mo)l(dulo)g Fv(\030)d FA(\(CR)p Fv(\030)p FA(\))h(if)79 b Fv(\031)73 b(\022)51 b(#)2478 3848 y Fp(\030)2533 3836 y FA(.)558 3936 y(3.)41 b Fv(A)28 b FA(is)g Fs(almost)i(Chur)l(ch-R)l(osser)g(mo)l (dulo)g Fv(\030)e FA(\(A)n(CR)p Fv(\030)p FA(\))f(if)2498 3906 y Fp(\003)2532 3936 y Fv( )g(\001)c(\030)g(\001)28 b(!)2910 3906 y Fp(\003)3026 3936 y Fv(\022)51 b(#)3184 3948 y Fp(\030)3239 3936 y FA(.)558 4036 y(4.)41 b Fv(A)28 b FA(is)g Fs(c)l(on\015uent)g(mo)l(dulo)j Fv(\030)c FA(\(CON)p Fv(\030)p FA(\))g(if)2040 4005 y Fp(\003)2074 4036 y Fv( )46 b(\001)g(!)2355 4005 y Fp(\003)2472 4036 y Fv(\022)k(#)2629 4048 y Fp(\030)2684 4036 y FA(.)558 4135 y(5.)41 b Fv(A)28 b FA(is)g Fs(lo)l(c)l(al)t(ly)j(c)l(on\015uent)e(mo)l(dulo)h Fv(\030)d FA(\(LCON)p Fv(\030)p FA(\))h(if)79 b Fv( )23 b(\001)g(!)74 b(\022)50 b(#)2820 4147 y Fp(\030)2876 4135 y FA(.)558 4235 y(6.)41 b Fv(A)28 b FA(is)g Fs(str)l(ongly)f FA(LCON)p Fv(\030)g FA(\(SLCON)p Fv(\030)p FA(\))h(if)79 b Fv( )23 b(\001)g(!)74 b(\022)22 b(!)2529 4205 y Fw(=)2608 4235 y Fv(\001)h(\030)f(\001)2806 4205 y Fp(\003)2840 4235 y Fv( )27 b FA(.)558 4335 y(7.)41 b Fv(A)28 b FA(is)g Fs(c)l(oher)l(ent)i(with)g Fv(`)-28 b(a)27 b FA(\(COH)p Fv(`)-28 b(a)p FA(\))28 b(if)56 b Fv(`)-28 b(a)46 b(\001)g(!)2182 4304 y Fp(\003)2271 4335 y Fv(\022)k(#)2428 4347 y Fp(\030)2484 4335 y FA(.)558 4434 y(8.)41 b Fv(A)28 b FA(is)g Fs(lo)l(c)l(al)t(ly)j (c)l(oher)l(ent)f(with)g Fv(`)-28 b(a)28 b FA(\(LCOH)p Fv(`)-28 b(a)p FA(\))28 b(if)55 b Fv(`)-28 b(a)28 b(\001)23 b(!)74 b(\022)50 b(#)2676 4446 y Fp(\030)2732 4434 y FA(.)558 4534 y(9.)41 b Fv(A)28 b FA(is)g Fs(str)l(ongly)h(c)l(oher)l (ent)h(with)h Fv(`)-28 b(a)27 b FA(\(SCOH)p Fv(`)-28 b(a)p FA(\))28 b(if)56 b Fv(`)-28 b(a)27 b(\001)d(!)2498 4504 y Fp(\003)2559 4534 y Fv(\001)51 b(\030)73 b(\022)50 b(#)2928 4546 y Fp(\030)2984 4534 y FA(.)517 4633 y(10.)40 b Fv(A)28 b FA(is)g Fs(c)l(omp)l(atible)j(with)f Fv(`)-28 b(a)28 b FA(\(COM)p Fv(`)-28 b(a)o FA(\))28 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