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y(client-serv)o(er)h(interaction:)25 b(when)20 b(a)i(client)f(requests)f(some)h(service,)f(s/he)h(may)g(naturally)e (wish)i(the)450 4649 y(computation)f(on)i(the)g(serv)o(er')-5 b(s)22 b(side)g(to)g(terminate)g(and)f(return)g(an)h(answer)-5 b(.)32 b(SN)22 b(is)i(thus)e(a)g(basic)h(re-)450 4757 y(quirement)15 b(for)m(,)h(say)-5 b(,)16 b(interaction)f(between)h (banks)g(and)g(their)g(customers.)23 b(As)17 b(another)e(e)o(xample,)g (the)450 4865 y(resource)21 b(preserv)n(ation)g(guaranteed)g(by)h(SN)h (has)g(been)f(one)g(of)h(the)f(main)h(reasons)f(for)g(Gunter)g(and)450 4973 y(his)i(colleagues)e(to)h(de)n(v)o(elop)f(their)h(typed)f (programming)e(language)h(for)i(acti)n(v)o(e)f(netw)o(orks)h(\(PLAN\)) 1928 5074 y FY(1)p eop %%Page: 2 2 2 1 bop 450 -257 a FX(2)971 b FW(Y)n(OSHID)m(A,)17 b(BERGER)f(AND)h (HOND)m(A)450 -4 y Ge([26,)12 b(56])26 b(on)g(the)g(basis)h(of)f(a)h (simply)f(typed)f F1(l)p Ge(-calculus.)42 b(Such)26 b(languages)f(w)o (ould)h(in)g(general)g(re-)450 104 y(quire)c(primiti)n(v)o(es)f(for)h (communication)e(and)i(concurrenc)o(y)-5 b(.)28 b(This)23 b(suggests)f(a)h(systematic)g(ef)n(fort)e(to)450 212 y(e)o(xtend)k(the)i(accumulated)e(theories)h(of)h(functional)e(SN)i (types)f(to)h(the)g(realm)f(of)h(interacti)n(vity)e(is)j(a)450 320 y(w)o(orthwhile)19 b(endea)n(v)n(our)-5 b(.)533 428 y(W)e(e)27 b(are)f(thus)g(moti)n(v)n(ated)e(to)i(reposition)f(and)g (study)h(strong)f(normalisability)f(in)i(the)g(conte)o(xt)f(of)450 535 y(process)20 b(theory)-5 b(.)23 b(In)d(particular)m(,)f(is)i(there) f(a)g(basic)h(typed)e(process)h(calculus)g(in)g(which)g(strongly)f(nor) n(-)450 643 y(malising)f(functional)f(calculi)h(are)h(f)o(aithfully)e (embeddable?)22 b(By)d(f)o(aithful,)f(we)h(mean)f(that)g(typability)450 751 y(of)j(the)g(encoding)d(automatically)i(ensures)g(strong)g (normalisability)g(of)g(the)h(source)f(calculus.)27 b(More)450 859 y(ambitiously)-5 b(,)22 b(can)h(we)g(obtain)f(e)o(xact)h(semantic)g 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b(strongly)f(normalising)g F1(l)p Ge(-calculi)i(are)f(fully)h (abstractly)f(embeddable.)28 b(The)22 b(type)f(disci-)450 1831 y(pline)c(simply)f(adds)h(causal)g(chains)f(to)h(the)g(system)g (introduced)e(in)i([11])f(where)g(we)i(established)e(fully)450 1939 y(abstract)24 b(encodings)f(of)i(PCF/PCFv)h([25].)37 b(This)25 b(small)g(addition)e(radically)h(changes)f(the)i(class)h(of) 450 2046 y(typable)19 b(process)g(beha)n(viour)m(,)e(turning)h (possibly)h(di)n(v)o(er)o(ging)d(computation)h(into)j(a)g(strongly)e (normal-)450 2154 y(ising)i(one.)25 b(As)20 b(w)o(ould)g(be)g(imagined) f(by)g(the)h(embeddability)e(of)i(typed)f F1(l)p Ge(-calculi,)g(the)h (proof)f(of)h(SN)450 2262 y(is)f(non-tri)n(vial,)d(defying)g(nai)n(v)o (e)h(structural)g(induction.)22 b(W)-7 b(e)19 b(adapt)e(methods)g(de)n (v)o(eloped)e(for)j(strongly)450 2370 y(normalising)27 b F1(l)p Ge(-calculi)i([8,)13 b(24,)f(63],)31 b(combined)c(with)i (process-algebraic)e(reasoning)h(techniques)450 2478 y([11,)12 b(53,)h(55,)f(59,)h(68].)36 b(As)25 b(f)o(ar)f(as)h(we)f(kno) n(w)-5 b(,)24 b(this)g(is)h(the)f(\002rst)h(time)g(a)f(compositional)e (principle)h(for)450 2586 y(ensuring)18 b(SN)i(has)g(been)f (established)g(for)g(name)f(passing)i(processes)f(with)h(non-tri)n (vial)d(use)j(of)f(repli-)450 2694 y(cation.)41 b(The)26 b(proof)e(method)g(for)i(SN)g(is)h(applicable)d(to)i(signi\002cant)g(e) o(xtensions)e(of)i(the)g(presented)450 2802 y(formalism,)f(including)e (state)j(and)f(polymorphism)d([12,)12 b(33,)h(36,)f(69,)h(70].)39 b(Further)24 b(discussions)h(on)450 2910 y(these)g(e)o(xtensions)f(are) g(found)f(in)i(Section)g(7.)38 b(In)25 b(the)g(follo)n(wing,)f(we)h (outline)f(k)o(e)o(y)g(technical)g(ideas)450 3018 y(and)c(relate)g(our) f(w)o(ork)h(to)g(the)g(e)o(xisting)g(literature.)450 3176 y F2(The)h F1(p)p F2(-Calculus.)82 b Ge(F)o(ollo)n(wing)18 b([11],)g(we)i(use)g(an)f(asynchronous)e(v)n(ariant)h(of)i(the)f F1(p)p Ge(-calculus)g([32].)450 3284 y(Computation)f(in)j(this)f 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y(the)i(follo)n(wing)d(is)k(a)f(client)f(which)g(requests,)h(via)f F0(a)p Ge(,)h(to)g(ha)n(v)o(e)e(a)i(v)n(alue)f(returned)f(via)i(a)f (pri)n(v)n(ate)g(name)p eop %%Page: 3 3 3 2 bop 1171 -257 a FW(STR)m(ONG)17 b(NORMALISA)-7 b(TION)16 b(IN)g(THE)g FP(p)p FW(-CALCULUS)722 b FX(3)450 -4 y F0(c)p 1762 109 42 4 v 1762 155 a(a)p FX(\()p F0(c)p FX(\))9 b F0(c)p FX(\()p F0(y)p FX(\))p FZ(:)p F0(P)450 314 y Ge(where)p 674 268 V 20 w F0(a)p FX(\()p F0(c)p FX(\))g F0(c)p FX(\()p F0(y)p FX(\))p FZ(:)p F0(P)22 b Ge(stands)f(for)g FX(\()p F1(n)p F0(c)p FX(\)\()p 1584 268 V F0(a)p FV(h)p F0(c)p FV(i)12 b(j)g F0(c)p FX(\()p F0(y)p FX(\))p FZ(:)p F0(P)p FX(\))22 b Ge(with)f FX(\()p F1(n)p F0(c)p FX(\))h Ge(being)e(a)h(restriction)g(operator)-5 b(.)26 b(Us-)450 422 y(ing)19 b(these)g(agents,)g F0(R)h Ge(belo)n(w)f(is)h(a)g(simple)f(w)o(ay)g(to)h(represent)e(what)h(may)g (be)g(re)o(garded)e(as)j(a)g(denial)f(of)450 530 y(service)h(at)h F0(c)p Ge(.)1541 689 y F0(R)1610 642 y FQ(def)1615 689 y FX(=)i 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FX(\()p F0(x)p FX(\))p FZ(:)p FR(Fw)o FV(h)p F0(bx)p FV(i)12 b(j)p 1936 1442 V 12 w F0(a)o FX(\()p F0(c)p FX(\))p FR(Fw)o FV(h)p F0(cb)p FV(i)g(j)p 2355 1421 V 12 w F0(b)450 1696 y Ge(After)20 b(one)g(step)g(reduction)e(via) j F0(a)p Ge(,)f(we)g(obtain)1648 1904 y FR(Fw)o FV(h)p F0(bc)p FV(i)12 b(j)g FR(Fw)n FV(h)p F0(cb)p FV(i)g(j)p 2201 1837 V 12 w F0(b)450 2112 y Ge(which)20 b(e)o(xhibits)f(di)n(v)o (er)o(gence.)450 2270 y F2(T)-6 b(ype)20 b(Discipline)h(f)n(or)e(SN.)83 b Ge(The)20 b(type)f(discipline)g(of)h(this)g(paper)f(is)i(a)f(simple)g (re\002nement)e(of)i([11].)450 2378 y(Concretely)-5 b(,)18 b(the)j(system)f(is)h(based)f(on)g(tw)o(o)g(central)g(ideas:)533 2553 y(\(1\))46 b F0(Linear)26 b(types)g Ge([23,)12 b(43,)h(45,)f(68],) 27 b(which)e(ensure)g(that)h(a)h(channel)d(is)j(used)f(e)o(xactly)f (once)g(for)450 2661 y(input/output)19 b(and,)j(for)f(a)h(replicated)f (channel,)g(an)h(input)f(occurs)g(e)o(xactly)g(once)g(and)g(output)g (occurs)450 2769 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4515 y Gb(?)3342 4549 y Ge(\(1\))450 4757 y(This)23 b(type)f(means)h(that)g(a)g(unique)e (replicated)h(input)g(\(serv)o(er\))f(e)o(xists)i(at)h F0(a)p Ge(,)f(and)f F0(b)h Ge(is)h(a)f(channel)e(that)450 4865 y(is)28 b(used)f(for)f(service)h(requests)g(from)f(a)h(unique)f (replicated)g(serv)o(er)g(at)i F0(b)f Ge([10,)12 b(11,)h(28,)f(59])27 b(\(a)g(name)450 4973 y(of)c(type)f(with)h(mode)f(?)28 b(can)22 b(appear)g(man)o(y)g(times)h(in)g(the)g(process\).)32 b(When)23 b(composing)e(processes,)p eop %%Page: 4 4 4 3 bop 450 -257 a FX(4)971 b FW(Y)n(OSHID)m(A,)17 b(BERGER)f(AND)h (HOND)m(A)450 -4 y Ge(cut)24 b(elimination)e(occurs)h(between)f(input)h (and)g(output)f(on)h(a)h(shared)f(name)g(with)h(dual)f(types.)34 b(Here)450 104 y(!)26 b(and)21 b(?)26 b(are)21 b(dual)g(to)h(each)f (other)m(,)f(cf.)i([23],)e(so)i(that)g(cut)f(elimination)f(occurs)h (between)g FX(\(\))3125 74 y Gb(!)3176 104 y Ge(and)g FX(\(\))3382 74 y Gb(?)3418 104 y Ge(,)450 212 y(resulting)c(in)h FX(\(\))906 182 y Gb(!)954 212 y Ge(since)h(the)f(serv)o(er)f(can)h(al) o(w)o(ays)g(consume)f(a)i(client)f(request.)23 b(Thus)18 b(the)g(composition)450 320 y(of)i FR(Fw)o FV(h)p F0(ab)p FV(i)g Ge(and)g FR(Fw)o FV(h)p F0(bc)p FV(i)h Ge(is)g(typed)e(as:)1313 529 y FV(`)f FR(Fw)o FV(h)p F0(ab)p FV(i)12 b(j)g FR(Fw)n FV(h)p F0(bc)p FV(i)g FZ(.)g F0(a)d Ge(:)g FX(\(\))2106 494 y Gb(!)2134 529 y FZ(;)g F0(b)g Ge(:)g FX(\(\))2313 494 y Gb(!)2342 529 y FZ(;)g F0(c)g Ge(:)g FX(\(\))2516 494 y Gb(?)2553 529 y FZ(:)766 b Ge(\(2\))450 738 y(The)29 b(ideas)g(similar)g(to)g(the)f(re\002nement)g(abo)o(v)o(e)f(were)i (already)f(presented)f(in)i([10,)13 b(11,)f(28,)h(43,)f(59].)450 846 y(But)17 b(none)e(of)h(those)g(typing)f(disciplines)h(ensures)g (termination)f(of)h(processes.)23 b(In)16 b(f)o(act,)h(the)f(di)n(v)o (er)o(ging)450 954 y(process)k(in)g(\(1\))g(is)h(still)g(typable)e(as)i (follo)n(ws.)1435 1163 y FV(`)d FR(Fw)o FV(h)p F0(bc)p FV(i)12 b(j)g FR(Fw)n FV(h)p F0(cb)p FV(i)g FZ(.)g F0(b)d Ge(:)g FX(\(\))2223 1128 y Gb(!)2250 1163 y FZ(;)g F0(c)g Ge(:)g FX(\(\))2424 1128 y Gb(!)3342 1163 y Ge(\(3\))450 1372 y(In)18 b(the)h(light)f(of)g(such)g(e)o(xamples,)g(the)g(second)g (re\002nement)f(introduces)g(the)i(idea)f(to)h(record)e(causality)450 1480 y(of)j(beha)n(viour)e(in)i(types.)25 b(F)o(or)20 b(e)o(xample,)e FR(Fw)p FV(h)p F0(ab)p FV(i)i Ge(is)h(no)n(w)f(typed)f (as)i(follo)n(ws:)1521 1689 y FV(`)d FR(Fw)o FV(h)p F0(ab)p FV(i)12 b FZ(.)g F0(a)d Ge(:)g FX(\(\))2038 1654 y Gb(!)2084 1689 y FV(!)18 b F0(b)9 b Ge(:)g FX(\(\))2332 1654 y Gb(?)450 1898 y Ge(Here)26 b F0(a)12 b Ge(:)g FX(\(\))791 1867 y Gb(!)842 1898 y FV(!)21 b F0(b)12 b Ge(:)g FX(\(\))1099 1867 y Gb(?)1162 1898 y Ge(indicates)25 b(that)h(the)g(process)f (repeatedly)f(inputs)h(at)i F0(a)f Ge(and)f(then)g(outputs)g(at)450 2006 y F0(b)p Ge(.)51 b(Cut)30 b(elimination)e(no)n(w)g(occurs)g (between)h(dual)f(input)g(and)h(output)f(by)g(k)o(eeping)g(the)h (causality)450 2113 y(between)19 b(channels.)24 b(F)o(or)c(e)o(xample,) 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(typable)g(processes,)h(the)g(\002rst)h(idea)f(w)o(ould)450 3584 y(be,)e(in)h(the)f(light)h(of)f(the)g(pre)n(vious)f(e)o(xamples,)g (to)i(sho)n(w)f(that)g(reduction)f(steps)i(follo)n(w)e(a)i (non-circular)450 3692 y(ordering)j(on)i(free)h(channels.)37 b(F)o(or)24 b(e)o(xample,)g(the)g(reductions)f(of)p 2426 3645 42 4 v 24 w F0(a)p FV(j)p FR(Fw)o FV(h)p F0(ab)p FV(ij)p FR(Fw)o FV(h)p F0(bc)p FV(i)i Ge(proceed)d(at)k F0(a)p Ge(,)450 3799 y F0(b)d Ge(and)g F0(c)h Ge(in)f(this)h(order)m(,) e(and)h(e)n(v)o(entually)e(terminate.)34 b(But)24 b(reductions)d(of)p 2655 3753 V 23 w F0(a)p FV(j)p FR(Fw)o FV(h)p F0(ab)p FV(ij)p FR(Fw)o FV(h)p F0(ba)p FV(i)i Ge(repeat)450 3907 y(between)e F0(a)i Ge(and)f F0(b)g Ge(due)g(to)g(an)g(ob)o(vious)f (circularity)g(between)g F0(a)i Ge(and)e F0(b)p Ge(.)32 b(Ho)n(we)n(v)o(er)m(,)20 b(due)i(to)g(creation)450 4015 y(of)f(ne)n(w)f(links)h(and)g(replication)e(of)i(terms,)g(both)f(being) g(crucial)g(features)h(of)f F1(p)p Ge(-calculi,)g(considering)450 4123 y(only)f(simple)h(name)f(ordering)e(is)k(infeasible,)e(at)h(least) h(in)f(its)h(nai)n(v)o(e)e(form.)k(T)-7 b(o)20 b(see)h(this,)f (consider)e(the)450 4231 y(follo)n(wing)h(process)g(which)h(only)f (adds)h(one)g(name)f(restriction)h(\223)p 2362 4185 V F0(a)o FX(\()p F0(c)p FX(\))p Ge(\224)h(to)f(CCS)i(term:)1292 4440 y(!)t F0(a)p FX(\()p F0(x)p FX(\))p FZ(:)p FX(\()p 1522 4394 38 4 v F0(x)p FV(j)p 1582 4394 V F0(x)q FX(\))12 b FV(j)p 1699 4394 42 4 v 12 w F0(a)n FX(\()p F0(c)p FX(\))q FR(Fw)o FV(h)p F0(cb)p FV(i)g(j)g Ge(!)p F0(b)m FZ(:)p FX(\()p 2240 4394 V F0(a)m FV(h)p F0(d)t FV(ij)p 2412 4394 V F0(a)p FV(h)p F0(d)t FV(i)p FX(\))746 b Ge(\(4\))450 4649 y(\(this)19 b(process)f(is)h(typable)f(by)g F0(a)8 b Ge(:)g FX(\(\(\))1502 4619 y Gb(?)1537 4649 y FX(\))1569 4619 y Gb(!)1598 4649 y FZ(;)h F0(b)f Ge(:)g FX(\(\))1775 4619 y Gb(!)1820 4649 y FV(!)17 b F0(d)12 b Ge(:)c FX(\(\))2069 4619 y Gb(?)2123 4649 y Ge(as)19 b(we)g(shall)g(see)g(later\).)24 b(The)19 b(process)f(o)n(wns)450 4757 y(reductions)24 b(\002rst)j(at)f F0(a)p Ge(,)h(then)e(at)h F0(b)p Ge(,)h(then)e(at)h F0(a)g Ge(again.)41 b(Further)m(,)25 b(the)h(number)e(of)h(rede)o(x)o (es)g(increases)450 4865 y(e)o(xponentially)16 b(in)j(its)h(course,)e F0(b)n(ut)i Ge(the)f(computation)e(terminates.)24 b(Such)18 b(beha)n(viour)f(occurs)h(when)g(a)450 4973 y(process)g(requests)h(the) f(same)h(resource)f(more)f(than)i(once)f(in)g(an)h(interaction,)e(e.g.) i(in)f(an)h(encoding)e(of)p eop %%Page: 5 5 5 4 bop 1171 -257 a FW(STR)m(ONG)17 b(NORMALISA)-7 b(TION)16 b(IN)g(THE)g FP(p)p FW(-CALCULUS)722 b FX(5)450 -4 y Ge(the)20 b F1(l)p Ge(-term)f F1(l)p F0(xyz)p FZ(:)p FX(\(\()p F0(xz)p FX(\)\()p F0(yz)p FX(\)\))j Ge([51].)i(The)c(dif)n (\002culty)f(in)h(analysing)f(\(4\))h(can)g(be)g(seen)g(by)f (considering)450 104 y(the)h(follo)n(wing)f(subterm)g(of)h(a)h(one)e (step)i(descendant)d(of)i(\(4\).)1637 281 y FX(\()p F1(n)10 b F0(c)p FX(\)\()p 1823 235 37 4 v F0(c)i FV(j)p 1907 235 V 12 w F0(c)f FV(j)h FR(Fw)n FV(h)p F0(cb)p FV(i)p FX(\))450 459 y Ge(It)29 b(contains)e(a)i(chain)f F0(c)23 b FV(!)g F0(b)p Ge(,)30 b(which)e(is)h(dif)n(\002cult)f(to)h(determine) e(before)g F0(c)i Ge(is)g(passed.)49 b(But)29 b(if)g(we)450 567 y(nai)n(v)o(ely)18 b(represent)h(causality)h(incorporating)c(bound) i(names)i(in)g(\(4\),)f(there)g(is)i(a)f(circular)f(chain)g F0(a)f FV(!)450 675 y F0(c)i FV(!)f F0(b)g FV(!)h F0(a)p Ge(,)j(although)e(this)i(c)o(ycle)f(ne)n(v)o(er)f(arises)i(in)f(actual) h(interaction.)30 b(Ho)n(w)22 b(can)g(we)h(then)f(pro)o(v)o(e)450 782 y(termination?)h(Simple)c(structural)f(inductions)g(w)o(ould)g(not) h(be)g(usable)f(for)h(the)g(same)g(reason)f(the)o(y)h(do)450 890 y(not)h(w)o(ork)f(in)i(typed)e F1(l)p Ge(-calculi)g([8,)13 b(21].)533 998 y(The)25 b(idea)h(we)g(use)g(is)g(suggested)f(by)g(SN)h (proofs)f(for)g(typed)f F1(l)p Ge(-calculi,)i(due)f(to,)i(among)d (others,)450 1106 y(T)-7 b(ait)19 b([63].)k(His)c(method)e(emplo)o(ys)g (a)i(semantic)f(interpretation)e(of)i(each)g(type)f FX([)-9 b([)p F1(s)p FX(])g(])19 b Ge(as)g(a)f(collection)g(of)450 1214 y(strongly)i(normalising)g F1(l)p Ge(-terms,)h(and)g(sho)n(ws)g (that)h(all)g(typable)e(terms)i(are)f(indeed)g(in)g(these)h(sets.)30 b(A)450 1322 y(k)o(e)o(y)15 b(step)h(is)g(to)g(pro)o(v)o(e)d(that)j F1(l)p F0(x)e Ge(:)g F1(s)p FZ(:)p F0(M)j FV(2)e FX([)-9 b([)p F1(s)14 b FV(!)g F1(t)p FX(])-9 b(])16 b Ge(for)f(each)g F0(M)j Ge(:)c F1(t)i Ge(\(for)f(which)g(by)g(induction)e F0(M)18 b FV(2)c FX([)-9 b([)p F1(t)p FX(])g(])p Ge(\),)450 1430 y(which)20 b(means,)f(by)h(de\002nition,)f FX(\()p F1(l)p F0(x)p FZ(:)p F0(M)s FX(\))p F0(N)24 b FV(2)19 b FX([)-9 b([)p F1(t)p FX(])g(])20 b Ge(for)g(each)f F0(N)24 b FV(2)19 b FX([)-9 b([)p F1(s)p FX(])g(])p Ge(.)25 b(But)c(all)f(semantic)g(types)g(ha)n(v)o(e)450 1538 y(the)i(property)d(that)j F0(M)s FV(f)p F0(N)5 b FZ(=)p F0(x)p FV(g)19 b(2)g FX([)-9 b([)p F1(t)p FX(])g(])23 b Ge(and)e FX(\()p F1(l)p F0(x)p FZ(:)p F0(M)s FX(\))p F0(N)k FV(\000)-14 b(!)19 b F0(M)s FV(f)p F0(N)5 b FZ(=)p F0(x)p FV(g)22 b Ge(imply)f FX(\()p F1(l)p F0(x)p FZ(:)p F0(M)s FX(\))p F0(N)k FV(2)20 b FX([)-9 b([)p F1(t)p FX(])g(])p Ge(,)22 b(hence)450 1646 y(we)k(ha)n(v)o(e)e(only)h(to)g (sho)n(w)g F0(M)s FV(f)p F0(N)5 b FZ(=)p F0(x)p FV(g)21 b(2)g FX([)-9 b([)p F1(t)p FX(])g(])p Ge(.)41 b(T)-7 b(o)25 b(be)h(able)f(to)g(do)g(this)h(we)f(strengthen)f(the)h (induction)450 1754 y(hypothesis)f F0(M)h FV(2)d FX([)-9 b([)p F1(t)p FX(])g(])26 b Ge(to)g F0(M)f FV(2)d FX([)-9 b([)p F1(t)p FX(])g(])1510 1766 y Gc(r)1575 1754 y Ge(for)25 b(each)g(en)m(vironment)e F1(r)p Ge(,)k(mapping)d(each)i(v)n(ariable)e (of)i(type)450 1862 y F1(s)31 b Ge(to)f(some)g(term)g(in)g FX([)-9 b([)p F1(s)p FX(])g(])p Ge(.)55 b(No)n(w)30 b(the)g(result)h (is)g(immediate)e([8,)13 b(21].)54 b(While)30 b(we)h(cannot)e(use)h(an) 450 1970 y(identical)e(frame)n(w)o(ork)f(due)h(to)h(the)f(dif)n(ferent) f(nature)h(of)g(reduction)f(in)i(the)g F1(p)p Ge(-calculus,)g(a)g (similar)450 2078 y(technique)22 b(w)o(orks)h(\223for)f(the)h (induction)f(to)h(go)g(through\224.)32 b(A)24 b(k)o(e)o(y)f(observ)n (ation)e(concerns)h(the)h(close)450 2186 y(correspondence)17 b(between)j(the)g(substitution)g F0(M)s FV(f)p F0(N)5 b FZ(=)p F0(x)p FV(g)20 b Ge(and)g(the)h(consumption)d(of)i(a)h (message)p 3300 2140 38 4 v 20 w F0(x)q FV(h)p F0(v)p FV(i)450 2293 y Ge(by)i(a)h(replicated)e(process)h(!)p F0(x)p FX(\()p F0(y)p FX(\))p FZ(:)p F0(Q)p Ge(.)35 b(Thus,)23 b(at)h(each)f(induction)f(step,)i(we)g(pro)o(v)o(e)d(that)i F0(P)p FV(j)p FX(\()p F0(R)3171 2305 y Gb(1)3206 2293 y FV(j)p FZ(:::)p FV(j)p F0(R)3372 2305 y FT(n)3407 2293 y FX(\))450 2401 y Ge(con)m(v)o(er)o(ges)18 b(for)i(each)g(possible)h (\223en)m(vironment\224)c F0(R)1949 2413 y Gb(1)1984 2401 y FV(j)p FZ(:::)p FV(j)p F0(R)2150 2413 y FT(n)2206 2401 y Ge(which)j(complements)f F0(P)p Ge(.)27 b(The)21 b(semantic)450 2509 y(types)j(of)f(processes)h(are)g(formalised)e(via)i (type-directed)d(predicates)i(which)h(are)f(suggested)g(partly)450 2617 y(by)31 b(the)h(original)e(method)g(by)h(T)-7 b(ait)33 b([63])d(and)h(partly)g(by)g(the)h(duality-based)d(method)h(of)i([1,)12 b(23].)450 2725 y(T)-6 b(ermination)24 b(beha)n(viour)f(is)k(then)e (calculated)g(via)h(the)f(ne)n(w)h(reduction)d(relation)i(\(called)g(e) o(xtended)450 2833 y(reduction\),)17 b(which)h(is)i(suggested)d(by)i (strong)f(bisimilarity)g(and)g(replication)g(theorems)f([11,)c(53,)f (59].)450 2941 y(Finally)20 b(ac)o(yclicity)f(in)i(causality)f(yields)g (strong)f(normalisation.)450 3099 y F2(Summary)k(of)g(Contrib)n (utions.)82 b Ge(The)23 b(follo)n(wing)e(summaries)h(main)g(technical)h (contrib)n(utions)e(of)450 3207 y(the)f(present)g(paper)-5 b(.)24 b(\(4\))c(solv)o(es)g(an)g(open)f(problem)f(in)j([51])e(for)g (the)i(simple)f(type)f(hierarchy)-5 b(.)533 3351 y(\(1\))46 b(Introduction)17 b(of)k(a)g F1(p)p Ge(-calculus)e(with)i(the)g(linear) f(typing)f(in)i(which)f(where)g(strong)g(normalis-)450 3459 y(ability)g(is)h(ensured)e(by)h(typability)-5 b(.)533 3568 y(\(2\))46 b(Establishment)23 b(of)g(the)h(proof)e(methodology)f (for)i(strong)g(normalisability)f(of)i(typable)e(pro-)450 3676 y(cesses,)17 b(combining)c(ideas)i(from)f(traditional)h(SN)g (proofs)f(for)h(typed)f F1(l)p Ge(-calculi)g(with)i(process-theoretic) 450 3783 y(reasoning.)23 b(W)-7 b(e)21 b(also)e(sho)n(w)h(the)f(result) g(e)o(xtends)g(to)g(the)h(linear)f F1(p)p Ge(-calculus)f(with)i(free)f (name)g(passing)450 3891 y(via)h(encoding.)533 4000 y(\(3\))46 b(Establishment)29 b(of)h(the)h(\002nite)g(axiomatisation)d(of)j(the)f (weak)g(bisimilarity)g(in)g(linear)g(pro-)450 4108 y(cesses)25 b(as)f(a)g(consequence)d(of)i(strong)g(normalisability)-5 b(.)33 b(The)23 b(axiomatisation)f(yields)h(an)h(ef)n(fecti)n(v)o(e)450 4216 y(procedure)18 b(to)i(compute)f(equality)g(o)o(v)o(er)g(linear)h (processes)f(via)i(their)f(normal)f(forms.)533 4325 y(\(4\))46 b(Embedding,)35 b(using)f(Milner')-5 b(s)35 b(encoding)e([51],)k(of)d (the)h(simply)f(typed)g F1(l)p Ge(-calculus)f(with)450 4433 y(sums)28 b(and)f(products)f(\()p F1(l)1185 4445 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)1379 4433 y Ge(\))i(into)f(our)g (typed)g F1(p)p Ge(-calculus.)46 b(The)27 b(embedding)e(is)k(fully)e (abstract)450 4541 y(w)-5 b(.r)g(.t.)21 b(the)h(observ)n(ational)d (congruence)g(of)j F1(l)1769 4553 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)1963 4541 y Ge(,)h(justifying)d(all)i(commutati)n(v)o(e)e (con)m(v)o(ersions)f(and)450 4648 y F1(h)p Ge(-equations)g([6,)12 b(19,)h(20,)g(24],)19 b(automatically)g(leading)g(to)h(SN)h(in)f(the)g (source)g(calculus.)533 4757 y(\(5\))46 b(Establishment)21 b(of)g(a)i(basic)f(interaction-based)d(li)n(v)o(eness)j(property)d(in)j (linear)g(processes)f(via)450 4865 y(their)31 b(strong)g (normalisability)-5 b(,)32 b(bridging)d(the)i(traditional)g(notion)f (of)h(SN)h(and)f(one)g(of)g(the)g(basic)450 4973 y(properties)19 b(in)h(concurrent,)e(interacti)n(v)o(e)g(computation.)p eop %%Page: 6 6 6 5 bop 450 -257 a FX(6)971 b FW(Y)n(OSHID)m(A,)17 b(BERGER)f(AND)h (HOND)m(A)450 -4 y F2(Related)j(W)-6 b(ork.)82 b Ge(Strong)20 b(normalisation)f(for)h(typed)g F1(l)p Ge(-calculi)f(has)i(been)f (studied)g(e)o(xtensi)n(v)o(ely)f(in)450 104 y(the)i(past;)g(detailed)f (surv)o(e)o(ys)g(can)g(be)h(found)e(in)i([8,)13 b(21].)26 b(The)20 b(present)g(paper)g(sho)n(ws)h(that)f(traditional)450 212 y(methods)f(for)h(pro)o(ving)d(SN)k(can)f(be)g(adapted)f(to)i (interacting)e(processes.)533 323 y(Abramsk)o(y)26 b(e)o(xtends)h(the)g (Curry-Ho)n(w)o(ard)e(correspondence)g(to)j(linear)f(logic)g([23])g (using)g(proof)450 431 y(e)o(xpressions)16 b(\(which)g(are)g(proof)g (nets)h(in)g(term)f(form\),)g(and)h(pro)o(v)o(es)e(SN)i([1],)g(guiding) e(our)h(present)g(us-)450 539 y(age)j(of)f(ac)o(yclicity)g(in)h(names.) 24 b(This)19 b(programme)e(is)i(tak)o(en)g(further)e(with)i (realisability)g(semantics)g(of)450 647 y(linear)e(logic)f(in)i([5])e (where)g(CCS)j(processes)e(act)g(as)h(realisers,)g(using)e(renaming)f (operators)h(for)g(typed)450 754 y(process)21 b(composition)f([28].)28 b(The)22 b(operational)d(structure)i(of)h([5])f(follo)n(ws)g(his)h(o)n (wn)f F1(p)p Ge(-calculus)g(en-)450 862 y(coding)e(of)h(proof)g(nets)g ([2],)g(of)n(fering)f(a)i(process-algebraic)d(understanding)f(of)k (semantics)f(of)h(linear)450 970 y(logic.)28 b(The)20 b(appeal)h(of)g(realisability)g(lies)h(in)f(treating)f(semantics)i(and) e(syntax)h(uniformly)e(on)h(a)i(log-)450 1078 y(ical)27 b(basis.)44 b(In)26 b(the)g(conte)o(xt)f(of)h(SN)h(types)f(for)g(the)g F1(p)p Ge(-calculus,)h(sharing)e(of)h(names)g(and)g(dynamic)450 1186 y(link)g(creation)f(in)h(the)g F1(p)p Ge(-calculus)e(mak)o(e)i(it) h(dif)n(\002cult)e(to)h(directly)f(apply)g(the)h(frame)n(w)o(ork)e(in)i ([1,)13 b(5],)450 1294 y(especially)j(when)g(we)h(consider)f(imperati)n (v)o(e)f(e)o(xtensions.)22 b(In)17 b(comparison,)e(the)h(present)g(w)o (ork)g(of)n(fers)450 1402 y(a)27 b(basic)f(type)g(discipline)g(which)g (does)g(not)g(directly)f(correspond)f(to)j(kno)n(wn)e(logical)h (systems)g(b)n(ut)450 1510 y(which)f(is)i(based)e(on)h(the)g(standard)e (operators)h(of)g(processes)h(and)f(simple)h(operational)e(principles,) 450 1618 y(resulting)18 b(in)h(a)h(ne)n(w)f(ef)n(fecti)n(v)o(e)e (method)h(to)h(ensure)f(SN)i(for)e(name)h(passing)f(processes,)h(e)o (xtensible)f(to)450 1726 y(a)j(broad)d(class)k(of)d(interacti)n(v)o(e)g (beha)n(viours.)533 1837 y(As)26 b(our)f(initial)g(e)o(xample)f(of)h (serv)o(er)n(-client)f(interaction)g(suggests,)i(SN)g(in)f(processes)g (is)i(closely)450 1945 y(related)19 b(to)g(li)n(v)o(eness.)24 b(Y)-9 b(oshida)19 b([68])f(presents)g(a)i(typed)e F1(p)p Ge(-calculus)g(with)i(a)f(local)g(li)n(v)o(eness)g(property)-5 b(.)450 2053 y(K)m(obayashi)17 b(and)h(colleagues)f([39,)12 b(41\22643])k(propose)h(se)n(v)o(eral)h(typing)f(systems)h(which)g (ensure)f(dif)n(fer)n(-)450 2161 y(ent)k(forms)e(of)i(li)n(v)o(eness;)f (for)g(e)o(xample)f(in)i([42])e(time)i(quotas)f(are)g(assigned)g(to)h (communications)d(for)450 2269 y(this)g(purpose.)23 b(Sangior)o(gi)15 b([58])i(proposes)f(a)i(typing)e(system)i(to)g(guarantee)e(what)h(he)h (calls)g(recepti)n(v)o(e-)450 2377 y(ness,)k(which)f(means)h(that)g(an) f(appropriate)e(input)i(pre\002x)g(is)i(al)o(w)o(ays)f(a)n(v)n (ailable.)29 b(Unlik)o(e)21 b(the)h(present)450 2484 y(w)o(ork,)31 b(these)f(and)f(other)g(preceding)e(typing)i(systems)h (for)f F1(p)p Ge(-calculi)f([11,)13 b(27,)f(28,)h(57,)f(59])29 b(neither)450 2592 y(guarantee)18 b(SN)i(nor)f(the)h(associated)g(li)n (v)o(eness)f(properties)f(for)h(processes)h(in)m(v)n(olving)d(non-tri)n (vial)h(use)450 2700 y(of)k(replication.)29 b(As)23 b(a)g(result,)f (embeddability)e(of,)i(say)-5 b(,)22 b F1(l)2158 2712 y Gd(!)2246 2700 y Ge(in)g(these)h(systems)f(does)g(not)g(guarantee)450 2808 y(the)e(SN)h(of)f(the)g(source)g(calculus.)533 2919 y(Since)k(the)f(present)g(w)o(ork)g(w)o(as)h(reported)e(in)i([70],)f (Sangior)o(gi)e(has)j(proposed)d(an)j(alternati)n(v)o(e)e(ap-)450 3027 y(proach)c(to)n(w)o(ards)g(termination)f(of)i(interacti)n(v)o(e)f (processes)h([60].)k(He)d(e)o(xplicitly)e(adds)g(a)i(global)e(name)450 3135 y(ordering)j([60])h(to)h(ensure)g(strong)f(normalisation.)32 b(This)24 b(ordering)d(is)j(close)f(to)g(a)h(property)d(deri)n(v)o(ed) 450 3243 y(in)26 b(our)e(typing)g(system,)j(cf.)e(Proposition)f(2.1.)40 b(His)26 b(proofs)e(are)i(similar)f(to)h(ours)f(which)g(use)g(type-)450 3351 y(directed)d(predicates)h(for)g(termination.)32 b(Y)-8 b(et)24 b(his)g(types)f(do)g(not)g(seem)h(to)f(ensure)g(li)n(v)o (eness)g(at)h(linear)450 3459 y(channels.)g(A)c(fully)g(abstract)f (embedding)f(of)h(e)o(xisting)h(calculi)f(or)h(a)g(\002nite)h (axiomatisation)d(of)i(weak)450 3567 y(bisimilarity)h(is)h(not)f (reported)f(in)h([60].)28 b(On)21 b(the)h(other)e(hand,)h(his)g(system) h(can)f(type)g(processes)g(with)450 3675 y(\002rst-order)16 b(state.)24 b(Our)17 b(corresponding)c(result)k(is)h(discussed)f(in)g ([70,)f(Section)h(6];)h(see)f(also)h(Section)e(7)450 3783 y(and)h(the)g(ne)o(xt)g(paragraph.)22 b(Although)15 b(the)j(constructions)e(in)h([60])g(assume)g(a)h(global)f(name)g (ordering,)450 3891 y(the)i(present)g(paper)f(sho)n(ws)h(that)h(this)f (is)h(not)f(necessary:)24 b(name)19 b(orderings)e(can)i(be)h (speci\002ed)e(locally)-5 b(,)450 3999 y(composing)19 b(processes)h(one)g(by)g(one.)26 b(This)21 b(local)g(type)f(checking)f (mak)o(es)h(it)i(possible)e(to)h(add)f(richer)450 4106 y(type)g(structures)f(on)h(top)g(of)g(the)g(core)g(SN-calculus,)f(as)i (discussed)f(belo)n(w)-5 b(.)533 4218 y(A)18 b(basic)f(feature)g(of)g (our)g(approach)e(is)j(that)g(we)f(construct)g(an)g(inte)o(grated)f (calculus)h(combining)e(re-)450 4325 y(stricted)j(calculi)f(with)h (clear)f(beha)n(vioural)f(articulations)h(in)g(a)h(bottom-up)e(f)o (ashion,)h(cf.)g([10,)12 b(36].)24 b(F)o(or)450 4433 y(this)19 b(purpose,)e(this)i(paper)e(starts)j(from)d(in)m(v)o (estigations)f(of)j(one)f(of)g(the)g(most)h(restricti)n(v)o(e)f(beha)n (vioural)450 4541 y(properties,)28 b(namely)e(con\003uent,)i(strongly)e (normalising)g(name)h(passing.)47 b(This)28 b(approach)d(allo)n(ws)450 4649 y(generalisation:)d(starting)c(with)g(this)g(core)g(calculus)f (that)h(e)o(xactly)f(encapsulates)g(the)h(notion)f(of)h(types)450 4757 y(originally)d(found)g(in)i(terminating)e(functions)g(as)i(types)g (for)f(terminating)f(processes,)i(the)f(frame)n(w)o(ork)450 4865 y(opens)h(a)i(po)n(werful)e(proof)f(methodology)f(for)j(strong)f (normalisability)g(which)g(smoothly)g(e)o(xtends)g(to)450 4973 y(other)28 b(classes)h(of)f(beha)n(viours)f(such)h(as)h(stateful,) i(nondeterministic,)d(polymorphic)d(and)j(concur)n(-)p eop %%Page: 7 7 7 6 bop 1171 -257 a FW(STR)m(ONG)17 b(NORMALISA)-7 b(TION)16 b(IN)g(THE)g FP(p)p FW(-CALCULUS)722 b FX(7)450 -4 y Ge(rent)21 b(computation.)k(F)o(or)c(e)o(xample,)f([12])g(establishes)h (strong)g(normalisability)e(of)i(linear)g(processes)450 104 y(with)g(second-order)d(polymorphism)f(by)k(e)o(xtending)d(the)j (present)f(proof)g(method)f(with)i(reducibility)450 212 y(candidates)e(induced)g(by)h(double-ne)o(gation)15 b(closure,)20 b(cf.)g([23].)k(Similarly)-5 b(,)19 b(by)h(adding)e(recent)i(proof)450 320 y(techniques)26 b(for)g(termination)f(in)i(Classical)i(Logic)d ([44,)12 b(64],)28 b([69])e(obtains)g(strong-normalisation)450 428 y(of)c(\002rst-order)g(state,)h(non-determinism)d(and)i(concurrenc) o(y)-5 b(.)28 b(These)23 b(results)g(can)f(be)h(augmented)d(to)450 535 y(pro)o(ving)h(li)n(v)o(eness)h(in)h(the)g(presence)f(of)h (non-termination)c(and)k(non-determinism)d(by)i(mixing)g(type)450 643 y(structures)h([69].)33 b(This)23 b(incremental)f(nature)h(of)g (our)f(type)h(structure)g(leads)g(to)h(signi\002cant)e(applica-)450 751 y(tions)27 b(of)g(SN)h(to)g(the)f(semantics)g(of)g(processes.)46 b(F)o(or)27 b(e)o(xample,)h([71])e(reports)g(a)i(ne)n(w)f(bisimilarity) 450 859 y(method)d(associated)i(with)f(the)h(linear)f(type)g(structure) g(and)g(strong)g(normalisability)-5 b(,)25 b(and)g(presents)450 967 y(applications)17 b(to)i(the)g(semantics)f(of)g(secrec)o(y)g(in)h (programming)c(languages)i([17,)12 b(61,)h(62].)24 b(In)18 b(another)450 1075 y(paper)23 b([36],)h(we)h(adapt)f(these)g(results)h (in)f(a)h(practical)e(direction,)h(proposing)e(and)i(v)o(erifying)e(a)j (ne)n(w)450 1183 y(typing)e(systems)i(for)f(secure)g(programming)d (languages)i(based)h(on)g(linear/af)n(\002ne)f(typed)g F1(p)p Ge(-calculi,)450 1291 y(where)d(strong)f(normalisability)f(and)i (linearity)g(play)f(a)i(fundamental)d(role)h(in)i(the)f(analysis.)533 1399 y(One)e(aspect)g(of)g(our)f(type)h(structure,)f F0(input-output)f(modes)h Ge(\(cf.)24 b([4,)13 b(38,)g(52,)f(55]\),)17 b(has)i(an)f(incarna-)450 1507 y(tion)i(in)g(the)f(conte)o(xt)g(of)h (Linear)f(Logic,)f(yielding)h(a)h(v)n(ariant)f(called)h F0(P)-7 b(olarised)19 b(Linear)h(Lo)o(gic)g Ge(\(LLP\))450 1615 y([49,)12 b(50],)17 b(studied)g(by)h(Laurent.)k(Proof)17 b(nets)h(for)f(LLP)h(are)f(f)o(aithfully)g(embeddable)e(in)j(the)f (replicated)450 1723 y(fragment)g(of)i(the)g(present)f(calculus)g (\(i.e.)h(the)g(sub-calculus)e(which)i(only)f(uses)h(!)t(-?)k(types\).) h(Ac)o(yclic-)450 1831 y(ity)31 b(in)f(name)g(usage)g(in)g(the)h (presented)e(type)h(discipline)f(corresponds)f(to)j(the)f(so-called)g (Lafont-)450 1939 y(Danos-Re)o(gnier)19 b(condition)h(in)h(proof)f (nets.)29 b(These)21 b(connections)e(shed)j(light)f(on)g(the)g (constructions)450 2047 y(in)26 b(the)g(present)f(paper)g(from)g(a)h (logical)g(vie)n(wpoint.)41 b(A)26 b(k)o(e)o(y)f(dif)n(ference)f(is)j (that)f(the)g(constructions)450 2155 y(in)21 b(LLP)g(bear)f(logical)g (signi\002cance,)g(making)f(it)i(an)g(ef)n(fecti)n(v)o(e)e(medium)g(to) i(relate)g(computation)d(and)450 2263 y(proofs;)g(whereas)g(the)g (present)g(type)g(discipline)g(captures)g(SN)h(in)f(the)h(frame)n(w)o (ork)d(of)j(basic)f(process-)450 2371 y(theoretic)h(operators)f (\(parallel)i(composition,)d(hiding)i(and)g(pre\002x\).)24 b(This)c(process-based)f(approach)450 2479 y(leads)32 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(guarantees)g(uncomposability)d(of)k(linear)f(channels;)k(for)c(e)o (xample,)h(if)g F0(x)p FZ(:)p F2(0)g Ge(has)g(a)450 535 y FV(#)p Ge(-mode)17 b(and)p 862 489 V 18 w F0(x)j Ge(has)f(a)g FV(")p Ge(-mode,)f(then)g F0(x)p FZ(:)p F2(0)10 b FV(j)p 1707 489 V 10 w F0(x)20 b Ge(has)f FJ(l)p Ge(-mode)f(at)h F0(x)p Ge(.)25 b(The)19 b FJ(l)p Ge(-mode)f(at)h F0(x)h Ge(indicates)e(that)h(the)450 643 y(process)h F0(x)p FZ(:)p F2(0)11 b FV(j)p 868 597 V 12 w F0(x)20 b Ge(cannot)f(be)h (composed)f(with)h(an)o(y)g(process)f(that)i(has)f F0(x)h Ge(as)g(a)f(free)g(name.)533 751 y(W)-7 b(e)25 b(let)30 b F0(p)p FZ(;)9 b F0(q)p FZ(;)g(:)g(:)g(:)25 b Ge(range)e(o)o(v)o(er)f (action)i(modes.)35 b(If)29 b F0(p)21 b FV(6)p FX(=)f FJ(l)p Ge(,)25 b(we)f(write)p 2551 705 48 4 v 30 w F0(p)h Ge(for)e(the)h F0(dual)f Ge(of)29 b F0(p)p Ge(,)c(a)f(self-)450 859 y(in)m(v)o(erse)c(map)g(on)h(the)g(action)f(modes)h(such)f(that)p 1881 791 42 4 v 21 w FV(#)f FX(=)p FV(")h Ge(and)p 2211 794 32 4 v 21 w(!)j FX(=)18 b Ge(?)t(.)28 b(The)21 b(four)e(modes)i (correspond)d(to)450 967 y(!)478 979 y Gb(1)513 967 y FZ(;)9 b Ge(?)582 979 y Gb(1)616 967 y FZ(;)g Ge(!)676 979 y Gc(w)742 967 y Ge(and)18 b(?)918 979 y Gc(w)984 967 y Ge(introduced)e(in)j([11],)f(e)o(xcept)f(that)i(the)g(present)f (modes)g(indicate)h(true)f(linearity)g(for)450 1075 y(linear)i (channels)f(\(i.e.)h(input)f(and)h(output)f(interact)h(precisely)f (once\))h(rather)f(than)h(af)n(\002nity)f(\(i.e.)h(input)450 1183 y(and)g(output)f(interact)g(at)i(most)f(once\))f(and)h(lack)g(of)g (di)n(v)o(er)o(gence)d(for)j(replicated)f(channels.)450 1341 y F2(Channel)32 b(T)-6 b(ypes.)83 b Ge(Ne)o(xt)31 b(we)i(de\002ne)e F0(c)o(hannel)f(types)i Ge(by)f(the)h(follo)n(wing)e (grammar)-5 b(.)58 b(Belo)n(w)38 b F0(p)3420 1353 y FH(I)450 1449 y Ge(\(resp.)25 b F0(p)706 1461 y FH(O)741 1449 y Ge(\))c(denotes)e(input)h(\(resp.)f(output\))g(modes.)1019 1646 y F1(t)50 b Ge(::)p FX(=)g F1(t)1303 1658 y FH(I)1343 1646 y FV(j)21 b F1(t)1424 1658 y FH(O)1480 1646 y FV(j)g FJ(l)183 b F1(t)1781 1658 y FH(I)1850 1646 y Ge(::)p FX(=)50 b(\()2026 1642 y FZ(~)2043 1646 y F1(t)2080 1658 y FH(O)2116 1646 y FX(\))2153 1615 y FT(p)2185 1624 y FH(I)2390 1646 y F1(t)2427 1658 y FH(O)2513 1646 y Ge(::)p FX(=)f(\()2688 1642 y FZ(~)2705 1646 y F1(t)2742 1658 y FH(I)2762 1646 y FX(\))2799 1615 y FT(p)2831 1624 y FH(O)450 1840 y Ge(The)33 b(IO-alternation)d(constraint)i(\(names)h (used)f(for)h(input)f(carry)g(only)g(output)g(names)g(and)g(vice)450 1948 y(v)o(ersa\))h(comes)g(from)g(game)f(semantics)i([4,)13 b(35,)f(38].)65 b(This)34 b(condition)e(is)i(not)g(essential)g(for)f (SN)450 2056 y(b)n(ut)25 b(simpli\002es)g(presentation)f(and)g(proofs.) 38 b(F)o(or)25 b(characterising)e(sequential)h(interaction,)h(we)g(may) 450 2164 y(add)35 b(further)g(constraints)g(as)h(in)g([10];)43 b(we)36 b(do)g(not)f(do)h(so)g(here)f(since)i(the)e(proof)g(structure)g (of)450 2272 y(strong)26 b(normalisability)g(does)h(not)g(change)f(by)h (ha)n(ving)f(this)i(constraint.)45 b(Let)27 b FM(md)p FX(\()p F1(t)p FX(\))i Ge(be)e(the)g(out-)450 2380 y(ermost)d(mode)g (of)h F1(t)p Ge(;)j(for)c FJ(l)i Ge(we)f(set)g FM(md)p FX(\()p FJ(l)q FX(\))c(=)f FJ(l)q Ge(.)p 1927 2329 37 4 v 39 w F1(t)26 b Ge(is)f(de\002ned)f(by)g(dualising)g(the)h(modes)f (of)g(types:)p 450 2432 293 4 v 450 2504 a FX(\()p F1(t)518 2516 y Gb(1)554 2504 y FZ(::)p F1(t)636 2516 y FT(n)671 2504 y FX(\))708 2480 y FT(p)765 2457 y FQ(def)770 2504 y FX(=)j(\()p 894 2453 72 4 v F1(t)930 2516 y Gb(1)966 2504 y FZ(::)p 1012 2453 V F1(t)1048 2516 y FT(n)1083 2504 y FX(\))p 1115 2440 40 3 v 1120 2474 a FT(p)1155 2504 y Ge(,)i(and)p 1354 2438 38 4 v 28 w FJ(l)f Ge(is)h(unde\002ned.) 46 b(W)-7 b(e)28 b(de\002ne)g FV(\014)f Ge(as)i(the)f(least)g (commutati)n(v)o(e)e(partial)450 2612 y(operation)18 b(on)i(channel)f(types)h(such)g(that:)583 2875 y(\(1\))119 b F1(t)12 b FV(\014)p 925 2824 37 4 v 12 w F1(t)18 b FX(=)g FJ(l)100 b Ge(\()p FM(md)p FX(\()p F1(t)p FX(\))19 b(=)p FV(#)p Ge(\))264 b(\(2\))99 b F1(t)12 b FV(\014)g F1(t)18 b FX(=)g F1(t)j Ge(and)e F1(t)12 b FV(\014)p 2630 2824 V 12 w F1(t)18 b FX(=)p 2767 2824 V 18 w F1(t)100 b Ge(\()p FM(md)p FX(\()p F1(t)p FX(\))19 b(=)f Ge(?)t(\))450 3082 y(Intuiti)n(v)o(ely)-5 b(,)27 b(\(1\))g(says)h(that)g(once)f(we)g (compose)g(input-output)d(linear)k(channels,)g(the)f(channel)g(be-)450 3190 y(comes)h(uncomposable.)48 b(\(2\))28 b(says)h(that)g(a)g(serv)o (er)f(should)g(be)h(unique,)g(b)n(ut)f(an)h(arbitrary)e(number)450 3298 y(of)g(clients)h(can)f(request)f(interactions.)45 b(Note)27 b(that)h(other)e(compositions)g(are)h(unde\002ned.)44 b(F)o(or)27 b(e)o(x-)450 3405 y(ample,)f(!)p F0(x)p FZ(:)p F2(0)13 b FV(j)g Ge(!)p F0(x)p FZ(:)p F2(0)25 b Ge(is)i(ne)n(v)o(er)d (typable)g(because)h FX(\(\))1947 3375 y Gb(!)1990 3405 y FV(\014)13 b FX(\(\))2132 3375 y Gb(!)2187 3405 y Ge(is)26 b(unde\002ned,)e(while)p 2854 3359 38 4 v 26 w F0(x)13 b FV(j)p 2940 3359 V 13 w F0(x)27 b Ge(is)f(typable)f(by)450 3513 y F0(x)9 b Ge(:)g FX(\(\))592 3483 y Gb(?)628 3513 y Ge(,)21 b(and)e(!)p F0(x)p FZ(:)p F2(0)11 b FV(j)p 985 3467 V 11 w F0(x)h FV(j)p 1068 3467 V 11 w F0(x)21 b Ge(by)f F0(x)9 b Ge(:)g FX(\(\))1372 3483 y Gb(!)1401 3513 y Ge(.)26 b(This)20 b(partial)g(algebra)f(of)g(channel)g(types)h (ensures,)g(among)e(others,)450 3621 y(determinac)o(y)g(of)i (computation)e(in)i(typable)f(processes)h(by)g(controlling)e(their)i (composability)-5 b(.)450 3779 y F2(Action)19 b(T)-6 b(ypes)19 b(and)h(their)f(Algebra.)81 b Ge(Channel)19 b(types)f(are)h(assigned)g(to)g(free)f(names)h(of)f(a)i(process)450 3887 y(to)h(specify)f(possible)h(usage)g(of)g(names.)27 b(Action)20 b(types,)h(on)f(the)i(other)e(hand,)g(carry)g(causality)g (infor)n(-)450 3995 y(mation)e([68])g(and)h(witness)h(the)f(real)g (usage)g(of)g(channels.)k(W)-7 b(e)21 b(\002rst)e(de\002ne)g(action)g (types.)24 b(An)19 b F0(action)450 4103 y(type)p Ge(,)h(denoted)e F0(A)p FZ(;)9 b F0(B)p FZ(;)g(:)g(:)g(:)q Ge(,)20 b(is)h(a)g(\002nite)f (directed)g(graph)e(with)j(nodes)e(of)h(the)g(form)g F0(x)9 b Ge(:)g F1(t)p Ge(,)21 b(such)f(that:)533 4265 y FV(\017)33 b Ge(no)20 b(name)f(occurs)h(twice;)g(and)533 4379 y FV(\017)33 b Ge(edges)20 b(are)g(of)g(the)g(form)f F0(x)9 b Ge(:)g F1(t)19 b FV(!)g F0(y)9 b Ge(:)g F1(t)1683 4349 y Gd(0)1726 4379 y Ge(such)20 b(that)g(either)g(\(1\))f FM(md)p FX(\()p F1(t)p FX(\))g(=)p FV(#)h Ge(and)g FM(md)o FX(\()p F1(t)3033 4349 y Gd(0)3055 4379 y FX(\))f(=)p FV(")h Ge(or)450 4487 y(\(2\))g FM(md)o FX(\()p F1(t)p FX(\))g(=)e Ge(!)24 b(and)c FM(md)p FX(\()p F1(t)1239 4457 y Gd(0)1260 4487 y FX(\))f(=)f Ge(?)t(.)450 4649 y(W)-7 b(e)19 b(write)g F0(x)d FV(!)h F0(y)i Ge(if)f F0(x)8 b Ge(:)f F1(t)17 b FV(!)g F0(y)7 b Ge(:)g F1(t)1383 4619 y Gd(0)1424 4649 y Ge(for)17 b(some)h F1(t)i Ge(and)d F1(t)1963 4619 y Gd(0)1985 4649 y Ge(.)25 b(If)18 b F0(x)g Ge(occurs)g(in)g F0(A)h Ge(and)e(for)h(no)g F0(y)g Ge(we)h(ha)n(v)o(e)f F0(y)e FV(!)h F0(x)450 4757 y Ge(then)24 b(we)h(say)g F0(x)g Ge(is)h F0(active)e(in)h(A)p Ge(.)38 b FV(j)p F0(A)p FV(j)25 b Ge(\(resp.)f FM(fn)p FX(\()p F0(A)p FX(\))p Ge(,)i FM(activ)n(e)o FX(\()p F0(A)p FX(\))p Ge(,)g FM(md)p FX(\()p F0(A)p FX(\))p Ge(\))f(denotes)f(the)g(set)i(of) e(nodes)450 4865 y(\(resp.)j(names,)i(acti)n(v)o(e)e(names,)i(modes\))e (in)h F0(A)p Ge(.)47 b(W)-7 b(e)29 b(often)e(write)h F0(x)14 b Ge(:)f F1(t)23 b FV(2)g F0(A)28 b Ge(instead)f(of)g F0(x)14 b Ge(:)f F1(t)23 b FV(2)g(j)p F0(A)p FV(j)p Ge(,)450 4973 y(and)30 b(write)g F0(A)p FX(\()p F0(x)p FX(\))h Ge(for)f(the)g(channel)f(type)h(assigned)g(to)g F0(x)h Ge(in)g F0(A)p Ge(.)55 b F0(A)p FZ(=)-10 b(~)-32 b F0(x)31 b Ge(is)g(the)f(result)h(of)f(taking)f(of)n(f)p eop %%Page: 10 10 10 9 bop 450 -257 a FX(10)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)450 -4 y Ge(nodes)k(with)h(names)g(in)11 b FZ(~)-32 b F0(x)23 b Ge(from)e F0(A)p Ge(.)30 b F0(A)p FZ(;)9 b F0(B)22 b Ge(is)h(the)e(graph)g(union)f(of)i F0(A)g Ge(and)f F0(B)p Ge(,)h(with)g(the)g(condition)e(that)450 104 y FM(fn)o FX(\()p F0(A)p FX(\))10 b FV(\\)g FM(fn)o FX(\()p F0(B)p FX(\))17 b(=)986 102 y F1(/)977 104 y(0)p Ge(.)25 b F0(x)8 b Ge(:)f F1(t)17 b FV(!)f F0(A)j Ge(is)g(a)f(result)h (of)f(adding)e F0(x)8 b Ge(:)f F1(t)20 b Ge(to)e F0(A)g Ge(with)h(an)f(edge)f(from)h F0(x)8 b Ge(:)f F1(t)19 b Ge(to)f(all)h(of)f F0(A)p Ge(')-5 b(s)450 212 y(acti)n(v)o(e)20 b(nodes.)25 b(W)-7 b(e)21 b(assume)g(that)f(\223)p FV(!)p Ge(\224)h(is)g(stronger)e(than)h(\223)p FZ(;)p Ge(\224:)26 b(for)20 b(e)o(xample,)e F0(a)9 b Ge(:)g FX(\(\))2890 182 y Gd(#)2945 212 y FV(!)18 b F0(b)9 b Ge(:)g FX(\(\))3193 182 y Gd(")3229 212 y FZ(;)g F0(c)g Ge(:)g FX(\(\))3403 182 y Gd(")450 320 y Ge(means)20 b FX(\()p F0(a)9 b Ge(:)g FX(\(\))862 289 y Gd(#)916 320 y FV(!)18 b F0(b)9 b Ge(:)g FX(\(\))1164 289 y Gd(")1200 320 y FX(\))p FZ(;)g F0(c)g Ge(:)g FX(\(\))1406 289 y Gd(")1442 320 y Ge(.)533 428 y(It)16 b(is)g(sometimes)f(useful)f(to)i(write)f(do)n(wn)g(action)f (types)h(syntactically)-5 b(,)15 b(in)h(which)e(case)i(we)g(generate) 450 536 y(action)k(types)g(from)f(the)h(follo)n(wing)f(grammar:)1020 748 y F0(A)f Ge(::)p FX(=)1228 746 y F1(/)1219 748 y(0)41 b FV(j)g F0(a)9 b Ge(:)g F1(t)42 b FV(j)g F0(A)p FZ(;)9 b F0(B)41 b FV(j)g F0(a)9 b Ge(:)g F1(t)19 b FV(!)g FX(\()p F0(b)2145 760 y Gb(1)2189 748 y Ge(:)9 b F1(t)2257 760 y Gb(1)2292 748 y FZ(;)g F0(b)2366 760 y Gb(2)2410 748 y Ge(:)g F1(t)2478 760 y Gb(2)2514 748 y FZ(;)g(:::;)g F0(b)2689 760 y FT(n)2733 748 y Ge(:)g F1(t)2801 760 y FT(n)2836 748 y FX(\))450 961 y Ge(where)20 b(we)h(assume,)f(in)h F0(a)9 b Ge(:)g F1(t)20 b FV(!)f FX(\()p F0(b)1479 973 y Gb(1)1523 961 y Ge(:)9 b F1(t)1591 973 y Gb(1)1627 961 y FZ(;)g F0(b)1701 973 y Gb(2)1745 961 y Ge(:)g F1(t)1813 973 y Gb(2)1849 961 y FZ(;)g(:::;)g F0(b)2024 973 y FT(n)2068 961 y Ge(:)g F1(t)2136 973 y FT(n)2172 961 y FX(\))p Ge(,)21 b(that)g F1(t)g Ge(is)h(of)e(mode)g FV(#)h Ge(or)f(!)25 b(and,)20 b(accord-)450 1069 y(ingly)-5 b(,)17 b F1(t)691 1081 y FT(i)732 1069 y Ge(is)j(of)e(mode)g FV(")g Ge(or)g(?)24 b(with)18 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gr % arrowhead 3.750 slw n 8377 1973 m 8347 1853 l 8317 1973 l col0 s % Polyline 7.500 slw n 8174 1987 m 8241 1987 l gs col0 s gr % Polyline gs clippath 9285 1770 m 9345 1770 l 9345 1618 l 9315 1738 l 9285 1618 l cp 9345 1380 m 9285 1380 l 9285 1532 l 9315 1412 l 9345 1532 l cp eoclip n 9315 1395 m 9315 1755 l gs col0 s gr gr % arrowhead n 9345 1532 m 9315 1412 l 9285 1532 l col0 s % arrowhead n 9285 1618 m 9315 1738 l 9345 1618 l col0 s % Polyline n 1733 1080 m 1800 1080 l gs col0 s gr % Polyline n 4680 1125 m 4747 1125 l gs col0 s gr % Polyline n 1823 3465 m 1890 3465 l gs col0 s gr % Polyline n 1823 4320 m 1890 4320 l gs col0 s gr % Polyline n 1778 1935 m 1845 1935 l gs col0 s gr % Polyline n 4725 3510 m 4792 3510 l gs col0 s gr % Polyline n 8145 1395 m 8212 1395 l gs col0 s gr /Times-Roman ff 180.00 scf sf 3278 3991 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 4544 3668 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 4560 4523 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 225.00 scf sf 4658 4530 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Roman ff 300.00 scf sf 3105 4043 m gs 1 -1 sc (b) col0 sh gr /Times-Roman ff 300.00 scf sf 4380 3691 m gs 1 -1 sc (d) col0 sh gr /Times-Roman ff 300.00 scf sf 4402 4545 m gs 1 -1 sc (e) col0 sh gr /Times-Roman ff 180.00 scf sf 390 3924 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 1656 3601 m gs 1 -1 sc (:) col0 sh gr /Times-Roman ff 300.00 scf sf 1476 3631 m gs 1 -1 sc (b) col0 sh gr /Symbol ff 180.00 scf sf 1672 4456 m gs 1 -1 sc (:) col0 sh gr /Times-Roman ff 300.00 scf sf 1485 4479 m gs 1 -1 sc (c) col0 sh gr /Symbol ff 225.00 scf sf 457 3954 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 1731 3616 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 1747 4463 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 3367 4021 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Roman ff 180.00 scf sf 3196 1613 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 4462 1290 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 4478 2145 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 225.00 scf sf 4576 2152 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Roman ff 300.00 scf sf 4298 1313 m gs 1 -1 sc (d) col0 sh gr /Times-Roman ff 300.00 scf sf 4320 2167 m gs 1 -1 sc (e) col0 sh gr /Times-Roman ff 300.00 scf sf 113 1583 m gs 1 -1 sc (a) col0 sh gr /Times-Roman ff 180.00 scf sf 308 1546 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 1574 1223 m gs 1 -1 sc (:) col0 sh gr /Times-Roman ff 300.00 scf sf 1394 1253 m gs 1 -1 sc (b) col0 sh gr /Symbol ff 180.00 scf sf 1590 2078 m gs 1 -1 sc (:) col0 sh gr /Times-Roman ff 300.00 scf sf 1403 2101 m gs 1 -1 sc (c) col0 sh gr /Symbol ff 225.00 scf sf 375 1576 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 1649 1238 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 1665 2085 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 3285 1643 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 4567 1291 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Bold ff 225.00 scf sf 728 3847 m gs 1 -1 sc (!) col0 sh gr /Times-Bold ff 225.00 scf sf 3631 3945 m gs 1 -1 sc (!) col0 sh gr /Times-Bold ff 225.00 scf sf 1958 3502 m gs 1 -1 sc (?) col0 sh gr /Times-Bold ff 225.00 scf sf 1988 4372 m gs 1 -1 sc (?) col0 sh gr /Times-Bold ff 225.00 scf sf 4898 3540 m gs 1 -1 sc (?) col0 sh gr /Times-Bold ff 225.00 scf sf 4890 4425 m gs 1 -1 sc (?) col0 sh gr /Times-Roman ff 300.00 scf sf 3023 1665 m gs 1 -1 sc (b) col0 sh gr /Times-Roman ff 300.00 scf sf 195 3961 m gs 1 -1 sc (a) col0 sh gr /Times-Roman ff 300.00 scf sf 8984 1658 m gs 1 -1 sc (b) col0 sh gr /Symbol ff 180.00 scf sf 9157 1605 m gs 1 -1 sc (:) col0 sh gr /Times-Roman ff 300.00 scf sf 6496 1643 m gs 1 -1 sc (a) col0 sh gr /Times-Roman ff 300.00 scf sf 7762 1125 m gs 1 -1 sc (d) col0 sh gr /Times-Roman ff 300.00 scf sf 7770 1598 m gs 1 -1 sc (e) col0 sh gr /Times-Roman ff 300.00 scf sf 7777 2190 m gs 1 -1 sc (c) col0 sh gr /Symbol ff 180.00 scf sf 6659 1612 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 7957 1065 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 7964 1560 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 180.00 scf sf 7979 2152 m gs 1 -1 sc (:) col0 sh gr /Symbol ff 225.00 scf sf 6749 1612 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 8039 1087 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 8031 1575 m gs 1 -1 sc (\(t\)) col0 sh gr /Symbol ff 225.00 scf sf 8061 2160 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Bold ff 225.00 scf sf -540 1575 m gs 1 -1 sc (\(1\)) col0 sh gr /Times-Bold ff 225.00 scf sf -495 3915 m gs 1 -1 sc (\(2\)) col0 sh gr /Symbol ff 225.00 scf sf 4635 3645 m gs 1 -1 sc (\(t\)) col0 sh gr /Times-Roman ff 300.00 scf sf 8640 1665 m gs 1 -1 sc (,) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1426 1371 a FK(FIG.)15 b(2.)50 b FW(Composition)19 b(of)e(Action)h(T)-5 b(ypes)p 450 1479 V 1526 1731 a F3(2.3.)94 b(Linear)32 b(T)m(yping)450 1839 y Ge(W)-7 b(e)19 b(are)e(no)n(w)g(ready)g(to)g(present)g(the)h(typing)e(rules)i (for)f(strong)f(normalisability)-5 b(.)22 b(The)c(rules)f(are)h(gi)n(v) o(en)450 1947 y(in)23 b(Figure)g(3,)h(using)e(sequent)h(of)g(the)g (form)f FV(`)e F0(P)i FZ(.)f F0(A)p Ge(.)2029 1917 y Gb(1)2098 1947 y Ge(In)i(the)h(typing)e(rules)h(we)g(use)h(the)f(follo) n(wing)450 2055 y(notations:)533 2245 y FV(\017)33 b F0(A)651 2215 y FS(~)-23 b FT(y)p Gb(:)690 2212 y FS(~)703 2215 y Gc(t)755 2245 y Ge(is)21 b F0(A)f Ge(in)h(which)e(each)h F0(y)1421 2257 y FT(i)1451 2245 y Ge(:)9 b F1(t)1519 2257 y FT(i)1562 2245 y Ge(in)h FZ(~)-31 b F0(y)9 b Ge(:)1709 2241 y FZ(~)1726 2245 y F1(t)21 b Ge(occurs.)533 2377 y FV(\017)33 b F0(A)659 2347 y F9(-)o FT(x)736 2377 y Ge(is)21 b F0(A)f Ge(such)g(that)h F0(x)d FV(62)h FM(fn)o FX(\()p F0(A)p FX(\))p Ge(.)533 2510 y FV(\017)39 b F0(pA)20 b Ge(means)g F0(A)g Ge(such)g(that)h FM(md)o FX(\()p F0(A)p FX(\))e(=)24 b F0(p)p Ge(.)450 2700 y(W)-7 b(e)17 b(say)e F0(P)h(is)h(typable)d(under)h(A)p Ge(,)h(or)f F0(P)h(has)f(action)g(type)g(A)p Ge(,)i(if)f FV(`)d F0(P)k FZ(.)e F0(A)h Ge(is)h(deri)n(v)n(able.)k(Brief)16 b(illustration)450 2808 y(of)k(each)g(rule)g(in)g(Figure)f(3)i(follo)n(ws.)533 2998 y FM(\(Zero\))32 b Ge(starts)21 b(from)e(the)h(empty)f(action)h (type.)p 450 3198 299 4 v 516 3253 a FD(1)545 3277 y FW(W)-5 b(e)18 b(prefer)i(the)f(format)h FB(`)15 b FO(P)h FN(.)i FO(A)f FW(to)i FO(A)c FB(`)h FO(P)p FW(.)24 b(This)18 b(is)g(because)j FO(A)c FW(in)i FB(`)c FO(P)i FN(.)h FO(A)f FW(abstracts)k(the)e(beha)o(viour)h(of)f FO(P)e FW(rather)j(than)450 3356 y(its)d(en)m(vironment.)24 b(This)16 b(point)i(w)o(ould)g(be)g(elucidated)i(when)e(we)f(discuss)g (translation)j(of)d FP(l)p FW(-calculus)j(in)d(Section)i(5.)p 450 3565 2989 5 v 500 3884 a FM(\(Zero\))500 3992 y FV(\000)p 475 4053 274 5 v 500 4167 a(`)f F2(0)i FZ(.)p 699 4167 25 4 v 960 3770 a FM(\(P)m(ar\))960 3878 y FV(`)e F0(P)1072 3890 y FT(i)1114 3878 y FZ(.)i F0(A)1227 3890 y FT(i)1331 3878 y F4(\()p F7(i)c F4(=)8 b F9(1)p F6(;)g F9(2)p F4(\))960 3986 y F0(A)1011 3998 y Gb(1)1064 3986 y FV(\020)18 b F0(A)1198 3998 y Gb(2)p 935 4048 687 5 v 960 4161 a FV(`)p F0(P)1054 4173 y Gb(1)1088 4161 y FV(j)p F0(P)1154 4173 y Gb(2)1209 4161 y FZ(.)i F0(A)1322 4173 y Gb(1)1359 4161 y FV(\014)r F0(A)1477 4173 y Gb(2)1696 3770 y FM(\(Res\))1696 3878 y FV(`)e F0(P)j FZ(.)f F0(A)1950 3848 y FT(x)p Gb(:)p Gc(t)1696 3986 y FM(md)p FX(\()p F1(t)p FX(\))f FV(2)g(f)p FJ(l)p FZ(;)9 b Ge(!)t FV(g)p 1671 4048 615 5 v 1696 4161 a(`)18 b FX(\()p F1(n)10 b F0(x)f Ge(:)g F1(t)p FX(\))p F0(P)22 b FZ(.)e F0(A)p FZ(=)p F0(x)2607 3770 y FM(\(W)n(eak\))2607 3878 y FV(`)f F0(P)h FZ(.)g F0(A)2861 3848 y F9(-)p FT(x)2607 3986 y FM(md)p FX(\()p F1(t)p FX(\))g FV(2)e(f)p FJ(l)p FZ(;)9 b Ge(?)t FV(g)p 2583 4048 541 5 v 2607 4161 a(`)19 b F0(P)h FZ(.)g F0(A)9 b FZ(;)19 b F0(x)9 b Ge(:)g F1(t)500 4428 y FM(\(In)588 4398 y Gd(#)622 4428 y FM(\))500 4536 y FV(`)18 b F0(P)i FZ(.)10 b(~)-31 b F0(y)9 b Ge(:)764 4532 y FZ(~)781 4536 y F1(t)h FZ(;)f FV(")18 b F0(A)970 4506 y F9(-)p FT(x)1026 4536 y FZ(;)9 b Ge(?)t F0(B)1150 4506 y F9(-)p FT(x)p 475 4597 997 5 v 500 4711 a FV(`)18 b F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)9 b Ge(:)699 4707 y FZ(~)716 4711 y F1(t)q FX(\))p FZ(:)p F0(P)21 b FZ(.)f FX(\()p F0(x)9 b Ge(:)g FX(\()1067 4707 y FZ(~)1084 4711 y F1(t)r FX(\))1154 4681 y Gd(#)1189 4711 y FV(!)g F0(A)p FX(\))p FZ(;)g F0(B)1696 4428 y FM(\(In)1784 4398 y Gb(!)1812 4428 y FM(\))1696 4536 y FV(`)18 b F0(P)j FZ(.)9 b(~)-31 b F0(y)9 b Ge(:)1960 4532 y FZ(~)1977 4536 y F1(t)q FZ(;)g Ge(?)t F0(A)2138 4506 y F9(-)p FT(x)p 1671 4597 862 5 v 1696 4711 a FV(`)p Ge(!)g F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)9 b Ge(:)1914 4707 y FZ(~)1931 4711 y F1(t)q FX(\))p FZ(:)p F0(P)20 b FZ(.)h F0(x)9 b Ge(:)g FX(\()2250 4707 y FZ(~)2267 4711 y F1(t)q FX(\))2336 4681 y Gb(!)2365 4711 y FV(!)g F0(A)2607 4427 y FM(\(Out\))2607 4537 y FV(`)19 b F0(P)h FZ(.)g F0(A)2853 4507 y FS(~)-23 b FT(y)p Gb(:)2892 4504 y FS(~)2905 4507 y Gc(t)3020 4537 y F0(A)9 b FV(\020)g F0(x)g Ge(:)g FX(\()3247 4533 y FZ(~)3264 4537 y F1(t)q FX(\))3338 4507 y FT(p)3369 4516 y Fz(o)p 2583 4598 861 5 v 2607 4712 a FV(`)p 2677 4666 38 4 v 19 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)21 b FZ(.)f F0(A)p FZ(=)-10 b(~)-32 b F0(y)10 b FV(\014)i F0(x)d Ge(:)g FX(\()3259 4708 y FZ(~)3276 4712 y F1(t)q FX(\))3350 4682 y FT(p)3381 4691 y Fz(o)1547 4865 y FK(FIG.)15 b(3.)50 b FW(Linear)18 b(T)-5 b(yping)17 b(Rules)p 450 4973 2989 5 v eop %%Page: 12 12 12 11 bop 450 -257 a FX(12)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)533 -4 y FM(\(P)m(ar\))32 b Ge(uses)24 b FV(\020)f Ge(and)g FV(\014)g Ge(for)g(controlling)e (composition)h(\(which)g(in)i(ef)n(fect)e(ensures)h(both)g(determi-)450 104 y(nac)o(y)f(and)h(strong)f(normalisability\).)32 b(F)o(or)23 b(e)o(xample,)f(if)i F0(P)f Ge(has)h(type)e F0(x)11 b Ge(:)g FX(\(\))2649 74 y Gd(")2708 104 y Ge(and)23 b F0(Q)h Ge(has)f(type)g F0(x)11 b Ge(:)g FX(\(\))3383 74 y Gd(")3418 104 y Ge(,)450 212 y(then)20 b F0(P)11 b FV(j)h F0(Q)20 b Ge(is)h(not)f(typable)f(because)h FX(\(\))1607 182 y Gd(")1661 212 y FV(6\020)e FX(\(\))1808 182 y Gd(")1843 212 y Ge(.)533 331 y FM(\(Res\))33 b Ge(allo)n(ws)28 b(hiding)e(of)h(a)h(name)f(only)g(when)g(its)h(action)f (mode)g(is)h FJ(l)h Ge(or)e(!)32 b(\(which)27 b(intuiti)n(v)o(ely)450 439 y(says)22 b(that)f(channels)f(of)h(modes)f FV(")p Ge(,)h FV(#)g Ge(or)g(?)26 b(should)20 b(al)o(w)o(ays)h(be)g (compensated)e(by)i(their)g(duals)g(before)450 547 y(the)o(y)e(are)i (restricted\).)533 666 y FM(\(W)n(eak\))32 b Ge(weak)o(ens)e FJ(l)i Ge(and)f(?)36 b(since)31 b(we)h(allo)n(w)f(the)g(possibility)f (of)h(ha)n(ving)f(no)h(action)g(at)g(these)450 774 y(channels.)23 b(F)o(ormally)15 b(the)h(weak)o(ening)f(of)i(these)f(nodes)g(is)h (necessary)f(for)g(ha)n(ving)f(subject)h(reduction.)533 894 y FM(\(In)621 864 y Gd(#)655 894 y FM(\))33 b Ge(records)20 b(the)g(causality)h(from)e(linear)h(input)g(type)g F0(x)10 b Ge(:)f FX(\()2260 890 y FZ(~)2277 894 y F1(t)q FX(\))2346 864 y Gd(#)2402 894 y Ge(to)21 b(linear)f(output)g(types.)26 b(The)20 b(side)450 1002 y(condition)j F0(A)840 971 y Gd(\000)p FT(x)945 1002 y Ge(and)i F0(B)1142 971 y Gd(\000)p FT(x)1246 1002 y Ge(ensure)g(linearity)f(\(i.e.)h(unique)e (occurrence\))g(of)i F0(x)p Ge(.)40 b(F)o(or)25 b(IO-alternation,)450 1110 y(we)c(let)f(all)h(free)f(names)g(under)f(an)h(input)f(be)h (outputs)g([10,)12 b(11,)g(36,)h(71].)533 1229 y FM(\(In)621 1199 y Gb(!)649 1229 y FM(\))33 b Ge(records)e(the)g(causality)h(from)e (replicated)h(input)g(type)g(to)h(?)t(-types.)59 b(The)31 b(side)h(condition)450 1337 y F0(A)501 1307 y Gd(\000)p FT(x)607 1337 y Ge(is)26 b(required)f(to)h(ensure)f(ac)o(yclicity)-5 b(.)41 b(Of)26 b(course)f(we)h(cannot)f(allo)n(w)h FV(")p Ge(-types)f(in)h(the)g(body)-5 b(,)25 b(for)450 1445 y(otherwise)j(linearity)f(w)o(ould)h(be)g(lost.)50 b(F)o(or)28 b(e)o(xample,)h(if)g F0(z)g Ge(is)g(linear)f(channel,)h(then)f(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p FX(\()p 3262 1399 33 4 v F0(z)15 b FV(j)g F0(Q)p FX(\))450 1553 y Ge(should)k(be)h(untypable)e(because)i F0(z)h Ge(is)g(copied)e(at)i (each)f(interaction.)533 1672 y FM(\(Out\))32 b Ge(does)18 b(not)g(suppress)g(the)h(body)d(by)i(pre\002x)g(since)g(output)f(is)j (asynchronous.)h(Essentially)d(the)450 1780 y(rule)24 b(composes)f(the)h(output)e(pre\002x)i(and)f(the)h(body)f(in)h (parallel.)36 b(This)24 b(rule)g(can)f(be)h(understood)e(by)450 1888 y(translating)p 825 1842 38 4 v 23 w F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)25 b Ge(to)f FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p 1315 1842 V F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)p FV(i)13 b(j)g F0(P)p FX(\))p Ge(:)35 b(suppose)23 b F0(P)i Ge(has)g(a)f(type)g F0(A)p Ge(.)38 b(First)25 b(we)g(check)f F0(A)c FV(\020)h F0(x)12 b Ge(:)g FX(\()3260 1884 y FZ(~)3277 1888 y F1(t)p FX(\))3350 1858 y FT(p)3381 1867 y Fz(o)3418 1888 y Ge(,)450 1996 y(then)20 b(if)g(de\002ned,)f(we)i(hide)9 b FZ(~)-32 b F0(y)21 b Ge(from)e F0(A)12 b FV(\014)g F0(x)d Ge(:)g FX(\()1725 1992 y FZ(~)1742 1996 y F1(t)p FX(\))1815 1966 y FT(p)1846 1975 y Fz(o)1883 1996 y Ge(,)21 b(whence)p 2199 1950 V 19 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)21 b Ge(has)f(type)g F0(A)p FZ(=)-10 b(~)-32 b F0(y)11 b FV(\014)h F0(x)d Ge(:)g FX(\()3015 1992 y FZ(~)3032 1996 y F1(t)p FX(\))3105 1966 y FT(p)3136 1975 y Fz(o)3174 1996 y Ge(.)533 2223 y FG(Example)31 b(2.2.)533 2401 y FV(\017)i Ge(A)23 b F0(copy-cat)g Ge(copies)f(all)h(information) e(from)g(one)i(channel)e(to)i(another)e([4,)13 b(38],)22 b(tw)o(o)h(instances)450 2509 y(of)f(which)g(already)f(appeared)g(in)h (\247)g(2.1.)31 b(W)-7 b(e)24 b(sho)n(w)-5 b(,)22 b(step)g(by)g(step,)h (ho)n(w)f FX([)p F0(u)d FV(!)h F0(x)p FX(])2861 2479 y Gc(t)2915 2509 y Ge(with)i F1(t)e FX(=)g(\(\(\))3322 2479 y Gd(")3357 2509 y FX(\))3389 2479 y Gb(!)3418 2509 y Ge(,)450 2617 y(can)g(be)g(typed:)500 2818 y(1:)99 b FV(`)18 b F2(0)i FZ(.)867 2816 y F1(/)858 2818 y(0)500 2926 y Ge(2:)99 b FV(`)p 733 2880 42 4 v 18 w F0(a)20 b FZ(.)h F0(a)9 b Ge(:)g FX(\(\))1005 2896 y Gd(")500 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FZ(=)p F0(a)18 b FX(=)g F0(x)9 b Ge(:)p 2484 3199 V 9 w F1(t)q Ge(\))450 3483 y(In)19 b(this)g(deri)n(v)n (ation,)f(the)h(length)f(of)h(paths)g(in)g(action)f(types)h(does)g(not) g(e)o(xceed)f(1)h(e)n(v)o(en)f(when)h(the)g(term)450 3591 y(gets)j(bigger)f(and)g(bigger)g(in)h(size.)30 b(In)22 b(f)o(act,)g(all)h(paths)f(in)g(action)f(types)h(of)f(deri)n(v)n(able)g (sequents)g(ha)n(v)o(e)450 3699 y(length)e(0)i(or)e(1.)533 3818 y FV(\017)33 b Ge(First)21 b(we)f(ha)n(v)o(e:)25 b FV(`)18 b F0(a)p FZ(:)p FX(\()p 1267 3751 42 4 v F0(b)11 b FV(j)p 1355 3772 37 4 v 12 w F0(c)o FX(\))21 b FZ(.)g F0(a)9 b Ge(:)g FX(\(\))1654 3788 y Gd(#)1707 3818 y FV(!)19 b FX(\()p F0(b)9 b Ge(:)g FX(\(\))1988 3788 y Gd(")2023 3818 y FZ(;)g F0(c)g Ge(:)g FX(\(\))2197 3788 y Gd(")2233 3818 y FX(\))21 b Ge(and)f FV(`)e F0(b)m FZ(:)p 2558 3751 46 4 v F0(d)24 b FZ(.)c F0(b)9 b Ge(:)g FX(\(\))2833 3788 y Gd(#)2887 3818 y FV(!)18 b F0(d)c Ge(:)9 b FX(\(\))3140 3788 y Gd(")3175 3818 y Ge(.)25 b(Then)500 4023 y(1:)99 b 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FM(In)3136 4835 y Gb(!)3164 4865 y Ge(\))g(and)f(by)450 4973 y(de\002nition)e(of)h FV(\020)p Ge(,)g(respecti)n(v)o(ely)-5 b(.)p eop %%Page: 13 13 13 12 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(13)533 154 y F0(Remark.)140 b Ge(\(type)31 b(inference\))60 b(Basically)32 b(type)f(inference)e(is)j(as)g(dif)n(\002cult)e(as)i (its)g(functional)450 262 y(counterpart)23 b(\(i.e.)h(the)h(simply)g (typed)f F1(l)p Ge(-calculus\),)h(as)g(already)f(observ)o(ed)f(in)i ([66].)39 b(In)24 b(our)h(typing)450 369 y(system,)h(gi)n(v)o(en)e (modes)g(of)h(all)g(free)g(names)f(in)i(addition)d(to)i(type)g (annotation)e(on)h(bound)g(names)g(in)450 477 y F0(P)p Ge(,)f FV(`)d F0(P)h FZ(.)g F0(A)i Ge(w)o(ould)f(be)g(linearly)g (decidable.)31 b(W)m(ithout)22 b(information)e(of)i(modes)g(the)g (system)h(has)g(no)450 585 y(principal)c(type:)25 b(for)19 b(e)o(xample,)p 1397 539 38 4 v 19 w F0(x)i Ge(could)e(ha)n(v)o(e)h FX(\(\))1899 555 y Gd(")1955 585 y Ge(and)g FX(\(\))2160 555 y Gb(?)2196 585 y Ge(.)25 b(An)20 b(interesting)g(topic)f(is)j(to)e (in)m(v)o(estigate)450 693 y(tractable)g(\(partial\))f(type)g (inference)g(algorithm)g(for)g(our)h(typing)f(systems.)533 1009 y F0(Remark.)130 b Ge(In)21 b([10,)12 b(11,)h(33,)f(68])20 b(as)h(well)h(as)f(in)g(the)f(early)h(v)o(ersion)e(of)h(this)i(paper)d ([70],)h(we)h(used)450 1117 y(the)h(tw)o(o-sided)f(sequent)g F1(G)f FV(`)f F0(P)i FZ(.)g F0(A)h Ge(where)g F1(G)g Ge(is)h(a)f(standard)f(en)m(vironment)e(which)j(maps)f(channels)450 1225 y(to)28 b(pair)f(types)h(\(of)f(the)h(form)f FV(h)p F1(t)p FZ(;)p 1449 1174 37 4 v 9 w F1(t)q FV(i)p Ge(\))h(and)g F0(A)g Ge(records)e(the)i(action)f(modes)g(attached)g(to)h(names)g(and) 450 1333 y(causality)20 b(between)f(names.)25 b(F)o(or)20 b(e)o(xample,)e(the)i(cop)o(y)g(cat)g(in)h(Example)e(2.2)g(\(1\))h(is)h (typed)e(as)1333 1542 y F0(y)9 b Ge(:)g FV(h)p F1(t)p FZ(;)p 1530 1491 V 28 w F1(t)q FV(i)p FZ(;)g F0(x)g Ge(:)g FV(h)p F1(t)p FZ(;)p 1829 1491 V 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14 14 13 bop 450 -257 a FX(14)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)450 -4 y Ge(then)25 b(the)g(corresponding)d (maximal)i(non-c)o(ycles)g(do)g(not)h(o)o(v)o(erlap)f(in)h(names,)h (again)e(by)h(induction)450 104 y(on)d(the)g(typing)e(rules.)30 b(The)22 b(k)o(e)o(y)f(case)i(is)g FM(\(P)m(ar\))o Ge(,)g(the)f(only)f (rule)g(which)h(e)o(xtends)f(the)h(chain.)29 b(Assume)450 212 y F0(x)487 224 y Gb(1)536 212 y FV(!)15 b F0(x)671 224 y Gb(2)705 212 y FZ(;)k F0(x)784 224 y Gb(2)833 212 y FV(!)c F0(x)968 224 y Gb(3)1002 212 y FZ(;)9 b(:)g(:)g(:)h F0(x)1168 224 y FT(n)p Gd(\000)p Gb(1)1296 212 y FV(!)k F0(x)1430 224 y FT(n)1481 212 y Ge(in)j F0(A)8 b FV(])g F0(B)15 b Ge(and)g F0(x)1924 224 y Gb(1)1973 212 y FV(!)g F0(x)2108 224 y FT(n)2159 212 y Ge(in)h F0(A)8 b FV(\014)g F0(B)p Ge(.)22 b(By)16 b(inducti)n(v)o(e)f(hypothesis)f(there)450 320 y(are)20 b(the)g(corresponding)c(maximal)j(non-c)o(ycles.)k(In)c (them,)g(names)h(used)f(in)h(dif)n(ferent)e(non-c)o(ycles)g(in)450 428 y F0(A)e Ge(\(resp.)e(in)i F0(B)p Ge(\))f(ne)n(v)o(er)f(o)o(v)o (erlap)f(with)j(each)f(other)-5 b(.)23 b(Further)m(,)14 b(since)i(intermediate)e(names)h(in)g(these)h(non-)450 535 y(c)o(ycles)22 b(ha)n(v)o(e)g(either)g(mode)f(!)27 b(or)22 b(mode)f FJ(l)q Ge(,)i(these)g(names)f(do)g(not)g(o)o(v)o (erlap)e(between)h F0(A)i Ge(and)f F0(B)h Ge(either)-5 b(.)450 643 y(Thus)24 b(the)h(result)f(of)h(connecting)d(all)j(these)g (non-c)o(ycles)d(again)i(gi)n(v)o(es)g(a)h(non-c)o(ycle,)e(which)h (clearly)450 786 y(corresponds)18 b(to)i F0(x)994 798 y Gb(1)1047 786 y FV(!)f F0(x)1186 798 y FT(n)1221 786 y Ge(,)h(as)h(required.)p 1732 786 50 50 v 533 994 a F0(Remark.)119 b Ge(In)24 b(Proposition)e(2.1)i(\(2\),)g(the)g(notion)f (of)h(chain)f(does)h(not)f(include)g(the)h(case)h(where)450 1102 y(an)e(intermediate)f(channel)g(is)i(restricted)f(\(unlik)o(e)f ([1]\).)33 b(While)24 b(such)f(cases)h(can)f(be)g(included,)f(the)o(y) 450 1210 y(are)j(not)f(necessary)g(in)g(the)h(proof)e(of)h(strong)g (normalisability)f(gi)n(v)o(en)g(later)m(,)i(cf.)g(Lemma)f(3.3.)37 b(Also)450 1317 y(note)20 b(that)g(this)h(property)d(is)j(deri)n(v)o (ed)e F0(a)h(posteriori)h Ge(by)e(de\002ning)g(a)i(composition)d (operator)h(on)h(types,)450 1425 y(in)g(contrast)g(to)g([60])f(which)h (assumes)g(this)h(global)f(condition)e F0(a)i(priori)p Ge(.)450 1691 y(Ne)o(xt)g(we)g(list)i(basic)e(properties)f(of)g(the)h (reduction)f(relation)g(in)h(typed)f(processes.)25 b(In)20 b(\(3\))f(belo)n(w)h(and)450 1799 y(henceforth)e(we)i(use)h(the)f (follo)n(wing)e(notations.)533 1993 y FV(\017)33 b F0(P)18 b FV(+)g F0(Q)868 1946 y FQ(def)864 1993 y FV(,)57 b F0(P)19 b FV(\000)-15 b(!)1207 1961 y Gd(\003)1261 1993 y F0(Q)18 b FV(6\000)-14 b(!)p Ge(.)533 2127 y FV(\017)33 b F0(P)18 b FV(+)753 2080 y FQ(def)748 2127 y FV(,)40 b(9)p F0(Q)p FZ(:)p F0(P)18 b FV(+)g F0(Q)p Ge(.)25 b(Further)m(,)19 b F0(P)f FV(*)1691 2080 y FQ(def)1686 2127 y FV(,)58 b(8)p F0(n)17 b FV(2)i Fy(N)t FZ(:)26 b F0(P)18 b FV(\000)-14 b(!)2313 2097 y FT(n)2348 2127 y Ge(.)533 2261 y FV(\017)33 b FM(SN)p FX(\()p F0(P)p FX(\))889 2214 y FQ(def)884 2261 y FV(,)58 b(:)9 b F0(P)19 b FV(*)p Ge(.)533 2399 y FV(\017)33 b FM(CSN)o FX(\()p F0(P)p FX(\))943 2352 y FQ(def)938 2399 y FV(,)58 b FM(SN)p FX(\()p F0(P)p FX(\))12 b FV(^)g FX(\()p F0(P)18 b FV(+)g F0(Q)1607 2411 y Gb(1)p FS(;)p Gb(2)1710 2399 y FV(\))j F0(Q)1874 2411 y Gb(1)1927 2399 y FV(\021)d F0(Q)2070 2411 y Gb(2)2105 2399 y FX(\))p Ge(.)533 2580 y FG(Pr)n(oposition)31 b(2.2.)62 b FF(L)l(et)29 b FV(`)19 b F0(P)h FZ(.)g F0(A)p FF(.)533 2761 y Ge(\(1\))13 b(\(subject)19 b(reduction\))27 b FF(If)j F0(P)19 b FV(\000)-15 b(!)1588 2729 y Gd(\003)1642 2761 y F0(Q)29 b FF(then)h FV(`)18 b F0(Q)j FZ(.)f F0(A)p FF(.)533 2883 y Ge(\(2\))13 b(\(strong)20 b(con\003uence\))28 b FF(If)k F0(P)19 b FV(\000)-15 b(!)19 b F0(Q)1690 2895 y FT(i)1743 2883 y Ge(\()p F0(i)f FX(=)h Ge(1)p FZ(;)9 b Ge(2\))p FF(,)31 b(then)g(either)g F0(Q)2579 2895 y Gb(1)2633 2883 y FV(\021)19 b F0(Q)2777 2895 y 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b(only)f(when)g(a)h(replicated)f(input)f(is)j(shared)e (which)g(does)h(not)f(change)f(its)j(shape.)23 b(See)17 b(Appendix)450 3691 y(A.3.)25 b(\(3\))19 b(is)i(standard,)e(cf.)h([1,) 13 b(7,)g(37,)g(40],)19 b(all)i(using)e(Proposition)g(2.2)g(\(2\).)p 2724 3691 V 1218 3972 a F3(3.)95 b(STR)m(ONG)32 b(NORMALISA)-8 b(TION)450 4095 y Ge(This)20 b(section)g(pro)o(v)o(es)f(the)h(follo)n (wing)f(result.)533 4276 y FG(Theorem)31 b(3.1.)62 b Ge(\()p F2(main)20 b(theor)o(em,)f(str)o(ong)g(normalisation)p Ge(\))58 b FV(`)18 b F0(P)i FZ(.)h F0(A)46 b FV(\))g FM(CSN)o FX(\()p F0(P)p FX(\))p FF(.)553 4433 y Ge(A)20 b(fe)n(w)f(signi\002cant)g(consequences)f(of)h(the)h(theorem)e(will)i (be)f(discussed)g(in)h(Sections)f(4,)h(5,)f(6)h(and)450 4541 y(7.)49 b(In)28 b(the)g(proof,)g(we)h(\002rst)g(introduce)d(the)i F0(e)n(xtended)f(r)m(eduction)g(r)m(elation)g FV(7!)p Ge(,)k(which)c(eliminates)450 4649 y(all)e F0(cuts)g Ge(\(mutually)d(dual)i(channels\))f(in)i(a)f(typed)g(process.)37 b(Ne)o(xt)24 b(we)g(de\002ne)g F0(semantic)g(types)h FX([)-9 b([)p F0(A)p FX(])g(])p Ge(,)450 4757 y(which)18 b(are)g(sets)h(of)e(typed)h(terms)g(that)g(con)m(v)o(er)o(ge)d(when)i (composed)f(with)j(all)f(necessary)g(\223resources\224)450 4865 y(\(i.e.)24 b(complementary)d(processes\).)36 b(Finally)24 b(we)h(pro)o(v)o(e)d(that)i(each)g(typable)f(process)h(is)h(in)g(the)f (cor)n(-)450 4973 y(responding)c(semantic)i(type.)32 b(This)23 b(part)f(is)i(di)n(vided)d(into)h(tw)o(o)h(stages.)33 b(W)-7 b(e)23 b(start)h(with)e(sho)n(wing)g(all)p eop %%Page: 15 15 15 14 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(15)450 -4 y Ge(normal)16 b(forms)g(are)h(in)g(their)f(semantic)h(types.)24 b(Then)16 b(we)h(establish)g(that)g(each)g(typable)f(process)g(com-)450 104 y(bined)21 b(with)i(resources)f(al)o(w)o(ays)g(reaches)g(a)h (normal)e(form,)h(which)g(implies)g(strong)g(normalisability)450 212 y(of)30 b FV(\000)-14 b(!)p Ge(.)55 b(In)30 b(the)g(second)f(stage) 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Ge(in)f(a)h(gi)n(v)o(en)e(term.)533 2897 y FG(Pr)n(oposition)31 b(3.1.)92 b FF(L)l(et)29 b(al)t(l)i(pr)l(o)l(c)l (esses)f(b)l(e)g(typ)l(e)l(d)g(b)l(elow.)533 3057 y Ge(\(1\))13 b FF(If)30 b FV(`)18 b F0(P)i FZ(.)h F0(A)29 b FF(and)h F0(P)19 b FV(7!)f F0(P)1397 3027 y Gd(0)1448 3057 y FF(then)29 b FV(`)18 b F0(P)1752 3027 y Gd(0)1794 3057 y FZ(.)i F0(A)p FF(.)533 3170 y Ge(\(2\))13 b(\(CR\))59 b FF(If)30 b F0(P)18 b FV(7!)1108 3139 y Gd(\003)1162 3170 y F0(Q)1222 3182 y FT(i)1273 3170 y FF(then)29 b F0(Q)1517 3182 y FT(i)1557 3170 y FV(7!)1640 3139 y Gd(\003)1693 3170 y F0(R)h Ge(\()p F0(i)18 b FX(=)g Ge(1)p FZ(;)9 b Ge(2\).)533 3282 y(\(3\))k(\(determinac)o(y\))68 b FF(If)36 b F0(P)22 b FV(7!)g F0(P)1502 3252 y Gd(0)1559 3282 y FF(and)36 b FM(SN)1830 3294 y FT(e)1861 3282 y FX(\()p F0(P)1944 3252 y Gd(0)1965 3282 y FX(\))g FF(then)g FM(SN)2328 3294 y FT(e)2359 3282 y FX(\()p F0(P)p FX(\))p FF(.)58 b(Thus)36 b F0(P)22 b FV(+)2897 3294 y FT(e)2964 3282 y FF(i\013)36 b FM(SN)3180 3294 y FT(e)3211 3282 y FX(\()p F0(P)p FX(\))g FF(i\013)450 3390 y FM(CSN)608 3402 y FT(e)639 3390 y FX(\()p F0(P)p FX(\))p FF(.)533 3503 y Ge(\(4\))13 b(\(con)m(v)o(er)o(gence\))66 b(\(i\))35 b F0(P)13 b FV(j)h F0(Q)22 b FV(+)1532 3515 y FT(e)1599 3503 y FF(implies)37 b F0(P)21 b FV(+)2012 3515 y FT(e)2079 3503 y FF(and)37 b F0(Q)21 b FV(+)2379 3515 y FT(e)2410 3503 y FF(,)38 b Ge(\(ii\))d FF(if)i F0(P)21 b FV(+)2820 3515 y FT(e)2851 3503 y FF(,)38 b(then)e FX(\()p F1(n)9 b F0(x)p FX(\))p F0(P)22 b FV(+)3382 3515 y FT(e)3413 3503 y FF(,)450 3611 y(and)30 b Ge(\(iii\))f F0(P)19 b FV(+)886 3623 y FT(e)947 3611 y FF(i\013)30 b F0(a)p FX(\()-10 b FZ(~)-32 b F0(x)o FX(\))p FZ(:)p F0(P)19 b FV(+)1339 3623 y FT(e)1370 3611 y FF(,)30 b Ge(\(i)n(v\))f F0(P)18 b FV(+)1693 3623 y FT(e)1754 3611 y FF(i\013)30 b Ge(!)p F0(a)p FX(\()-10 b FZ(~)-32 b F0(x)o FX(\))p FZ(:)p F0(P)19 b FV(+)2174 3623 y FT(e)2205 3611 y FF(,)30 b(and)g Ge(\(v\))f F0(P)18 b FV(+)2668 3623 y FT(e)2729 3611 y FF(i\013)p 2835 3565 42 4 v 30 w F0(a)o FX(\()-10 b FZ(~)-32 b F0(x)q FX(\))p F0(P)18 b FV(+)3098 3623 y FT(e)3129 3611 y FF(.)533 3810 y F0(Pr)l(oof)o(.)65 b Ge(See)17 b(Appendix)d(B.1.)24 b(The)16 b(proof)f(of)h(Church-Rosser) f(proceeds)g(by)h(`postponing')d(appli-)450 3952 y(cations)20 b(of)g FV(7!)879 3964 y Fx(g)914 3952 y Ge(.)p 1018 3952 50 50 v 450 4110 a(Note)g(the)g(Church-Rosser)f(property)f(is)j(no)f (longer)f(one-step.)533 4218 y(Let)31 b(us)f(say)h(a)g(process)f F0(P)h Ge(is)g F0(prime)g(with)g(subject)f(x)p Ge(,)j(or)d(simply)g F0(prime)p Ge(,)j(if)e(either)f F0(P)h Ge(is)g(input)450 4325 y(with)g(subject)h F0(x)f Ge(or)g F0(P)25 b FV(\021)p 1231 4279 38 4 v 24 w F0(x)p FX(\()p F0(y)1337 4337 y Gb(1)1372 4325 y FZ(::)p F0(y)1455 4337 y FT(n)1490 4325 y FX(\))p F1(P)1586 4337 y FT(i)p Gd(2)p FT(I)1671 4325 y F0(P)1714 4337 y FT(i)1767 4325 y Ge(such)31 b(that)g(each)g F0(P)2334 4337 y FT(i)2387 4325 y Ge(is)h(prime)f(with)g(subject)g F0(y)3183 4337 y FT(i)3236 4325 y Ge(where)450 4433 y F1(P)514 4445 y FT(i)p Gd(2)p FT(I)599 4433 y F0(P)642 4445 y FT(i)694 4433 y Ge(denotes)e(the)h F0(n)p Ge(-ary)f(parallel)h (composition)e(of)i FV(f)p F0(P)2222 4445 y FT(i)2242 4433 y FV(g)2284 4445 y FT(i)p Gd(2)p FT(I)2400 4433 y Ge(\(if)g F0(I)e FX(=)2662 4431 y F1(/)2653 4433 y(0)i Ge(then)g F1(P)2963 4445 y FT(i)p Gd(2)p FT(I)3048 4433 y F0(P)3091 4445 y FT(i)3136 4433 y FX(=)23 b F2(0)p Ge(\).)55 b(In)450 4541 y(the)22 b(follo)n(wing)e(proofs)h(we)h(use)g (a)h(v)n(ariant)e(of)g(the)h(typing)f(rule)g(for)h(output)e(pre\002x)o (es)h(which)h(is)h(gi)n(v)o(en)450 4649 y(by)d(adding)f(the)h (condition)e(\223)p F0(P)g FV(\021)g F1(P)p F0(P)1553 4661 y FT(i)1595 4649 y Ge(with)i F0(P)1806 4661 y FT(i)1847 4649 y Ge(prime)g(with)g(subject)g F0(y)2523 4661 y FT(i)2544 4649 y Ge(\224)h(in)f(the)h(premise)e(of)h FM(\(Out\))g Ge(in)450 4757 y(Figure)d(3.)24 b(W)-7 b(e)19 b(call)f(this)g(system,)g F0(alternative)f(typing)g(system)p Ge(.)25 b(Note)17 b(that,)h(in)g(the)f(alternati)n(v)o(e)g(typing)450 4865 y(system,)j(we)g(can)f(assume)g(acti)n(v)o(e)g(names)h(under)e(an)h (output)f(pre\002x)h(are)h(bound)d(by)i(that)h(pre\002x.)k(W)m(ith)450 4973 y(the)c(same)h(proof)d(as)j(in)f(Appendix)f(D)h(of)g([11],)f(we)i (can)f(easily)g(check:)p eop %%Page: 16 16 16 15 bop 450 -257 a FX(16)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)533 -4 y FG(Pr)n(oposition)31 b(3.2.)59 b FF(If)24 b FV(`)15 b F0(P)i FZ(.)g F0(A)24 b FF(is)g(derivable)j(in)d(the)g(system)g(in)g(Figur)l(e)g(3,)i(then)e (for)g(some)450 104 y F0(P)493 116 y Gb(0)546 104 y FV(\021)18 b F0(P)29 b FF(we)h(have)h FV(`)18 b F0(P)1135 116 y Gb(0)1190 104 y FZ(.)j F0(A)29 b FF(in)h(the)g(alternative)g(typing)h (system.)561 262 y Ge(Proposition)26 b(3.2)h(says)h(that)g(we)f(can)h (assume,)h(without)e(loss)h(of)f(generality)-5 b(,)27 b(that)h(all)g(pre\002x)o(ed)450 369 y(processes)h(are)g(primes)f (whene)n(v)o(er)f(we)j(are)f(discussing)f(properties)g(in)m(v)n(ariant) f(under)h FV(\021)h Ge(\(such)g(as)450 477 y(strong)22 b(normalisability\).)31 b(F)o(or)23 b(this)g(reason)f(the)h(follo)n (wing)e(con)m(v)o(ention)f(does)j(not)g(lose)g(generality)450 585 y(in)d(our)g(technical)f(de)n(v)o(elopment.)533 742 y FG(Convention)31 b(3.1.)71 b FF(Her)l(e)l(after)40 b(in)f(this)h(se)l(ction)f(we)h(assume)f(al)t(l)h(typ)l(e)l(d)g(pr)l(o) l(c)l(esses)f(ar)l(e)450 850 y(derive)l(d)31 b(in)f(this)g(alternative) h(typing)f(system.)561 1008 y Ge(When)d(we)h(w)o(ork)f(in)g(the)h (alternati)n(v)o(e)e(typing)g(system,)j(we)f(restrict)g FV(\021)f Ge(so)h(that)g(it)g(is)g(generated)450 1116 y(without)19 b FM(\(S7\))h Ge(and)g FM(\(S9\))g Ge(in)g(Figure)g(1)g (\(for)f(ha)n(ving)g(closure)h(of)g(typability)f(under)g FV(\021)h Ge(and)g FV(\000)-15 b(!)p Ge(\).)533 1224 y(Among)31 b(others)g(the)h(alternati)n(v)o(e)f(typing)f(gi)n(v)o(es)i (a)g(simple)g(inducti)n(v)o(e)e(characterisation)g(of)i(e)o(x-)450 1332 y(tended)19 b(normal)g(forms)h(which)f(we)i(shall)f(use)h(in)f (the)g(proof.)k(Belo)n(w)c(and)g(henceforth)e(we)i(write)h FM(NF)3407 1344 y FT(e)450 1453 y Ge(for)f(the)g(set)h(of)f(e)o (xtended)e(normal)h(forms:)24 b FM(NF)1810 1465 y FT(e)1860 1406 y FQ(def)1865 1453 y FX(=)e FV(f)p F0(P)e FV(j)h(9)p F0(A)p FZ(:)28 b FV(`)18 b F0(P)i FZ(.)h F0(A)9 b Ge(and)f F0(P)18 b FV(67!g)p Ge(.)533 1610 y FG(Pr)n(oposition)31 b(3.3.)66 b FM(NF)1344 1622 y FT(e)1410 1610 y FF(c)l(oincides)36 b(with,)g(up)e(to)h FV(\021)p FF(,)g(the)f(set)g(of)i(the)e(pr)l(o)l(c) l(esses)h(induc-)450 1717 y(tively)c(gener)l(ate)l(d)f(by)g(the)g(fol)t (lowing)i(rules:)533 1874 y FV(\017)20 b F2(0)e FV(2)h FM(NF)829 1886 y FT(e)860 1874 y Ge(,)533 1986 y FV(\017)h Ge(If)g F0(P)f FV(2)f FM(NF)914 1998 y FT(e)966 1986 y Ge(then)i F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)18 b Ge(:)1279 1982 y FZ(~)1296 1986 y F1(t)p FX(\))p FZ(:)p F0(P)-11 b FZ(;)30 b Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)18 b Ge(:)1657 1982 y FZ(~)1674 1986 y F1(t)p FX(\))p FZ(:)p F0(P)-11 b FZ(;)p 1858 1940 38 4 v 30 w F0(x)p FX(\()h FZ(~)-32 b F0(y)19 b Ge(:)2008 1982 y FZ(~)2024 1986 y F1(t)q FX(\))p F0(P)f FV(2)h FM(NF)2335 1998 y FT(e)2367 1986 y Ge(.)533 2098 y FV(\017)h Ge(If)g F0(P)714 2110 y FT(i)753 2098 y FV(2)f FM(NF)927 2110 y FT(e)979 2098 y FX(\()p F0(i)g FV(2)f F0(I)23 b FV(6)p FX(=)1269 2096 y F1(/)1260 2098 y(0)p FX(\))p Ge(,)d F0(P)1418 2110 y FT(i)1459 2098 y Ge(is)i(prime,)d(and)g F0(P)1954 2110 y FT(i)1975 2098 y FV(j)p F0(P)2050 2110 y FT(j)2091 2098 y FV(67!)i Ge(\()p F0(i)d FV(6)p FX(=)30 b F0(j)r Ge(\))21 b(then)e F1(P)2660 2110 y FT(i)p Gd(2)p FT(I)2746 2098 y F0(P)2789 2110 y FT(i)2828 2098 y FV(2)g FM(NF)3001 2110 y FT(e)3033 2098 y Ge(.)533 2255 y FF(wher)l(e)30 b(we)h(implicitly)g(assume)f(typ)l(ability)i(in)d(e)l(ach)i(rule.)533 2401 y F0(Pr)l(oof)o(.)89 b Ge(First)29 b(the)f(set)g(generated)e(is)j (immediately)e(a)h(subset)g(of)f FM(NF)2636 2413 y FT(e)2696 2401 y Ge(by)g(de\002nition.)47 b(F)o(or)27 b(the)450 2509 y(re)n(v)o(erse)18 b(direction,)f(we)i(use)g(induction)e(on)i (tying)e(rules)i(of)g(the)f(\(alternati)n(v)o(e\))f(typing)h(system,)h (noting)450 2617 y(if)25 b F0(P)c FV(2)g FM(NF)774 2629 y FT(e)831 2617 y Ge(then)k(its)h(subterms)e(are)h(also)g(in)g FM(NF)1905 2629 y FT(e)1936 2617 y Ge(.)40 b(F)o(or)25 b FM(\(Res\))o Ge(,)i(assume)e FX(\()p F1(n)9 b F0(x)p FX(\))p F0(P)26 b Ge(is)g(deri)n(v)o(ed.)37 b(Since)450 2725 y F0(P)20 b FV(2)h FM(NF)697 2737 y FT(e)728 2725 y Ge(,)26 b(if)e F0(x)h Ge(has)f(mode)f FJ(l)q Ge(,)i(this)g F0(x)f Ge(is)h(the)g(result)f(of)f(weak)o(ening,)h(hence)f(the)h (hiding)f FX(\()p F1(n)10 b F0(x)p FX(\))25 b Ge(can)e(be)450 2833 y(tak)o(en)h(a)o(w)o(ay)h(by)f FV(\021)p Ge(.)38 b(Further)24 b F0(x)h Ge(cannot)f(ha)n(v)o(e)g(mode)f(!)29 b(since)c(if)g(so)g(it)g(w)o(ould)f(result)h(in)g(a)g FV(7!)p Ge(-rede)o(x)450 2941 y(\(of)f(rule)g FM(\(E3\))p Ge(\).)37 b(F)o(or)25 b FM(\(P)m(ar\))f Ge(assume)g F0(P)1594 2953 y Gb(1)1629 2941 y FV(j)p F0(P)1695 2953 y Gb(2)1754 2941 y Ge(is)i(deri)n(v)o(ed.)36 b(Since)24 b(each)h F0(P)2570 2953 y FT(i)2611 2941 y FV(2)c FM(NF)2787 2953 y FT(e)2818 2941 y Ge(,)26 b(by)e(induction)f(it)i(is)450 3049 y(\(up)d(to)g FV(\021)p Ge(\))g(deri)n(v)o(ed)f(from)g(one)h(of)g (the)h(three)f(rules)g(abo)o(v)o(e.)30 b(If)22 b(either)g(is)h(deri)n (v)o(ed)e(from)g(the)i(\002rst)g(one)450 3157 y(we)d(ha)n(v)o(e)g (nothing)e(to)i(pro)o(v)o(e.)j(If)d(not,)f(then)g(each)h(is)h(deri)n(v) o(ed)d(from)h(the)h(second)f(or)g(the)h(third)g(rule)f(\(up)450 3294 y(to)e FV(\021)p Ge(\),)h(and)e(the)o(y)h(do)f(not)h(share)g(a)h (complementary)c(channel)i(by)g F0(P)2386 3306 y Gb(1)2421 3294 y FV(j)p F0(P)2487 3306 y Gb(2)2537 3294 y FV(2)g FM(NF)2707 3306 y FT(e)2739 3294 y Ge(,)i(thus)f(as)h(required.)p 3389 3294 50 50 v 533 3452 a(This)30 b(proposition)e(says)i(that)g(a)h (process)e(is)i(in)f FM(NF)2075 3464 y FT(e)2137 3452 y Ge(if)n(f)g(either:)44 b(\(1\))30 b(it)g(is)h(inaction,)g(\(2\))e(it) i(is)g(a)450 3560 y(pre\002x)20 b(of)g(an)h(ENF)-7 b(,)21 b(or)g(\(3\))f F0(n)p Ge(-ary)f(parallel)i(composition)d(of)j(ENFs)g (without)f(complementary)e(input)450 3668 y(and)j(output.)28 b(Note)21 b(this)h(also)g(says)g(that)f(an)h(ENF)g(does)f(not)g(ha)n(v) o(e)g(substantial)g(hiding)g(\(i.e.)g(a)h(hiding)450 3775 y FX(\()p F1(n)10 b F0(x)p FX(\))p F0(P)20 b Ge(such)g(that)h F0(x)d FV(2)h FM(fn)o FX(\()p F0(P)p FX(\))p Ge(\).)1491 3979 y F3(3.2.)94 b(Seman)m(tic)31 b(T)m(yp)s(es)450 4087 y Ge(Semantic)f(types)g(are)g(pro)o(v)n(ably)e(strongly)h (normalising)f(typed)i(terms)g(of)g(some)g(kind.)55 b(W)-7 b(e)31 b(need)450 4194 y(some)20 b(preliminaries.)533 4365 y FV(\017)33 b Fw(cl)o FX(\()p F0(A)p FX(\))798 4318 y FQ(def)803 4365 y FX(=)23 b FV(f)p F0(x)970 4377 y FT(i)1000 4365 y Ge(:)p 1032 4314 58 4 v 9 w F1(t)1068 4377 y FT(i)1110 4365 y FV(j)e F0(x)1191 4377 y FT(i)1221 4365 y Ge(:)9 b F1(t)1289 4377 y FT(i)1330 4365 y FV(2)19 b F0(A)p FZ(;)29 b FM(md)p FX(\()p F1(t)1679 4377 y FT(i)1701 4365 y FX(\))19 b FV(2)f(f")p FZ(;)9 b Ge(?)t FV(gg)p Ge(.)533 4492 y FV(\017)33 b Ge(Let)20 b F0(A)e FV(\020)g F0(B)j Ge(and)e F0(A)12 b FV(\014)g F0(B)17 b FX(=)d F0(C)r FZ(;)o(~)-32 b F0(x)9 b Ge(:)g FJ(l)22 b Ge(where)d FJ(l)28 b FZ(=)-51 b FV(2)19 b FM(md)p FX(\()l F0(C)r FX(\))p Ge(.)26 b(Then)19 b F0(A)12 b FV(\001)g F0(B)2594 4445 y FQ(def)2599 4492 y FX(=)19 b F0(C)r Ge(.)450 4649 y(By)29 b Fw(cl)p FX(\()p F0(A)p FX(\))p Ge(,)i(called)e(the)f F0(complement)g(of)g(A)p Ge(,)j(we)e(indicate)f(the)h(\(type)f(of)g (the\))g(en)m(vironment)e(which)450 4757 y(gi)n(v)o(es)21 b(complementary)d(linear)i(and)h(replicated)f(inputs)h(for)f(all)i (free)e(output)g(channels)g(in)i F0(A)p Ge(.)28 b F0(A)12 b FV(\001)g F0(B)20 b Ge(is)450 4865 y(a)d(\223semantic)f(v)o (ersion\224)f(of)h F0(A)8 b FV(\014)g F0(B)p Ge(,)16 b(where)g(we)g(for)o(get)f(inessential)h FJ(l)q Ge(-channels.)22 b(Hence)16 b(by)g(de\002nition,)450 4973 y FM(md)p FX(\()p F0(A)c FV(\014)g F0(B)p FX(\))17 b(=)h Ge(!)s(.)26 b(W)-7 b(e)21 b(can)f(no)n(w)g(de\002ne)g(semantic)f(types.)p eop %%Page: 17 17 17 16 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(17)533 -4 y FG(Definition)34 b(3.2.)42 b Ge(Recall)14 b FZ(~)-32 b F0(y)11 b Ge(:)1464 -8 y FZ(~)1481 -4 y F1(t)21 b FX(=)f F0(y)1660 8 y Gb(1)1706 -4 y Ge(:)11 b F1(t)1776 8 y Gb(1)1812 -4 y FZ(;)e(:::;)g F0(y)1982 8 y FT(n)2028 -4 y Ge(:)i F1(t)2098 8 y FT(n)2134 -4 y Ge(.)36 b(The)23 b F0(semantic)h(type)f FX([)-9 b([)p F0(A)p FX(])g(])24 b Ge(and)f(the)h F0(prime)450 104 y(semantic)c(type)g FV(h)-9 b(h)f FZ(~)-32 b F0(x)9 b Ge(:)1041 100 y FZ(~)1058 104 y F1(t)q FV(i)-9 b(i)21 b Ge(are)f(de\002ned)f(by)h(the)g(follo)n (wing)f(rules:)1019 328 y FX([)-9 b([)p F0(A)p FX(])g(])1193 281 y FQ(def)1198 328 y FX(=)54 b FV(f)20 b(`)f F0(P)h FZ(.)g F0(A)37 b FV(j)g(8)p F0(Q)17 b FV(2)i(h)-9 b(h)p Fw(cl)p FX(\()p F0(A)p FX(\))p FV(i)g(i)p FZ(:)21 b F0(P)p FV(j)p F0(Q)d FV(+)2457 340 y FT(e)2507 328 y F0(R)g FV(2)h(h)-9 b(h)p 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b(proof)450 2115 y(proceeds)d(by)h(the)h(rule)f(induction)f (on)h(the)h(generation)d(of)j FV(`)d F0(P)j FZ(.)f F0(A)h Ge(with)g(respect)f(to)h(the)f(rules)h(gi)n(v)o(en)450 2223 y(in)f(Proposition)f(3.3.)24 b(Belo)n(w)d(gi)n(v)o(en)e F0(A)f FX(=)g(\()p F0(x)1748 2235 y Gb(1)1792 2223 y Ge(:)9 b F1(t)1860 2235 y Gb(1)1896 2223 y FZ(;)g(:::;)g F0(x)2066 2235 y FT(n)2110 2223 y Ge(:)g F1(t)2178 2235 y FT(n)2214 2223 y FX(\))p Ge(,)20 b(we)h(write)p 2596 2157 51 4 v 20 w F0(A)g Ge(for)e F0(x)2822 2235 y Gb(1)2866 2223 y Ge(:)p 2898 2172 72 4 v 9 w F1(t)2934 2235 y Gb(1)2970 2223 y FZ(;)9 b(:::;)g F0(x)3140 2235 y FT(n)3184 2223 y Ge(:)p 3216 2172 V 9 w F1(t)3252 2235 y FT(n)3288 2223 y Ge(.)450 2355 y F2(\(Inaction\))p Ge(.)39 b(By)26 b Fw(cl)p FX(\()1083 2353 y F1(/)1074 2355 y(0)p FX(\))21 b(=)1264 2353 y F1(/)1255 2355 y(0)p Ge(,)26 b(if)g F0(Q)21 b FV(2)g(h)-9 b(h)p Fw(cl)p FX(\()1731 2353 y F1(/)1722 2355 y(0)p FX(\))p FV(i)g(i)p Ge(,)27 b(then)e F0(Q)c FV(\021)g F2(0)p Ge(.)40 b(Hence)25 b F2(0)p FV(j)p F0(Q)c FV(\021)g F2(0)g FV(+)2922 2367 y FT(e)2974 2355 y F2(0)g FV(2)g(h)-9 b(h)3177 2353 y F1(/)3168 2355 y(0)p FV(i)g(i)26 b Ge(with)450 2462 y Fw(cl)p FX(\()548 2460 y F1(/)539 2462 y(0)p FX(\))12 b FV(\001)669 2460 y F1(/)660 2462 y(0)17 b FX(=)811 2460 y F1(/)802 2462 y(0)p Ge(,)j(so)g(that)h(we)f (ha)n(v)o(e)g F2(0)e FV(2)g FX([)-9 b([)1596 2460 y F1(/)1587 2462 y(0)p FX(])g(])p Ge(.)450 2594 y F2(\(Linear)30 b(Input\))p Ge(.)54 b(Assume)29 b F0(P)24 b FV(2)g FX([)-9 b([)f FZ(~)-32 b F0(y)14 b Ge(:)1609 2590 y FZ(~)1625 2594 y F1(t)q FZ(;)9 b FV(")23 b F0(A)p FZ(;)9 b Ge(?)t F0(B)p FX(])-9 b(])p Ge(.)54 b(W)-7 b(e)30 b(sho)n(w)g F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)14 b Ge(:)2529 2590 y FZ(~)2546 2594 y F1(t)p FX(\))p FZ(:)p F0(P)24 b FV(2)g FX([)-9 b([\()p F0(x)15 b Ge(:)f FX(\()2964 2590 y FZ(~)2981 2594 y F1(t)q FX(\))3050 2564 y Gd(#)3109 2594 y FV(!)23 b F0(A)p FX(\))p FZ(;)9 b F0(B)p FX(])-9 b(])p Ge(.)450 2702 y(Let)25 b F1(t)d FX(=)f(\()745 2698 y FZ(~)762 2702 y F1(t)q FX(\))831 2672 y Gd(#)866 2702 y Ge(.)40 b(First)26 b(we)g(note)f Fw(cl)o FX(\(\()p F0(x)12 b Ge(:)g F1(t)23 b FV(!)e F0(A)p FX(\))p FZ(;)9 b F0(B)p FX(\))21 b(=)g(\()p 2101 2636 51 4 v F0(A)p FZ(;)p 2184 2638 V 9 w F0(B)p FX(\))p Ge(.)41 b(Let)25 b F0(Q)d FV(2)f(h)-9 b(h)p 2678 2636 V F0(A)p FZ(;)p 2761 2638 V 9 w F0(B)p FV(i)g(i)26 b Ge(and)f F0(R)c FV(2)g(h)-9 b(h)f FZ(~)-32 b F0(y)12 b Ge(:)3309 2681 y FZ(~)p 3326 2651 37 4 v 3326 2702 a F1(t)q FV(i)-9 b(i)p Ge(.)450 2810 y(W)h(.l.o.g.)20 b(we)g(assume)533 2982 y FV(\017)33 b F0(Q)18 b FV(\021)g F0(Q)829 2994 y Gb(1)864 2982 y FV(j)p F0(Q)947 2994 y Gb(2)1000 2982 y FV(2)h FM(NF)1173 2994 y FT(e)1226 2982 y Ge(such)h(that)g F0(Q)1604 2994 y Gb(1)1657 2982 y FV(\021)e F1(P)1804 2994 y FT(i)1825 2982 y F0(Q)1885 2994 y Gb(1)p FT(i)1928 2982 y Ft(h#)6 b Fs(a)2015 2991 y Fp(i)2031 2982 y Ft(i)18 b FV(2)g(h)-9 b(h)p 2200 2916 51 4 v F0(A)p FV(i)g(i)21 b Ge(and)f F0(Q)2528 2994 y Gb(2)2581 2982 y FV(\021)e F1(P)2737 2994 y FT(j)2760 2982 y F0(Q)2820 2994 y Gb(2)9 b FT(j)2873 2982 y Ft(h)p FQ(!)n Fs(b)2948 2991 y Fp(j)2965 2982 y Ft(i)18 b FV(2)h(h)-9 b(h)p 3135 2918 V F0(B)o FV(i)g(i)p Ge(;)22 b(and)533 3118 y FV(\017)33 b F0(R)14 b FV(\021)g F0(R)803 3130 y Gb(1)845 3118 y FV(j)8 b F0(R)927 3130 y Gb(2)976 3118 y FV(2)15 b FM(NF)1145 3130 y FT(e)1193 3118 y Ge(such)h(that)g F0(R)1554 3130 y Gb(1)1603 3071 y FQ(def)1608 3118 y FX(=)j F1(P)1756 3131 y FT(k)1788 3118 y F0(R)1839 3131 y Gb(1)p FT(k)1893 3118 y Ft(h#)6 b Fs(z)1974 3127 y Fp(k)1997 3118 y Ft(i)16 b Ge(with)g F0(R)2251 3131 y Gb(1)p FT(k)2305 3118 y Ft(h#)6 b Fs(z)2386 3127 y Fp(k)2409 3118 y Ft(i)14 b FV(2)h(h)-9 b(h)p F0(z)2603 3131 y FT(k)2641 3118 y Ge(:)p 2669 3067 69 4 v 5 w F1(t)2705 3131 y FT(k)2738 3118 y FV(i)g(i)17 b Ge(and)e F0(R)2997 3130 y Gb(2)3047 3071 y FQ(def)3051 3118 y FX(=)k F1(P)3199 3131 y FT(l)3223 3118 y F0(R)3274 3131 y Gb(2)p FT(l)3320 3118 y Ft(h)p FQ(!)n Fs(w)3399 3127 y Fp(l)3416 3118 y Ft(i)450 3226 y Ge(with)h F0(R)669 3239 y Gb(2)p FT(l)715 3226 y Ft(h)p FQ(!)n Fs(w)794 3235 y Fp(l)811 3226 y Ft(i)e FV(2)h(h)-9 b(h)p F0(w)1036 3239 y FT(l)1070 3226 y Ge(:)p 1102 3175 61 4 v 9 w F1(t)1138 3239 y FT(l)1163 3226 y FV(i)g(i)21 b Ge(with)f FV(f)-10 b FZ(~)-32 b F0(y)p FV(g)18 b FX(=)g FV(f)-15 b FZ(~)-27 b F0(z)n FZ(~)-40 b F0(w)p FV(g)p Ge(.)25 b(Let)450 3398 y(By)c(induction)d(hypothesis,)1158 3603 y F0(P)p FV(j)p FX(\()p F0(Q)p FV(j)p F0(R)1398 3615 y Gb(1)1444 3603 y FV(j)12 b F0(R)1530 3615 y Gb(2)1564 3603 y FX(\))19 b FV(+)1666 3615 y FT(e)1716 3603 y F0(Q)1776 3615 y Gb(2)1810 3603 y FV(j)p F0(R)1884 3615 y Gb(2)1938 3603 y FV(2)f(h)-9 b(hf)p F0(w)2163 3616 y FT(l)2197 3603 y Ge(:)p 2229 3553 V 9 w F1(t)2265 3616 y FT(l)2290 3603 y FV(g)2332 3615 y FT(y)2359 3628 y Fn(l)2379 3615 y FU(=)p FT(w)2468 3628 y Fn(l)2493 3603 y FZ(;)p 2525 3540 51 4 v 9 w F0(B)p FV(i)g(i)690 b Ge(\(5\))474 3809 y(Hence)22 b(by)h(Proposition)e(3.1)h(\(4-i\),)h(we)g(kno)n(w)f F0(P)p FV(j)p F0(Q)e FV(+)2073 3821 y FT(e)2124 3809 y F0(P)2175 3779 y Gd(0)2196 3809 y FV(j)p F0(Q)2279 3821 y Gb(2)2337 3809 y Ge(where)i FM(fn)p FX(\()p F0(P)2709 3779 y Gd(0)2730 3809 y FX(\))e FV(\022)f(f)-10 b FZ(~)-32 b F0(y)p FV(g)p Ge(.)33 b(By)24 b(the)f(de\002-)450 3917 y(nition)c(of)h FV(7!)g Ge(this)h(implies)f F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)p FV(j)p F0(Q)18 b FV(+)1627 3929 y FT(e)1676 3917 y F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)1888 3886 y Gd(0)1909 3917 y FV(j)p F0(Q)1992 3929 y Gb(2)2027 3917 y Ge(.)26 b(W)-7 b(e)21 b(no)n(w)e(sho)n(w)h F0(P)2609 3886 y Gd(0)2648 3917 y FV(2)e FX([)-9 b([)f FZ(~)-32 b F0(y)9 b Ge(:)2819 3913 y FZ(~)2836 3917 y F1(t)p FX(])-9 b(])p Ge(,)20 b(which)g(implies,)450 4025 y(by)f(de\002nition)f(of)g FX([)-9 b([)p FV(\001)p FX(])g(])p Ge(,)19 b F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(P)1329 3994 y Gd(0)1367 4025 y FV(2)18 b(h)-9 b(h)p F0(x)8 b Ge(:)g F1(t)p FV(i)-9 b(i)p Ge(.)26 b(W)-7 b(e)20 b(already)e(kno)n(w)g F0(P)p FV(j)p F0(Q)p FV(j)p F0(R)2511 4037 y Gb(1)2546 4025 y FV(j)p F0(R)2620 4037 y Gb(2)2672 4025 y FV(7!)2755 3994 y Gd(\003)2807 4025 y F0(P)2858 3994 y Gd(0)2879 4025 y FV(j)p F0(Q)2962 4037 y Gb(2)2997 4025 y FV(j)p F0(R)3071 4037 y Gb(1)3105 4025 y FV(j)p F0(R)3179 4037 y Gb(2)3214 4025 y Ge(,)h(while)450 4132 y(by)25 b(\(5\))f(abo)o(v)o (e,)g(we)h(ha)n(v)o(e)g F0(P)p FV(j)p F0(Q)p FV(j)p F0(R)1434 4144 y Gb(1)1468 4132 y FV(j)p F0(R)1542 4144 y Gb(2)1598 4132 y FV(7!)1681 4102 y Gd(\003)1737 4132 y F0(P)1788 4102 y Gd(0)1808 4132 y FV(j)p F0(Q)1891 4144 y Gb(2)1926 4132 y FV(j)p F0(R)2000 4144 y Gb(1)2035 4132 y FV(j)p F0(R)2109 4144 y Gb(2)2165 4132 y FV(+)2216 4144 y FT(e)2268 4132 y F0(Q)2328 4144 y Gb(2)2363 4132 y FV(j)p F0(R)2437 4144 y Gb(2)2471 4132 y Ge(.)40 b(Note)25 b(that)g FM(fn)o FX(\()p F0(Q)3023 4144 y Gb(2)3058 4132 y FX(\))h Ge(and)e FM(fn)p FX(\()p F0(R)p FX(\))450 4240 y Ge(are)d(disjoint,)h(hence)e (there)i(is)g(no)f(interaction)f(between)h F0(Q)2188 4252 y Gb(2)2245 4240 y Ge(and)g F0(P)2438 4210 y Gd(0)2459 4240 y FV(j)p F0(R)2533 4252 y Gb(1)2567 4240 y FV(j)p F0(R)2641 4252 y Gb(2)2676 4240 y Ge(.)29 b(No)n(w)22 b(by)f(CR)i(of)e FV(7!)h Ge(we)450 4348 y(kno)n(w)d F0(P)704 4318 y Gd(0)725 4348 y FV(j)p F0(R)799 4360 y Gb(1)834 4348 y FV(j)p F0(R)908 4360 y Gb(2)961 4348 y FV(+)1012 4360 y FT(e)1061 4348 y F0(R)1112 4360 y Gb(2)1165 4348 y FV(2)g(h)-9 b(hf)p F0(w)1391 4361 y FT(l)1425 4348 y Ge(:)p 1457 4297 61 4 v 9 w F1(t)1493 4361 y FT(l)1518 4348 y FV(g)1560 4360 y FT(y)1587 4373 y Fn(l)1607 4360 y FU(=)p FT(w)1696 4373 y Fn(l)1720 4348 y FV(i)g(i)p Ge(.)26 b(This)21 b(sho)n(ws)f F0(P)2268 4318 y Gd(0)2307 4348 y FV(2)f FX([)-9 b([)f FZ(~)-32 b F0(y)9 b Ge(:)2479 4344 y FZ(~)2496 4348 y F1(t)p FX(])-9 b(])p Ge(,)21 b(as)g(required.)450 4494 y F2(\(Replicated)f(Input\))p Ge(.)25 b(Similar)20 b(to)h(the)f(pre)n(vious)f(case.)450 4626 y F2(\(Linear)h(Output\))p Ge(.)25 b(Similar)20 b(to)g(and)g(simpler)g(than)g(the)g(ne)o(xt)f(case.)450 4757 y F2(\(Replicated)35 b(Output\))p Ge(.)70 b(Assume)35 b F0(P)27 b FV(2)g FX([)-9 b([)l F0(C)r FZ(;)9 b F0(x)18 b Ge(:)f FX(\()1965 4753 y FZ(~)1982 4757 y F1(t)q FX(\))2051 4727 y Gb(?)2087 4757 y FX(])-9 b(])36 b Ge(with)31 b F0(C)r FZ(=)-10 b(~)-32 b F0(y)26 b FX(=)p FV(")g F0(A)p FZ(;)9 b Ge(?)t F0(B)2809 4727 y Gd(\000)p FT(x)2888 4757 y Ge(.)71 b(Let)36 b F1(t)27 b FX(=)g(\()3297 4753 y FZ(~)3314 4757 y F1(t)p FX(\))3382 4727 y Gb(?)3418 4757 y Ge(.)450 4865 y(W)-7 b(e)28 b(ha)n(v)o(e)e(to)h(sho)n(w)p 1059 4819 38 4 v 26 w F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)22 b Ge(:)1216 4861 y FZ(~)1233 4865 y F1(t)p FX(\))p F0(P)g FV(2)h FX([)-9 b([)p F0(A)p FZ(;)9 b F0(B)p FZ(;)g F0(x)k Ge(:)g F1(t)p FX(])-9 b(])p Ge(.)45 b(First)27 b(we)g(note)g(that)f Fw(cl)p FX(\()p F0(A)p FZ(;)9 b F0(B)p FZ(;)g F0(x)k Ge(:)g F1(t)p FX(\))23 b(=)e Fw(cl)p FX(\()l F0(C)r FZ(;)9 b F0(x)k Ge(:)g F1(t)p FX(\))23 b(=)450 4973 y(\()p 482 4907 51 4 v F0(A)p FZ(;)p 565 4909 V 9 w F0(B)p FZ(;)9 b F0(x)j Ge(:)p 732 4922 37 4 v 12 w F1(t)p FX(\))p Ge(.)39 b(Assume)25 b F0(Q)c FV(2)g(h)-9 b(h)p 1365 4907 51 4 v F0(A)p FZ(;)p 1448 4909 V 9 w F0(B)p FZ(;)9 b F0(x)j Ge(:)p 1615 4922 37 4 v 12 w F1(t)p FV(i)-9 b(i)p Ge(.)39 b(W)-8 b(.l.o.g.)24 b(we)h(can)f(write)h F0(Q)c FV(\021)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)o FX(\))p FZ(:)p F0(Q)2903 4943 y Gd(0)2903 4997 y Gb(0)2948 4973 y FV(j)9 b F0(Q)3040 4985 y Gb(1)3084 4973 y FV(j)g F0(Q)3176 4985 y Gb(2)3236 4973 y Ge(where)p eop %%Page: 19 19 19 18 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(19)450 -4 y Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)699 -34 y Gd(0)699 20 y Gb(0)752 -4 y FV(2)19 b(h)-9 b(h)p F0(x)9 b Ge(:)p 959 -55 37 4 v 9 w F1(t)q FV(i)-9 b(i)p Ge(,)21 b F0(Q)1153 8 y Gb(1)1206 -4 y FV(\021)d F1(P)1353 8 y FT(i)1374 -4 y F0(Q)1434 8 y Gb(1)p FT(i)1477 -4 y Ft(h#)6 b Fs(a)1564 5 y Fp(i)1580 -4 y Ft(i)20 b Ge(and)g F0(Q)1824 8 y Gb(2)1877 -4 y FV(\021)e F1(P)2033 8 y FT(j)2056 -4 y F0(Q)2116 8 y Gb(2)9 b FT(j)2169 -4 y Ft(h)p FQ(!)n Fs(b)2244 5 y Fp(j)2261 -4 y Ft(i)o Ge(.)26 b(Then)19 b(we)i(ha)n(v)o(e:)p 1192 165 38 4 v 1192 211 a F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)9 b FV(j)g F0(Q)37 b FV(\000)-14 b(!)37 b FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)2009 177 y Gd(0)2009 232 y Gb(0)2044 211 y FX(\))9 b FV(j)g Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(Q)2367 177 y Gd(0)2367 232 y Gb(0)2411 211 y FV(j)9 b F0(Q)2503 223 y Gb(1)2547 211 y FV(j)g F0(Q)2639 223 y Gb(2)2674 211 y FZ(:)450 427 y Ge(By)27 b(induction)f(hypothesis,)g F0(P)p FV(j)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)1643 397 y Gd(0)1643 451 y Gb(0)1678 427 y FV(j)p F0(Q)1761 439 y Gb(1)1796 427 y FV(j)p F0(Q)1879 439 y Gb(2)1936 427 y FV(+)1987 439 y FT(e)2040 427 y F0(P)2091 397 y Gd(0)2112 427 y FV(j)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)2384 397 y Gd(0)2384 451 y Gb(0)2419 427 y FV(j)p F0(Q)2502 439 y Gb(2)2564 427 y Ge(such)27 b(that)g F0(P)2947 397 y Gd(0)2990 427 y FV(2)c FX([)-9 b([)f FZ(~)-32 b F0(y)12 b Ge(:)3173 423 y FZ(~)3190 427 y F1(t)q FX(])-9 b(])27 b Ge(with)450 535 y FM(md)p FX(\()p F1(t)622 547 y FT(i)644 535 y FX(\))19 b FV(2)g(f)p Ge(!)p FZ(;)9 b FV(#g)p Ge(.)28 b(Hence)21 b(we)h(can)f(write)h F0(P)1738 505 y Gd(0)1778 535 y FV(\021)c F1(P)1925 548 y FT(k)1958 535 y F0(R)2009 548 y Gb(1)p FT(k)2062 535 y Ft(h#)6 b Fs(z)2143 544 y Fp(k)2166 535 y Ft(i)12 b FV(j)g F1(P)2300 548 y FT(l)2324 535 y F0(R)2375 548 y Gb(2)p FT(l)2420 535 y Ft(h)p FQ(!)n Fs(w)2499 544 y Fp(l)2517 535 y Ft(i)21 b Ge(with)h FV(f)-10 b FZ(~)-32 b F0(y)o FV(g)19 b FX(=)g FV(f)-15 b FZ(~)-27 b F0(z)n FZ(~)-41 b F0(w)q FV(g)p Ge(.)28 b(W)-7 b(e)23 b(also)450 643 y(note)d(that)g F0(Q)819 613 y Gd(0)819 667 y Gb(0)872 643 y FV(2)f(h)-9 b(h)f FZ(~)-32 b F0(y)9 b Ge(:)1062 639 y FZ(~)1079 643 y F1(t)q FV(i)-9 b(i)p Ge(.)26 b(Hence,)19 b(by)h(assumption,)894 858 y FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)1131 824 y Gd(0)1152 858 y FV(j)p F0(Q)1235 824 y Gd(0)1235 879 y Gb(0)1270 858 y FX(\))18 b FV(7!)1403 824 y Gd(\003)1403 879 y Fx(l)1457 858 y FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\)\()p F1(P)1706 871 y FT(l)1731 858 y F0(R)1782 871 y Gb(2)p FT(l)1827 858 y Ft(h)p FQ(!)n Fs(w)1906 867 y Fp(l)1923 858 y Ft(i)11 b FV(j)h F0(Q)2052 824 y Gd(00)2052 879 y Gb(0)2090 858 y FX(\))19 b FV(7!)2224 824 y Gd(\003)2277 858 y FX(\()p F1(n)o FZ(~)-31 b F0(y)o FX(\)\()p F1(P)2526 871 y FT(l)2551 858 y F0(R)2602 871 y Gb(2)p FT(l)2648 858 y Ft(h)p FQ(!)m Fs(w)2726 867 y Fp(l)2744 858 y Ft(i)o FX(\))19 b FV(7!)2900 824 y Gd(\003)2900 879 y Fx(g)2953 858 y F2(0)450 1074 y Ge(No)n(w)h(by)g(CR,)h(we)g(ha)n(v)o(e)e F0(P)12 b FV(j)g F0(Q)17 b FV(+)1404 1086 y FT(e)1436 1074 y Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)o FX(\))p FZ(:)p F0(Q)1684 1044 y Gd(0)1684 1098 y Gb(0)1731 1074 y FV(j)12 b F0(Q)1826 1086 y Gb(2)1879 1074 y FV(2)19 b(h)-9 b(h)p 2008 1010 51 4 v F0(B)o FZ(;)9 b F0(x)g Ge(:)p 2168 1023 37 4 v 9 w F1(t)r FV(i)-9 b(i)p Ge(,)20 b(as)h(desired.)450 1229 y F2(\(Composition)16 b(of)g(Primes\))p Ge(.)24 b(Gi)n(v)o(en)16 b F1(P)1636 1241 y FT(i)p Gd(2)p FT(I)1722 1229 y F0(P)1765 1241 y FT(i)1785 1229 y Ge(,)i(assume)f(by)f (induction)f F0(P)2560 1241 y FT(i)2596 1229 y FV(2)g FX([)-9 b([)p F0(A)2754 1241 y FT(i)2775 1229 y FX(])g(])17 b Ge(\()p F0(i)e FV(2)h F0(I)t Ge(\))g(and)g F0(P)3222 1241 y FT(i)3243 1229 y FV(j)p F0(P)3318 1241 y FT(j)3356 1229 y FV(67!)450 1337 y Ge(for)29 b F0(i)23 b FV(6)p FX(=)35 b F0(j)r Ge(.)54 b(Note)29 b(that,)i(for)e(each)g F0(x)23 b FV(2)h FM(fn)o FX(\()p F0(A)1784 1349 y FT(i)1806 1337 y FX(\))15 b FV(\\)g FM(fn)o FX(\()p F0(A)2077 1349 y FT(j)2100 1337 y FX(\))30 b Ge(\()p F0(i)23 b FV(6)p FX(=)35 b F0(j)r Ge(\))c(we)e(ha)n(v)o(e)g F0(x)14 b Ge(:)h F1(t)24 b FV(2)f(j)p F0(A)3031 1349 y FT(i)3052 1337 y 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2980 310 46 4 v 3 w FX(\)\()p Ge(!)p F0(d)p 3117 310 V 4 w FZ(:)p 3186 250 37 4 v F0(y)p FX(\))j FV(j)p 3302 250 V 12 w F0(y)f FV(j)p 3385 250 V 12 w F0(y)o FX(\))1613 404 y FV(7!)1696 373 y Gb(2)1696 424 y Fx(g)1781 404 y Ge(!)p F0(a)p FX(\()p F0(x)p FX(\))p FZ(:)p FX(\()p 2007 358 38 4 v F0(x)h FV(j)p 2091 358 V 12 w F0(x)o FX(\))g FV(j)p 2206 336 42 4 v 12 w F0(b)p 2206 418 V 11 w FV(j)p 2294 336 V 12 w F0(b)p 2294 418 V 10 w FV(j)g Ge(!)p F0(b)p 2409 418 V -1 w FZ(:)p FX(\()p 2505 358 37 4 v F0(y)f FV(j)p 2588 358 V 12 w F0(y)p FX(\))1613 512 y FV(7!)1696 482 y Gb(2)1696 532 y Fx(r)1781 512 y Ge(!)p F0(a)p FX(\()p F0(x)p FX(\))p FZ(:)p FX(\()p 2007 466 38 4 v F0(x)h FV(j)p 2091 466 V 12 w F0(x)o FX(\))g FV(j)p 2206 466 37 4 v 12 w F0(y)f FV(j)p 2289 466 V 12 w F0(y)g FV(j)p 2372 466 V 12 w F0(y)g FV(j)p 2455 466 V 12 w F0(y)g FV(j)h Ge(!)p F0(b)m FZ(:)p FX(\()p 2660 466 V F0(y)e FV(j)p 2742 466 V 12 w F0(y)o FX(\))450 722 y Ge(The)27 b(proof)f(follo)n(ws.)46 b(Belo)n(w)28 b(we)g(say)f(an)h(output)e(channel)g F0(x)d FV(2)g FM(fn)o FX(\()p F0(A)p FX(\))28 b Ge(is)g F0(complemented)e(by)h(R)h Ge(if)450 830 y FV(`)18 b F0(R)j FZ(.)f Fw(cl)p FX(\()p F0(A)p FX(\))p Ge(.)533 997 y F0(Pr)l(oof)o(.)66 b Ge(By)21 b(rule)f(induction)e(on)i(the)g(typing)f(rules.)450 1128 y F2(Case)j FM(\(Zero\))p Ge(.)53 b(Suppose)21 b FV(`)e F2(0)i FZ(.)1431 1126 y F1(/)1422 1128 y(0)p Ge(.)31 b(Then)21 b Fw(cl)p FX(\()1807 1126 y F1(/)1798 1128 y(0)p FX(\))e(=)1984 1126 y F1(/)1975 1128 y(0)p Ge(.)31 b(Since)22 b(for)g(all)g F0(Q)e FV(2)f FX([)-9 b([)2701 1126 y F1(/)2692 1128 y(0)p FX(])g(])p Ge(,)22 b(we)h(ha)n(v)o(e)e F0(Q)f FV(+)3241 1140 y FT(e)3292 1128 y F2(0)i Ge(by)450 1236 y(Corollary)d(3.1)h(\(1\),)f F2(0)11 b FV(j)h F0(Q)18 b FV(+)1269 1248 y FT(e)1318 1236 y F2(0)p Ge(,)i(as)h(desired.)450 1368 y F2(Case)f Ge(\()p FM(Res)p Ge(\).)25 b(W)-7 b(e)21 b(do)f(case)h(analysis)f(based)g(on)g(the)g(mode)f(of)h(the)g(hidden)f (channel.)450 1500 y F2(Subcase:)61 b FV(`)20 b FX(\()p F1(n)10 b F0(x)p FX(\))p F0(P)23 b FZ(.)f F0(A)j(is)h(derived)f(fr)l (om)g FV(`)c F0(P)h FZ(.)g F0(A)p FZ(;)9 b F0(x)j Ge(:)g FJ(l)q F0(.)40 b Ge(W)-7 b(e)26 b(sho)n(w)-5 b(,)25 b(for)g(each)f (complementing)450 1607 y(process)19 b F0(Q)f FV(2)h FX([)-9 b([)p Fw(cl)o FX(\()p F0(A)p FX(\)])g(])p Ge(,)20 b(we)g(ha)n(v)o(e)f FX(\()p F1(n)10 b F0(x)p FX(\))p F0(P)p FV(j)p F0(Q)18 b FV(+)1805 1619 y FT(e)1836 1607 y Ge(.)26 b(By)20 b(induction)e(hypothesis,)g(for)h(each)g F0(R)f FV(2)g FX([)-9 b([)p Fw(cl)p FX(\()p F0(A)p FZ(;)9 b F0(x)g Ge(:)450 1715 y FJ(l)p FX(\)])-9 b(])p Ge(,)20 b(we)f(ha)n(v)o(e)f F0(P)p FV(j)p F0(R)f FV(+)1079 1727 y FT(e)1110 1715 y Ge(.)25 b(Note)19 b(that)g Fw(cl)o FX(\()p F0(A)p FZ(;)9 b F0(x)f Ge(:)g FJ(l)q FX(\))17 b(=)g Fw(cl)p FX(\()p F0(A)p FX(\))i Ge(by)g(de\002nition.)k(Hence,)18 b(ob)o(viously)-5 b(,)17 b(we)i(ha)n(v)o(e)450 1823 y F0(P)p FV(j)p F0(Q)e FV(+)652 1835 y FT(e)702 1823 y Ge(for)h(each)g F0(Q)f FV(2)h FX([)-9 b([)p Fw(cl)o FX(\()p F0(A)p FX(\)])g(])p Ge(.)25 b(This)19 b(in)f(turn)g(implies)h FX(\()p F1(n)10 b F0(x)p FX(\))p F0(P)p FV(j)p F0(Q)17 b FV(\021)f FX(\()p F1(n)10 b F0(x)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\))17 b FV(+)2903 1835 y FT(e)2954 1823 y Ge(by)h(Proposition)450 1931 y(3.1)i(\(4-ii\),)f(hence)g(done.)450 2063 y F2(Subcase:)25 b FV(`)c FX(\()p F1(n)9 b F0(x)p FX(\))p F0(P)23 b FZ(.)f F0(A)j(is)h(derived)e(fr)l(om)h FV(`)c F0(P)h FZ(.)g F0(B)j(suc)o(h)f(that)h FM(md)p FX(\()p F0(B)p FX(\()p F0(x)p FX(\)\))c(=)g Ge(!)t F0(.)64 b Ge(W)m(ithout)24 b(loss)i(of)450 2171 y(generality)-5 b(,)21 b(we)h(set)h F0(B)c FX(=)g F0(A)1260 2183 y Gb(0)1307 2171 y FV(\014)12 b F0(x)f Ge(:)f F1(t)19 b FV(!)h Ge(?)t F0(B)1715 2183 y Gb(0)1772 2171 y Ge(and)i F0(A)1966 2183 y Gb(0)2013 2171 y FV(\014)12 b Ge(?)s F0(B)2181 2183 y Gb(0)2236 2171 y FX(=)19 b F0(A)p Ge(.)31 b(Again,)21 b(by)h(de\002nition,)f(we)h(kno)n(w)450 2279 y Fw(cl)p FX(\()p F0(A)p FX(\))c(=)g Fw(cl)p FX(\()p F0(B)p FX(\))p Ge(.)26 b(The)20 b(rest)g(is)h(similar)g(to)f(the)g(abo)o(v)o(e)f (case.)450 2410 y F2(Case)h Ge(\()p FM(W)n(eak)p Ge(\).)k(T)m(ri)n (vial)c(by)f(inducti)n(v)o(e)g(hypothesis.)450 2542 y F2(Case)g Ge(\()p FM(In)731 2512 y Gd(#)765 2542 y Ge(\).)25 b(Assume)19 b FV(`)d F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(P)19 b FZ(.)g F0(A)g Ge(is)h(deri)n(v)o(ed)d(from)h FV(`)f F0(P)i FZ(.)9 b(~)-32 b F0(y)8 b Ge(:)2335 2538 y FZ(~)2352 2542 y F1(t)i FZ(;)f FV(")17 b F0(A)2540 2512 y F9(-)p FT(x)2540 2567 y Gb(0)2596 2542 y FZ(;)9 b Ge(?)t F0(B)2720 2512 y F9(-)p FT(x)2720 2567 y Gb(0)2796 2542 y Ge(with)19 b F0(A)e FX(=)g(\()p F0(x)8 b Ge(:)g FX(\()3236 2538 y FZ(~)3253 2542 y F1(t)q FX(\))3322 2512 y Gd(#)3356 2542 y FV(!)450 2650 y F0(A)501 2662 y Gb(0)536 2650 y FX(\))p FZ(;)h F0(B)651 2662 y Gb(0)686 2650 y Ge(.)53 b(Let)25 b F0(C)h FX(=)12 b FZ(~)-31 b F0(y)14 b Ge(:)1136 2646 y FZ(~)1153 2650 y F1(t)c FZ(;)f F0(A)1282 2662 y Gb(0)1317 2650 y FZ(;)g F0(B)1400 2662 y Gb(0)1435 2650 y Ge(.)53 b(By)29 b(induction)f(hypothesis,)i(for)e (each)h F0(Q)24 b FV(2)g FX([)-9 b([)p Fw(cl)o FX(\()l F0(C)r FX(\)])g(])p Ge(,)32 b(we)e(ha)n(v)o(e)450 2758 y F0(P)13 b FV(j)h F0(Q)21 b FV(+)683 2770 y FT(e)715 2758 y Ge(,)27 b(which)e(implies)h F0(P)c FV(+)1387 2770 y FT(e)1439 2758 y F0(P)1490 2727 y Gd(0)1533 2758 y FV(2)f FM(NF)1709 2770 y FT(e)1767 2758 y Ge(by)k(Proposition)f(3.1)i (\(4-i\).)40 b(Then)25 b(by)h(construction)e(of)450 2866 y FM(NF)550 2878 y FT(e)581 2866 y Ge(,)33 b(we)e(kno)n(w)e F0(a)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)1193 2835 y Gd(0)1238 2866 y FV(2)24 b FM(NF)1417 2878 y FT(e)1448 2866 y Ge(,)33 b(hence)d(by)f(Lemma)h(3.2,)i(we)e(kno)n(w)g F0(a)p FX(\()-10 b FZ(~)-32 b F0(y)o FX(\))p FZ(:)p F0(P)2840 2835 y Gd(0)2885 2866 y FV(2)25 b FX([)-9 b([)p F0(A)p FX(])g(])p Ge(.)55 b(No)n(w)30 b(by)450 2973 y(Lemma)19 b(3.1)h(\(3\),)f(we)i(ha)n(v)o(e)e F0(a)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)18 b FV(2)h FX([)-9 b([)p F0(A)p FX(])g(])p Ge(.)25 b(Then)19 b(by)h(Lemma)f(3.1)h(\(1\),)f F0(a)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)18 b FV(+)2877 2985 y FT(e)2908 2973 y Ge(,)j(as)g(desired.)450 3105 y F2(Case)f Ge(\()p FM(In)732 3075 y Gb(!)760 3105 y Ge(\).)46 b(Similar)20 b(to)h(\()p FM(In)1297 3075 y Gd(#)1330 3105 y Ge(\).)450 3237 y F2(Case)28 b Ge(\()p FM(Out)p Ge(\).)f(Assume)h FV(`)p 1243 3191 38 4 v 22 w F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)23 b FZ(.)g F0(A)28 b Ge(is)h(deri)n(v)o(ed)d(from)h FV(`)c F0(P)g FZ(.)c F0(C)2414 3206 y F9(-)p FT(x)2499 3237 y Ge(such)28 b(that)g FM(activ)n(e)o FX(\()l F0(C)r FX(\))23 b(=)12 b FZ(~)-31 b F0(y)28 b Ge(and)446 3344 y F0(C)r FZ(=)-10 b(~)-32 b F0(y)18 b FX(=)g F0(A)p Ge(.)25 b(Let)20 b F0(A)p FX(\()p F0(x)p FX(\))f(=)f F1(t)p Ge(.)450 3476 y F2(Subcase:)39 b FM(md)p FX(\()p F1(t)p FX(\))15 b(=)p FV(")p F0(.)f Ge(By)i(induction)e(hypothesis,)g(for)h(each)g F0(Q)f FV(2)g FX([)-9 b([)p Fw(cl)p FX(\()l F0(C)r FX(\)])g(])p Ge(,)16 b(we)g(ha)n(v)o(e)f F0(P)7 b FV(j)g F0(Q)14 b FV(+)3204 3488 y FT(e)3249 3476 y F0(P)3300 3446 y Gd(0)3328 3476 y FV(j)7 b F0(Q)3418 3446 y Gd(0)450 3584 y Ge(with)17 b F0(P)666 3554 y Gd(0)702 3584 y FV(2)f FX([)-9 b([)f FZ(~)-32 b F0(y)15 b Ge(:)888 3580 y FZ(~)901 3584 y F1(r)o FX(])-9 b(])17 b Ge(where)g F1(t)f FX(=)f(\()1373 3580 y FZ(~)1385 3584 y F1(r)o FX(\))1462 3554 y Gd(")1498 3584 y Ge(.)24 b(Assume)17 b F0(R)e FV(2)h FX([)-9 b([)p Fw(cl)o FX(\()p F0(A)p FX(\)])g(])p Ge(.)25 b(Then)16 b(by)g(the)h(shape)g(of)f(the)h(action)g(type)450 3708 y(and)k(by)g(de\002nition,)g(we)h(can)f(set)i F0(R)1500 3661 y FQ(def)1504 3708 y FX(=)h(\()p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(R)1837 3678 y Gd(0)1858 3708 y FX(\))p FV(j)p F0(Q)23 b Ge(such)e(that)h F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(R)2529 3678 y Gd(0)2569 3708 y FV(2)20 b FX([)-9 b([)p F0(x)19 b Ge(:)p 2779 3657 37 4 v 19 w F1(t)q FX(])-9 b(])22 b Ge(and)f F0(Q)e FV(2)g FX([)-9 b([)p Fw(cl)p FX(\()l F0(C)r FX(\)])g(])p Ge(.)450 3816 y(W)i(e)21 b(can)f(no)n(w)g (calculate:)p 938 3985 38 4 v 938 4031 a F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FV(j)p FX(\()p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(R)1394 3996 y Gd(0)1415 4031 y FX(\))p FV(j)p F0(Q)19 b FV(7!)1632 3996 y Gd(\003)p 1685 3985 V 1685 4031 a F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1874 3996 y Gd(0)1895 4031 y FV(j)p FX(\()p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(R)2162 3996 y Gd(0)2183 4031 y FX(\))p FV(j)p F0(Q)2298 3996 y Gd(0)2338 4031 y FV(7!)19 b FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\)\()p F0(P)2676 3996 y Gd(0)2697 4031 y FV(j)p F0(R)2771 3996 y Gd(0)2792 4031 y FX(\))p FV(j)p F0(Q)2907 3996 y Gd(0)2928 4031 y FZ(:)450 4245 y Ge(By)18 b(de\002nition)f F0(R)953 4215 y Gd(0)990 4245 y FV(2)f FX([)-9 b([)f FZ(~)-32 b F0(y)16 b Ge(:)p 1191 4172 46 4 v 1179 4241 a FZ(~)1191 4245 y F1(r)o FX(])-9 b(])p Ge(,)19 b(we)f(ha)n(v)o(e)f F0(P)1651 4215 y Gd(0)1672 4245 y FV(j)p F0(R)1746 4215 y Gd(0)1783 4245 y FV(+)1834 4257 y FT(e)1865 4245 y Ge(.)25 b(Also)18 b(by)f F0(Q)f FV(2)h FX([)-9 b([)p Fw(cl)o FX(\()l F0(C)r FX(\)])g(])p Ge(,)19 b(we)f(ha)n(v)o(e)f F0(Q)2969 4215 y Gd(0)3007 4245 y FV(+)3058 4257 y FT(e)3089 4245 y Ge(.)25 b(Note)17 b(that)450 4353 y FM(fn)o FX(\()p F0(Q)604 4323 y Gd(0)626 4353 y FX(\))23 b Ge(is)g(disjoint)f(from)f FM(fn)o FX(\()p F0(P)1360 4323 y Gd(0)1381 4353 y FV(j)p F0(R)1455 4323 y Gd(0)1476 4353 y FX(\))i Ge(so)g(that)f(there)g(is)h (no)f(further)e FV(+)2447 4365 y FT(e)2501 4353 y Ge(from)h FX(\()p F1(n)p FZ(~)-32 b F0(y)p FX(\)\()p F0(P)2922 4323 y Gd(0)2943 4353 y FV(j)p F0(R)3017 4323 y Gd(0)3038 4353 y FX(\))p FV(j)p F0(Q)3153 4323 y Gd(0)3174 4353 y Ge(.)32 b(Hence)450 4461 y(we)21 b(ha)n(v)o(e)e FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)979 4431 y Gd(0)1000 4461 y FV(j)p F0(R)1074 4431 y Gd(0)1095 4461 y FX(\))p FV(j)p F0(Q)1210 4431 y Gd(0)1249 4461 y FV(+)1300 4473 y FT(e)1331 4461 y Ge(,)21 b(as)g(required.)450 4593 y F2(Subcase:)46 b FM(md)p FX(\()p F1(r)p FX(\))18 b(=)g Ge(?)t F0(.)j Ge(Similar)f(to)g(the)h(subcase)f(abo)o(v)o(e.)450 4724 y F2(Case)25 b Ge(\()p FM(P)m(ar)p Ge(\).)65 b(Suppose)24 b FV(`)d F0(P)1324 4736 y FT(i)1368 4724 y FZ(.)h F0(A)1483 4736 y FT(i)1529 4724 y Ge(with)k F0(i)21 b FX(=)g Ge(1)p FZ(;)9 b Ge(2)25 b(such)g(that)g F0(A)2353 4736 y Gb(1)2409 4724 y FV(\020)c F0(A)2546 4736 y Gb(2)2606 4724 y Ge(and)j(let)i F0(A)21 b FX(=)g F0(A)3069 4736 y Gb(1)3117 4724 y FV(\014)13 b F0(A)3246 4736 y Gb(2)3280 4724 y Ge(.)41 b(By)450 4846 y(induction)17 b(hypothesis)h F0(P)1201 4858 y Gb(1)1253 4846 y FV(+)1304 4858 y FT(e)1353 4846 y F0(P)1404 4815 y Gd(0)1396 4870 y Gb(1)1450 4846 y Ge(and)g F0(P)1632 4858 y Gb(2)1684 4846 y FV(+)1735 4858 y FT(e)1784 4846 y F0(P)1835 4815 y Gd(0)1827 4870 y Gb(2)1861 4846 y Ge(.)26 b(Let)19 b F0(P)2106 4799 y FQ(def)2111 4846 y FX(=)j F0(P)2249 4815 y Gd(0)2241 4870 y Gb(1)2275 4846 y FV(j)p F0(P)2349 4815 y Gd(0)2341 4870 y Gb(2)2376 4846 y Ge(.)j(Then)18 b F0(P)g FV(\021)f F0(Q)2823 4858 y Gb(1)2857 4846 y FV(j)p FZ(::)p FV(j)p F0(Q)3009 4858 y FT(n)3064 4846 y Ge(where)i(each)450 4973 y F0(Q)510 4985 y FT(i)553 4973 y Ge(is)j(prime.)27 b(If)20 b F0(n)f FX(=)f Ge(0)j(there)g(is)h(nothing)d(to)i(pro)o(v)o(e.)26 b(Assume)21 b F0(n)e Fy(\015)f Ge(0)j(and)g(let)g F0(X)2838 4926 y FQ(def)2843 4973 y FX(=)i FV(f)p Ge(1)p FZ(;)9 b Ge(2)p FZ(;)g(::;)g F0(n)p FV(g)p Ge(.)26 b(W)-7 b(e)p eop %%Page: 21 21 21 20 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(21)450 -4 y Ge(de\002ne)20 b(the)g(relation)f FV(&)i Ge(on)f F0(X)28 b Ge(as)21 b(follo)n(ws:)1250 201 y F0(i)d FV(&)31 b F0(j)1493 154 y FQ(def)1488 201 y FV(,)58 b(9)p F0(x)18 b FV(2)h FM(fn)p FX(\()p F0(Q)1959 213 y FT(i)1980 201 y FX(\))p FZ(;)g F0(y)f FV(2)h FM(fn)o FX(\()p F0(Q)2346 213 y FT(j)2369 201 y FX(\))p FZ(:)i F0(x)e Fy(y)f F0(y)450 394 y Ge(F)o(or)j(e)o(xample,)f(tak)o(e)h(the)g(process)g F0(P)d FV(\021)p Ge(!)p F0(a)p FZ(:)p 1689 327 42 4 v F0(b)10 b FV(j)p 1776 348 V 12 w F0(a)i FV(j)g Ge(!)p F0(b)m FZ(:)p 1955 348 37 4 v F0(c)e FV(j)i Ge(!)p F0(c)p FZ(:)p F2(0)21 b Ge(discussed)g(just)g(before)f(the)h(proof)f(of)h (this)450 502 y(lemma,)h(then)h(we)f(ha)n(v)o(e:)30 b(1)19 b FV(&)h Ge(3)p FZ(;)32 b Ge(2)19 b FV(&)h Ge(1)p FZ(;)9 b Ge(3)19 b FV(&)h Ge(4.)32 b(As)23 b(in)g(this)g(e)o(xample,)e FV(&)2725 472 y Gd(\003)2783 502 y Ge(ne)n(v)o(er)h(collapses)g(tw)o(o) 450 610 y(names.)43 b(In)26 b(f)o(act,)i(if)e F0(i)c FV(&)1201 580 y FU(+)1287 610 y F0(j)j FV(&)1418 580 y FU(+)1491 610 y F0(i)i Ge(then)f(there)g(is)h(a)g(c)o(ycle)f(of)g (the)g(form)f F0(x)d Fy(y)2805 580 y FU(+)2879 610 y F0(x)27 b Ge(in)g(the)f(sense)h(of)450 718 y(Proposition)20 b(2.1)g(\(2\).)28 b(Thus)21 b(the)g(relation)g FV(&)1797 687 y Gd(\003)1853 718 y Ge(is)h(al)o(w)o(ays)g(a)g(partial)f(order)f (on)h F0(X)8 b Ge(.)28 b(W)-7 b(e)22 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(type,)h(then)f(both)g(input)g(and)h(output)450 3472 y(actions)22 b(\(resp.)f(output\))f(at)i F0(x)h Ge(should)e(be)g(e)o (xcluded)f(since)i(such)g(actions)f(can)h(ne)n(v)o(er)f(be)g(observ)o (ed)f(in)450 3580 y(a)i(typed)e(conte)o(xt)g(\(cf.)h(Section)f(4.2)h (and)g(Appendix)e(E)i(of)g([11]\).)27 b(Among)20 b(the)h(remaining)e (rules,)i(the)450 3688 y(\002rst)27 b(rule)f(says)h(that)g(the)f (transition)g(is)h(de\002ned)f(on)g(processes)g(modulo)e FV(\021)p Ge(.)44 b(As)27 b(we)g(shall)g(discuss)450 3796 y(later)d(we)h(can)f(dispense)g(with)g(this)h(rule)f(by)g(adding)f (tw)o(o)i(transition)e(rules)h(for)g(output)f(pre\002x.)37 b(The)450 3903 y(induced)19 b(transition)g(is)i(well-de\002ned)e(in)h (the)g(follo)n(wing)f(sense.)533 4082 y FG(Pr)n(oposition)31 b(4.1.)74 b FF(If)42 b FV(`)25 b F0(P)g FZ(.)g F0(A)41 b FF(and)i F0(P)1887 4052 y FT(A)2010 4035 y(l)1953 4082 y FV(\000)-14 b(!)25 b F0(Q)2172 4052 y FT(B)2255 4082 y FF(is)43 b(derivable)h(fr)l(om)e(Figur)l(e)h(4)f(then)450 4190 y FV(`)18 b F0(Q)j FZ(.)f F0(B)p FF(.)533 4371 y F0(Pr)l(oof)o(.)66 b Ge(Simple)20 b(inspection)f(of)h(each)g(rule)g(in) g(Figure)g(4.)p 2304 4371 50 50 v 533 4529 a(In)27 b(the)h(light)g(of)f (Proposition)f(4.1,)j(we)f(hereafter)e(safely)h(assume)h FV(`)22 b F0(P)h FZ(.)g F0(A)28 b Ge(and)f FV(`)c F0(Q)g FZ(.)g F0(B)28 b Ge(hold)450 4652 y(whene)n(v)o(er)18 b(we)j(write)f F0(P)1150 4622 y FT(A)1267 4606 y(l)1210 4652 y FV(\000)-14 b(!)18 b F0(Q)1422 4622 y FT(B)1464 4652 y Ge(.)25 b(W)-7 b(e)22 b(also)e(observ)o(e:)533 4815 y FG(Pr)n(oposition)31 b(4.2.)62 b FF(L)l(et)29 b FV(`)19 b F0(P)h FZ(.)g F0(A)p FF(.)39 b(Then)30 b F0(P)18 b FV(\000)-14 b(!)18 b F0(Q)30 b FF(i\013)g F0(P)2385 4785 y FT(A)2498 4768 y Gc(t)2445 4815 y FV(\000)-14 b(!)18 b F0(Q)2657 4785 y FT(A)2699 4815 y FF(.)p eop %%Page: 23 23 23 22 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(23)533 -4 y F0(Pr)l(oof)o(.)66 b Ge(Standard.)24 b(In)c(detail,)g(see)g (Appendix)f(C.1.)p 2159 -4 50 50 v 450 154 a(Finally)i(we)h(present)f (the)g(tw)o(o)h(rules)f(for)g(asynchronous)e(output)h(which)h(allo)n(w) g(us)h(to)f(dispense)g(with)450 262 y(\()p FV(\021)p Ge(\))f(from)f(Figure)g(4,)h(which)g(becomes)f(useful)h(in)g(our)g (proof)e(later)-5 b(.)545 460 y F0(P)596 419 y FT(A)634 431 y FD(1)588 485 y Gb(1)742 413 y FT(l)685 460 y FV(\000)-14 b(!)19 b F0(P)889 419 y FT(A)927 431 y FD(2)881 485 y Gb(2)1042 460 y FM(n)p FX(\()p F0(l)t FX(\))12 b FV(\\)g(f)-10 b FZ(~)-32 b F0(y)o FV(g)18 b FX(=)1484 458 y F1(/)1475 460 y(0)p 521 521 1284 5 v 545 587 38 4 v 545 634 a F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)755 589 y FT(A)793 601 y FD(1)822 589 y FS(=)-8 b(~)-23 b FT(y)p Gd(\014)p FT(x)p Gb(:)p FU(\()983 586 y FS(~)996 589 y Gc(t)p FU(\))1051 567 y Fn(p)727 659 y Gb(1)1159 587 y FT(l)1102 634 y FV(\000)-14 b(!)p 1254 587 V 18 w F0(x)p FX(\()k FZ(~)-32 b F0(y)q FX(\))p F0(P)1450 589 y FT(A)1488 601 y FD(2)1517 589 y FS(=)-8 b(~)-23 b FT(y)p Gd(\014)p FT(x)p Gb(:)p FU(\()1678 586 y FS(~)1691 589 y Gc(t)p FU(\))1746 567 y Fn(p)1436 659 y Gb(2)1995 469 y F0(P)2046 428 y FT(A)2084 440 y FD(1)2038 494 y Gb(1)2152 414 y FT(x)p FU(\()-11 b FS(~)-20 b FT(z)p FU(\))2135 469 y FV(\000)-15 b(!)19 b F0(P)2338 428 y FT(A)2376 440 y FD(2)2330 494 y Gb(2)p 1970 530 1302 5 v 1995 590 38 4 v 1995 636 a F0(x)2032 637 y FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)2191 592 y FT(A)2229 604 y FD(1)2257 592 y FS(=)-8 b(~)-23 b FT(y)p Gd(\014)p FT(x)p Gb(:)p FU(\()2418 589 y FS(~)2431 592 y Gc(t)p FU(\))2486 570 y Fn(p)2176 662 y Gb(1)2591 589 y Gc(t)2537 636 y FV(\000)-14 b(!)2690 637 y FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\))p F0(P)2886 649 y Gb(2)2921 637 y FV(f)-10 b FZ(~)-32 b F0(y)o FZ(=)-15 b(~)-27 b F0(z)p FV(g)3122 606 y FT(A)3160 618 y FD(2)3188 606 y FS(=)-11 b(~)-20 b FT(z)3342 535 y Ge(\(6\))450 805 y(These)29 b(rules)f(materialise)h(asynchronous)c (nature)j(of)h(the)f(output)g(in)h(transition)f(\(the)g(second)g(rule) 450 913 y(needs)g(renaming)f(to)i(a)n(v)n(oid)f(clash)h(of)g(bound)e (names\).)49 b(The)29 b(transition)f(system)h(which)f(adds)g(the)450 1021 y(rules)20 b(in)h(\(6\))e(to)i(the)f(rules)h(in)f(Figure)g(4)g (replacing)f FV(\021)h Ge(in)h(\()p FV(\021)p Ge(\))f(by)g FV(\021)2422 1033 y Gc(a)2464 1021 y Ge(,)h(is)g(called)f F0(syntactic)g(tr)o(ansition)450 1129 y(system)p Ge(.)48 b(The)28 b(transition)f(system)h(which)f(simply)g(replaces)g FV(\021)h Ge(in)g(\()p FV(\021)p Ge(\))f(by)g FV(\021)2794 1141 y Gc(a)2865 1129 y Ge(from)g(the)h(rules)f(in)450 1237 y(Figure)20 b(4,)g(is)h(called)f F0(prime)g(syntactic)g(tr)o (ansition)f(system)p Ge(.)26 b(W)-7 b(e)22 b(observ)o(e:)533 1383 y FG(Pr)n(oposition)31 b(4.3.)533 1529 y Ge(\(1\))13 b F0(If)20 b(P)766 1499 y FT(A)883 1482 y(l)826 1529 y FV(\000)-14 b(!)18 b F0(Q)1038 1499 y FT(B)1101 1529 y F0(in)i(the)g(syntactic)g(tr)o(ansition)f(system,)i(so)g(is)g(in)f (the)h(original)e(system.)533 1654 y Ge(\(2\))13 b F0(If)22 b(P)768 1624 y FT(A)885 1607 y(l)828 1654 y FV(\000)-14 b(!)20 b F0(Q)1042 1624 y FT(B)1105 1654 y F0(in)i(the)g(original)f(tr) o(ansition)g(system,)i(then)e(P)2431 1624 y FT(A)2549 1607 y(l)2492 1654 y FV(\000)-14 b(!)19 b F0(Q)2705 1624 y FT(B)2705 1678 y Gb(0)2769 1654 y F0(suc)o(h)i(that)h(Q)3154 1666 y Gb(0)3208 1654 y FV(\021)d F0(Q)j(in)450 1762 y(the)e(syntactic)g(tr)o(ansition)g(system.)533 1886 y Ge(\(3\))13 b F0(Let)23 b(P)824 1856 y FT(A)888 1886 y F0(be)f(derived)g(under)f(Con)m(vention)g(3.1.)30 b(Then)22 b(P)2276 1856 y FT(A)2394 1839 y(l)2337 1886 y FV(\000)-14 b(!)19 b F0(Q)2550 1856 y FT(B)2615 1886 y F0(in)j(the)h(original)e(tr) o(ansition)450 2009 y(system)g(if)o(f)g(P)832 1979 y FT(A)948 1962 y(l)891 2009 y FV(\000)-14 b(!)19 b F0(Q)1104 1979 y FT(B)1104 2033 y Gb(0)1166 2009 y F0(suc)o(h)h(that)g(Q)1548 2021 y Gb(0)1601 2009 y FV(\021)e F0(Q)j(in)f(the)g(prime)h(syntactic)e (tr)o(ansition)h(system.)533 2228 y(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(C.2.)p 1515 2228 50 50 v 450 2386 a(Note)23 b(\(3\))f(indicates)g(that)h(the)f(prime)g(syntactic)h (transition)f(is)h(precisely)f(the)h(transition)f(which)g(cor)n(-)450 2494 y(responds)d(to)h(Con)m(v)o(ention)e(3.1.)533 2602 y(Based)28 b(on)f(typed)f(transition,)j(we)e(de\002ne)g(a)h (bisimulation.)46 b(Let)27 b(us)h(say)g(a)g(relation)e(o)o(v)o(er)g (typed)450 2710 y(processes)19 b(is)h F0(typed)g Ge(if)g(it)g(only)e (relates)h(processes)g(with)g(identical)g(action)f(type.)24 b(A)c(typed)e(relation)g(is)450 2817 y(a)24 b F0(typed)e(congruence)f Ge(when)i(it)g(is)h(a)g(typed)e(equi)n(v)n(alence)f(which)i(contains)f FV(\021)h Ge(and)g(which)g(is)h(closed)450 2955 y(under)18 b(each)i(typing)f(rule)g(\(allo)n(wing,)g(as)h(a)h(result,)e(weak)o (ening)g(of)g(bases,)h(cf.)g([11,)12 b(55]\).)24 b(Belo)n(w)3368 2893 y Gb(\210)3362 2908 y FT(l)3305 2955 y FX(=)-14 b FV(\))450 3063 y Ge(denotes)19 b(the)i(standard)e(abstracted)g (transition.)533 3220 y FG(Definition)31 b(4.1.)69 b Ge(\(typed)19 b(bisimulation\))g(A)i(typed)f(relation)g F2(R)h Ge(is)h(a)f F0(weak)g(bisimulation)p Ge(,)e(or)i(a)450 3344 y F0(bisimulation)p Ge(,)j(if)h F0(P)1039 3303 y FT(A)1077 3315 y FD(1)1031 3370 y Gb(1)1109 3344 y F2(R)p F0(Q)1229 3303 y FT(A)1267 3315 y FD(1)1229 3370 y Gb(1)1325 3344 y Ge(implies:)34 b(whene)n(v)o(er)22 b F0(P)2022 3303 y FT(A)2060 3315 y FD(1)2014 3370 y Gb(1)2170 3297 y FT(l)2113 3344 y FV(\000)-14 b(!)21 b F0(P)2319 3303 y FT(A)2357 3315 y FD(2)2311 3370 y Gb(2)2414 3344 y Ge(then)j(there)g(is)i(a)f(typed)e(transition)450 3489 y(sequence)i F0(Q)841 3448 y FT(A)879 3460 y FD(1)841 3514 y Gb(1)996 3427 y(\210)990 3442 y FT(l)933 3489 y FX(=)-14 b FV(\))21 b F0(Q)1148 3448 y FT(A)1186 3460 y FD(2)1148 3514 y Gb(2)1245 3489 y Ge(such)26 b(that)g F0(P)1626 3448 y FT(A)1664 3460 y FD(2)1618 3514 y Gb(2)1696 3489 y F2(R)p F0(Q)1816 3448 y FT(A)1854 3460 y FD(2)1816 3514 y Gb(2)1887 3489 y Ge(,)h(as)g(well)f(as)h(the)f(symmetric)e (case.)43 b(By)26 b(replacing)513 3572 y Gb(\210)507 3587 y FT(l)450 3634 y FX(=)-14 b FV(\))25 b Ge(with)839 3587 y FT(l)782 3634 y FV(\000)-14 b(!)p Ge(,)26 b(we)f(obtain)f(a)h F0(str)l(ong)g(bisimulation)p Ge(.)38 b(If)24 b F0(P)2221 3603 y FT(A)2272 3634 y F2(R)9 b F0(Q)2401 3603 y FT(A)2468 3634 y Ge(for)24 b(some)h(weak)g(\(resp.)f(strong\))450 3742 y(bisimulation)19 b F2(R)p Ge(,)h(we)h(write)f F0(P)1347 3711 y FT(A)1407 3742 y FV(\031)e F0(Q)1550 3711 y FT(A)1613 3742 y Ge(\(resp.)h F0(P)1871 3711 y FT(A)1931 3742 y FV(\030)f F0(Q)2074 3711 y FT(A)2115 3742 y Ge(\).)556 3899 y(W)-7 b(e)23 b(often)e(omit)h F0(A)g Ge(from)f F0(P)1363 3869 y FT(A)1405 3899 y Ge(,)h(writing)g F0(P)d FV(\031)g F0(Q)p Ge(,)k(if)f F0(A)g Ge(is)h(clear)f(from)f(the)h(conte) o(xt.)29 b(By)23 b(de\002nition,)450 4007 y FV(\031)31 b Ge(\(resp.)e FV(\030)p Ge(\))h(is)i(the)e(union)g(of)g(all)h(weak)f (bisimulations)g(\(resp.)55 b(strong)29 b(bisimulations\),)j(which)450 4115 y(is)d(in)f(f)o(act)g(the)g(lar)o(gest)g(weak)f(\(resp.)48 b(strong\))27 b(bisimulation,)i(and)e(is)i(called)f F0(weak)h Ge(\(resp.)f F0(str)l(ong)p Ge(\))450 4223 y F0(bisimilarity)p Ge(.)46 b(The)27 b(follo)n(wing)e(technical)i(de)n(v)o(elopment)d (focusses)j(on)f(weak)h(bisimilarity)-5 b(,)28 b(which)450 4331 y(we)j(hereafter)e(simply)h(call)g F0(bisimilarity)p Ge(.)56 b FV(\031)31 b Ge(is)g(clearly)f(an)g(equi)n(v)n(alence)e (relation.)55 b(Since)30 b FV(\021)g Ge(is)450 4439 y(easily)g(a)h (bisimulation,)g(by)f(Proposition)f(4.3,)j(it)f(is)g(enough)d(to)i(use) h(the)f(syntactic)g(transition)f(to)450 4547 y(deri)n(v)o(e)19 b F0(P)f FV(\031)g F0(Q)j Ge(\(and)e(the)h(prime)g(one)f(if)i(we)f(are) g(under)f(Con)m(v)o(ention)f(3.1\).)1665 4747 y F3(4.2.)94 b(Axioms)450 4865 y Ge(Let)25 b F1(\301)f Ge(\()p F1(\301)752 4835 y Gd(0)773 4865 y Ge(,...\))37 b(denote)23 b(a)i(formal)e (\(equational\))f(theory)h(o)o(v)o(er)g(typed)g(processes,)i(which)f (is)h(a)g(set)g(of)450 4973 y(axioms)e(and)f(rules)h(with)g(formulae)f (of)h(the)g(form)f F0(P)1991 4943 y FT(A)2052 4973 y FX(=)e F0(Q)2197 4943 y FT(A)2238 4973 y Ge(.)35 b(In)23 b F0(P)2438 4943 y FT(A)2499 4973 y FX(=)d F0(Q)2644 4943 y FT(A)2685 4973 y Ge(,)k F0(P)2781 4943 y FT(A)2846 4973 y Ge(and)f F0(Q)3050 4943 y FT(A)3115 4973 y Ge(should)f(be)p eop %%Page: 24 24 24 23 bop 450 -257 a FX(24)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)450 -4 y Ge(well-typed:)25 b(we)c(shall)g(ho)n(we)n(v)o(er)e(not)h(mention)g(types)g(unless)h(the) o(y)f(are)h(necessary)-5 b(,)19 b(writing)h F0(P)f FX(=)f F0(Q)p Ge(.)450 104 y(If)k F0(P)e FX(=)f F0(Q)k Ge(is)h(pro)o(v)n(able) c(in)j F1(\301)p Ge(,)h(we)f(write)f F1(\301)e FV(`)g F0(P)f FX(=)h F0(Q)p Ge(.)32 b F1(\301)12 b FX(+)g F1(\301)2277 74 y Gd(0)2322 104 y Ge(is)23 b(the)g(result)g(of)f(adding)f(the)i (axioms)450 212 y(and)d(rules)g(from)f(tw)o(o)h(theories.)25 b(W)-7 b(e)21 b(e)o(xtend)e(this)i(to)f(an)g(arbitrary)f(f)o(amily)g (of)h(theories.)450 369 y F2(Axioms)d(I:)h(\(Pr)o(e\)Congruence)d (Rules.)83 b Ge(W)-7 b(e)18 b(consider)e(the)h(standard)f(equi)n(v)n (alence)f(rules)i(and)g(clo-)450 477 y(sure)h(under)f(well-typed)g (conte)o(xts.)23 b(This)18 b(theory)f(is)j(denoted)c F1(\301)2320 489 y FT(c)2352 477 y Ge(.)24 b(W)-7 b(e)20 b(also)e(de\002ne)g(its)h(subtheory)d F1(\301)3404 489 y FT(p)450 585 y Ge(by)k(remo)o(ving)e FM(\(C1\))i Ge(from)f F1(\301)1297 597 y FT(c)1328 585 y Ge(.)799 790 y FM(\(C1\))41 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h F0(Q)18 b FX(=)g F0(P)358 b FM(\(C2\))40 b F0(P)18 b FX(=)g F0(Q)p FZ(;)30 b F0(Q)19 b FX(=)f F0(R)46 b FV(\))g F0(P)18 b FX(=)g F0(R)799 922 y FM(\(C3\))41 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h F0(P)11 b FV(j)h F0(R)18 b FX(=)g F0(Q)12 b FV(j)g F0(R)163 b FM(\(C4\))40 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h F0(R)12 b FV(j)g F0(P)17 b FX(=)h F0(R)12 b FV(j)g F0(Q)799 1053 y FM(\(C5\))41 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h FX(\()p F1(n)9 b F0(x)p FX(\))p F0(P)19 b FX(=)f(\()p F1(n)9 b F0(x)p FX(\))p F0(Q)51 b FM(\(C6\))40 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)18 b FX(=)g F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)799 1185 y FM(\(C7\))41 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))p 1374 1139 38 4 v 47 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)18 b FX(=)p 1664 1139 V 18 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(Q)82 b FM(\(C8\))40 b F0(P)18 b FX(=)g F0(Q)46 b FV(\))h Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)o FX(\))p FZ(:)p F0(P)19 b FX(=)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)o FX(\))p FZ(:)p F0(Q)450 1440 y F2(Axioms)28 b(II:)g(Structural)g(Rules.)83 b Ge(Let)28 b F1(\301)1762 1452 y FT(s)1818 1440 y Ge(denote)f(the)h(set) g(of)g(rules)f(deri)n(v)o(ed)f(from)h(the)h(axioms)450 1548 y FM(\(S0)p Ge(\226)p FM(9\))19 b Ge(in)h(Figure)g(1.)25 b(Hence)20 b F0(P)e FV(\021)g F0(Q)i Ge(stands)h(for)e F1(\301)1974 1560 y FT(c)2017 1548 y FX(+)12 b F1(\301)2151 1560 y FT(s)2197 1548 y FV(`)18 b F0(P)g FX(=)g F0(Q)p Ge(.)450 1706 y F2(Axioms)k(III:)h(Con)m(v)o(ersion)e(Rules.)83 b Ge(Con)m(v)o(ertibility)20 b(is)j(induced)d(by)i(the)g(e)o(xtended)e (reduction)g(re-)450 1814 y(lation,)j(taking)f FM(\(E1)p Ge(\226)p FM(3\))g Ge(from)g(De\002nition)g(3.1)h(as)g(rules.)34 b F1(\301)2202 1826 y FT(e)2256 1814 y Ge(denotes)23 b(the)g(theory)-5 b(.)31 b(Note)23 b F0(P)d FV(7!)g F0(Q)j Ge(if)n(f)450 1922 y F1(\301)512 1934 y FT(p)558 1922 y FX(+)12 b F1(\301)692 1934 y FT(s)731 1922 y FX(+)g F1(\301)865 1934 y FT(e)913 1922 y FV(`)19 b F0(P)f FX(=)g F0(Q)p Ge(.)533 2079 y FG(Definition)32 b(4.2.)41 b Ge(The)22 b(typed)e(congruence)f FV( )-47 b(!)22 b Ge(is)h(de\002ned)e(by)g(the)g (follo)n(wing)g(logical)g(equi)n(v-)450 2187 y(alence:)k F0(P)18 b FV( )-47 b(!)18 b F0(Q)j Ge(if)n(f)f F1(\301)1152 2199 y FT(c)1195 2187 y FX(+)12 b F1(\301)1329 2199 y FT(s)1368 2187 y FX(+)g F1(\301)1502 2199 y FT(e)1550 2187 y FV(`)19 b F0(P)f FX(=)g F0(Q)p Ge(.)554 2345 y(In)i(other)f(w)o (ords,)h FV( )-47 b(!)20 b Ge(is)h(the)g(symmetric)e(and)g(transiti)n (v)o(e)h(closure)f(of)h FV(7!)f([)g(\021)p Ge(.)1186 2559 y F3(4.3.)94 b(Characterisation)33 b(and)f(its)f(Pro)s(of)533 2667 y Ge(W)-7 b(e)21 b(no)n(w)f(sho)n(w)g(that)g FV( )-47 b(!)21 b Ge(completely)d(characterises)i(bisimilarity)-5 b(.)533 2848 y FG(Theorem)31 b(4.1.)62 b FF(\(char)l(acterisation)31 b(of)g FV(\031)p FF(\))89 b FV( )-47 b(!)29 b FX(=)g FV(\031)p FF(.)556 3006 y Ge(W)-7 b(e)24 b(pro)o(v)o(e)c(Theorem)h(4.1) h(by)g(sho)n(wing)f(tw)o(o)i(inclusions,)f(\(1\))g FV( )-47 b(!)23 b(\032)g(\031)f Ge(and)g(\(2\))g FV( )-47 b(!)23 b(\033)f(\031)p Ge(.)32 b(W)-7 b(e)450 3114 y(call)19 b(the)g(\002rst)g(inclusion)f F0(soundness)f Ge(and)h(the)h(second)e (one)h F0(completeness)p Ge(.)24 b(F)o(or)18 b(soundness,)g(we)h (\002rst)450 3222 y(sho)n(w)h FV(\031)g Ge(is)h(a)g(typed)e (congruence.)533 3403 y FG(Pr)n(oposition)31 b(4.4.)62 b FV(\031)29 b FF(is)h(a)h(typ)l(e)l(d)f(c)l(ongruenc)l(e.)533 3583 y F0(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(C.3)p 1495 3583 50 50 v 533 3741 a(Ne)o(xt)i(we)g(sho)n(w:)533 3922 y FG(Pr)n(oposition)31 b(4.5.)62 b FF(If)30 b F1(\301)1384 3934 y FT(e)1434 3922 y FV(`)18 b F0(P)g FX(=)g F0(Q)30 b FF(then)g F0(P)18 b FV(\031)g F0(Q)p FF(.)533 4103 y F0(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(C.4.)p 1515 4103 V 533 4261 a(Since)i FV( )-47 b(!)21 b Ge(is)g(the)f (congruent)e(closure)h(of)h F1(\301)1836 4273 y FT(e)1868 4261 y Ge(,)g(by)g(Propositions)f(4.4)g(and)h(4.5)g(we)g(conclude:)533 4442 y FG(Cor)n(ollar)-5 b(y)33 b(4.1.)91 b FV( )-47 b(!)30 b(\032)f(\031)p FF(.)554 4599 y Ge(F)o(or)20 b(the)g(re)n(v)o (erse)f(inclusion,)g(we)h(reduce)f(the)i(equality)e(by)h FV( )-47 b(!)20 b Ge(to)h(those)f(o)o(v)o(er)e(normal)h(forms.)533 4757 y FG(Definition)28 b(4.3.)61 b Ge(Let)19 b(us)h(write)f F0(P)e FV(\021)1723 4727 y Gd(0)1761 4757 y F0(Q)i Ge(for)g F1(\301)2014 4769 y FT(c)2056 4757 y FX(+)10 b FV(f)p FM(\(S0,S2,S3,S5\2269\))-5 b FV(g)17 b(`)g F0(P)g FX(=)g F0(Q)i Ge(and)g F0(P)e Fl(B)g F0(Q)450 4865 y Ge(for)23 b F1(\301)633 4877 y FT(p)681 4865 y FX(+)13 b F1(\301)816 4877 y FT(s)864 4865 y FV(`)20 b F0(P)h FX(=)f F0(Q)k Ge(\(note)g Fl(B)g Ge(is)h(a)f F0(pr)m(econgruence)p Ge(\).)34 b(W)-7 b(e)25 b(say)g F0(P)f Ge(is)h(in)f Fl(B)p Ge(-)p F0(normal)f(form)i Ge(if:)33 b(\(1\))450 4973 y F0(P)18 b FV(2)h FM(NF)693 4985 y FT(e)745 4973 y Ge(and)g(\(2\))h F0(P)e Fl(B)g F0(Q)j Ge(implies)f F0(P)e FV(\021)1635 4943 y Gd(0)1674 4973 y F0(Q)p Ge(.)p eop %%Page: 25 25 25 24 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(25)556 -4 y Ge(Note)23 b F0(P)c FV(\021)876 -34 y Gd(0)916 -4 y F0(Q)k Ge(means)f(that)h F0(P)g Ge(and)f F0(Q)h Ge(are)f(essentially) h(identical)f(without)g(changing)e(the)i(size)i(of)450 104 y(terms.)h(F)o(or)20 b Fl(B)p Ge(-normal)e(forms)h(we)i(observ)o (e:)533 272 y FG(Lemma)30 b(4.1.)533 440 y Ge(\(1\))54 b F0(A)21 b(pr)l(ocess)f(in)h FM(NF)1212 452 y FT(e)1264 440 y F0(is)h(a)e Fl(B)p F0(-normal)f(form)h(if)h(it)g(does)f(not)g (contain)f F2(0)h F0(as)h(its)g(pr)l(oper)f(subterm.)533 555 y Ge(\(2\))13 b F0(If)20 b FV(`)e F0(P)12 b FZ(.)g F0(A)19 b(then)h(ther)m(e)g(is)h(a)g Fl(B)p F0(-normal)d(form)j(Q)g (suc)o(h)e(that)h(P)e FV(7!)2546 525 y Gd(\003)2600 555 y F0(Q.)533 670 y Ge(\(3\))13 b F0(The)25 b(set)h(of)f Fl(B)p F0(-normal)f(forms)i(coincide)d(with)j(those)f(pr)l(ocesses)h(g) o(ener)o(ated)d(by)i(the)h(rules)f(in)450 778 y(Pr)l(oposition)19 b(3.3.)533 893 y Ge(\(4\))54 b F0(If)21 b FV(`)d F0(P)11 b FZ(.)h F0(A)20 b(and)f(P)h(is)i(a)e Fl(B)p F0(-normal)f(form)h(then)g (P)e FV(\021)2196 862 y Gd(0)2235 893 y F0(P)2278 907 y Gd(#)2312 893 y FV(j)p F0(P)2378 907 y Gd(")2413 893 y FV(j)p F0(P)2479 905 y Gb(!)2507 893 y FV(j)p F0(P)2573 905 y Gb(?)2629 893 y F0(wher)m(e:)1174 1096 y(P)1217 1110 y Gd(#)1301 1096 y FX(=)49 b F1(P)1479 1108 y FT(i)p Gd(2)p FT(I)1555 1121 y Fk(#)1589 1096 y F0(y)1626 1108 y FT(i)1647 1096 y FX(\()-8 b FZ(~)-34 b F0(z)1711 1108 y FT(i)1733 1096 y FX(\))p FZ(:)p F0(P)1831 1108 y FT(i)2059 1096 y F0(P)2102 1110 y Gd(")2186 1096 y FX(=)49 b F1(P)2364 1108 y FT(i)p Gd(2)p FT(I)2440 1121 y Fk(")p 2474 1050 37 4 v 2474 1096 a F0(y)2511 1116 y FT(i)2532 1096 y FX(\()-8 b FZ(~)-34 b F0(z)2596 1108 y FT(i)2618 1096 y FX(\))p F0(P)2693 1108 y FT(i)1180 1207 y F0(P)1223 1219 y Gb(!)1301 1207 y FX(=)49 b F1(P)1479 1219 y FT(i)p Gd(2)p FT(I)1555 1231 y FD(!)1585 1207 y Ge(!)p F0(y)1650 1219 y FT(i)1671 1207 y FX(\()-8 b FZ(~)-34 b F0(z)1735 1219 y FT(i)1756 1207 y FX(\))p FZ(:)p F0(P)1854 1219 y FT(i)2058 1207 y F0(P)2101 1219 y Gb(?)2186 1207 y FX(=)49 b F1(P)2364 1219 y FT(i)p Gd(2)p FT(I)2440 1231 y FD(?)p 2475 1161 V 2475 1207 a F0(y)2512 1227 y FT(i)2533 1207 y FX(\()-8 b FZ(~)-34 b F0(z)2597 1219 y FT(i)2619 1207 y FX(\))p F0(P)2694 1219 y FT(i)450 1403 y F0(Her)m(e)17 b(I)656 1417 y Gd(#)691 1403 y FZ(;)9 b F0(I)749 1417 y Gd(")784 1403 y FZ(;)g F0(I)842 1415 y Gb(!)870 1403 y FZ(;)g F0(I)928 1415 y Gb(?)981 1403 y F0(partition)16 b(the)h(\002nite)g(set)h(I)j(suc)o(h)c(that)f Ge(\(i\))h F0(for)h(all)f(i)p FZ(;)k F0(j)d FV(2)e F0(I)c FV(n)d F0(I)2678 1415 y Gb(?)2713 1403 y F0(:)24 b(i)15 b FV(6)p FX(=)27 b F0(j)20 b(implies)e(x)3233 1415 y FT(i)3269 1403 y FV(6)p FX(=)d F0(x)3395 1415 y FT(j)3418 1403 y F0(,)450 1511 y Ge(\(ii\))k F0(for)g(all)g(i)f FV(2)f F0(I)933 1523 y Gb(!)972 1511 y FV([)10 b F0(I)1063 1523 y Gb(?)1118 1511 y F0(and)18 b(all)31 b(j)20 b FV(2)e F0(I)1523 1525 y Gd(#)1568 1511 y FV([)10 b F0(I)1659 1525 y Gd(")1694 1511 y F0(:)24 b(x)1783 1523 y FT(i)1822 1511 y FV(6)p FX(=)17 b F0(x)1950 1523 y FT(j)1992 1511 y F0(and)h Ge(\(iii\))h F0(P)2323 1523 y FT(i)2363 1511 y F0(is)h(in)f Fl(B)p F0(-normal)e(form)j(for)f(all)g(i)f FV(2)f F0(I)t(.)450 1619 y(Furthermor)m(e)o(,)i(P)959 1633 y Gd(#)994 1619 y F0(,)h(P)1078 1633 y Gd(")1113 1619 y F0(,)g(P)1197 1631 y Gb(!)1246 1619 y F0(and)f(P)1434 1631 y Gb(?)1490 1619 y F0(ar)m(e)h(unique)f(up)h(to)g FV(\021)2119 1589 y Gd(0)2140 1619 y F0(.)533 1758 y Ge(\(5\))54 b F0(If)21 b FV(`)d F0(P)11 b FZ(.)h F0(A)20 b(is)h(a)f Fl(B)p F0(-normal)f(form)h(and)g(P)1953 1711 y FT(l)1896 1758 y FV(\000)-14 b(!)18 b F0(Q)j(is)g(a)f(tr)o(ansition,) f(then)h(l)j FV(6)p FX(=)18 b F1(t)p F0(.)533 1999 y(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(C.5.)p 1515 1999 50 50 v 533 2156 a(Let)25 b F0(P)h Ge(be)f(a)h Fl(B)p Ge(-normal)d(form.)39 b(Then)24 b F0(P)d FV(\021)1822 2126 y Gd(0)1864 2156 y F0(P)1907 2170 y Gd(#)1941 2156 y FV(j)p F0(P)2007 2170 y Gd(")2041 2156 y FV(j)p F0(P)2107 2168 y Gb(!)2135 2156 y FV(j)p F0(P)2201 2168 y Gb(?)2262 2156 y Ge(by)k(Lemma)f(4.1)h (\(4\).)39 b(The)25 b(right-hand)450 2264 y(side)30 b(of)g(this)g (equation)e(is)j(called)e F0(normal)g(form)i(decomposition)c Ge(of)j F0(P)p Ge(,)i(with)e F0(P)2891 2278 y Gd(#)2925 2264 y Ge(,)i F0(P)3021 2278 y Gd(")3056 2264 y Ge(,)g F0(P)3152 2276 y Gb(!)3211 2264 y Ge(and)d F0(P)3404 2276 y Gb(?)450 2372 y Ge(being,)19 b(respecti)n(v)o(ely)-5 b(,)18 b(its)j FV(#)p Ge(-)p F0(component)p Ge(,)c FV(")p Ge(-)p F0(component)i Ge(!)t(-)p F0(component)g Ge(and)g(?)t(-)p F0(component)p Ge(.)533 2541 y FG(Lemma)30 b(4.2.)62 b FF(L)l(et)29 b F0(P)1221 2510 y FT(i)1213 2568 y Gd(#)1247 2541 y FV(j)p F0(P)1321 2510 y FT(i)1313 2568 y Gd(")1348 2541 y FV(j)p F0(P)1422 2510 y FT(i)1414 2566 y Gb(!)1443 2541 y FV(j)p F0(P)1517 2510 y FT(i)1509 2566 y Gb(?)1573 2541 y FF(b)l(e)h(a)g(normal)h(form)f(de)l(c)l(omp)l(osition)i F0(P)2817 2510 y FT(i)2867 2541 y FF(\()p F0(i)19 b FX(=)f Ge(1)p FZ(;)9 b Ge(2)p FF(\).)533 2728 y Ge(\(1\))52 b FF(Assume)39 b(that)h F0(P)1234 2698 y Gb(1)1226 2754 y Gd(#)1292 2728 y FX(=)23 b F1(P)1444 2698 y FT(m)1453 2753 y(j)r FU(=)p Gb(1)1554 2728 y F0(y)1591 2740 y FT(i)1613 2728 y FX(\()-8 b FZ(~)-34 b F0(z)1677 2740 y FT(i)1698 2728 y FX(\))p FZ(:)p F0(P)1804 2698 y Gb(1)1805 2751 y FT(j)1879 2728 y FF(and)40 b F0(P)2101 2698 y Gb(2)2093 2754 y Gd(#)2159 2728 y FX(=)23 b F1(P)2311 2698 y FT(n)2320 2753 y(j)r FU(=)p Gb(1)2422 2728 y F0(a)2464 2740 y FT(i)2484 2728 y FX(\()2512 2707 y FZ(~)2516 2728 y F0(b)2558 2740 y FT(i)2579 2728 y FX(\))p FZ(:)p F0(P)2685 2698 y Gb(2)2686 2751 y FT(j)2720 2728 y FF(.)68 b(Then)41 b F0(P)3091 2698 y Gb(1)3083 2754 y Gd(#)3149 2728 y FV(\031)23 b F0(P)3288 2698 y Gb(2)3280 2754 y Gd(#)3362 2728 y FF(i\013)450 2854 y F0(m)16 b FX(=)g F0(n)26 b FF(and)g(ther)l(e)h(is)f(a)h(p)l (ermutation)f F1(s)g FF(of)h FV(f)p Ge(1)p FZ(;)9 b(:)g(:)g(:)g(;)g F0(n)p FV(g)25 b FF(such)h(that)g F0(y)2563 2866 y FT(i)2584 2854 y FX(\()-8 b FZ(~)-34 b F0(z)2648 2866 y FT(i)2670 2854 y FX(\))p FZ(:)p F0(P)2776 2824 y Gb(1)2768 2877 y FT(i)2827 2854 y FV(\031)16 b F0(a)2950 2871 y Gc(s)p FU(\()p FT(i)p FU(\))3056 2854 y FX(\()3126 2834 y FZ(~)3088 2854 y F0(b)3130 2871 y Gc(s)p FU(\()p FT(i)p FU(\))3235 2854 y FX(\))p FZ(:)p F0(P)3341 2824 y Gb(2)3333 2883 y Gc(s)p FU(\()p FT(i)p FU(\))450 2989 y FF(for)31 b(al)t(l)f F0(i)p FF(.)39 b(Similarly)32 b(for)e F0(P)1327 2948 y Gb(1)p FS(;)p Gb(2)1319 3016 y Gd(")1409 2989 y FF(,)h F0(P)1516 2948 y Gb(1)p FS(;)p Gb(2)1508 3014 y(!)1628 2989 y FF(and)f F0(P)1840 2948 y Gb(1)p FS(;)p Gb(2)1832 3014 y(?)1922 2989 y FF(.)533 3115 y Ge(\(2\))42 b F0(P)724 3085 y Gb(1)777 3115 y FV(\031)18 b F0(P)911 3085 y Gb(2)976 3115 y FF(i\013)30 b F0(P)1133 3085 y Gb(1)1125 3141 y Gd(#)1186 3115 y FV(\031)18 b F0(P)1320 3085 y Gb(2)1312 3141 y Gd(#)1354 3115 y FF(,)31 b F0(P)1461 3085 y Gb(1)1453 3141 y Gd(")1514 3115 y FV(\031)18 b F0(P)1648 3085 y Gb(2)1640 3141 y Gd(")1682 3115 y FF(,)30 b F0(P)1788 3085 y Gb(1)1780 3139 y(!)1841 3115 y FV(\031)18 b F0(P)1975 3085 y Gb(2)1967 3139 y(!)2040 3115 y FF(and)30 b F0(P)2252 3085 y Gb(1)2244 3139 y(?)2305 3115 y FV(\031)18 b F0(P)2439 3085 y Gb(2)2431 3139 y(?)2473 3115 y FF(.)533 3347 y F0(Pr)l(oof)o(.)81 b Ge(F)o(or)25 b(\(1\),)g(the)g(cases)h(for)f F0(P)1606 3306 y Gb(1)p FS(;)p Gb(2)1598 3374 y Gd(#)1688 3347 y Ge(,)i F0(P)1787 3306 y Gb(1)p FS(;)p Gb(2)1779 3374 y Gd(")1895 3347 y Ge(and)d F0(P)2091 3306 y Gb(1)p FS(;)p Gb(2)2083 3372 y(!)2199 3347 y Ge(are)h(immediate)g(by)f (considering)f(traces.)450 3476 y(F)o(or)17 b F0(P)633 3435 y Gb(1)p FS(;)p Gb(2)625 3502 y(?)716 3476 y Ge(,)h(we)g(proceed)e (by)h(contradiction.)22 b(Assume)17 b(w)-5 b(.l.o.g.)16 b F0(P)2334 3446 y Gb(1)2326 3500 y(?)2385 3476 y FV(\021)2450 3446 y Gd(0)p 2486 3430 38 4 v 2486 3476 a F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)2667 3488 y Gb(1)2702 3476 y FV(j)p 2725 3430 V F0(x)p FX(\()-15 b FZ(~)-27 b F0(z)q FX(\))p F0(P)2902 3488 y Gb(2)2936 3476 y FV(j)p F0(P)3010 3446 y Gd(0)3049 3476 y Ge(while)18 b F0(P)3303 3446 y Gb(2)3295 3500 y(?)3353 3476 y FV(\021)3418 3446 y Gd(0)p 450 3538 V 450 3584 a F0(x)p FX(\()-8 b FZ(~)-34 b F0(a)p FX(\))p F0(Q)p FV(j)p F0(Q)736 3554 y Gd(0)776 3584 y Ge(such)19 b(that)f(neither)g F0(P)1392 3554 y Gd(0)1432 3584 y Ge(nor)g F0(Q)1622 3554 y Gd(0)1662 3584 y Ge(contain)g F0(x)h Ge(as)h(an)e(acti)n(v)o(e)h(name.)k(By)d (Lemma)e(3.2,)g(all)h(acti)n(v)o(e)450 3692 y(names)k(in)g F0(P)817 3704 y Gb(1)851 3692 y FZ(;)9 b F0(P)926 3704 y Gb(2)985 3692 y Ge(and)22 b F0(Q)i Ge(are)f(in)g FV(f)-10 b FZ(~)-32 b F0(y)o FV(g)p Ge(,)24 b FV(f)-15 b FZ(~)-27 b F0(z)p FV(g)p Ge(,)23 b(and)g FV(f)-8 b FZ(~)-34 b F0(a)o FV(g)p Ge(,)24 b(respecti)n(v)o(ely)-5 b(.)31 b(T)-7 b(yping)22 b(then)h(ensures)g(that)g(all)450 3800 y(these)g(acti)n(v)o(e)f(names)g(are)h(inputs.)32 b(By)23 b(Lemma)f(4.1)g(\(5\),)h(no)f(process)g(in)h Fl(B)p Ge(-normal)e(form)g (can)i(ha)n(v)o(e)450 3928 y(a)17 b F1(t)p Ge(-transition.)23 b(Hence)16 b F0(P)1207 3898 y Gb(1)1199 3952 y(?)1258 3928 y Ge(and)g F0(P)1446 3898 y Gb(2)1438 3952 y(?)1498 3928 y Ge(cannot)f(ha)n(v)o(e)h(the)g(same)h(set)g(of)f(traces.)24 b(\(2\))16 b(follo)n(ws)g(from)f(\(1\).)p 3389 3928 50 50 v 450 4086 a(W)-7 b(e)26 b(no)n(w)d(pro)o(v)o(e)g(the)i(k)o(e)o(y)e (lemma)h(for)g(completeness.)37 b(F)o(or)24 b(\(2\).)37 b(we)25 b(can)f(indeed)f(sho)n(w)h Fl(B)p Ge(-normal)450 4194 y(forms)19 b(are)i(a)f(class)h(of)f(processes)g(where)g FV(\031)p Ge(,)g FV(\030)p Ge(,)g FV(\021)1956 4164 y Gd(0)1997 4194 y Ge(and)g FV(\021)g Ge(all)h(coincide.)533 4362 y FG(Lemma)30 b(4.3.)91 b FF(L)l(et)30 b F0(P)f FF(and)h F0(Q)g FF(b)l(e)g Fl(B)p FF(-normal)g(forms.)39 b(Then)30 b F0(P)18 b FV(\031)g F0(Q)30 b FF(i\013)g F0(P)18 b FV(\021)2979 4332 y Gd(0)3018 4362 y F0(Q)p FF(.)533 4514 y F0(Pr)l(oof)o(.)78 b Ge(The)24 b F0(size)h Ge(of)f F0(P)p Ge(,)h Fw(size)p FX(\()p F0(P)p FX(\))p Ge(,)g(is)g(the)f(number)f(of)h(constructors)e(in)i F0(P)p Ge(.)37 b Fw(size)p FX(\()p F0(P)p FX(\))25 b Ge(is)g(in)m(v)n(ariant) 450 4622 y(under)g FV(\021)731 4592 y Gd(0)751 4622 y Ge(.)44 b(By)26 b(induction)f(on)g Fw(size)p FX(\()p F0(P)p FX(\))14 b(+)g Fw(size)o FX(\()p F0(Q)p FX(\))27 b Ge(we)f(sho)n(w)g FV(\031\032\021)2515 4592 y Gd(0)2535 4622 y Ge(.)43 b(The)26 b(base)g(case,)i Fw(size)o FX(\()p F0(P)p FX(\))14 b(+)450 4730 y Fw(size)p FX(\()p F0(Q)p FX(\))21 b(=)g Ge(2,)27 b(is)f(immediate.)40 b(The)25 b(inducti)n(v)o(e)f(step)h(uses)h(Lemma)f(4.2)g(\(1,2\))f(to)h(reduce)g (the)g(ar)o(gu-)450 4838 y(ment)g(to)g(each)g(prime)f(component)f(for)h (which,)i(after)f(stripping)f(of)n(f)g(the)h(common)f(pre\002x,)h(we)h (can)450 4973 y(al)o(w)o(ays)16 b(use)g(induction.)21 b(Since)16 b FV(\021)1446 4943 y Gd(0)1482 4973 y Ge(is)h(easily)f(a)g (bisimulation)e(we)i(also)g(ha)n(v)o(e)f FV(\021)2746 4943 y Gd(0)2766 4973 y FV(\032\031)p Ge(,)h(hence)f(done.)p 3389 4973 V eop %%Page: 26 26 26 25 bop 450 -257 a FX(26)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)p 450 -71 2989 5 v 658 171 a FX([)p F0(V)-9 b(ar)o FX(])83 b F0(E)6 b FZ(;)j F0(x)g Ge(:)g F0(T)30 b FV(`)18 b F0(x)h Ge(:)f F0(T)376 b FX([)p F0(Unit)p FX(])83 b F0(E)24 b FV(`)18 b FX(\(\))h Ge(:)g Fj(unit)658 389 y FX([)p F0(Lam)o FX(])1048 347 y F0(E)6 b FZ(;)j F0(x)g Ge(:)g F0(T)30 b FV(`)18 b F0(M)k Ge(:)c F0(T)1549 313 y Gd(0)p 944 370 730 4 v 944 441 a F0(E)25 b FV(`)18 b F1(l)p F0(x)g Ge(:)g F0(T)5 b FZ(:)p F0(M)22 b Ge(:)c F0(T)i FV(\))9 b F0(T)1652 407 y Gd(0)1733 389 y FX([)p F0(App)o FX(])2006 356 y F0(E)24 b FV(`)18 b F0(M)k Ge(:)d F0(T)h FV(\))9 b F0(T)2498 322 y Gd(0)2602 356 y F0(E)24 b FV(`)18 b F0(N)24 b Ge(:)18 b F0(T)p 2006 370 918 4 v 2257 441 a(E)24 b FV(`)18 b F0(M)s(N)25 b Ge(:)18 b F0(T)2651 407 y Gd(0)658 643 y FX([)p F0(P)-7 b(air)p FX(])962 597 y F0(E)25 b FV(`)18 b F0(M)1176 609 y FT(i)1216 597 y Ge(:)g F0(T)1297 609 y FT(i)1337 597 y FX(\()p F0(i)g FX(=)g Ge(1)p FZ(;)9 b Ge(2)p FX(\))p 938 624 728 4 v 938 689 a F0(E)24 b FV(`)18 b(h)p F0(M)1183 701 y Gb(1)1218 689 y FZ(;)9 b F0(M)1319 701 y Gb(2)1355 689 y FV(i)18 b Ge(:)h F0(T)1487 701 y Gb(1)1524 689 y FV(\002)r F0(T)1631 701 y Gb(2)1733 643 y FX([)p F0(Pr)l(oj)p FX(])2129 605 y F0(E)25 b FV(`)18 b F0(M)13 b Ge(:)c F0(T)2428 617 y Gb(1)2465 605 y FV(\002)r F0(T)2572 617 y Gb(2)p 1972 624 792 4 v 1972 690 a F0(E)24 b FV(`)19 b F1(p)2163 702 y FT(i)2183 690 y FX(\()p F0(M)s FX(\))h Ge(:)e F0(T)2420 702 y FT(i)2460 690 y FX(\()p F0(i)h FX(=)f Ge(1)p FZ(;)9 b Ge(2)p FX(\))658 899 y([)p F0(Inl)o FX(])1058 861 y F0(E)25 b FV(`)18 b F0(M)k Ge(:)c F0(T)1375 873 y Gb(1)p 889 880 691 4 v 889 945 a F0(E)24 b FV(`)18 b FR(inl)o FX(\()p F0(M)s FX(\))h Ge(:)g F0(T)1401 957 y Gb(1)1438 945 y FX(+)r F0(T)1545 957 y Gb(2)1733 899 y FX([)p F0(Case)p FX(])2076 857 y F0(E)24 b FV(`)18 b F0(M)13 b Ge(:)c F0(T)2374 869 y Gb(1)2412 857 y FX(+)r F0(T)2519 869 y Gb(2)2592 857 y F0(E)d FZ(;)j F0(x)2718 869 y FT(i)2749 857 y Ge(:)g F0(T)2821 869 y FT(i)2861 857 y FV(`)18 b F0(M)2999 869 y FT(i)3029 857 y Ge(:)9 b F0(T)3118 823 y Gd(0)p 1994 880 1227 4 v 1994 951 a F0(E)25 b FV(`)18 b FR(case)h F0(M)24 b FR(of)c FV(f)p FR(in)2664 963 y FT(i)2685 951 y FX(\()p F0(x)2754 963 y FT(i)2794 951 y Ge(:)e F0(T)2875 963 y FT(i)2896 951 y FX(\))p FZ(:)p F0(M)3020 963 y FT(i)3042 951 y FV(g)g Ge(:)g F0(T)3200 917 y Gd(0)1480 1133 y FK(FIG.)e(5.)49 b FW(T)-5 b(yping)17 b(Rules)h(for)f FP(l)2232 1143 y Fk(!)p Fm(;)p Fk(\002)p Fm(;)p FC(+)p 450 1241 2989 5 v 533 1486 a Ge(W)-7 b(e)19 b(can)e(no)n(w)g(conclude)f(the)i(proof)e (of)h(Theorem)f(4.1)h(by)g(establishing)g(the)h(completeness,)e FV( )-47 b(!)18 b(\033)450 1594 y(\031)p Ge(,)j(and)f(combining)f(it)i (with)g(Proposition)f(4.1.)26 b(Assume)21 b F0(P)d FV(\031)h F0(Q)p Ge(.)27 b(By)21 b(Lemma)f(4.1)h(\(2\))f(we)h(can)g(\002nd)450 1702 y Fl(B)p Ge(-normal)15 b(forms)i F0(P)1050 1715 y FT(nf)1119 1702 y Ge(and)g F0(Q)1317 1715 y FT(nf)1387 1702 y Ge(of)g F0(P)h Ge(and)f F0(Q)p Ge(,)h(respecti)n(v)o(ely)-5 b(,)16 b(such)h(that)g F0(P)f FV(7!)2674 1672 y Gd(\003)2725 1702 y F0(P)2768 1715 y FT(nf)2837 1702 y Ge(and)h F0(Q)f FV(7!)3134 1672 y Gd(\003)3184 1702 y F0(Q)3244 1715 y FT(nf)3296 1702 y Ge(.)25 b(By)450 1810 y(Corollary)d(4.1,)h(we)h (kno)n(w)e F0(P)1311 1823 y FT(nf)1383 1810 y FV(\031)e F0(Q)1528 1823 y FT(nf)1580 1810 y Ge(.)35 b(But)23 b(Lemma)g(4.3)g (implies)g(that)h FV(\031)f Ge(restricted)g(to)g Fl(B)p Ge(-normal)450 1918 y(forms)c(is)h(contained)e(in)h FV( )-47 b(!)p Ge(,)20 b(hence)f F0(P)e FV(7!)1691 1888 y Gd(\003)1744 1918 y F0(P)1787 1931 y FT(nf)1856 1918 y FV( )-47 b(!)17 b F0(Q)2052 1931 y FT(nf)2124 1918 y Ge(and)i F0(Q)e FV(7!)2424 1888 y Gd(\003)2477 1918 y F0(Q)2537 1931 y FT(nf)2609 1918 y Ge(which)i(means)g F0(P)e FV( )-47 b(!)18 b F0(Q)p Ge(,)i(as)450 2026 y(required.)877 2259 y F3(5.)95 b(FULL)-8 b(Y)33 b(ABSTRA)m(CT)g(EMBEDDING)e(OF)h F1(l)2817 2271 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)1313 2358 y F3(5.1.)94 b(The)32 b(F)-8 b(unctional)32 b(Calculus)450 2466 y Ge(W)-7 b(e)20 b(use)f(the)g(simply)f(typed)g F1(l)p Ge(-calculus)g(with)h(products)e(and)i(sums)g(\(written)f F1(l)2773 2478 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)2987 2466 y Ge(from)g(no)n(w)g(on\))450 2574 y(as)k(a)g(testbed)g(for)f(the) g(e)o(xpressi)n(v)o(eness)f(of)i(the)f(presented)g(calculus,)g (establishing)g(its)i(fully)e(abstract)450 2682 y(embeddability)15 b(in)j(the)g F1(p)p Ge(-calculus.)k(W)-7 b(e)19 b(ha)n(v)o(e)e(chosen)g F1(l)2130 2694 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)2343 2682 y Ge(because)g(of)g(its)i(rich)e(type)g(structures)450 2790 y(and)24 b(non-tri)n(vial)f(equational)g(theory)-5 b(.)37 b(F)o(or)24 b(simplicity)h(we)g(omit)f(base)h(types)g(other)f (than)g(unit.)38 b(W)-7 b(e)450 2898 y(re)n(vie)n(w)20 b(the)g(syntax)f(of)h(types)g(and)g(terms)g(belo)n(w)-5 b(,)19 b(with)h F0(i)h Ge(ranging)e(o)o(v)o(er)f FV(f)p Ge(1)p FZ(;)9 b Ge(2)p FV(g)p Ge(.)540 3106 y F0(T)60 b Ge(::)p FX(=)50 b Fj(unit)22 b FV(j)f F0(T)1081 3118 y Gb(1)1134 3106 y FV(!)d F0(T)1275 3118 y Gb(2)1331 3106 y FV(j)j F0(T)1415 3118 y Gb(1)1461 3106 y FV(\002)12 b F0(T)1578 3118 y Gb(2)1633 3106 y FV(j)20 b F0(T)1716 3118 y Gb(1)1763 3106 y FX(+)12 b F0(T)1880 3118 y Gb(2)524 3238 y F0(M)53 b Ge(::)p FX(=)d F0(x)21 b FV(j)f FX(\(\))i FV(j)e F1(l)p F0(x)9 b Ge(:)g F0(T)c FZ(:)p F0(M)24 b FV(j)d(h)p F0(M)s FZ(;)9 b F0(N)c FV(i)22 b(j)f F1(p)1712 3250 y FT(i)1733 3238 y FX(\()p F0(M)s FX(\))h FV(j)e FR(in)2021 3250 y FT(i)2043 3238 y FX(\()p F0(M)s FX(\))h FV(j)g FR(case)e F0(L)i FR(of)f FV(f)p FR(in)2743 3250 y FT(i)2764 3238 y FX(\()p F0(x)2833 3250 y FT(i)2873 3238 y Ge(:)e F0(T)2954 3250 y FT(i)2976 3238 y FX(\))p FZ(:)p F0(M)3100 3250 y FT(i)3121 3238 y FV(g)3163 3254 y FT(i)p Gd(2f)p Gb(1)p FS(;)p Gb(2)p Gd(g)450 3439 y Ge(W)-7 b(e)20 b(write)g F0(M)h FV(\021)924 3451 y Gc(a)984 3439 y F0(N)k Ge(for)18 b F1(a)p Ge(-equality)h(on)f(terms.)25 b(A)20 b(term)f(is)h F0(closed)f Ge(if)g(no)g(v)n(ariables)f(occur)h (free.)24 b(The)450 3547 y(typing)17 b(rules)h(are)g(standard,)f(which) g(we)i(list)g(in)f(Figure)f(5)i(\(cf.)e([25,)12 b(54]\).)23 b(W)-7 b(e)20 b(write)e F0(E)k FV(`)16 b F0(M)k Ge(:)d F0(T)29 b Ge(when)450 3655 y(a)20 b(term)f F0(M)k Ge(is)d(typable)e (with)i(type)e F0(T)31 b Ge(under)18 b(a)h(base)h F0(E)6 b Ge(.)25 b(W)-7 b(e)20 b(write)c F0(C)r FX([)j(])2490 3667 y FT(T)2545 3655 y Ge(:)8 b F0(T)2633 3625 y Gd(0)2674 3655 y Ge(for)18 b(a)i(\(typed\))e(conte)o(xt)g(of)450 3763 y(type)i F0(T)670 3733 y Gd(0)712 3763 y Ge(with)h(one)e(hole)h (of)g(type)g 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FT(i)2051 647 y FX(])41 b FV(7!)2198 659 y Fx(g)2273 647 y F2(0)450 843 y Ge(The)20 b(typed)g(transition)f (is)j(de\002ned)d(by)h(e)o(xtending)e(the)j(set)g(of)f(labels)h(with)f F0(x)p FR(in)2767 855 y FT(i)2788 843 y FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))21 b Ge(and)p 3051 797 V 20 w F0(x)p FR(in)3175 855 y FT(i)3197 843 y FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))21 b Ge(and)450 951 y(by)c(adding)f(the)i(rules)g(in)g(Figure)f(6.)24 b(The)17 b(weak)g(bisimilarity)g FV(\031)h Ge(is)h(then)e(de\002ned)g (by)g(the)g(same)h(clause)450 1059 y(as)j(in)f(De\002nition)g(4.1)f(in) i(Section)e(4)i(using)e(the)i(e)o(xtended)d(transition)h(relation.)533 1167 y(The)k(technical)g(de)n(v)o(elopment)e(for)j(the)f(full)h (calculus)f(is)i(identical)e(with)h(that)g(for)f(the)h(unary)e(cal-)450 1275 y(culus)e(in)g(the)h(preceding)d(sections,)i(e)o(xcept)f(for)g (the)i(follo)n(wing)d(minor)h(changes:)533 1443 y FV(\017)33 b Ge(In)20 b(Proposition)g(2.1:)25 b(In)c(\(1\),)f(\223precisely)g (once\224)g(for)g(a)h 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b(Appendix)d(D.1.)p 1520 775 50 50 v 450 911 a FV(\030)450 937 y FX(=)535 933 y Ge(and)i FV(\031)g Ge(are)g(related)g(in)g(the)g(follo)n(wing)f(w)o(ay)-5 b(.)533 1094 y FG(Pr)n(oposition)31 b(5.3.)92 b FV(\031)29 b Fy(\()1458 1072 y FV(\030)1458 1098 y FX(=)1523 1094 y FF(.)533 1241 y F0(Pr)l(oof)o(.)65 b Ge(By)19 b(Proposition)d(4.4,)i FV(\031)g Ge(is)h(a)g(typed)e(congruence)e(and)i(it)i(respects)f(con)m (v)o(er)o(gence)c(at)19 b Fy(B)36 b Ge(by)450 1349 y(de\002nition.)23 b(Since)1017 1327 y FV(\030)1017 1353 y FX(=)1100 1349 y Ge(is)c(the)f(maximum)f(such)g(this)i(sho)n(ws)f FV(\031\032)2313 1327 y(\030)2313 1353 y FX(=)2377 1349 y Ge(.)24 b(F)o(or)18 b(strictness,)h(tak)o(e)f FV(`)p 3127 1303 38 4 v 16 w F0(x)h FZ(.)f F0(x)7 b Ge(:)g FX(\(\))3381 1319 y Gb(?)3418 1349 y Ge(.)450 1490 y(By)21 b(Proposition)d(5.2)i(\(3\))f(this)i (process)f(is)1700 1468 y FV(\030)1700 1494 y FX(=)1764 1490 y Ge(-equal)g(to)g F2(0)g Ge(b)n(ut)h(clearly)p 2512 1444 V 19 w F0(x)e FV(6\031)f F2(0)o Ge(.)p 2796 1490 50 50 v 533 1648 a(Finally)23 b(we)h(list)g(processes)f(of)g (speci\002c)g(form)g(used)g(in)g(the)g(encoding)e(later)m(,)j(called)f F0(copycat)p Ge(.)33 b(A)450 1756 y(cop)o(y-cat)26 b(dynamically)g (links)h(tw)o(o)h(locations,)g(which)f(has)h(an)g(origin)e(in)i F0(forwar)m(der)g Ge(in)f(actors)h(as)450 1863 y(well)21 b(as)g(in)f(game)f(semantics.)1121 2072 y FX([)p F0(x)f FV(!)h F0(x)1338 2041 y Gd(0)1359 2072 y FX(])1382 2041 y FU(\()1393 2038 y FS(~)1406 2041 y Gc(t)p FU(\))1457 2019 y Fk(#)1540 2025 y FQ(def)1545 2072 y FX(=)54 b F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p 1825 2004 59 4 v F0(x)1862 2048 y Gd(0)1883 2072 y FX(\()1909 2051 y FZ(~)1915 2072 y F0(y)1952 2048 y Gd(0)1973 2072 y FX(\))p F1(P)2069 2084 y FT(i)2091 2072 y FX([)p F0(y)2151 2041 y Gd(0)2151 2095 y FT(i)2190 2072 y FV(!)19 b F0(y)2329 2084 y FT(i)2350 2072 y FX(])p 2373 2004 50 3 v -31 x Gc(t)2400 2052 y Fn(i)1118 2199 y FX([)p F0(x)g FV(!)f F0(x)1335 2169 y Gd(0)1356 2199 y FX(])1379 2169 y FU(\()1390 2166 y FS(~)1403 2169 y Gc(t)p FU(\))1454 2146 y FD(!)1540 2152 y FQ(def)1545 2199 y FX(=)54 b Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p 1853 2131 59 4 v F0(x)1890 2175 y Gd(0)1911 2199 y FX(\()1937 2178 y FZ(~)1943 2199 y F0(y)1980 2175 y Gd(0)2001 2199 y FX(\))p F1(P)2097 2211 y FT(i)2118 2199 y FX([)p F0(y)2178 2169 y Gd(0)2178 2222 y FT(i)2218 2199 y FV(!)18 b F0(y)2356 2211 y FT(i)2377 2199 y FX(])p 2400 2131 50 3 v -30 x Gc(t)2427 2180 y Fn(i)1051 2326 y FX([)p F0(x)g FV(!)h F0(x)1268 2296 y Gd(0)1289 2326 y FX(])1312 2296 y FU([)p Gb(&)1377 2307 y Fn(i)1382 2293 y FS(~)1395 2296 y Gc(t)1422 2307 y Fn(i)1440 2296 y FU(])1457 2274 y Fk(#)1540 2279 y FQ(def)1545 2326 y FX(=)54 b F0(x)p FX([)p Ge(&)1789 2338 y FT(i)1810 2326 y FX(\()-10 b FZ(~)-32 b F0(y)1879 2338 y FT(i)1900 2326 y FX(\))p FZ(:)p 1955 2259 59 4 v F0(x)1992 2302 y Gd(0)2013 2326 y FR(in)2100 2338 y FT(i)2122 2326 y FX(\()2148 2305 y FZ(~)2154 2326 y F0(y)2191 2302 y 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Gc(t)2770 2438 y Fn(i)e(j)2815 2457 y FX(])450 2645 y Ge(The)20 b(follo)n(wing)e(property)g(of)i(cop)o(y-cats)f(is)j(used)d(later)-5 b(.)533 2806 y FG(Pr)n(oposition)31 b(5.4.)533 2968 y Ge(\(1\))13 b FV(`)18 b FX([)p F0(x)g FV(!)h F0(y)p FX(])953 2938 y Gc(t)1005 2968 y FZ(.)h F0(x)9 b Ge(:)g F1(t)19 b FV(!)g FX(\()p F0(y)9 b Ge(:)p 1412 2917 37 4 v 9 w F1(t)q FZ(;)g Ge(?)t F0(A)p FX(\))30 b FF(for)g(e)l(ach)h(input)f(typ)l (e)f F1(t)i FF(and)f Ge(?)t F0(A)f FF(with)i F0(x)p FZ(;)9 b F0(y)18 b FV(62)h FM(fn)o FX(\()p F0(A)p FX(\))p FF(.)533 3081 y Ge(\(2\))13 b FX(\()p F1(n)c F0(y)p FX(\)\()p F0(P)p FV(j)p FX([)p F0(y)19 b FV(!)f F0(x)p FX(])1143 3051 y Gc(t)1174 3081 y FX(\))h FV(7!)1308 3051 y Gd(\003)1361 3081 y F0(P)p FV(f)p F0(x)p FZ(=)p F0(y)p FV(g)28 b FF(assuming)i(typ)l (ability.)533 3315 y F0(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(D.2.)p 1520 3315 50 50 v 1551 3569 a F3(5.3.)94 b(Sequen)m(tialit)m(y)450 3677 y Ge(One)21 b(of)g(the)h(basic)f(notions)g(we)g(shall)h(use)g(for) e(the)i(proof)d(of)j(the)f(full)g(abstraction,)f(is)j F0(sequentiality)p Ge(.)450 3785 y(\223Sequential\224)30 b(in)i(this)g(conte)o(xt)e(means)h(that)h(processes)f(ha)n(v)o(e)g(at)h (most)f(one)g(acti)n(v)o(e)g(thread:)47 b(The)450 3893 y(combination)19 b(with)i(the)h(sequential)e(type)h(discipline)f(in)i ([10])e(can)h(realise)g(this)h(beha)n(viour)d(in)j(linear)450 4001 y(processes.)35 b(While)24 b(the)g(full)g(abstraction)e(result)i (is)g(established)g(in)g(the)f(linear)h F1(p)p Ge(-calculus)e(without) 450 4109 y(the)f(sequentiality)f(constraint,)h(sequentiality)f(plays)h (a)g(crucial)g(role)g(in)g(se)n(v)o(eral)g(ar)o(guments.)26 b(Belo)n(w)450 4217 y(we)i(restrict)f(the)g(linear)g F1(p)p Ge(-calculus)g(to)g(its)h(sequential)f(subsystem)g(follo)n(wing) f([11])g(and)h(study)f(its)450 4325 y(basic)20 b(properties)f(used)h (in)g(the)h(subsequent)d(proofs.)533 4433 y(The)i(\002rst)h(constraint) 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FM(md)p FX(\()p F1(t)1216 4985 y FT(n)1252 4973 y FX(\))e(=)g FV(")p Ge(.)p eop %%Page: 30 30 30 29 bop 450 -257 a FX(30)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)p 450 -71 2989 5 v 504 248 a FM(\(Zero\))504 356 y FV(\000)p 479 418 289 5 v 504 531 a(`)555 543 y Fz(I)607 531 y F2(0)11 b FZ(.)p 718 531 25 4 v 979 134 a FM(\(P)m(ar\))979 242 y FV(`)1030 254 y Gc(f)1062 265 y Fn(i)1103 242 y F0(P)1146 254 y FT(i)1178 242 y FZ(.)h F0(A)1283 254 y FT(i)1386 242 y F4(\()p F7(i)k F4(=)8 b F9(1)p F6(;)g F9(2)p F4(\))979 350 y F0(A)1030 362 y Gb(1)1083 350 y FV(\020)18 b F0(A)1217 362 y Gb(2)1335 350 y F1(f)1378 362 y Gb(1)1431 350 y FV(\020)g F1(f)1557 362 y Gb(2)p 954 412 758 5 v 979 525 a FV(`)1030 537 y Gc(f)1062 549 y FD(1)1091 537 y Gd(\014)p Gc(f)1171 549 y FD(2)1204 525 y F0(P)1247 537 y Gb(1)1281 525 y FV(j)p F0(P)1347 537 y Gb(2)1393 525 y FZ(.)12 b F0(A)1498 537 y Gb(1)1534 525 y FV(\014)r F0(A)1652 537 y Gb(2)1923 134 y FM(\(Res\))1923 242 y FV(`)1974 254 y Gc(f)2028 242 y 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b(When)23 b F1(f)1307 2068 y Gb(1)1355 2056 y FV(\014)13 b F1(f)1476 2068 y Gb(2)1534 2056 y Ge(is)24 b(de\002ned)e(\(that)h(is,)h(if)g(the)o(y)e (are)h(not)g(simultaneously)e(output\),)450 2164 y(then)f(we)g(write)h F1(f)966 2176 y Gb(1)1019 2164 y FV(\020)d F1(f)1145 2176 y Gb(2)1180 2164 y Ge(.)533 2272 y(The)23 b(typing)g(rules)g(are)h (gi)n(v)o(en)e(in)i(Figure)f(7)g(\(the)g(sequential)g(v)o(ersion)g(of)g FM(\(Br)o(a\))g Ge(and)g FM(\(Sel\))g Ge(follo)n(w)450 2380 y FM(\(In\))29 b Ge(and)f FM(\(Out\))o Ge(\).)52 b(The)28 b(use)h(of)g(IO-modes)e(in)j(in)f FM(\(P)m(ar\))f Ge(ensures)h(single)f(threading)f(since)j Fz(O)14 b FV(\014)g Fz(O)29 b Ge(is)450 2488 y(unde\002ned.)45 b(An)28 b(output)f(\(resp.)f (input\))h(can)h(only)e(pre\002x)h(a)h(body)f(in)h(input)f(\(resp.)f (output\))h(mode,)450 2596 y(resulting)f(in)i(output)e(\(resp.)g (input\))g(mode.)45 b F2(0)28 b Ge(starts)g(from)e Fz(I)p Ge(.)46 b(Other)27 b(rules,)i FM(\(Res)o(,W)n(eak\))n Ge(,)g(do)e(not)450 2704 y(change)19 b(IO-modes.)k(F)o(or)d(this)h (system)f(we)h(observ)o(e:)533 2880 y FG(Pr)n(oposition)31 b(5.5.)533 3055 y Ge(\(1\))13 b F0(If)18 b FV(`)764 3067 y Gc(f)817 3055 y F0(P)9 b FZ(.)h F0(A)18 b(and)f(P)f FV(\000)-14 b(!)17 b F0(Q)h(then)g FV(`)1650 3067 y Gc(f)1702 3055 y F0(Q)10 b FZ(.)g F0(A.)24 b(Similarly)18 b(if)h FV(`)2359 3067 y Gc(f)2411 3055 y F0(P)10 b FZ(.)g F0(A)17 b(and)h(P)e FV(7!)h F0(Q)h(then)g FV(`)3194 3067 y Gc(f)3247 3055 y F0(Q)10 b FZ(.)g F0(A.)533 3172 y Ge(\(2\))j F0(If)20 b FV(`)766 3184 y Gc(f)820 3172 y F0(P)12 b FZ(.)g F0(A)19 b(and)h(P)e FV(\000)-15 b(!)19 b F0(Q)1434 3184 y Gb(1)p FS(;)p Gb(2)1537 3172 y F0(then)h(Q)1761 3184 y Gb(1)1814 3172 y FV(\021)e F0(Q)1957 3184 y Gb(2)1992 3172 y F0(.)533 3290 y Ge(\(3\))13 b F0(If)20 b FV(`)766 3302 y Gc(f)820 3290 y F0(P)12 b FZ(.)g F0(A)19 b(then)h FM(CSN)p FX(\()p F0(P)p FX(\))h F0(and)e FM(CSN)1768 3302 y FT(e)1799 3290 y FX(\()p F0(P)p FX(\))p F0(.)533 3504 y(Pr)l(oof)o(.)66 b Ge(\(1\))19 b(follo)n(ws)g(the)h(proof)e(of)h(Proposition)f(2.2)h (\(presented)f(in)i(Appendix)e(A.2)h(using)g(A.4\),)450 3612 y(incorporating)30 b(IO-modes)h(in)j(addition.)62 b(\(2\))32 b(is)i(because)f(there)f(is)i(at)g(most)f(one)f(acti)n(v)o (e)h(output)450 3753 y(in)20 b(a)h(sequential)e(process.)25 b(\(3\))19 b(is)j(immediate)d(since)h FV(`)2053 3765 y Gc(f)2107 3753 y F0(P)12 b FZ(.)g F0(A)19 b Ge(implies)i FV(`)d F0(P)i FZ(.)g F0(A)h Ge(by)f(de\002nition.)p 3360 3753 50 50 v 450 3960 a F0(Remark.)126 b Ge(Proposition)25 b(5.5)h(\(2\))g(indicates)g(the)g(sequential)g(nature)f(of)h(dynamics)g (in)g(sequential)450 4068 y(linear)21 b(processes:)28 b(in)22 b(spite)g(of)f(this,)h FV(7!)g Ge(gi)n(v)o(es)f(a)h(w)o(ay)g (of)f(computing)f(normal)g(forms)h(of)g(sequential)450 4176 y(processes)f(by)g(parallel)f(reduction.)555 4334 y(A)j(signi\002cant)f(property)e(is)k(that)e(linear)g(processes)h (typed)e(under)g(sequential)h(channel)f(types)h(are)450 4442 y(already)e(sequential)g(from)h(a)g(semantic)g(vie)n(wpoint.)533 4599 y FG(Definition)35 b(5.2.)41 b Ge(An)24 b(action)g(type)g F0(A)h Ge(is)g F0(sequential)e Ge(if)i(all)f(channel)g(types)g(used)g (in)g F0(A)h Ge(are)f(se-)450 4707 y(quential)19 b(and,)h(moreo)o(v)o (er)m(,)c(it)21 b(does)f(not)g(contain)f(tw)o(o)i(linear)f(output)f (channels.)564 4865 y(Ha)n(ving)30 b(at)h(most)g(one)f(linear)g(output) f(in)i(an)g(action)f(type)g(\(cf.)g([11,)i(Appendix)d(F]\))h(mak)o(es)h (it)450 4973 y(possible)20 b(to)g(ha)n(v)o(e)g(inducti)n(v)o(e)e (de\002nition)h(of)h(sequentialisation,)f(gi)n(v)o(en)g(ne)o(xt.)p eop %%Page: 31 31 31 30 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(31)533 -4 y FG(Pr)n(oposition)31 b(5.6.)59 b Ge(\(sequentialisation\))46 b FF(Given)26 b FV(`)15 b F0(P)j FZ(.)f F0(A)24 b FF(such)h(that)g F0(A)f FF(is)h(se)l(quential,)h F0(P)16 b FV(2)450 104 y FM(NF)550 116 y FT(e)611 104 y FF(and)31 b F0(P)f FF(do)l(es)h(not)f 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FV(\017)p FX(\()p 607 610 38 4 v F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(\))828 626 y FS(])875 609 y FQ(def)879 656 y FX(=)p 967 610 V 23 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1156 626 y FS(])1214 656 y FF(if)31 b FV("2)18 b FM(md)p FX(\()p F0(A)p FX(\))p FF(,)30 b FX(\()p 1716 610 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(\))1937 626 y FS(])1984 609 y FQ(def)1989 656 y FX(=)23 b F2(0)29 b FF(if)h FV("62)19 b FM(md)o FX(\()p F0(A)p FX(\))p FF(,)31 b(similarly)g(for)g(sele)l (ction.)450 827 y(Then)f(we)g(have)h FV(`)1031 839 y Gc(f)1086 827 y F0(P)1137 796 y FS(])1176 827 y FZ(.)12 b F0(A)28 b FF(for)j(some)f F1(f)g FF(and,)h(mor)l(e)l(over,)g F0(P)2367 805 y FV(\030)2367 831 y FX(=)2450 827 y F0(P)2501 796 y FS(])2529 827 y FF(.)533 1007 y F0(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(D.3.)p 1520 1007 50 50 v 450 1165 a(Using)24 b(sequentialisation)e(we)j(establish)f(a)g(re\002ned)f (conte)o(xt)g(lemma.)35 b(W)-7 b(e)25 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b(base)h(case)g(is)g FR(unit)2772 3695 y Gd(\016)2828 3725 y FX(=)21 b(\(\(\))3010 3695 y Gd(")3045 3725 y FX(\))3077 3695 y Gb(!)3106 3725 y Ge(,)27 b(which)e(is)450 3833 y(immediate.)e(F)o(or)c(induction,)e FX(\()p F0(T)1406 3845 y Gb(1)1449 3833 y FV(\))8 b F0(T)1580 3845 y Gb(2)1614 3833 y FX(\))1646 3803 y Gd(\016)1698 3833 y FX(=)17 b(\()p 1812 3767 V F0(T)1869 3804 y Gd(\016)1852 3858 y Gb(1)1904 3833 y FX(\()p F0(T)1993 3803 y Gd(\016)1976 3857 y Gb(2)2027 3833 y FX(\))2059 3803 y Gd(")2094 3833 y FX(\))2126 3803 y Gb(!)2175 3833 y Ge(is)i(sequential)f(if)n(f)h F0(T)2757 3803 y Gd(\016)2741 3857 y Gb(1)p FS(;)p Gb(2)2842 3833 y Ge(are)g(sequential)f(and)450 3970 y(ha)n(v)o(e)f(mode)f(!)t(,)i (which)e(is)i(the)f(induction)f(hypothesis.)22 b(Similarly)17 b(for)f FX(\()p F0(T)2552 3982 y Gb(1)2596 3970 y FV(\002)9 b F0(T)2710 3982 y Gb(2)2744 3970 y FX(\))2776 3940 y Gd(\016)2829 3970 y Ge(and)17 b FX(\()p F0(T)3039 3982 y Gb(1)3082 3970 y FX(+)9 b F0(T)3196 3982 y Gb(2)3230 3970 y FX(\))3262 3940 y Gd(\016)3297 3970 y Ge(.)p 3389 3970 50 50 v 533 4140 a FG(Pr)n(oposition)31 b(5.8.)59 b Ge(\(syntactic)16 b(soundness\))47 b FF(If)25 b F0(E)d FV(`)15 b F0(M)k Ge(:)d F0(T)35 b FF(then)24 b FV(`)15 b FX([)-9 b([)p F0(M)19 b Ge(:)d F0(T)11 b FX(])-9 b(])2927 4152 y FT(u)2979 4140 y FZ(.)18 b F0(u)6 b Ge(:)g F0(T)3173 4110 y Gd(\016)3223 4140 y FV(!)16 b F0(E)3379 4110 y Gd(\016)3413 4140 y FF(.)533 4321 y F0(Pr)l(oof)o(.)66 b Ge(See)21 b(Appendix)d(D.4.)p 1520 4321 V 533 4479 a(Note)25 b(also)g FX([)-9 b([)p F0(M)25 b Ge(:)c F0(T)11 b FX(])-9 b(])1148 4491 y FT(u)1208 4479 y Ge(has)25 b(al)o(w)o(ays)g(the)g(shape)g(!)p F0(u)p FX(\()-15 b FZ(~)-27 b F0(z)p FX(\))p FZ(:)p F0(P)p Ge(.)39 b(Further)24 b FX([)-9 b([)p F0(M)25 b Ge(:)c F0(T)11 b FX(])-9 b(])2779 4491 y FT(u)2839 4479 y Ge(is)26 b(sequentially)e(ty-)450 4587 y(pable,)30 b(though)d(we)i(do)f(not)h(use)g(this)g(property)d(in) j(our)f(subsequent)f(proof.)49 b(This)29 b(concludes)e(the)450 4695 y(v)o(eri\002cation)19 b(of)h(static)h(properties)d(of)i(the)g (encoding.)533 4803 y(F)o(or)g(dynamics,)f(we)h(obtain:)533 4973 y FG(Pr)n(oposition)31 b(5.9.)92 b FF(If)30 b F0(E)24 b FV(`)18 b F0(M)k Ge(:)d F0(T)40 b FF(and)30 b F0(M)22 b Fl( )d F0(M)2146 4943 y Gd(0)2197 4973 y FF(then)30 b FX([)-9 b([)p F0(M)s FX(])g(])2528 4985 y FT(u)2581 4973 y FV(7!)2664 4943 y FU(+)2734 4973 y FX([)g([)p F0(M)2843 4943 y Gd(0)2865 4973 y FX(])g(])2902 4985 y FT(u)2936 4973 y FF(.)p eop %%Page: 32 32 32 31 bop 450 -257 a FX(32)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)p 450 -71 2989 5 v 475 226 a Ff(\(T)-6 b(ype\))125 b Fe(unit)958 199 y Ft(\016)1027 184 y FQ(def)1035 226 y F4(=)43 b(\(\(\))1223 199 y Ft(")1256 226 y F4(\))1285 199 y FQ(!)1729 226 y F4(\()p F7(T)1794 239 y FQ(1)1835 226 y FJ(\))8 b F7(T)1954 239 y FQ(2)1987 226 y F4(\))2016 199 y Ft(\016)2085 184 y FQ(def)2093 226 y F4(=)43 b(\()p 2223 165 85 3 v F7(T)2274 200 y Ft(\016)2259 251 y FQ(1)2307 226 y F4(\()p F7(T)2387 199 y Ft(\016)2372 250 y FQ(2)2421 226 y F4(\))2450 199 y Ft(")2483 226 y F4(\))2512 199 y FQ(!)806 345 y F4(\()p F7(T)871 358 y FQ(1)914 345 y FJ(\002)10 b F7(T)1018 358 y FQ(2)1052 345 y F4(\))1081 318 y Ft(\016)1149 303 y FQ(def)1157 345 y F4(=)44 b(\(\()p F7(T)1368 318 y Ft(\016)1353 369 y FQ(1)1401 345 y F7(T)1452 318 y Ft(\016)1437 369 y FQ(2)1485 345 y F4(\))1514 318 y Ft(")1547 345 y F4(\))1576 318 y FQ(!)1729 345 y F4(\()p F7(T)1794 358 y FQ(1)1837 345 y F4(+)10 b F7(T)1941 358 y FQ(2)1975 345 y F4(\))2004 318 y Ft(\016)2072 303 y FQ(def)2080 345 y F4(=)44 b(\([)p F7(T)2282 318 y Ft(\016)2268 369 y FQ(1)2326 345 y FJ(\010)10 b F7(T)2445 318 y Ft(\016)2430 369 y FQ(2)2479 345 y F4(])2500 318 y Ft(")2532 345 y F4(\))2561 318 y FQ(!)475 488 y Ff(\(Base\))814 487 y F8(/)806 488 y(0)843 461 y Ft(\016)911 446 y FQ(def)920 488 y F4(=)1029 487 y F8(/)1021 488 y(0)671 b F4(\()p F7(E)5 b F6(;)j F7(x)h F9(:)f F7(T)g F4(\))1988 461 y Ft(\016)2057 446 y FQ(def)2065 488 y F4(=)43 b F7(E)2217 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4973 y FX(])3439 4943 y Gd(")3474 4973 y Ge(.)p eop %%Page: 33 33 33 32 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(33)450 -1 y Ge(Again)24 b(by)h(Proposition)e(3.3)i F0(P)1376 -32 y Gd(0)1368 23 y Gb(1)1423 -1 y FX(=)p 1509 -47 33 4 v 21 w F0(z)p FR(in)1629 11 y FT(i)1650 -1 y FX(\()p F0(w)p FX(\))p F0(P)1820 -32 y Gd(0)1812 23 y Gb(2)1873 -1 y Ge(with)g FV(`)c F0(P)2169 -32 y Gd(0)2161 23 y Gb(2)2218 -1 y FZ(.)h F0(w)12 b Ge(:)g FX(\(\(\))2480 -32 y Gd(")2516 -1 y FX(\))2548 -32 y Gb(!)2577 -1 y Ge(.)40 b(This)25 b(w)o(ay)g(we)h(reach)e F0(P)d FV(\021)450 136 y Ge(!)p F0(u)p FX(\()p F0(z)p FX(\))p FZ(:)p 639 90 V F0(zin)12 b(j)771 148 y FT(i)793 136 y FX(\()p F0(w)p FX(\))p Ge(!)p F0(w)p FX(\()p F0(v)p FX(\))p FZ(:)p 1119 90 37 4 v F0(v)p Ge(.)p 1260 136 50 50 v 533 313 a FG(Lemma)30 b(5.2.)71 b Ge(\(computational)26 b(adequac)o(y\))36 b FF(L)l(et)j F0(M)27 b Ge(:)c Fy(B)2292 330 y Gc(l)2375 313 y FF(b)l(e)39 b(close)l(d.)67 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FV(+)14 b FX([)-9 b([)p Fw(false)p FX(])g(])2912 1896 y FT(u)2946 1884 y Ge(,)17 b(which)f(contra-)450 2027 y(dicts)21 b(the)f(CR)h(of)f FV(7!)p Ge(,)h(hence)e(done.)p 1575 2027 V 450 2185 a(By)i(the)f(standard)f(ar)o(gument)f(we)i (obtain:)533 2362 y FG(Cor)n(ollar)-5 b(y)33 b(5.2.)60 b Ge(\(soundness\))26 b FX([)-9 b([)p F0(E)23 b FV(`)17 b F0(M)k Ge(:)c F0(T)11 b FX(])-9 b(])2017 2374 y FT(u)2069 2340 y FV(\030)2069 2366 y FX(=)2151 2362 y([)g([)p F0(E)23 b FV(`)17 b F0(N)23 b Ge(:)17 b F0(T)11 b FX(])-9 b(])2542 2374 y FT(u)2604 2362 y FF(implies)30 b F0(E)23 b FV(`)17 b F0(M)3119 2340 y FV(\030)3119 2366 y FX(=)3183 2379 y Gc(l)3238 2362 y F0(N)23 b Ge(:)18 b F0(T)10 b FF(.)1536 2731 y F3(5.5.)94 b(Completeness)450 2839 y Ge(W)-7 b(e)30 b(no)n(w)f(tackle)g(a)h(harder)e(direction,)i(the)f(equational)f (completeness)g(of)h(the)g(encoding.)50 b(While)450 2947 y(preceding)25 b(studies)i(of)g(types)g(for)g(the)g F1(p)p 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F0(F)57 b Ge(::)p FX(=)49 b(\(\))61 b FV(j)39 b F0(x)g FV(j)h F1(l)p F0(x)p FZ(:)p F0(F)45 b FV(j)39 b(h)p F0(F)1725 4457 y Gb(1)1760 4445 y FZ(;)9 b F0(F)1835 4457 y Gb(2)1869 4445 y FV(i)40 b(j)f FR(in)2090 4457 y FT(i)2111 4445 y FX(\()p F0(F)7 b FX(\))40 b FV(j)f FR(case)19 b F0(x)i FR(of)f FV(f)p FR(in)2824 4457 y FT(i)2846 4445 y FX(\()p F0(x)2915 4457 y FT(i)2936 4445 y FX(\))p FZ(:)p F0(F)3034 4457 y FT(i)3055 4445 y FV(g)944 4553 y(j)93 b FR(let)20 b FX(\(\))f(=)f F0(z)j FR(in)f F0(F)46 b FV(j)39 b FR(let)19 b F0(x)g FX(=)f F0(zF)28 b FR(in)20 b F0(F)2264 4523 y Gd(0)2325 4553 y FV(j)39 b FR(let)19 b FV(h)p F0(x)p FZ(;)9 b F0(y)p FV(i)19 b FX(=)f F0(z)j FR(in)f F0(F)450 4757 y Ge(FCFs)i(use)g(three)e(additional)f(constructs,)h FR(let)h FX(\(\))e(=)f F0(N)27 b FR(in)21 b F0(M)46 b Ge(\(let-unit\),)20 b FR(let)g F0(x)f FX(=)f F0(N)3010 4769 y Gb(1)3045 4757 y F0(N)3098 4769 y Gb(2)3152 4757 y Ge(:)h F0(S)j FR(in)e F0(M)450 4865 y Ge(\(let-app\))f(and)h FR(let)g FV(h)p F0(x)p 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b(we)h(can)f(no)n(w)g(encode)e(FCFs)k(into)e(the)g F1(p)p Ge(-calculus.)533 1235 y(Another)15 b(w)o(ay)h(to)g(map)f(FCFs)i (into)f(the)g F1(p)p Ge(-calculus)f(is)i(to)f(directly)f(encode)f(FCFs) k(to)e(ENFs.)24 b(Belo)n(w)450 1356 y(we)d(set,)f(w)-5 b(.l.o.g.:)24 b(for)19 b Fj(let)p Ge(s,)j FX([)-9 b([)p F0(F)7 b FX(])-9 b(])1444 1368 y FT(u)1497 1309 y FQ(def)1502 1356 y FX(=)t Ge(!)p F0(u)p FX(\()o FZ(~)-41 b F0(w)p FX(\))p FZ(:)p F0(P)p Ge(;)20 b(and,)g(for)f Fj(case)p Ge(,)j FX([)-9 b([)p F0(F)2447 1368 y FT(i)2468 1356 y FX(])g(])2505 1368 y FT(u)2558 1309 y FQ(def)2563 1356 y FX(=)t Ge(!)p F0(u)p FX(\()o FZ(~)-41 b F0(w)p FX(\))p FZ(:)p F0(P)2887 1368 y FT(i)2908 1356 y Ge(.)1610 1573 y FV(h)-9 b(h)p FX(\(\))p FV(i)g(i)1784 1585 y FT(u)1870 1526 y FQ(def)1875 1573 y FX(=)54 b Ge(!)p F0(u)p FX(\()p F0(c)p FX(\))p FZ(:)p 2188 1527 37 4 v F0(c)1638 1700 y FV(h)-9 b(h)p F0(x)p FV(i)g(i)1785 1712 y FT(u)1870 1653 y FQ(def)1875 1700 y FX(=)54 b([)p F0(u)18 b FV(!)g F0(x)p FX(])1511 1827 y FV(h)-9 b(h)p F1(l)p F0(x)p FZ(:)p F0(F)7 b FV(i)-9 b(i)1785 1839 y FT(u)1870 1780 y FQ(def)1875 1827 y FX(=)54 b Ge(!)p F0(u)p FX(\()p F0(xc)p FX(\))p FZ(:)p 2225 1781 V F0(c)o FX(\()12 b F0(f)g FX(\))p FV(h)-9 b(h)p F0(F)8 b FV(i)-9 b(i)2550 1839 y FT(f)1444 1955 y FV(h)g(h)p FR(in)1587 1967 y FT(i)1608 1955 y FX(\()p F0(F)7 b FX(\))p FV(i)-9 b(i)1785 1967 y FT(u)1870 1908 y FQ(def)1875 1955 y FX(=)54 b Ge(!)p F0(u)p FX(\()p F0(c)p FX(\))p FZ(:)p 2188 1909 V F0(c)o FR(in)2311 1967 y FT(i)2333 1955 y FX(\()12 b F0(f)g FX(\))p FV(h)-9 b(h)p F0(F)8 b FV(i)-9 b(i)2622 1967 y FT(f)1424 2082 y FV(h)g(hh)p F0(F)1554 2094 y Gb(1)1588 2082 y FZ(;)9 b F0(F)1663 2094 y Gb(2)1698 2082 y FV(ii)-9 b(i)1785 2094 y FT(u)1870 2035 y FQ(def)1875 2082 y FX(=)54 b Ge(!)p F0(u)p FX(\()p F0(c)p FX(\))p FZ(:)p 2188 2036 V F0(c)o FX(\()12 b F0(f)2293 2094 y Gb(1)2341 2082 y F0(f)2366 2094 y Gb(2)2401 2082 y FX(\)\()p FV(h)-9 b(h)p F0(F)8 b FV(i)-9 b(i)2643 2094 y FT(f)2662 2106 y FD(1)2695 2082 y 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2317 V 12 w F0(E)1091 2356 y Gd(\016)1124 2380 y FZ(;)51 b FV(`)1249 2392 y Fz(O)1301 2380 y F0(S)12 b FZ(.)g F0(u)d Ge(:)p 1492 2317 V 9 w F0(T)1547 2356 y Gd(\016)1582 2380 y FZ(;)g F0(v)g Ge(:)g Fy(B)j FZ(:)912 2488 y F0(C)965 2500 y Fa(f)992 2488 y FX([\()p F1(n)e F0(x)-10 b FZ(~)-32 b F0(y)p FX(\)\([)-9 b([)p F0(M)1344 2500 y Gb(1)1379 2488 y FX(])g(])1416 2500 y FT(u)1451 2488 y FV(j)p F0(R)p FV(j)p F0(S)q FX(\)])1646 2458 y FT(w)1646 2509 y(v)1708 2488 y FV(+)1759 2500 y FT(e)1809 2488 y FX([)g([)p Fw(true)p FX(])g(])2021 2500 y FT(w)2066 2488 y FZ(;)44 b F0(C)2186 2500 y Fa(f)2214 2488 y FX([\()p F1(n)9 b F0(x)-10 b FZ(~)-32 b F0(y)p FX(\)\([)-9 b([)p F0(M)2565 2500 y Gb(2)2601 2488 y FX(])g(])2638 2500 y FT(u)2672 2488 y FV(j)p F0(R)p FV(j)p F0(S)q FX(\)])2867 2458 y FT(w)2867 2509 y(v)2930 2488 y FV(+)2981 2500 y FT(e)3031 2488 y FX([)g([)p Fw(false)o FX(])g(])3258 2500 y FT(w)3303 2488 y FZ(:)916 2596 y Ge(\(Proposition)18 b(5.13)h(\(2\)\))571 2733 y FV(,)96 b(9)18 b(`)865 2745 y Fz(I)917 2733 y F0(R)12 b FZ(.)p 1034 2669 V 12 w F0(E)1091 2709 y Gd(\016)1124 2733 y FZ(;)51 b FV(`)1249 2745 y Fz(O)1301 2733 y F0(S)12 b FZ(.)g F0(u)d Ge(:)p 1492 2669 V 9 w F0(T)1547 2709 y Gd(\016)1582 2733 y FZ(;)g F0(v)g Ge(:)g Fy(B)j FZ(:)916 2840 y FX(\()p F1(n)e F0(x)-10 b FZ(~)-32 b F0(y)p FX(\)\([)-9 b([)p F0(M)1245 2852 y Gb(1)1280 2840 y FX(])g(])1317 2852 y FT(u)1351 2840 y FV(j)p F0(R)p FV(j)l F0(C)1497 2852 y Fa(f)1525 2840 y FX([)p F0(S)q FX(])1614 2810 y FT(w)1614 2861 y(v)1658 2840 y FX(\))18 b FV(+)1759 2852 y FT(e)1809 2840 y FX([)-9 b([)p Fw(true)p FX(])g(])2021 2852 y FT(w)2066 2840 y FZ(;)48 b FX(\()p F1(n)10 b F0(x)-10 b FZ(~)-32 b F0(y)p FX(\)\([)-9 b([)p F0(M)2466 2852 y Gb(2)2501 2840 y FX(])g(])2538 2852 y FT(u)2573 2840 y FV(j)p F0(R)p FV(j)l F0(C)2719 2852 y Fa(f)2746 2840 y FX([)p F0(S)q FX(])2835 2810 y FT(w)2835 2861 y(v)2879 2840 y FX(\))19 b FV(+)2981 2852 y FT(e)3031 2840 y FX([)-9 b([)p Fw(false)o FX(])g(])3258 2852 y FT(w)3303 2840 y FZ(:)916 2948 y Ge(\(Proposition)18 b(5.2)i(\(2-b\)\))571 3085 y FV(\))96 b(9)p FZ(:)18 b FV(`)g F0(N)959 3097 y FT(i)999 3085 y Ge(:)h F0(T)1081 3097 y FT(i)1102 3085 y FZ(;)51 b F0(u)18 b Ge(:)g F0(T)30 b FV(`)18 b F0(L)h Ge(:)f Fy(B)11 b FZ(:)916 3193 y FX(\()p F1(n)f F0(x)-10 b FZ(~)-32 b F0(y)p FX(\)\([)-9 b([)p F0(M)1245 3205 y Gb(1)1280 3193 y FX(])g(])1317 3205 y FT(u)1351 3193 y FV(j)p F1(P)p FX([)g([)p F0(N)1528 3205 y FT(i)1549 3193 y FX(])g(])1586 3205 y FT(y)1613 3216 y Fn(i)1636 3193 y FV(j)p FX([)g([)p F0(L)p FX(])g(])1779 3205 y FT(w)1824 3193 y FX(\))p FV(+)1907 3205 y FT(e)1938 3193 y FX([)g([)p Fw(true)p FX(])g(])2150 3205 y FT(w)2195 3193 y FZ(;)30 b FX(\()p F1(n)10 b F0(x)-10 b FZ(~)-32 b F0(y)o FX(\)\([)-9 b([)p F0(M)2576 3205 y Gb(2)2612 3193 y FX(])g(])2649 3205 y FT(u)2683 3193 y FV(j)p F1(P)p FX([)g([)p F0(N)2860 3205 y FT(i)2881 3193 y FX(])g(])2918 3205 y FT(y)2945 3216 y Fn(i)2968 3193 y FV(j)p FX([)g([)p F0(L)p FX(])g(])3111 3205 y FT(w)3156 3193 y FX(\))p FV(+)3239 3205 y FT(e)3270 3193 y FX([)g([)p Fw(false)p FX(])g(])3498 3205 y FT(w)3543 3193 y FZ(:)916 3301 y Ge(\(Corollary)39 b(5.3\))571 3437 y FV(,)96 b(9)p FZ(:)18 b FV(`)g F0(N)959 3449 y FT(i)999 3437 y Ge(:)h F0(T)1081 3449 y FT(i)1102 3437 y FZ(;)51 b F0(u)18 b Ge(:)g F0(T)30 b FV(`)18 b F0(L)h Ge(:)f Fy(B)11 b FZ(:)916 3546 y FX(\()p F1(l)p F0(u)p FZ(:)p F0(L)p FX(\)\(\()p F1(l)p F0(y)1284 3558 y Gb(1)1318 3546 y FZ(::)p F0(y)1401 3558 y FT(n)1436 3546 y FZ(:)p F0(M)1528 3558 y Gb(1)1563 3546 y FX(\))1592 3528 y FZ(~)1595 3546 y F0(N)6 b FX(\))18 b FV(+)h Fw(true)p FZ(;)48 b FX(\()p F1(l)p F0(u)p FZ(:)p F0(L)p FX(\)\(\()p F1(l)p F0(y)2353 3558 y Gb(1)2388 3546 y FZ(::)p F0(y)2471 3558 y FT(n)2506 3546 y FZ(:)p F0(M)2598 3558 y Gb(2)2633 3546 y FX(\))2662 3528 y FZ(~)2665 3546 y F0(N)5 b FX(\))19 b FV(+)f Fw(false)p FZ(:)916 3654 y Ge(\(Lemma)h(5.2\))575 3762 y FQ(def)571 3809 y FV(,)96 b F0(M)819 3821 y Gb(1)872 3809 y FV(6)872 3787 y(\030)872 3813 y FX(=)937 3826 y Gc(l)993 3809 y F0(M)1062 3821 y Gb(2)1097 3809 y FZ(;)450 4038 y Ge(as)21 b(required.)p 920 4038 50 50 v 533 4196 a F0(Remark.)h Ge(By)h(Corollary)e(5.3,)h(the)g(embedding)e(is)j(in)g (addition)e(fully)g(complete)g(\(in)i(the)f(sense)h(of)450 4304 y([3]\))c(up)h(to)784 4281 y FV(\030)784 4308 y FX(=)849 4304 y Ge(.)664 4526 y F3(6.)95 b(LINEAR)31 b F1(p)p F3(-CALCULUS)i(WITH)f(FREE)f(NAME)g(P)-8 b(ASSING)450 4649 y Ge(In)18 b(the)h(pre)n(vious)e(sections,)h(we)h(ha)n(v)o(e)f(in) m(v)o(estigated)f(the)h(properties)f(of)i(the)f(linear)g F1(p)p Ge(-calculus)g(whose)450 4757 y(outputs)30 b(are)g(restricted)g (to)h(those)g(which)f(pass)h(only)f(bound)f(names.)55 b(Using)31 b(bound)e(names)h(has)450 4865 y(signi\002cance)21 b(in)h(making)e(the)i(representation)e(of)i(computational)d(beha)n (viour)h(as)i(tight)g(as)h(possible:)450 4973 y(gi)n(v)o(en)k(some)h (beha)n(viour)f(which)h(we)h(wish)f(to)h(model,)g(the)g(w)o(ay)f(of)h (representing)d(it)j(in)g(the)f(typed)p eop %%Page: 37 37 37 36 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(37)450 -4 y Ge(calculus)28 b(becomes)g(strongly)f(constrained)g(and)i(thus,)h (for)e(e)o(xample,)h(we)g(o)n(wn)g(a)g(f)o(airly)f(tractable)450 104 y(notion)e(of)h(inhabitants)f(in)h(each)g(type)g(\(Theorem)e (5.1\).)44 b(Ho)n(we)n(v)o(er)m(,)27 b(a)h(natural)e(question)g (remains:)450 212 y(can)17 b(we)g(impose)f(beha)n(vioural)f (constraints)h(of)h(the)g(similar)g(kind)f(on)g(terms)h(with)g(free)g (name)f(passing,)450 320 y(i.e.)21 b(using)f(the)h(standard)e(syntax)i (for)f(the)g(asynchronous)e F1(p)p Ge(-calculus?)26 b(And)20 b(if)h(we)g(can,)g(does)f(it)i(add)450 428 y(an)o(y)k(e)o(xpressi)n(v)o (e)f(po)n(wer?)45 b(This)27 b(is)g(not)g(only)f(intellectual)h (curiosity)-5 b(.)43 b(Apart)27 b(from)f(the)h(simplicity)450 535 y(of)d(the)h(presentation)e(\(by)g(mo)o(ving)g(to)i(free)f(name)g (passing)g(we)h(can)f(get)g(rid)h(of)f(a)h(couple)e(of)i(added)450 643 y(structural)31 b(rules\),)j(free)e(name)f(passing)h(mak)o(es)f (the)h(computation)e(more)h(tractable:)48 b(it)33 b(also)f(has)450 751 y(technical)19 b(adv)n(antages)g(in)h(the)h(second-order)c(setting) j([12].)533 859 y(This)28 b(section)f(studies)h(these)g(questions,)g(e) o(xtending)e(the)h(syntax)g(to)h(free)f(outputs)g(while)h(using)450 967 y(precisely)20 b(the)g(same)g(type)g(structures.)25 b(The)20 b(typing)f(rules)h(do)g(not)g(change)f(e)o(xcept)g(for)h(free) g(outputs.)450 1075 y(The)j(embedding)f(of)h(terms)h(of)f(the)h(system) g(with)g(bound)e(outputs)h(into)g(the)h(system)g(with)g(free)f(out-)450 1183 y(puts)k(is)h(essentially)g(subset)f(inclusion.)45 b(After)27 b(presenting)e(the)i(translation,)h(we)g(sho)n(w)f(these)g (tw)o(o)450 1291 y(maps)22 b(not)g(only)f(preserv)o(e)g(types)h(b)n(ut) g(also)g(the)g(semantics:)30 b(the)o(y)21 b(do)h(not)g(change)e(the)j (beha)n(viour)d(of)450 1399 y(processes)i(up)g(to)h(the)f(canonical)f (equalities.)32 b(This)22 b(result)h(also)f(sho)n(ws)h(that)f(the)h (uni)n(v)o(erse)e(of)h(linear)450 1507 y(terms)e(with)f(free)g(outputs) g(is)h(semantically)f(equi)n(v)n(alent)f(to)i(its)g(strict)g(subset)g (which)f(use)h(only)e(bound)450 1615 y(outputs,)29 b(thus)f(answering)f (the)h(question)f(posed)h(abo)o(v)o(e.)47 b(The)28 b(e)o(xtended)e (reduction)g(is)j(used)f(as)h(a)450 1723 y(tractable)20 b(tool)g(to)g(pro)o(v)o(e)e(their)i(correspondence.)1002 1928 y F3(6.1.)94 b(Linear)32 b(T)m(yping)h(with)e(F)-8 b(ree)32 b(Name)e(P)m(assing)450 2036 y Ge(The)22 b(syntax)f(of)h (processes)g(with)h(free)f(name)f(passing)h(is)h(the)f(standard)g (asynchronous)d(polyadic)i F1(p)p Ge(-)450 2144 y(calculus)g(with)h (branchings)e(and)h(selections.)30 b(W)-7 b(e)23 b(tak)o(e)e(of)n(f)h (the)f(bound)f(output)h(and)g(selection)g(from)450 2252 y(the)f(syntax)g(in)g(Section)g(5)g(and)g(replace)f(them)h(with)g(the)h (follo)n(wing)d(tw)o(o.)1466 2451 y F0(P)74 b Ge(::)p FX(=)f FZ(:::)37 b FV(j)p 1941 2405 38 4 v 37 w F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)p FV(i)37 b(j)p 2176 2405 V 37 w F0(x)p FR(in)2300 2463 y FT(i)2321 2451 y FV(h)-10 b FZ(~)-32 b F0(y)q FV(i)450 2649 y Ge(The)18 b(bound)e(output)p 1054 2603 V 16 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)19 b Ge(can)e(be)h(reco)o(v)o(ered)d(as)k FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p 2109 2603 V F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)p FV(ij)p F0(P)p FX(\))p Ge(,)19 b(so)f(that)g(the)g(second)f(syntax)g(in)h(f)o(act)450 2757 y(subsumes)i(the)g(\002rst)h(one.)j(F)o(or)c(the)g(reduction)f (relation)g FV(\000)-14 b(!)p Ge(,)20 b(we)h(replace)e(the)h(axioms)g (with:)1622 2950 y F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)21 b FV(j)p 1899 2904 V 21 w F0(x)p FV(h)-10 b FZ(~)-32 b F0(v)p FV(i)50 b(\000)-14 b(!)50 b F0(P)p FV(f)-10 b FZ(~)-32 b F0(v)o FZ(=)-10 b(~)-32 b F0(y)o FV(g)1348 3058 y F0(x)p FX([)p Ge(&)1473 3070 y FT(i)1494 3058 y FX(\()-10 b FZ(~)-32 b F0(y)1563 3070 y FT(i)1584 3058 y FX(\))p FZ(:)p F0(P)1682 3070 y FT(i)1703 3058 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v 1896 3834 a(`)1947 3847 y Gb(f)p 1990 3788 38 4 v 1990 3834 a F0(x)p FR(in)2123 3846 y FT(j)2146 3834 y FV(h)-10 b FZ(~)-32 b F0(y)2224 3846 y FT(j)2247 3834 y FV(i)12 b FZ(.)g F0(x)d Ge(:)g FX([)p FV(\010)2511 3846 y FT(i)2514 3830 y FZ(~)2531 3834 y F1(t)2567 3846 y FT(i)2589 3834 y FX(])2617 3804 y FT(p)2649 3813 y FH(O)2690 3834 y FV(\014)h FZ(~)-40 b F0(y)2813 3846 y FT(j)2844 3834 y Ge(:)p 2876 3761 69 4 v 2875 3830 a FZ(~)2876 3834 y F1(t)2921 3846 y FT(j)450 4023 y Ge(In)20 b FM(\(Out\))o Ge(,)h(we)g(assume)f F1(t)1171 4035 y FT(i)1211 4023 y FX(=)f F1(t)1340 4035 y FT(j)1384 4023 y Ge(if)i F0(y)1493 4035 y FT(i)1532 4023 y FX(=)d F0(y)1661 4035 y FT(j)1684 4023 y Ge(.)26 b(Similarly)20 b(for)g FM(\(Sel\))o Ge(.)26 b(Note)20 b(the)h(types)f(for)g(object)g(names)450 4131 y(in)i(the)g(abo)o(v)o(e)e(tw)o(o)i(rules)g(are)g(dualised.)29 b(The)22 b(resulting)f(system)h(is)h(denoted)d FM(FNP)i Ge(and)g(the)g(original)450 4238 y(system)e(is)i(denoted)c FM(BNP)p Ge(.)25 b(F)o(or)20 b(clarity)g(we)g(hereafter)f(write)i FV(`)2305 4251 y Gb(b)2358 4238 y F0(P)11 b FZ(.)h F0(A)19 b Ge(for)h(the)g(typability)f(in)i FM(BNP)o Ge(.)533 4346 y(The)i(rules)g(for)g(outputs)f(gi)n(v)o(en)g(abo)o(v)o(e,)g(are)h (best)h(understood)c(in)k(terms)f(of)g(the)g(follo)n(wing)f(repre-)450 4454 y(sentation)e(of)f(free)h(outputs)f(in)i(the)f(realm)g(of)g(bound) e(outputs.)p 1468 4620 38 4 v 1468 4666 a F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)1561 4629 y FS(~)1574 4632 y Gc(t)1605 4666 y FV(i)1646 4632 y Gd(\016)1718 4619 y FQ(def)1723 4666 y FX(=)p 1829 4620 V 41 w F0(x)q FX(\()o FZ(~)-41 b F0(w)p FX(\))p F1(P)2050 4678 y FT(i)2071 4666 y FX([)p F0(w)2149 4678 y FT(i)2189 4666 y FV(!)19 b F0(y)2328 4678 y FT(i)2349 4666 y FX(])2372 4632 y Gc(t)2399 4643 y Fn(i)450 4865 y Ge(\(the)26 b(same)h(e)o(xpression)e(already)g (appeared)g(as)i Fw(Msg)p FV(h)p F0(x)o FZ(~)-41 b F0(w)q FV(i)27 b Ge(in)g(Figure)e(8\).)44 b(The)26 b(annotation)f(of)h(free) 450 4973 y(objects)d(do)g(not)g(lose)g(generality)f(since,)i(when)f (processes)f(are)i(typed,)e(we)i(can)f(al)o(w)o(ays)h(restore)e(the)p eop %%Page: 38 38 38 37 bop 450 -257 a FX(38)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)450 -4 y Ge(original)f(type)h(information.)k (The)c(abo)o(v)o(e)e(representation)g(says)j(that)f(a)h(free)e(name)h (output)f(is)i(a)f(bound)450 104 y(name)h(output)g(in)i(which)e(all)i (e)o(xported)d(names)i(are)g(\223equated\224)e(with)i(the)h(mentioned)d (free)h(names.)25 b(In)450 212 y(this)f(representation,)d F0(w)1164 224 y FT(i)1209 212 y Ge(is)j(used)f(as)h F1(t)1593 224 y FT(i)1638 212 y Ge(and,)f(as)h(a)f(result,)h F0(y)2223 224 y FT(i)2268 212 y Ge(is)g(used)f(as)p 2615 161 37 4 v 23 w F1(t)2652 224 y FT(i)2673 212 y Ge(,)h(illustrating)f(the)g (typing)450 320 y(rules)d(gi)n(v)o(en)f(abo)o(v)o(e.)533 441 y(Let)27 b FV(`)722 454 y Gb(b)779 441 y F0(P)14 b FZ(.)f F0(A)27 b Ge(and)g(de\002ne)f F0(P)1406 411 y Gd(\016)1468 441 y 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y(is)h(by)f(reducing)e(them)i(to)h(those)f(of)g FM(BNP)p Ge(.)40 b(W)-7 b(e)27 b(\002rst)f(de\002ne)f(the)g(e)o (xtended)e(reduction)h(for)g(the)i(free)450 1441 y(output)19 b(calculus)h(as)h(follo)n(ws.)878 1633 y FX(\()p Fw(E1)1002 1645 y Fa(f)1028 1633 y FX(\))129 b F0(C)r FX([)p 1269 1587 V F0(x)p FV(h)-10 b FZ(~)-32 b F0(v)1375 1645 y Gb(1)1410 1633 y FV(i)p FX(])p FZ(::)p FX([)p 1534 1587 V F0(x)q FV(h)-10 b FZ(~)-32 b F0(v)1641 1645 y FT(n)1676 1633 y FV(i)p FX(])20 b FV(j)h F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)60 b FV(7!)2159 1645 y Fx(l)2230 1633 y F0(C)r FX([)p F0(Q)p FV(f)-10 b FZ(~)-32 b F0(v)2449 1645 y Gb(1)2483 1633 y FZ(=)-10 b(~)-32 b F0(y)p FV(g)p FX(])p FZ(::)p FX([)p F0(Q)p FV(f)-10 b FZ(~)-32 b F0(v)2835 1645 y FT(n)2868 1633 y FZ(=)-10 b(~)-32 b F0(y)p FV(g)p FX(])878 1753 y(\()p Fw(E2)1002 1765 y Fa(f)1028 1753 y FX(\))443 b F0(C)r FX([)p 1583 1707 V F0(x)p FV(h)-10 b FZ(~)-32 b F0(v)p FV(i)p FX(])p FV(j)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b 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(as)j(those)f(ideas)g(in)h FM(BNP)o Ge(.)533 4478 y FG(Cor)n(ollar)-5 b(y)33 b(6.1.)533 4643 y Ge(\(1\))13 b(\(reduction\))54 b F0(If)28 b FV(`)1200 4656 y FT(f)1244 4643 y F0(P)14 b FZ(.)g F0(A,)30 b(then)d Ge(\(i\))h F0(P)23 b FV(\000)-15 b(!)23 b F0(Q)29 b(implies)f FV(`)2383 4656 y FT(f)2427 4643 y F0(Q)14 b FZ(.)g F0(A;)32 b Ge(\(ii\))c F0(P)22 b FV(\000)-14 b(!)23 b F0(Q)3088 4655 y Gb(1)p FS(;)p Gb(2)3199 4643 y F0(implies)450 4751 y(either)d(Q)724 4763 y Gb(1)778 4751 y FV(\021)e F0(Q)921 4763 y Gb(2)976 4751 y F0(or)j(Q)1131 4763 y Gb(1)p FS(;)p Gb(2)1232 4751 y FV(\000)-15 b(!)19 b F0(R)h(for)h(some)f(R;)h(and)e Ge(\(iii\))h FM(CSN)p FX(\()p F0(P)p FX(\))p F0(.)533 4865 y Ge(\(2\))13 b(\(e)o(xtended)26 b(reduction\))85 b F0(If)30 b FV(`)1560 4878 y FT(f)1604 4865 y F0(P)14 b FZ(.)g F0(A,)31 b(then)e Ge(\(i\))f F0(P)23 b FV(7!)h F0(Q)29 b(implies)g FV(`)2698 4878 y FT(f)2742 4865 y F0(Q)15 b FZ(.)f F0(A;)33 b Ge(\(ii\))c F0(P)23 b FV(7!)g F0(Q)3356 4877 y Gb(1)p FS(;)p Gb(2)450 4973 y F0(implies)e(Q)771 4985 y Gb(1)p FS(;)p Gb(2)872 4973 y FV(7!)955 4943 y Gd(\003)1008 4973 y F0(R)f(for)h(some)f(R;)h(and)e Ge(\(iii\))h FM(CSN)1936 4985 y FT(e)1967 4973 y FX(\()p F0(P)p FX(\))p F0(.)p eop %%Page: 39 39 39 38 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(39)533 -14 y F0(Pr)l(oof)o(.)73 b Ge(Direct)23 b(from)f(the)g(corresponding)d (results)k(in)g FM(BNP)p Ge(.)32 b(As)24 b(an)e(e)o(xample,)g(let)h FV(`)3084 -1 y Gb(f)3128 -14 y F0(P)12 b FZ(.)g F0(A)23 b Ge(and)450 94 y F0(P)c FV(\000)-14 b(!)20 b F0(Q)p Ge(.)32 b(Then)21 b FV(`)1031 107 y Gb(b)1086 94 y F0(P)1137 64 y Gd(\016)1183 94 y FZ(.)12 b F0(A)23 b Ge(by)f(Proposition)f(6.1.) 31 b(Further)21 b(let)i F0(P)c FV(\000)-14 b(!)20 b F0(Q)p Ge(.)32 b(By)22 b(Proposition)f(6.2)h(we)450 202 y(ha)n(v)o(e)e F0(P)676 172 y Gd(\016)729 202 y FV(7!)812 172 y Gd(\003)866 202 y F0(Q)926 172 y Gd(\016)961 202 y Ge(,)h(hence)f(by)g(subject)g (reduction)f(in)i 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FV(h)-10 b FZ(~)-32 b F0(y)q FV(i)2637 1083 y Gd(\016)2672 1075 y FS(?)2722 1091 y FV(\030)2722 1117 y FX(=)2786 1126 y Gb(f)p 2826 1067 V 2826 1113 a F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)p FV(i)p Ge(.)25 b(In)16 b(f)o(act,)i(using)450 1221 y(Lemma)h(6.1,)h(we) g(ha)n(v)o(e:)p 1199 1399 V 1199 1445 a F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)q FV(i)1338 1414 y Gd(\016)1373 1406 y FS(?)1457 1398 y FQ(def)1462 1445 y FX(=)54 b(\()p F1(n)9 b FZ(~)-41 b F0(w)p FX(\)\()p 1785 1399 V F0(x)q FV(h)o FZ(~)g F0(w)q FV(ij)p F1(P)2030 1457 y FT(i)2051 1445 y FX([)p F0(w)2129 1457 y FT(i)2169 1445 y FV(!)18 b F0(y)2307 1457 y FT(i)2328 1445 y FX(]\))2423 1422 y FV(\030)2423 1449 y FX(=)2487 1458 y Gb(f)p 2551 1399 V 2551 1445 a F0(x)p FV(h)-10 b FZ(~)-32 b F0(y)p FV(i)450 1682 y Ge(as)21 b(required.)p 920 1682 50 50 v 450 1840 a(Theorem)f(6.1)i(sho)n(ws)g(that)g(all)h(additional)e(terms)h(in)g FM(FNP)h Ge(which)e(do)h(not)g(e)o(xist)g(in)g FM(BNP)h Ge(are)f(in)g(f)o(act)450 1947 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b(to)g(carry)e(o)o(v)o (er)g(the)i(SN)g(type)f(discipline)g(and)g(related)g(results)h(in)g (imperati)n(v)o(e)e(computation)450 2969 y(in)m(v)n(olving)18 b(non-tri)n(vial)g(procedure)g(calls)j(in)f([36].)533 3083 y(F)o(or)i(the)h(incorporation)d(of)i(control)g(into)g(the)h (present)f(type)g(discipline,)h(all)g(we)g(need)f(is)i(to)f(elim-)450 3191 y(inate)h FV(#)p Ge(-)p FV(")g Ge(types)g(from)f(the)h(present)g (system.)37 b(In)24 b(other)f(w)o(ords,)i(the)f(system)g(presented)f (in)i(Section)450 3299 y(2)e(already)f(contains)g(the)g(calculus)h(for) f(\(linear\))g(control)f(as)j(its)f(subcalculus.)32 b(This)23 b(means,)g(among)450 3407 y(others,)e(the)h(calculus)f(satis\002es)i (all)f(syntactic)f(properties)f(we)i(ha)n(v)o(e)f(e)o(xplored)e(in)j (Sections)f(2)h(and)f(3,)450 3515 y(including)16 b(strong)i (normalisability)-5 b(.)22 b(W)-7 b(e)20 b(ha)n(v)o(e)d(v)o(eri\002ed)g (that)i(a)f(sequential)g(v)o(ersion)f(of)h(this)h(calculus)450 3623 y(can)k(fully)g(abstractly)g(embed)g(P)o(arigo')-5 b(s)23 b F1(l)p F0(\265)p Ge(-calculus.)33 b(Further)22 b(discussions)i(on)f(this)h(calculus)f(and)450 3730 y(its)e(e)o (xtensions)e(will)i(be)f(discussed)g(else)n(where.)450 3894 y F2(Second-order)j(and)i(Other)e(Extensions)i(in)f(T)-6 b(ype)25 b(Structur)o(e.)81 b Ge(Can)24 b(the)g(presented)f(results)h (be)450 4002 y(augmented)15 b(to)j(co)o(v)o(er)e(more)h(e)o(xpressi)n (v)o(e)f(notions)h(of)g(types)g(studied)g(in)h(functional)e(calculi?)24 b(Adding)450 4110 y(recursi)n(v)o(e)g(types)i([52,)12 b(66])25 b(easily)i(leads)f(to)g(a)g(system)g(that)g(is)h(not)e (strongly)g(normalising:)35 b(for)25 b(e)o(x-)450 4218 y(ample,)h(the)g(encoding,)e(follo)n(wing)g(Figure)h(8,)i(of)e FX(\()p F1(l)p F0(x)p FZ(:)p F0(xx)p FX(\)\()p F1(l)p F0(x)p FZ(:)p F0(xx)p FX(\))i Ge(becomes)e(typable.)40 b(Re)o(garding)450 4325 y(second-order)24 b(types,)29 b(our)d(recent)h(w)o(ork)g([12])f(demonstrates)g(that)h(such)g(e)o 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b(present)f(w)o(ork)h(adds)g(minimum)e(causality)i(to)h(the)f(system)g (in)g([11])450 104 y(to)e(ensure)f(SN)h(of)f(replicated)g(processes.)44 b(Ho)n(we)n(v)o(er)m(,)26 b(our)f(SN)j(proof)d(seems)i(to)g(be)f(able)h (to)f(cope,)450 212 y(without)21 b(signi\002cant)f(change,)g(with)i (more)e(comple)o(x)g(causality)h(relations:)27 b(for)21 b(e)o(xample,)f(we)h(could)450 320 y(relax)16 b(the)g(channel)f(type)h (constraints)f(and)h(e)o(xtend)f(action)h(types)g(to)g(\002nite)h (graph)e(structures)g(between)450 428 y(arbitrary)k(linear)i(nodes)f (as)i(in)f([68].)26 b(An)21 b(e)n(v)o(en)f(wider)h(class)h(of)f(SN)g (interactions)f(w)o(ould)g(be)h(typable)450 535 y(if)28 b(we)f(further)f(allo)n(wed)h(edges)g(of)g(the)g(more)g(general)f(form) 32 b F0(px)23 b FV(!)f F0(qy)p Ge(,)29 b(where)k F0(p)22 b FV(2)g(f#)p FZ(;)9 b FV(")p FZ(;)g Ge(?)s FV(g)28 b Ge(and)450 643 y F0(q)d FV(2)g(f)p Ge(!)s FZ(;)9 b FV(#)p FZ(;)g FV("g)32 b Ge(\(i.e.)g(replicated)f(and)g(linear)h(nodes)f(can)h (be)g(mix)o(ed\).)59 b(Di)n(v)o(erse)32 b(structures)f(w)o(ould)450 751 y(be)d(embeddable)e(in)i(such)f(an)h(e)o(xtension,)g(including)e (full)i(proof)f(nets)h([9].)47 b(The)28 b(status)h(of)e(strong)450 859 y(reduction)18 b(w)o(ould)i(become)f(subtle)h(in)g(this)h(setting,) f(cf.)g([23].)450 1017 y F2(Game)j(Semantics.)82 b Ge(In)23 b(game)g(semantics,)g(\223winning)f(strate)o(gies\224)h(represent)f (strong)h(normalisa-)450 1125 y(tion)29 b([3].)51 b(This)29 b(representation)e(ensures,)k(essentially)e(by)g(de\002nition,)h(that)f (composition)e(of)i(tw)o(o)450 1233 y(winning)g(strate)o(gies)h(will)h (ne)n(v)o(er)d(go)i(into)g(in\002nite)g F1(t)p Ge(-actions)g(\(which)f (w)o(ould)g(mak)o(e)h(the)g(strate)o(gy)450 1341 y(partial\).)24 b(This)c(e)o(xtensional)e(representation)g(of)h(SN)i(does)e(not)g (directly)g(suggest)h(concrete)e(type)h(dis-)450 1449 y(ciplines)26 b(to)h(ensure)f(SN)h(for)f(mobile)g(processes,)i(e)n(v)o (en)d(though)g(the)i(li)n(v)o(eness)f(property)f(discussed)450 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(innocent)g(function)f(for)i(each)g(process)g(is)450 2312 y(\002nite.)k(It)20 b(w)o(ould)e(be)h(interesting)g(to)g(use)h (the)f(frame)n(w)o(ork)e(introduced)g(in)j(the)f(present)g(paper)m(,)f (among)450 2420 y(others)f(typed)f(processes)h(and)g(their)g(beha)n (vioural)e(characterisation,)h(for)h(formulating)e(and)i(studying)450 2528 y(v)n(arious)k(notions)g(of)g(SN)i(and)e(related)g(ideas)h(in)g (game)f(semantics)h(\(for)f(e)o(xample)f(we)i(may)g(consider)450 2636 y(e)o(xplicit)e(incorporation)d(of)j(ac)o(yclicity)f (conditions\).)450 2794 y F2(T)-8 b(erm)25 b(Rewriting)f(and)g (Reduction)g(Strategies.)81 b Ge(The)24 b(proof)e(method)h(presented)g (in)h(this)g(paper)450 2902 y(uses)31 b(the)f(e)o(xtended)e(reduction)h FV(7!)i Ge(to)f(pro)o(v)o(e)e(not)i(only)g(SN)h(b)n(ut)f(also)g(other)g (results)g(including)f(a)450 3010 y(fully)f(abstract)g(embedding)e(of)j F1(l)1466 3022 y Gd(!)p FS(;)p Gd(\002)p FS(;)p FU(+)1660 3010 y Ge(.)51 b(One)28 b(of)g(the)h(merits)g(in)f(using)g FV(7!)i Ge(lies)f(in)g(the)f(potential)450 3118 y(applicability)d(of)i (v)n(arious)e(T)-6 b(erm)27 b(Re)n(writing)f(Lemmas)g(in)h(the)g(conte) o(xt)e(of)i(interacting)e(processes.)450 3226 y(In)j(f)o(act,)j (technically)c(speaking,)i(this)g(may)f(be)h(re)o(garded)d(as)j(one)f (of)g(the)g(main)h(dif)n(ferences)d(from)450 3334 y(other)17 b(studies)h(addressing)e(termination)g(and)i(other)f(related)g (properties)f(of)i(processes)f([42,)12 b(60].)24 b(Our)450 3442 y(recent)18 b(w)o(ork)g([69])f(partly)h(addresses)g(this)h(point.) 24 b(In)18 b(Section)h(3,)f(we)h(de\002ne)f(a)h(reduction)e(strate)o (gy)h(of)450 3550 y FV(7!)j Ge(to)g(pro)o(v)o(e)e(SN.)i(Lik)o(e)g(the)f (left-most)g(reduction)f(strate)o(gy)h(of)g(the)h F1(l)p Ge(-calculus,)e(this)i(strate)o(gy)f(could)450 3657 y(be)g(de\002ned)g (in)g(the)h(untyped)d(setting)i(in)h(general,)e(then)h(could)f(be)i (used)f(to)g(pro)o(v)o(e)f(the)h(normalisation)450 3765 y(theorem)i(\(i.e.)i(it)g(al)o(w)o(ays)h(deri)n(v)o(es)d(a)j(normal)d (form)h(if)h(it)h(e)o(xists\).)35 b(This)24 b(opens)g(possibility)f(to) h(study)450 3873 y(v)n(arious)18 b(reduction)g(strate)o(gies)h(in)g (the)h(name-passing)d(scenario,)h(which)h(had)g(not)g(been)g(in)m(v)o (estigated)450 3981 y(so)j(f)o(ar)h(due)e(to,)i(among)d(others,)i(e)o (xistence)f(of)h(structure)f(rules.)31 b(W)-7 b(e)23 b(may)f(hope)f(that,)i(through)d(such)450 4089 y(studies,)29 b(that)e(the)g(accumulated)f(ideas)h(from)f(functional)g(computation)f (such)h(as)i(optimality)e([46])450 4197 y(may)20 b(be)g(transferable)e (into)i(non-deterministic)e(and)i(non-terminating)c(interacti)n(v)o(e)j (computation.)1619 4453 y F3(REFERENCES)467 4606 y FW(1.)33 b(Abramsk)o(y)l(,)16 b(S.,)g(Computational)k(interpretation)i(of)17 b(linear)i(logic.)f FO(TCS)p FW(,)e(V)-9 b(ol.)18 b(111)f(\(1993\))g (3\22657,)h(1993.)467 4750 y(2.)33 b(Abramsk)o(y)l(,)16 b(S.,)g(Proofs)h(as)g(Processes,)g FO(TCS)p FW(,)g(V)-9 b(ol.)17 b(135)g(\(1994\))h(5\2269,)f(1994.)467 4894 y(3.)33 b(Abramsk)o(y)l(,)13 b(S.)g(and)h(Jagadeesan,)h(R.,)e(Games)h (and)g(Full)g(Completeness)h(for)f(Multiplicati)n(v)o(e)k(Linear)d (Logic.)e FO(JSL)p FW(,)g(V)-9 b(ol.)14 b(59,)550 4973 y(1994.)p eop %%Page: 44 44 44 43 bop 450 -257 a FX(44)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)467 -4 y(4.)33 b(Abramsk)o(y)l(,)20 b(S.,)g(Jagadeesan,)j(R.)d(and)h(Malacaria,)j(P)-7 b(.,)20 b(Full)h(Abstraction)i(for)d(PCF)-5 b(,)19 b FO(Info.)i(&)f(Comp.)g FW(163)h(\(2000\),)g(409-)550 75 y(470.)467 187 y(5.)33 b(Abramsk)o(y)l(,)14 b(S.,)f(Process)i(Realizability)l(,)k(A)13 b(T)m(utorial)j(W)-5 b(orkshop)14 b(on)g(Realizability)19 b(Semantics)d(and)e(Applications,)j(1999.)550 266 y(A)-5 b(v)n(ailable)20 b(at)d(web)m(.comlab)m(.ox.ac.uk/oucl/w)o(ork)q (/samson.ab)q(ramsk)o(y)l(.)467 378 y(6.)33 b(Altenkirch,)18 b(T)-5 b(.,)15 b(Dybjer)m(,)i(P)-7 b(.,)14 b(Hofmann,)i(M.)g(and)g (Scott,)h(P)-7 b(.,)15 b(Normalisation)j(by)e(Ev)n(aluation)j(for)d(T) -5 b(yped)16 b(Lambda)g(Calcu-)550 457 y(lus)h(with)g(Coproducts,)i FO(LICS'01)f FW(303\226310,)f(IEEE,)e(2001.)467 569 y(7.)33 b(Barendre)o(gt,)19 b(H.,)d FO(The)h(Lambda)g(Calculus:)23 b(Its)17 b(Syntax)i(and)e(Semantics)p FW(.)i(North)e(Holland,)h(1984.) 467 681 y(8.)33 b(Barendre)o(gt,)24 b(H.,)d(Lambda)g(Calculi)j(with)e (T)-5 b(ypes,)21 b(Handbook)i(of)e(Logic)h(in)f(Computer)h(Science,)i (V)-9 b(ol)22 b(2,)g(pp.118\226310,)550 760 y(Clarendon)d(Press,)d (Oxford,)h(1992.)467 872 y(9.)33 b(Bellin,)18 b(G.,)e(and)i(Scott,)f(P) -7 b(.J.,)15 b(On)i(the)h(Pi-calculus)h(and)f(linear)h(logic,)f FO(TCS)p FW(,)e(V)-9 b(ol.)18 b(135,)e(pp.)h(11-65,)g(1994.)433 985 y(10.)34 b(Ber)o(ger)m(,)28 b(M.,)d(Honda,)i(K.)d(and)i(Y)-7 b(oshida,)27 b(N.,)f(Sequentiality)j(and)c(the)h FP(p)p FW(-Calculus,)j FO(TLCA01)p FW(,)d(LNCS)e(2044,)j(29\22645,)550 1063 y(Springer)m(,)18 b(2001.)433 1176 y(11.)34 b(The)17 b(full)g(v)o(ersion)i(of)d([10].)h(A)-5 b(v)n(ailable)21 b(at)c(www)l(.dcs.qmw)l(.ac.uk/\230k)o(ohei.)433 1288 y(12.)34 b(Ber)o(ger)m(,)22 b(M.,)e(Honda,)h(K.)e(and)i(Y)-7 b(oshida,)22 b(N.,)d(Genericity)24 b(and)c(the)h FP(p)p FW(-Calculus,)j FO(F)-7 b(oSSaCs03)p FW(,)21 b(LNCS)f(2620,)h (103\226119,)550 1367 y(Springer)m(,)d(2003.)433 1479 y(13.)34 b(Boreale,)18 b(M.,)e(On)h(e)o(xpressi)n(v)o(eness)j(in)d (internal)i(mobility)g(of)e(name)h(passing)f(calculi,)j FO(TCS)p FW(,)c(195:206\226226,)j(1998.)433 1591 y(14.)34 b(Boreale,)j(M.)31 b(and)h(Sangior)o(gi,)k(D.,)e(Some)e(Congruence)i (Properties)g(for)d FP(p)p FW(-calculus)k(Bisimilarities,)j FO(TCS)p FW(,)31 b(1998,)550 1670 y(V)-9 b(ol.198,)17 b(pages)h(=)f(159\226176.)433 1782 y(15.)34 b(Boudol,)17 b(G.,)f(Asynchron)o(y)i(and)g(the)g(pi-calculus,)h(INRIA)e(Research)i (Report)g(1702,)e(1992.)433 1895 y(16.)34 b(Boudol,)17 b(G.,)f(The)h(pi-calculus)j(in)d(direct)i(style,)f FO(POPL)n('97)p FW(,)d(228\226241,)j(A)m(CM,)f(1997.)433 2007 y(17.)34 b(Boudol,)23 b(G.)e(and)i(Castellani,)j(I.,)c(Noninterference)j(for)d (Concurrent)j(Programs,)d FO(ICALP01)p FW(,)h(LNCS)e(2076,)i (382\226395,)550 2086 y(Springer)m(,)18 b(2001.)433 2198 y(18.)34 b(Chaki,)e(S.,)f(Rajamani,)i(S.)28 b(and)h(Rehof,)k(J.,)d(T)-5 b(ypes)28 b(as)h(Models:)46 b(Model)30 b(Checking)h(Message-P)o(assing) g(Programs,)550 2277 y FO(POPL)n('02)p FW(,)15 b(A)m(CM,)i(2002.)433 2389 y(19.)34 b(Ghani,)17 b(N.,)f FO(Adjoint)i(Re)o(writing)p FW(,)g(PhD)f(Thesis,)g(LFCS,)f(Uni)n(v)o(ersity)j(of)e(Edinb)o(ur)o (gh,)g(No)o(v)l(.)f(1995.)433 2501 y(20.)34 b(Ghani,)17 b(N.,)f FP(bh)p FW(-equality)k(from)d(coproducts,)i FO(TLCA)n('95)p FW(,)d(LNCS)g(902,)h(171\226185,)g(Springer)m(,)i(1995.)433 2613 y(21.)34 b(Gallier)m(,)23 b(J.)c(H.,)h(On)g(Girard')l(s)i (\223Candidats)h(de)d(Reductibilit)t(\264)-26 b(e\224,)26 b(123\226203,)c(Logic)f(and)g(Computer)h(Science,)h(Academic)550 2692 y(Press)16 b(Limited,)i(1990.)433 2804 y(22.)34 b(Gay)l(,)26 b(S.)d(and)i(Hole,)i(M.,)e(T)-5 b(ypes)24 b(and)h(Subtypes)h(for)e(Client-Serv)o(er)k(Interactions,)h FO(ESOP'99)p FW(,)d(LNCS)e(1576,)i(74\22690,)550 2883 y(Springer)m(,)18 b(1999.)433 2995 y(23.)34 b(Girard,)17 b(J.-Y)-9 b(.,)16 b(Linear)i(Logic,)f FO(TCS)p FW(,)g(V)-9 b(ol.)17 b(50,)g(1\226102,)g(1987.)433 3108 y(24.)34 b(Girard,)17 b(J.-Y)-9 b(.,)15 b(Lafont)j(Y)-9 b(.)16 b(and)h(T)-5 b(aylor)m(,)17 b(P)-7 b(.,)15 b FO(Pr)m(oofs)i(and)g(T)-5 b(ypes)p FW(,)17 b(v)o(ol.)f(7)h(of)f(Cambridge)j(T)n(racts)d(in)h (Theoretical)j(Computer)550 3187 y(Science,)e(CUP)-7 b(,)16 b(1989.)433 3299 y(25.)34 b(Gunter)m(,)18 b(C.A.,)d FO(Semantics)k(of)e(Pr)m(o)o(gr)o(amming)h(Langua)o(g)o(es:)23 b(Structur)n(es)18 b(and)f(T)-6 b(ec)o(hniques)p FW(,)19 b(MIT)d(Press,)h(1992.)433 3411 y(26.)34 b(Hicks,)29 b(M.,)f(Kakkar)m(,)i(P)-7 b(.,)28 b(Moore,)h(J.T)-5 b(.,)27 b(Gunter)m(,)k(C.A.)25 b(and)j(Nettles,)i(S.)c(PLAN:)g(A)h(P)o(ack)o (et)h(Language)h(for)d(Acti)n(v)o(e)550 3490 y(Netw)o(orks,)17 b FO(Pr)m(oceedings)i(of)e(the)h(International)i(Confer)n(ence)g(on)d (Functional)i(Pr)m(o)o(gr)o(amming)e(\(ICFP'98\))p FW(,)g(cf.)g([56].) 433 3602 y(27.)34 b(Honda,)17 b(K.,)f(T)-5 b(ypes)16 b(for)h(Dyadic)i(Interaction.)h FO(CONCUR'93)p FW(,)d(LNCS)f(715,)h (509-523,)h(1993.)433 3714 y(28.)34 b(Honda,)17 b(K.,)f(Composing)i (Processes,)f FO(POPL)n('96)p FW(,)e(344-357,)j(A)m(CM,)f(1996.)433 3827 y(29.)34 b(Honda,)17 b(K.,)f(Notes)h(on)g(Linear)h(T)-5 b(yping)17 b(for)g(Free)h(Outputs,)g(May)l(,)e(2001.)433 3939 y(30.)34 b(Honda,)17 b(K.,)f(Notes)h(on)g(the)h(linear)h FP(p)p FW(-calculus)h(and)d(LLP)-7 b(,)15 b(June,)i(2001.)433 4051 y(31.)34 b(Honda,)40 b(K.,)e(K)o(ubo,)i(M.)34 b(and)i(V)-7 b(asconcelos,)41 b(V)-9 b(.,)39 b(Language)e(Primiti)n(v)o(es)g(and)f (T)-5 b(ype)35 b(Discipline)j(for)d(Structured)550 4130 y(Communication-Based)21 b(Programming.)c FO(ESOP'98)p FW(,)f(LNCS)h(1381,)g(122\226138.)g(Springer)o(-V)-7 b(erlag,)19 b(1998.)433 4242 y(32.)34 b(Honda,)18 b(K.)f(and)h(T)-5 b(ok)o(oro,)18 b(M.,)f(An)h(Object)h(Calculus)h(for)e(Asynchronous)h (Communication.)h FO(ECOOP'91)p FW(,)d(LNCS)h(512,)550 4321 y(133\226147,)f(Springer)o(-V)-7 b(erlag)20 b(1991.)433 4433 y(33.)34 b(Honda,)15 b(K.)e(V)-7 b(asconcelos,)16 b(V)-9 b(.,)14 b(and)h(Y)-7 b(oshida,)15 b(N.)e(Secure)j(Information)g (Flo)n(w)f(as)f(T)-5 b(yped)14 b(Process)h(Beha)o(viour)m(,)i FO(ESOP)c('99)p FW(,)550 4512 y(LNCS)18 b(1782,)h(180\226199,)i (Springer)o(-V)-7 b(erlag,)21 b(2000.)e(Full)h(v)o(ersion)g(a)o(v)n (ailable)i(as)d(MCS)g(technical)j(report)f(01/2000,)f(Uni-)550 4591 y(v)o(ersity)e(of)f(Leicester)m(,)i(2000.)433 4703 y(34.)34 b(Honda,)15 b(K.)e(and)h(Y)-7 b(oshida,)15 b(N.,)f(On)f (Reduction-Based)19 b(Process)14 b(Semantics.)h FO(TCS)p FW(,)f(437\226486,)h(V)-9 b(ol.)15 b(151,)f(North-Holland,)550 4782 y(1995.)433 4894 y(35.)34 b(Honda,)27 b(K.)c(and)j(Y)-7 b(oshida,)27 b(N.)48 b(Game-theoretic)28 b(analysis)f(of)d(call-by-v)n (alue)29 b(computation.)e FO(TCS)e FW(V)-9 b(ol.)25 b(221)g(\(1999\),) 550 4973 y(393\226456,)17 b(North-Holland,)j(1999.)p eop %%Page: 45 45 45 44 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(45)433 -4 y FW(36.)34 b(Honda,)17 b(K.)f(and)i(Y)-7 b(oshida,)17 b(N.,)f(A)g(Uniform)i(T)-5 b(ype)16 b(Structure)j(for)f(Secure)g (Information)h(Flo)n(w)l(,)e FO(POPL)n('02)p FW(,)e(81\22692,)i(A)m(CM) 550 75 y(Press,)22 b(2002.)g(Full)g(v)o(ersion)h(as)f(a)g(DOC)f (technical)26 b(report,)d(Department)h(of)e(Computing,)i(Imperial)g (Colle)o(ge)g(London,)550 154 y(August,)17 b(2002.)g(A)-5 b(v)n(ailable)20 b(at:)i(www)l(.doc.ic.ac.uk/\230yoshida.)433 266 y(37.)34 b(Huet,)26 b(G.,)e(Con\003uent)h(reductions:)38 b(Abstract)25 b(properties)h(and)f(applications)i(to)d(term)h(re)n (writing)h(systems,)f FO(J)n(.,)g(Assoc,)550 345 y(Comput.)17 b(Mar)n(c)o(h)p FW(,)g(23\(1\),)g(1981,)g(11-21.)433 457 y(38.)34 b(Hyland,)17 b(M.)g(and)g(Ong,)g(L.,)e(\224On)i(Full)h (Abstraction)i(for)d(PCF\224:)k(I,)c(II)f(and)i(III.)f FO(Info.)g(&)f(Comp.)h FW(163)g(\(2000\),)g(285-408.)433 569 y(39.)34 b(Igarashi,)17 b(A.)g(and)g(K)n(obayashi,)h(N.,)e(A)g (generic)j(type)f(system)g(for)f(the)g(pi-calculus,)j FO(POPL)n('01)p FW(,)15 b(A)m(CM,)i(2001.)433 681 y(40.)34 b(Klop,)17 b(J.)g(W)-6 b(.,)15 b(T)-5 b(erm)18 b(Re)n(writing)i (Systems,)d(Handbook)i(of)e(Logic)h(in)f(Computer)i(Science,)g(V)-9 b(ol)18 b(2,)f(pp.2\226117,)h(Clarendon)550 760 y(Press,)e(Oxford,)h (1992.)433 872 y(41.)34 b(K)n(obayashi,)25 b(N.,)d(A)g(partially)k (deadlock-free)g(typed)e(process)f(calculus,)j FO(A)n(CM)c(T)o(OPLAS)p FW(,)f(V)-9 b(ol.)23 b(20,)g(No.)f(2,)i(436\226482,)550 951 y(1998.)433 1063 y(42.)34 b(K)n(obayashi,)c(N.,)f(T)-5 b(ype)27 b(Systems)g(for)g(Concurrent)j(Processes:)43 b(From)26 b(Deadlock-Freedom)31 b(to)d(Li)n(v)o(elock-Freedom,)550 1142 y(T)n(ime-Boundedness,)18 b FO(Pr)m(oc.)e(of)i(TCS2000)p FW(,)f(LNCS)f(1872,)i(pp.365-389,)f(Springer)m(,)h(2000.)433 1255 y(43.)34 b(K)n(obayashi,)26 b(N.,)e(Pierce,)i(B.,)e(and)g(T)m (urner)m(,)i(D.,)e(Linear)g(T)-5 b(ypes)23 b(and)h FP(p)p FW(-calculus,)k FO(POPL)n('96)p FW(,)23 b(358\226371,)j(A)m(CM)e (Press,)550 1333 y(1996.)433 1446 y(44.)34 b(Laird,)22 b(J.,)e(A)h(deconstruction)j(of)d(non-deterministic)k(classical)f(cut)e (elimination,)i FO(TLCA)n('01)p FW(,)d(LNCS)g(2044,)h(268\226282,)550 1525 y(Springer)m(,)c(2001.)433 1637 y(45.)34 b(Lafont,)17 b(Y)-9 b(.,)16 b(Interaction)k(Nets,)d FO(POPL)n('90)p FW(,)f(pp.)g(95\226108,)i(A)m(CM)f(press,)g(1990.)433 1749 y(46.)34 b(L)t(\264)-26 b(evy)l(,)17 b(J.-J.,)e(An)i(algebraic)k (interpretation)h(of)17 b(the)h FP(l)p FW(-)p FP(b)p FW(-)p FO(K)s FW(-calculus)k(and)c(a)g(labelled)i FP(l)p FW(-calculus,)g(pp.147\226165,)e(Proc.)f(of)550 1828 y FP(l)p FO(-calculus)i(and)e(Computer)i(Science)g(Theory)p FW(,)e(LNCS)g(37,)f(Springer)o(-V)-7 b(erlag,)20 b(1975.)433 1940 y(47.)34 b(Merro,)26 b(M.)e(and)h(Sangior)o(gi,)j(D.,)d(On)f (asynchron)o(y)j(in)e(name-passing)h(calculi,)j FO(ICALP'98)p FW(,)c(LNCS)f(1443,)j(856\226867,)550 2019 y(Springer)o(-V)-7 b(erlag,)19 b(1998.)433 2131 y(48.)34 b(Milner)m(,)18 b(R.,)e FO(A)g(Calculus)j(of)e(Communicating)j(Systems)p FW(,)e(LNCS)e(76,)h(Springer)o(-V)-7 b(erlag,)19 b(1980.)433 2243 y(49.)34 b(Laurent,)17 b(O.,)f(Polarized)k(games,)c(LICS)h(2002,)g (265-274,)g(IEEE,)e(2002.)433 2356 y(50.)34 b(Laurent,)17 b(O.,)f(In)m(vited)j(lecture:)24 b(Polarities)c(in)d(linear)i(logic.)f (Linear)g(Logic)f(W)-5 b(orkshop,)17 b(July)g(2002.)433 2468 y(51.)34 b(Milner)m(,)18 b(R.,)e(Functions)j(as)e(Processes.)g FO(MSCS)p FW(,)g(2\(2\),)g(119\226146,)h(CUP)-7 b(,)16 b(1992.)433 2580 y(52.)34 b(Milner)m(,)16 b(R.,)d(Polyadic)j FP(p)p FW(-Calculus:)23 b(a)14 b(tutorial.)i FO(Pr)m(oceedings)g(of)e (the)h(International)i(Summer)e(Sc)o(hool)g(on)f(Lo)o(gic)h(Alg)o(ebr)o (a)550 2659 y(of)i(Speci\002cation)p FW(,)j(Marktoberdorf,)f(1992.)433 2771 y(53.)34 b(Milner)m(,)25 b(R.,)e(P)o(arro)n(w)l(,)h(J.G.)d(and)i (W)-5 b(alk)o(er)m(,)25 b(D.J.,)e(A)f(Calculus)i(of)f(Mobile)h (Processes,)h FO(Information)f(and)f(Computation)550 2850 y FW(100\(1\),)17 b(1\22677,)g(1992.)433 2962 y(54.)34 b(Mitchell,)19 b(J.)d FO(F)-7 b(oundations)18 b(for)g(Pr)m(o)o(gr)o (amming)f(Langua)o(g)o(es)p FW(,)i(MIT)d(Press,)g(1996.)433 3074 y(55.)34 b(Pierce,)18 b(B.C.)e(and)i(Sangior)o(gi.)g(D,)e(T)-5 b(yping)17 b(and)h(subtyping)h(for)e(mobile)h(processes.)f FO(LICS'93)p FW(,)h(187\226215,)f(IEEE,)e(1993.)433 3187 y(56.)34 b(PLAN:)18 b(A)g(P)o(ack)o(et)i(Language)g(for)f(Acti)n(v)o(e) i(Netw)o(orks,)e(SwitchW)-5 b(are)21 b(Project,)f(Uni)n(v)o(ersity)h (of)d(Pennsylv)n(ania,)k(a)o(v)n(ailable)550 3265 y(from)16 b(http://www)l(.cis.upenn.edu/\230switchware)q(/.)433 3378 y(57.)34 b(Quaglia,)17 b(P)-7 b(.)14 b(and)j(W)-5 b(alk)o(er)m(,)17 b(D.,)d(On)i(Synchronous)h(and)f(Asynchronous)h (Mobile)g(Processes,)g FO(F)-7 b(oSSaCS)16 b(00)p FW(,)f(LNCS)g(1784,) 550 3457 y(283\226296,)i(Springer)m(,)i(2000.)433 3569 y(58.)34 b(Sangior)o(gi,)18 b(D.)23 b FP(p)p FW(-calculus,)d(internal)f (mobility)l(,)f(and)f(agent-passing)j(calculi.)26 b FO(TCS)p FW(,)17 b(167\(2\):235\226271,)i(North-Holland,)550 3648 y(1996.)433 3760 y(59.)34 b(Sangior)o(gi,)26 b(D.,)e(The)f(name)h (discipline)j(of)c(uniform)h(recepti)n(v)o(eness,)29 b FO(ICALP'97)p FW(,)24 b(LNCS)f(1256,)i(303\226313,)h(Springer)m(,)550 3839 y(1997.)433 3951 y(60.)34 b(Sangior)o(gi,)18 b(D.,)e(T)-5 b(ermination)19 b(of)e(Processes,)g(Draft,)h(December)m(,)g(2001.)433 4063 y(61.)34 b(Smith,)17 b(G.,)e(A)i(Ne)n(w)g(T)-5 b(ype)17 b(System)g(for)g(Secure)i(Information)f(Flo)n(w)l(,)f FO(CSFW'01)p FW(,)g(IEEE,)e(2001.)433 4175 y(62.)34 b(Smith,)c(G.)d (and)h(V)-9 b(olpano,)32 b(D.,)d(Secure)g(information)h(\003o)n(w)e(in) g(a)f(multi-threaded)32 b(imperati)n(v)o(e)e(language,)j(355\226364,) 550 4254 y FO(POPL)n('98)p FW(,)15 b(A)m(CM,)i(1998.)433 4366 y(63.)34 b(T)-5 b(ait,)17 b(W)-6 b(.,)15 b(Intensional)20 b(interpretation)h(of)c(functionals)j(of)d(\002nite)h(type,)f(I.)g FO(J)n(.)g(Symb)m(.)g(Lo)o(g)p FW(,)g(32,)g(198\226212,)g(1967.)433 4479 y(64.)34 b(Urban,)27 b(C.)e(and)h(Bierman,)i(G.M.,)e(Strong)g (Normalisation)i(of)d(Cut-Elimination)j(in)e(Classical)i(Logic.)d FO(Fundamenta)550 4558 y(Informaticae)p FW(,)19 b (45\(1-2\):123\226155,)g(2001.)433 4670 y(65.)34 b(V)-7 b(asconcelos,)18 b(V)-9 b(.,)16 b(T)-5 b(yped)17 b(concurrent)j (objects.)e FO(ECOOP'94)p FW(,)e(LNCS)h(821,)g(100\226117.)g(Springer)m (,)i(1994.)433 4782 y(66.)34 b(V)-7 b(asconcelos,)17 b(V)-9 b(.)16 b(and)g(Honda,)h(K.,)e(Principal)j(T)-5 b(yping)16 b(Scheme)h(for)f(Polyadic)i FP(p)p FW(-Calculus.)g FO(CONCUR'93)p FW(,)e(LNCS)g(715,)550 4861 y(524\226538,)h(Springer)o (-V)-7 b(erlag,)19 b(1993.)433 4973 y(67.)34 b(W)m(insk)o(el,)18 b(G.,)e FO(The)h(F)-7 b(ormal)17 b(Semantics)i(of)e(Pr)m(o)o(gr)o (amming)h(Langua)o(g)o(es:)k(An)17 b(Intr)m(oduction)p FW(,)i(MIT)e(Press,)f(1993.)p eop %%Page: 46 46 46 45 bop 450 -257 a FX(46)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)433 -4 y(68.)34 b(Y)-7 b(oshida,)27 b(N.,)f(Graph)f(T)-5 b(ypes)25 b(for)g(Monadic)i(Mobile)f(Processes,)i FO(FST/TCS'16)p FW(,)f(LNCS)d(1180,)k(371\226387,)f(Springer)o(-)550 75 y(V)-7 b(erlag,)17 b(1996.)g(Full)h(v)o(ersion)g(as)f(LFCS)f(T)-5 b(echnical)20 b(Report,)e(ECS-LFCS-96-350,)f(1996.)433 194 y(69.)34 b(Y)-7 b(oshida,)18 b(N.,)f(T)-5 b(ype-Based)19 b(Li)n(v)o(eness)g(Guarantee)i(in)d(the)h(Presence)g(of)f (Nontermination)j(and)e(Nondeterminism,)h(April)550 273 y(2002.)d(MCS)g(T)-5 b(echnical)19 b(Report,)f(2002-20,)g(Uni)n(v)o (ersity)h(of)e(Leicester)l(.)i(A)-5 b(v)n(ailable)20 b(at)e(www)l(.doc.ic.ac.uk/\230yoshida.)433 392 y(70.)34 b(Y)-7 b(oshida,)21 b(N.,)f(Ber)o(ger)m(,)j(M.)d(and)h(Honda,)g(K.,)f (Strong)i(Normalisation)h(in)e(the)g FP(p)p FW(-Calculus,)j FO(LICS'01)p FW(,)d(311\226322,)h(IEEE,)550 471 y(2001.)17 b(The)g(\002rst)g(full)g(v)o(ersion)h(as)f(MCS)g(technical)k(report,)c (2001-09,)h(Uni)n(v)o(ersity)h(of)e(Leicester)m(,)i(2001.)433 590 y(71.)34 b(Y)-7 b(oshida,)33 b(N.,)f(Honda,)i(K.)29 b(and)i(Ber)o(ger)m(,)j(M.)29 b(Linearity)k(and)d(Bisimulation,)36 b(Proc.)30 b(of)g(5th)h(International)j(Confer)o(-)550 669 y(ence,)18 b(F)o(oundations)h(of)f(Softw)o(are)h(Science)g(and)f (Computer)h(Structures)g(\(F)o(oSSaCs)f(2002\),)f(LNCS)g(2303,)h (pp.417\226433,)550 747 y(Springer)m(,)j(2002.)f(A)f(full)i(v)o(ersion) g(as)f(a)f(MCS)h(technical)j(report,)e(2001-48,)g(Uni)n(v)o(ersity)h (of)e(Leicester)m(,)i(2001.)e(A)-5 b(v)n(ailable)550 826 y(at)17 b(www)l(.doc.ic.ac.uk/\230yoshida.)1632 1086 y F3(APPENDIX)32 b(A)1226 1333 y(A.1.)94 b(PR)m(OOFS)32 b(F)m(OR)g(SECTION)f(2)741 1579 y(A.2.)94 b(PR)m(OOF)32 b(OF)g(SUBJECT)g(REDUCTION)g(THEOREM)450 1702 y Ge(W)-7 b(e)30 b(pro)o(v)o(e)d(Proposition)h(2.2)g(\(1\).)51 b(The)29 b(k)o(e)o(y)f(point)g(is)i(to)f(pro)o(v)o(e)e(basic)j (properties)d(of)i(algebra)f(on)450 1810 y(action)20 b(types.)k(W)-7 b(e)22 b(use)e(the)g(same)h(routine)e(as)i(in)f([11,)12 b(33,)h(68].)24 b(W)-7 b(e)21 b(\002rst)g(sho)n(w:)533 1992 y FG(Lemma)30 b(A.1.)62 b FF(Assume)29 b F0(A)p FZ(;)9 b F0(A)1491 2004 y Gb(1)1555 1992 y FF(and)31 b F0(A)1768 2004 y Gb(2)1832 1992 y FF(ar)l(e)f(action)h(typ)l(es.)533 2173 y Ge(\(1\))42 b F0(A)p FZ(=)-10 b(~)-32 b F0(y)29 b FF(is)h(an)g(action)h(typ)l(e.)533 2296 y Ge(\(2\))42 b FF(If)31 b F0(A)812 2308 y Gb(1)865 2296 y FV(\020)18 b F0(A)999 2308 y Gb(2)1033 2296 y FF(,)30 b(then)g F0(A)1324 2308 y Gb(1)1370 2296 y FV(\014)12 b F0(A)1498 2308 y Gb(2)1561 2296 y FF(is)30 b(an)g(action)h(typ)l(e.)533 2522 y F0(Pr)l(oof)o(.)65 b Ge(As)19 b(the)f(proof)e(in)i(Lemma)g(3.4)f (in)h([33].)23 b(\(1\))17 b(is)i(tri)n(vial.)24 b(The)18 b(case)g FM(fn)o FX(\()p F0(A)2863 2534 y Gb(1)2898 2522 y FX(\))9 b FV(\\)g FM(fn)q FX(\()p F0(A)3150 2534 y Gb(2)3185 2522 y FX(\))16 b(=)3323 2520 y F1(/)3314 2522 y(0)i Ge(in)450 2629 y(\(2\))d(is)h(ob)o(vious.)22 b(The)15 b(other)g(case)h(is)h(pro)o(v)o(ed)c(by)i(induction)f(on)h(the)h(size)g (of)f F0(A)2675 2641 y Gb(1)2726 2629 y Ge(and)g F0(A)2913 2641 y Gb(2)2964 2629 y Ge(using)g(the)g(BNF)450 2772 y(representation)j(of)i(action)g(types.)p 1534 2772 50 50 v 533 2954 a FG(Lemma)30 b(A.2.)62 b FF(L)l(et)30 b F0(A)1242 2966 y Gb(1)1276 2954 y FF(,)g F0(A)1382 2966 y Gb(2)1417 2954 y FF(,)g F0(A)1523 2966 y Gb(3)1587 2954 y FF(b)l(e)g(action)h(typ)l(es.)39 b(Then)30 b(we)g(have:)533 3135 y Ge(\(1\))36 b(\(commutati)n(vity\))20 b FF(Assume)j F0(A)1583 3147 y Gb(1)1632 3135 y FV(\020)15 b F0(A)1763 3147 y Gb(2)1798 3135 y FF(.)36 b(Then)24 b(we)g(have)h F0(A)2422 3147 y Gb(2)2471 3135 y FV(\020)15 b F0(A)2602 3147 y Gb(1)2660 3135 y FF(and)24 b F0(A)2866 3147 y Gb(1)2909 3135 y FV(\014)8 b F0(A)3033 3147 y Gb(2)3082 3135 y FX(=)14 b F0(A)3212 3147 y Gb(2)3255 3135 y FV(\014)8 b F0(A)3379 3147 y Gb(1)3413 3135 y FF(.)533 3258 y Ge(\(2\))39 b(\(associati)n(vity\))25 b FF(Assume)g F0(A)1522 3270 y Gb(1)1574 3258 y FV(\020)16 b F0(A)1706 3270 y Gb(2)1767 3258 y FF(and)27 b FX(\()p F0(A)2008 3270 y Gb(1)2052 3258 y FV(\014)10 b F0(A)2178 3270 y Gb(2)2212 3258 y FX(\))17 b FV(\020)f F0(A)2393 3270 y Gb(3)2428 3258 y FF(.)37 b(Then)28 b(we)e(have:)39 b Ge(\(1\))25 b F0(A)3221 3270 y Gb(1)3272 3258 y FV(\020)16 b F0(A)3404 3270 y Gb(3)450 3366 y FF(and)30 b F0(A)662 3378 y Gb(2)715 3366 y FV(\020)18 b F0(A)849 3378 y Gb(3)884 3366 y FF(,)60 b Ge(\(2\))28 b F0(A)1146 3378 y Gb(1)1199 3366 y FV(\020)18 b FX(\()p F0(A)1365 3378 y Gb(2)1412 3366 y FV(\014)12 b F0(A)1540 3378 y Gb(3)1573 3366 y FX(\))30 b FF(and)60 b Ge(\(3\))29 b FX(\()p F0(A)2036 3378 y Gb(1)2082 3366 y FV(\014)12 b F0(A)2210 3378 y Gb(2)2244 3366 y FX(\))g FV(\014)g F0(A)2416 3378 y Gb(3)2467 3366 y FX(=)18 b F0(A)2601 3378 y Gb(1)2647 3366 y FV(\014)12 b FX(\()p F0(A)2807 3378 y Gb(2)2853 3366 y FV(\014)g F0(A)2981 3378 y Gb(3)3014 3366 y FX(\))p FF(.)533 3586 y F0(Pr)l(oof)o(.)65 b Ge(\(1\))16 b(is)i(ob)o(vious)d(by)i(de\002nition.)22 b(\(2\))16 b(is)i(pro)o(v)o(ed)d(by)h(induction)f(on)i(the)g(size)g(of) g F0(A)3102 3598 y FT(i)3140 3586 y Ge(using)f(the)450 3685 y(BNF)28 b(representation)d(of)i(action)f(types.)45 b(This)27 b(is)h(pro)o(v)o(ed)d(as)j(\(a)f(special)g(case)g(of\))g (Lemma)f(3.5)g(in)450 3793 y([33].)p 692 3793 V 533 3975 a FG(Lemma)k(A.3.)533 4156 y Ge(\(1\))40 b FF(If)29 b F0(x)8 b Ge(:)g F1(t)18 b FV(2)g(j)p F0(A)p FV(j)27 b FF(and)h FM(md)p FX(\()p F1(t)p FX(\))18 b FV(2)g(f)p Ge(!)s FZ(;)9 b FV(#g)27 b FF(then)h(ther)l(e)g(is)g(no)g F0(y)8 b Ge(:)g F1(t)2458 4126 y Gd(0)2497 4156 y FV(2)18 b(j)p F0(A)p FV(j)27 b FF(such)h(that)g F0(y)8 b Ge(:)g F1(t)3161 4126 y Gd(0)3200 4156 y FV(!)17 b F0(x)8 b Ge(:)g F1(t)p FF(.)533 4279 y Ge(\(2\))42 b(?)t F0(B)19 b FV(\020)f Ge(?)t F0(B)29 b FF(and)h F0(B)12 b FV(\014)g F0(B)17 b FX(=)h F0(B)p FF(.)533 4403 y Ge(\(3\))41 b FF(If)29 b F0(A)18 b FV(\020)f F0(B)28 b FF(with)h F0(A)p FZ(=)-10 b(~)-32 b F0(x)17 b FX(=)g F0(A)1447 4415 y Gb(0)1482 4403 y FF(,)29 b F0(x)1573 4415 y FT(i)1603 4403 y Ge(:)9 b F1(t)1671 4415 y FT(i)1710 4403 y FV(2)18 b(j)p F0(A)p FV(j)p FF(,)29 b FM(md)p FX(\()p F1(t)2106 4415 y FT(i)2128 4403 y FX(\))17 b FV(2)i(f)p Ge(!)s FZ(;)9 b FV(#)-9 b FZ(;)9 b FJ(l)p FV(g)p FF(,)29 b(and)g FM(fn)o FX(\()p F0(B)p FX(\))11 b FV(\\)g(f)-10 b FZ(~)-32 b F0(x)p FV(g)17 b FX(=)3197 4401 y F1(/)3188 4403 y(0)p FF(,)29 b(then)450 4511 y F0(A)501 4523 y Gb(0)554 4511 y FV(\020)18 b F0(B)29 b FF(and)i FX(\()p F0(A)12 b FV(\014)g F0(B)p FX(\))p FZ(=)-10 b(~)-32 b F0(x)16 b FX(=)i F0(A)1363 4523 y Gb(0)1409 4511 y FV(\014)12 b F0(B)p FF(.)533 4634 y Ge(\(4\))36 b FF(If)23 b F0(A)15 b FV(\020)f F0(B)23 b FF(with)h F0(A)p FZ(=)-10 b(~)-32 b F0(x)14 b FX(=)h F0(A)1415 4646 y Gb(0)1449 4634 y FF(,)25 b F0(x)1536 4646 y FT(i)1563 4634 y Ge(:)6 b F1(t)1628 4646 y FT(i)1664 4634 y FV(2)15 b(j)p F0(A)p FV(j)p FF(,)25 b FM(md)p FX(\()p F1(t)2053 4646 y FT(i)2075 4634 y FX(\))15 b FV(2)g(f)p Ge(!)s FZ(;)9 b FV(#)-9 b FZ(;)9 b FJ(l)p FV(g)p FF(,)25 b(and)f F0(B)p FZ(=)-10 b(~)-32 b F0(x)14 b FX(=)g F0(B)2920 4646 y Gb(0)2955 4634 y FF(,)25 b(then)e F0(A)3234 4646 y Gb(0)3283 4634 y FV(\020)15 b F0(B)p FF(,)450 4742 y F0(A)j FV(\020)g F0(B)653 4754 y Gb(0)688 4742 y FF(,)30 b F0(A)794 4754 y Gb(0)847 4742 y FV(\020)18 b F0(B)981 4754 y Gb(0)1015 4742 y FF(,)31 b(and)f FX(\()p F0(A)12 b FV(\014)g F0(B)p FX(\))p FZ(=)-10 b(~)-32 b F0(x)16 b FX(=)i F0(A)1716 4754 y Gb(0)1762 4742 y FV(\014)12 b F0(B)1890 4754 y Gb(0)1924 4742 y FF(.)533 4865 y Ge(\(5\))46 b FF(Supp)l(ose)34 b F0(A)p FZ(=)-10 b(~)-32 b F0(x)20 b FX(=)h F0(A)1284 4877 y Gb(0)1318 4865 y FF(,)35 b F0(B)p FZ(=)-10 b(~)-32 b F0(x)20 b FX(=)g F0(B)1664 4877 y Gb(0)1733 4865 y FF(and)34 b F0(A)1949 4877 y Gb(0)2004 4865 y FV(\020)20 b F0(B)2140 4877 y Gb(0)2175 4865 y FF(.)50 b(Assume)33 b F0(x)2601 4877 y FT(i)2634 4865 y Ge(:)11 b F1(t)2704 4877 y FT(i)2746 4865 y FV(2)21 b(j)p F0(A)p FV(j)34 b FF(with)g FM(md)p FX(\()p F1(t)3309 4877 y FT(i)3331 4865 y FX(\))21 b FV(2)450 4973 y(f)p FJ(l)p FZ(;)9 b Ge(!)t FZ(;)g FV(#)-9 b(g)p FF(,)29 b(and)h(if)h F0(x)1033 4985 y FT(i)1063 4973 y Ge(:)9 b F1(t)1131 4943 y Gd(0)1131 4996 y FT(i)1172 4973 y FV(2)18 b(j)p F0(B)p FV(j)p FF(,)30 b(then)g F1(t)1618 4985 y FT(i)1658 4973 y FV(\020)18 b F1(t)1777 4943 y Gd(0)1777 4996 y FT(i)1799 4973 y FF(.)38 b(Then)31 b F0(A)18 b FV(\020)g F0(B)29 b FF(and)i FX(\()p F0(A)12 b FV(\014)g F0(B)p FX(\))p FZ(=)-10 b(~)-32 b F0(x)16 b FX(=)i F0(A)2957 4985 y Gb(0)3003 4973 y FV(\014)12 b F0(B)3131 4985 y Gb(0)3165 4973 y FF(.)p eop %%Page: 47 47 47 46 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(47)533 -15 y F0(Pr)l(oof)o(.)66 b Ge(\(1\))19 b(is)i(by)e(the)h(de\002nition)e (of)i(permissibility)f(of)g FV(!)p Ge(,)h(i.e.)25 b(there)20 b(is)g(no)g(edge)f(to)h(inputs)f(and)450 92 y FJ(l)p Ge(.)25 b(\(2\))18 b(is)h(ob)o(vious)e(by)g F1(t)10 b FV(\014)g F1(t)17 b FX(=)f F1(t)k Ge(with)e FM(md)p FX(\()p F1(t)p FX(\))f(=)f Ge(?)t(.)25 b(F)o(or)18 b(\(3\),)g(by)g(\(1\),)g(we) g(can)g(write)h F0(A)d FX(=)g FV(\014)3132 104 y FT(i)3153 92 y FX(\()p F0(x)3222 104 y FT(i)3251 92 y Ge(:)7 b F1(t)3317 104 y FT(i)3356 92 y FV(!)450 200 y F0(A)501 170 y Gd(0)501 223 y FT(i)522 200 y FX(\))p FZ(;)i F0(A)637 170 y Gd(0)674 200 y Ge(since)16 b FM(md)p FX(\()p F1(t)1033 212 y FT(i)1055 200 y FX(\))f FV(2)f(f)p Ge(!)s FZ(;)9 b FV(#)-9 b FZ(;)9 b FJ(l)q FV(g)15 b Ge(\(note)g F0(A)1674 212 y FT(i)1711 200 y Ge(may)h(be)1974 198 y F1(/)1965 200 y(0)p Ge(\).)24 b(Then)14 b(by)i F0(A)e FV(\020)g F0(B)p Ge(,)i(ob)o(viously)e F0(A)2992 212 y Gb(0)3041 200 y FX(=)g(\()p F0(A)3203 212 y FT(i)3224 200 y FZ(;)9 b F0(A)3307 170 y Gd(0)3328 200 y FX(\))14 b FV(\020)450 308 y F0(B)p Ge(.)25 b(Hence)19 b(we)h(ha)n(v)o(e)e FX(\()p F0(A)11 b FV(\014)g F0(B)p FX(\))p FZ(=)-10 b(~)-32 b F0(x)16 b FX(=)i(\()p F0(A)p FZ(=)-10 b(~)-32 b F0(x)10 b FV(\014)h F0(B)p FZ(=)-10 b(~)-32 b F0(x)o FX(\))17 b(=)h F0(A)2060 320 y Gb(0)2105 308 y FV(\014)11 b F0(B)p Ge(.)24 b(The)19 b(proof)f(of)h(\(4\))g(is)h(similar)-5 b(.)25 b(\(5\))19 b(uses)450 451 y(\(2\))h(and)f(\(4\).)p 909 451 50 50 v 533 658 a F0(Remark.)129 b Ge(If)27 b(we)g(delete)g (the)g(side)h(condition)d FM(md)p FX(\()p F1(t)2179 670 y FT(i)2201 658 y FX(\))d FV(2)h(f)p Ge(!)s FZ(;)9 b FV(#)-9 b FZ(;)9 b FJ(l)p FV(g)27 b Ge(in)h(\(5\),)g(the)f(property)e (does)450 766 y(not)k(hold.)52 b(F)o(or)30 b(countere)o(xample,)e(let)i F0(A)23 b FX(=)g F0(x)1847 778 y Gb(1)1897 766 y Ge(:)14 b F1(t)1970 778 y Gb(1)2029 766 y FV(!)24 b F0(x)2173 778 y Gb(2)2222 766 y Ge(:)p 2259 715 72 4 v 14 w F1(t)2295 778 y Gb(2)2361 766 y Ge(and)29 b F0(B)23 b FX(=)g F0(x)2710 778 y Gb(2)2760 766 y Ge(:)14 b F1(t)2833 778 y Gb(2)2892 766 y FV(!)24 b F0(x)3036 778 y Gb(1)3085 766 y Ge(:)p 3122 715 V 14 w F1(t)3158 778 y Gb(1)3194 766 y Ge(.)53 b(Then)450 874 y F0(A)p FZ(=)p F0(x)580 886 y Gb(1)614 874 y F0(x)651 886 y Gb(2)704 874 y FV(\020)18 b F0(B)p FZ(=)p F0(x)917 886 y Gb(1)951 874 y F0(x)988 886 y Gb(2)1023 874 y Ge(,)i(b)n(ut)h F0(A)12 b FV(\014)g F0(B)18 b Ge(is)j (unde\002ned.)533 1189 y FG(Lemma)30 b(A.4.)533 1346 y Ge(\(1\))42 b FV(`)19 b F0(P)h FZ(.)g F0(A)30 b FF(and)g F0(P)18 b FV(\021)g F0(Q)30 b FF(then)f FV(`)19 b F0(Q)h FZ(.)g F0(A)p FF(.)533 1458 y Ge(\(2\))37 b FV(`)15 b F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)7 b Ge(:)859 1454 y FZ(~)876 1458 y F1(t)q FX(\))p FZ(:)p F0(P)h FV(j)p 1059 1412 38 4 v 9 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)6 b Ge(:)1183 1454 y FZ(~)1200 1458 y F1(t)1236 1428 y Gd(0)1258 1458 y FX(\))p F0(Q)18 b FZ(.)g F0(A)24 b FF(implies)i FV(`)15 b FX(\()p F1(n)o FZ(~)-31 b F0(y)6 b Ge(:)1988 1454 y FZ(~)2005 1458 y F1(t)2041 1428 y Gd(00)2080 1458 y FX(\)\()p F0(P)j FV(j)g F0(Q)p FX(\))17 b FZ(.)h F0(A)24 b FF(with)h F1(t)2691 1428 y Gd(0)2691 1481 y FT(i)2728 1458 y FX(=)p 2808 1407 58 4 v 15 w F1(t)2844 1470 y FT(i)2890 1458 y FF(and)g F1(t)3082 1428 y Gd(00)3082 1481 y FT(i)3136 1458 y FX(=)15 b F1(t)3252 1470 y FT(i)3283 1458 y FV(\014)9 b F1(t)3393 1428 y Gd(0)3393 1481 y FT(i)3413 1458 y FF(.)533 1570 y Ge(\(3\))54 b FV(`)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)15 b Ge(:)906 1566 y FZ(~)923 1570 y F1(t)q FX(\))p FZ(:)p F0(P)g FV(j)p 1120 1524 38 4 v 16 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)15 b Ge(:)1262 1566 y FZ(~)1280 1570 y F1(t)1316 1540 y Gd(0)1337 1570 y FX(\))p F0(Q)25 b FZ(.)g F0(A)40 b FF(implies)j FV(`)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)15 b Ge(:)2129 1566 y FZ(~)2146 1570 y F1(t)q FX(\))p FZ(:)p F0(P)g FV(j)h FX(\()p F1(n)o FZ(~)-31 b F0(y)15 b Ge(:)2501 1566 y FZ(~)2518 1570 y F1(t)2554 1540 y Gd(00)2593 1570 y FX(\)\()p F0(P)g FV(j)h F0(Q)p FX(\))25 b FZ(.)f F0(A)41 b F1(t)3073 1540 y Gd(0)3073 1593 y FT(i)3119 1570 y FX(=)p 3209 1519 58 4 v 25 w F1(t)3245 1582 y FT(i)3307 1570 y FF(and)450 1678 y F1(t)486 1648 y Gd(00)486 1701 y FT(i)543 1678 y FX(=)18 b F1(t)662 1690 y FT(i)695 1678 y FV(\014)12 b F1(t)808 1648 y Gd(0)808 1701 y FT(i)829 1678 y FF(.)533 1875 y F0(Pr)l(oof)o(.)92 b Ge(The)28 b(proof)f(is)j(essentially)f(the)f(same)h(as)h(in)e([11,)13 b(33].)49 b(Assume)29 b FV(`)23 b F0(P)h FZ(.)f F0(A)p Ge(.)51 b(Then,)30 b(as)450 1983 y(in)c([11,)g(Proposition)e(1)i (\(ii\)],)h(there)e(e)o(xists)h(a)h(minimum)d(action)h(type)g F0(A)2623 1995 y Gb(0)2684 1983 y Ge(such)h(that)g F0(A)3065 1995 y Gb(0)3121 1983 y FV(\022)21 b F0(A)3258 1995 y Gb(1)3319 1983 y Ge(and)450 2091 y FV(`)f F0(P)i FZ(.)g F0(A)709 2103 y Gb(0)768 2091 y Ge(\(since)i(we)h(only)e(ha)n(v)o(e)h (to)g(use)h FM(\(W)n(eak\))e Ge(before)g(restriction)g(and)h(input)f (rules\).)36 b(Hence)24 b(in)450 2199 y(the)c(follo)n(wing)f(we)h(only) g(consider)f(the)h(minimum)f(action)g(types.)450 2307 y F2(\(1\))f Ge(By)g(rule)g(induction)e(on)i FV(\021)p Ge(.)24 b(The)18 b(case)h(of)f F0(P)9 b FV(j)h F2(0)16 b FV(\021)g F0(P)i Ge(is)h(easy)f(because)2603 2305 y F1(/)2594 2307 y(0)g Ge(is)h(a)g(unit)f(of)f FV(\014)p Ge(.)25 b(Similarly)450 2415 y(the)19 b(cases)g(of)f F0(P)10 b FV(j)g F0(Q)17 b FV(\021)g F0(Q)10 b FV(j)g F0(P)p Ge(,)19 b(and)f FX(\()p F0(P)10 b FV(j)g F0(Q)p FX(\))g FV(j)g F0(R)18 b FV(\021)e F0(P)10 b FV(j)g FX(\()p F0(Q)g FV(j)g 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y Gb(1)1623 3012 y FV(\014)12 b F0(A)1751 3024 y Gb(2)1805 3012 y FX(=)19 b F0(A)p Ge(.)30 b(By)23 b(strengthening)c(of)j(bases)g(we)h(can)f(set)g FV(f)-10 b FZ(~)-32 b F0(y)p FV(g)12 b(\\)450 3120 y FM(fn)o FX(\()p F0(A)595 3132 y Gb(2)630 3120 y FX(\))j(=)765 3118 y F1(/)756 3120 y(0)p Ge(.)24 b(From)15 b FV(`)p 1105 3074 V 15 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)17 b FZ(.)f F0(A)1420 3132 y Gb(1)1471 3120 y Ge(we)h(deduce)e FV(`)f F0(P)j FZ(.)f F0(A)2079 3089 y Gd(0)2079 3144 y Gb(1)2114 3120 y FZ(;)o(~)-32 b F0(y)5 b Ge(:)2200 3116 y FZ(~)2216 3120 y F1(t)18 b Ge(with)e F0(A)2485 3132 y Gb(1)2534 3120 y FX(=)e F0(A)2664 3089 y Gd(0)2664 3144 y Gb(1)2707 3120 y FV(\014)8 b F0(x)e Ge(:)f FX(\()2866 3116 y FZ(~)2883 3120 y F1(t)p FX(\))2956 3089 y FT(p)2988 3098 y FH(O)3027 3120 y Ge(.)24 b(By)16 b(Lemma)450 3227 y(A.3)j(\(5\))f(and)h(associati)n(vity)-5 b(,)18 b(we)i(ha)n(v)o(e)e FX(\()p F0(A)1669 3197 y Gd(0)1669 3252 y Gb(1)1704 3227 y FZ(;)o(~)-32 b F0(y)9 b 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b(by)f(\(i)n(v\),)g(we)h(ha)n(v)o(e)e F0(A)2355 4835 y Gd(0)2355 4889 y Gb(1)2407 4865 y FV(\020)f FX(\()p F0(A)2571 4835 y Gd(0)2571 4889 y Gb(2)2606 4865 y FZ(;)9 b F0(A)2689 4835 y Gd(00)2689 4889 y Gb(2)2727 4865 y FX(\))19 b Ge(and)f F0(A)2968 4835 y Gd(0)2968 4889 y Gb(1)3012 4865 y FV(\014)10 b FX(\()p F0(A)3170 4835 y Gd(0)3170 4889 y Gb(2)3204 4865 y FZ(;)f F0(A)3287 4835 y Gd(00)3287 4889 y Gb(2)3325 4865 y FX(\))17 b(=)450 4973 y F0(A)501 4985 y Gb(1)536 4973 y FZ(=)p F0(x)12 b FV(\014)g F0(A)755 4985 y Gb(2)787 4973 y FZ(=)p F0(x)p Ge(.)p eop %%Page: 48 48 48 47 bop 450 -257 a FX(48)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)533 -4 y Ge(Suppose)i FM(md)p FX(\()p F1(t)p FX(\))g(=)p FV(#)p Ge(.)25 b(Then)19 b(we)i(ha)n(v)o(e) 1476 196 y FV(`)d F0(P)11 b FV(j)h F0(Q)20 b FZ(.)h F0(y)9 b Ge(:)g FJ(l)p FZ(;)g FX(\()p F0(A)2015 208 y Gb(1)2050 196 y FZ(=)p F0(x)j FV(\014)g F0(A)2269 208 y Gb(2)2302 196 y FZ(=)p F0(x)p FX(\))450 396 y Ge(Hence)20 b(by)g FM(\(Res\))o 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b(\(2\),)g(\002rst)g(we)h(start)450 4424 y(from)29 b(the)i(follo)n(wing)e(lemma)h(about)f(garbage)f(collection)i (and)g(linear)g(reduction.)54 b(The)30 b(proof)e(is)450 4532 y(mechanical.)c(W)-7 b(e)21 b(assume)f(all)h(terms)f(are)g (typable.)533 4698 y FG(Lemma)30 b(B.1.)533 4865 y Ge(\(1\))13 b(\(postponement)i(of)k FV(7!)1328 4877 y Fx(g)1363 4865 y Ge(\))27 b FF(If)h F0(P)16 b FV(7!)1653 4877 y Fx(g)1706 4865 y F0(Q)h FV(7!)1866 4877 y Fx(l)1899 4865 y F0(R)p FF(,)28 b(then)f(for)i(some)e F0(S)q FF(,)h F0(P)17 b FV(7!)2772 4877 y Fx(l)2805 4865 y F0(S)g FV(7!)2947 4877 y Fx(g)3000 4865 y F0(R)p FF(.)37 b(Similarly)450 4973 y(if)31 b F0(P)18 b FV(7!)683 4985 y Fx(g)736 4973 y F0(Q)h FV(7!)898 4985 y Fx(r)939 4973 y F0(R)p FF(,)29 b(then)h(for)h(some)f F0(S)q FF(,)f F0(P)18 b FV(7!)1823 4985 y Fx(r)1864 4973 y F0(S)h FV(7!)2008 4943 y FU(+)2008 4994 y Fx(g)2078 4973 y F0(R)p FF(.)p eop %%Page: 49 49 49 48 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(49)533 -4 y Ge(\(2\))13 b(\(strong)21 b(con\003uence)f(of)i FV(7!)1459 8 y Fx(g)1494 -4 y Ge(\))31 b FF(If)h F0(P)19 b FV(7!)1795 8 y Fx(g)1850 -4 y F0(Q)1910 8 y FT(i)1962 -4 y FF(\()p F0(i)h FX(=)f Ge(1)p FZ(;)9 b Ge(2)p FF(\),)32 b(then)f F0(Q)2576 8 y Gb(1)2630 -4 y FV(\021)19 b F0(Q)2774 8 y Gb(2)2841 -4 y FF(or)32 b(ther)l(e)g(exists)f F0(R)450 104 y FF(such)f(that)g F0(Q)869 116 y FT(i)908 104 y FV(7!)991 116 y Fx(g)1045 104 y F0(R)p FF(.)533 229 y Ge(\(3\))13 b(\(strong)26 b(normalisation)g(of)h FV(7!)1572 241 y Fx(g)1607 229 y Ge(\))37 b FF(F)-6 b(or)37 b(al)t(l)i F0(P)p FF(,)g(ther)l(e)f(exists)e F0(Q)i FF(such)f(that)g F0(P)23 b FV(7!)3153 199 y Gd(\003)3153 249 y Fx(g)3210 229 y F0(Q)37 b FF(and)450 337 y F0(Q)18 b FV(67!)611 349 y Fx(g)646 337 y FF(.)533 462 y Ge(\(4\))13 b(\(strong)23 b(con\003uence)f(of)j FV(7!)1466 474 y Fx(l)1482 462 y Ge(\))34 b FF(If)g F0(P)21 b FV(7!)1790 474 y Fx(l)1827 462 y F0(Q)1887 474 y FT(i)1942 462 y FF(\()p F0(i)g 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y(occurrences)18 b(of)i(the)g(underlined)e F0(a)i Ge(in)h F0(P)1646 2718 y Gb(2)1701 2706 y Ge(to)f(obtain)f F0(P)2057 2718 y Gb(3)2092 2706 y Ge(.)25 b(The)20 b(rest)h(is)g(the)f (just)h(same)f(as)h(in)f([7].)p 3335 2706 50 50 v 533 2864 a(By)h(Lemma)f(B.2,)h(we)g(obtained)e(CR-property)g(of)i FV(7!)g Ge(\(Proposition)e(3.1)h(\(2\)\).)26 b(T)-7 b(o)21 b(pro)o(v)o(e)d(that)j(the)450 2972 y(\002rst)f(statement)g(in)g (Proposition)e(3.1)h(\(3\),)f(we)i(note)g(that)f F0(P)f FV(7!)2264 2984 y Fx(g)2317 2972 y F0(P)2368 2941 y Gd(0)2409 2972 y Ge(and)h FM(SN)2653 2984 y FT(e)2684 2972 y FX(\()p F0(P)2767 2941 y Gd(0)2788 2972 y FX(\))h Ge(does)g(not)f(normally)450 3080 y(imply)i FM(SN)770 3092 y FT(e)801 3080 y FX(\()p F0(P)p FX(\))i Ge(in)f(untyped)e(setting)h(\(e.g.)g F0(Q)e FV(*)1859 3092 y FT(e)1913 3080 y Ge(b)n(ut)j FX(\()p F1(n)9 b F0(x)p FX(\))p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(Q)19 b FV(7!)2545 3092 y Fx(g)2599 3080 y F2(0)p Ge(\).)30 b(Hence)21 b(we)h(shall)g(pro)o(v)o(e)450 3187 y(this)f(statement)f(using)f(postponement)f(of)i FV(7!)1790 3199 y Fx(g)1825 3187 y Ge(,)g(Lemma)g(B.1)g(\(1\).)533 3371 y FG(Lemma)30 b(B.3.)533 3554 y Ge(\(1\))42 b FF(If)31 b F0(P)18 b FV(7!)913 3566 y Gb(0)966 3554 y F0(P)1017 3524 y Gd(0)1068 3554 y FF(and)30 b FM(SN)1333 3566 y FT(e)1364 3554 y FX(\()p F0(P)1447 3524 y Gd(0)1468 3554 y FX(\))p FF(,)g(then)g FM(SN)1844 3566 y FT(e)1875 3554 y FX(\()p F0(P)p FX(\))p FF(.)533 3679 y Ge(\(2\))42 b FF(Supp)l(ose)30 b F0(P)19 b FV(67!)1142 3691 y Gb(0)1177 3679 y FF(.)38 b(Then)31 b F0(P)18 b FV(7!)1609 3691 y Fx(g)1662 3679 y F0(P)1713 3648 y Gd(0)1764 3679 y FF(and)30 b FM(SN)2029 3691 y FT(e)2060 3679 y FX(\()p F0(P)2143 3648 y Gd(0)2164 3679 y FX(\))g FF(implies)h FM(SN)2614 3691 y FT(e)2646 3679 y FX(\()p F0(P)p FX(\))p FF(.)533 3804 y Ge(\(3\))42 b FF(If)31 b F0(P)18 b FV(7!)g F0(P)982 3773 y Gd(0)1033 3804 y FF(and)30 b FM(SN)1298 3816 y FT(e)1329 3804 y FX(\()p F0(P)1412 3773 y Gd(0)1433 3804 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FT(e)1469 4349 y FX(\()p F0(P)1552 4319 y Gd(0)1573 4349 y FX(\))h Ge(implies)f FM(SN)1989 4361 y FT(e)2020 4349 y FX(\()p F0(P)p FX(\))p Ge(.)25 b(Then)16 b(by)g(Lemma)g(B.1)i(\(1\))e(there)g(e)o(xists)i(at) 450 4457 y(least)h(one)e(pass)i(such)f(that)g F0(P)1286 4427 y Gd(0)1323 4457 y FV(7!)1406 4427 y Gd(\003)1406 4481 y Gb(0)1458 4457 y F0(P)1501 4469 y Gb(1)1552 4457 y FV(7!)1635 4427 y Gd(\003)1635 4477 y Fx(g)1686 4457 y F0(R)e FV(67!)j Ge(with)f F0(P)2064 4469 y Gb(1)2115 4457 y FV(67!)2198 4469 y Gb(0)2233 4457 y Ge(.)25 b(Since)18 b FM(SN)2586 4469 y FT(e)2617 4457 y FX(\()p F0(R)p FX(\))p Ge(,)h(we)g(ha)n(v)o(e)e FM(SN)3164 4469 y FT(e)3195 4457 y FX(\()p F0(P)3270 4469 y Gb(1)3305 4457 y FX(\))i Ge(by)450 4565 y(\(2\).)k(No)n(w)17 b(by)g(applying)e(Lemma)h(B.1)h (\(1\))g(again,)f(we)h(ha)n(v)o(e)g(some)g F0(P)2447 4534 y Gd(0)2439 4589 y Gb(1)2490 4565 y Ge(such)g(that)g F0(P)f FV(7!)2952 4534 y Gd(\003)2952 4589 y Gb(0)3002 4565 y F0(P)3053 4534 y Gd(0)3045 4589 y Gb(1)3095 4565 y FV(7!)3178 4534 y Gd(\003)3178 4585 y Fx(g)3228 4565 y F0(P)3271 4577 y Gb(1)3321 4565 y FV(7!)3404 4534 y Gd(\003)3404 4585 y Fx(g)450 4673 y F0(R)g FV(67!)j Ge(with)f F0(P)836 4642 y Gd(0)828 4697 y Gb(1)879 4673 y FV(67!)962 4685 y Gb(0)997 4673 y Ge(.)24 b(W)-7 b(e)20 b(again)d(ha)n(v)o(e)g FM(SN)1644 4685 y FT(e)1675 4673 y FX(\()p F0(P)1758 4642 y Gd(0)1750 4697 y Gb(1)1785 4673 y FX(\))i Ge(by)e(\(2\),)h(from) f(which)g(we)i(can)f(obtain)f FM(SN)3054 4685 y FT(e)3086 4673 y FX(\()p F0(P)p FX(\))i Ge(by)e(\(1\),)450 4815 y(as)k(required.)p 920 4815 V 450 4973 a(The)f(rest)g(of)g(Proposition) f(3.1)h(\(3\))f(is)i(straightforw)o(ard)d(by)i(this)h(and)e(CR)j (property)c(of)i FV(7!)p Ge(.)p eop %%Page: 50 50 50 49 bop 450 -257 a FX(50)950 b FW(Y)n(OSHID)m(A,)16 b(BERGER)h(AND)g(HOND)m(A)1633 -4 y F3(APPENDIX)33 b(C)1360 231 y(PR)m(OOFS)f(F)m(OR)f(SECTION)h(4)1128 467 y(C.1.)94 b(PR)m(OOF)31 b(OF)h(PR)m(OPOSITION)g(4.2)533 619 y Ge(By)19 b(induction)d(on)i(generation)f(rules)h(of)g FV(\000)-14 b(!)p Ge(,)19 b(it)g(is)g(easy)g(to)f(check)g F0(P)e FV(\000)-14 b(!)17 b F0(Q)h Ge(implies)h F0(P)3108 589 y FT(A)3219 572 y Gc(t)3166 619 y FV(\000)-14 b(!)16 b F0(Q)3376 589 y FT(A)3418 619 y Ge(.)450 727 y(F)o(or)25 b(the)h(other)f(direction,)g(we)h(\002rst)g(sho)n(w)-5 b(,)26 b(by)g(rule)f(induction)f(on)h(transition)g(rules,)h(that,)h(if) f F0(x)g Ge(has)450 862 y(mode)20 b FV(#)h Ge(\(resp.)28 b(!)s(\),)22 b F0(P)1088 832 y FT(A)1164 807 y(x)p FU(\()-8 b FS(~)-23 b FT(y)p FU(\))1148 862 y FV(\000)-14 b(!)19 b F0(Q)1361 832 y FT(A)1424 862 y Ge(implies)i F0(P)e FV(\021)c F0(C)r FX([)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)2124 874 y Gb(1)2158 862 y FX(])22 b Ge(and)f F0(Q)d FV(\021)d F0(C)r FX([)p F0(P)2626 874 y Gb(1)2660 862 y FX(])22 b Ge(\(resp.)27 b F0(P)19 b FV(\021)14 b F0(C)r FX([)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)3381 874 y Gb(1)3416 862 y FX(])450 1001 y Ge(and)19 b F0(Q)f FV(\021)c F0(C)r FX([)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)1059 1013 y Gb(1)1093 1001 y FV(j)p F0(P)1159 1013 y Gb(1)1193 1001 y FX(])p Ge(\))20 b(where)c F0(C)r FX([)k(])g Ge(is)h(a)f(reduction)e(conte)o(xt.)24 b(Similarly)19 b(for)g F0(P)2881 970 y FT(A)p 2956 912 28 3 v 2956 946 a(x)q FU(\()-8 b FS(~)-23 b FT(y)p FU(\))2941 1001 y FV(\000)-15 b(!)18 b F0(Q)3152 970 y FT(A)3194 1001 y Ge(.)25 b(Using)450 1109 y(them)i(we)i(sho)n(w)-5 b(,)29 b(again)d(by)i(rule)f(induction)f(on)i(transition)f(rules,)i(that)f F0(P)2677 1078 y FT(A)2795 1062 y Gc(t)2741 1109 y FV(\000)-14 b(!)23 b F0(Q)2958 1078 y FT(A)3028 1109 y Ge(implies)28 b F0(P)22 b FV(\021)446 1216 y F0(C)r FX([)l F0(C)575 1228 y Gb(1)610 1216 y FX([\()p Ge(!)p FX(\))p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)929 1228 y Gb(1)964 1216 y FX(][)l F0(C)1059 1228 y Gb(2)1094 1216 y FX([)p 1117 1170 38 4 v F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1298 1228 y Gb(2)1333 1216 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FV(\000)-15 b(!)20 b F0(Q)775 1892 y FT(B)839 1922 y Ge(such)h(that)h F0(Q)d FV(\021)g F0(Q)1383 1934 y Gb(0)1418 1922 y Ge(.)30 b(The)21 b(proof)g(is)h (standard,)f(using)g(rule)h(induction)e(on)h(the)h(syntactic)450 2030 y(transition)29 b(system)g(together)f(with)i(inspection)e(of)h (the)h(structure)e(of)h(processes.)52 b(From)29 b(this)h(it)g(is)450 2138 y(easy)d(to)h(check)f(that)g(the)g(gi)n(v)o(en)g(statement)g (holds,)h(this)g(time)g(by)f(rule)g(induction)e(on)i(the)h(original)450 2246 y(transition)g(system.)50 b(Finally)28 b(for)g(\(3\))g(\223if)5 b(\224)28 b(is)i(by)e(\(1\))g(while)g(\223only)g(if)5 b(\224)28 b(is)i(by)e(\(2\),)i(noting,)f(under)450 2354 y(Con)m(v)o(ention)20 b(3.1,)i(the)h(transition)e(induced)g(by)h (syntactic)g(transition)g(system)h(and)f(the)g(one)g(induced)450 2462 y(by)e(prime)f(syntactic)h(transition)g(system)g(is)h(identical.)p 2136 2462 V 1128 2682 a F3(C.3.)94 b(PR)m(OOF)31 b(OF)h(PR)m(OPOSITION) g(4.4)533 2805 y Ge(Using)g(the)f(characterisation)f(in)i(Proposition)e (4.3)h(\(1\)\(2\),)h(it)g(is)h(enough)c(to)j(sho)n(w)f FV(\031)h Ge(deri)n(v)o(ed)450 2913 y(using)25 b(the)g(syntactic)g (transition)g(is)h(a)g(typed)e(congruence.)38 b(Input)24 b(pre\002x)o(es,)h(parallel)g(composition)450 3034 y(and)16 b(restriction)g(are)h(entirely)f(standard,)h(cf.)f([48].)23 b(F)o(or)16 b(output)g(pre\002x)g(we)i(de\002ne)e F2(R)2920 2987 y FQ(def)2925 3034 y FX(=)k F2(R)3070 3046 y Gb(1)3113 3034 y FV([)8 b F2(R)3236 3046 y Gb(2)3280 3034 y FV([)g F2(R)3403 3046 y Gb(3)450 3142 y Ge(where:)533 3308 y(\(1\))46 b F2(R)737 3320 y Gb(1)790 3261 y FQ(def)795 3308 y FX(=)t FV(\031)p Ge(;)533 3432 y(\(2\))g F2(R)737 3444 y Gb(2)790 3385 y FQ(def)795 3432 y FX(=)23 b FV(fh)p 957 3386 38 4 v F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1138 3444 y Gb(1)1172 3432 y FZ(;)p 1204 3386 V 9 w F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1386 3444 y Gb(2)1420 3432 y FV(i)21 b(j)g F0(P)1560 3444 y Gb(1)1613 3432 y FV(\031)d F0(P)1739 3444 y Gb(2)1773 3432 y FV(g)p Ge(;)i(and)533 3562 y(\(3\))46 b F2(R)737 3574 y Gb(3)790 3515 y FQ(def)795 3562 y FX(=)23 b FV(fh)p FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\))p F0(P)1153 3574 y Gb(1)1188 3562 y FV(f)-10 b FZ(~)-32 b F0(y)o FZ(=)-15 b(~)-27 b F0(z)p FV(g)p FZ(;)9 b FX(\()p F1(n)o FZ(~)-32 b F0(y)p FX(\))p F0(P)1610 3574 y Gb(2)1645 3562 y FV(f)-10 b FZ(~)-32 b F0(y)o FZ(=)-15 b(~)-27 b F0(z)p FV(gi)20 b(j)h F0(P)1978 3574 y Gb(1)2031 3562 y FV(\031)d F0(P)2157 3574 y Gb(2)2191 3562 y FV(g)p Ge(.)450 3728 y(In)25 b(\(3\))g(we)h(assume)f(the)h(mentioned)d(substitution)i(is)h (well-typed.)40 b(It)25 b(is)i(easy)e(to)h(check)e(whene)n(v)o(er)450 3836 y F0(P)p F2(R)561 3848 y Gb(1)609 3836 y FV([)13 b F2(R)737 3848 y Gb(2)772 3836 y F0(Q)25 b Ge(its)h(deri)n(v)n(ati)n (v)o(es)d(are)h(related)h(by)f F2(R)p Ge(.)39 b(F)o(or)24 b F2(R)2148 3848 y Gb(3)2208 3836 y Ge(we)h(sho)n(w)f(this)i(relation)d (coincides)h(with)450 3944 y FV(\031)p Ge(.)43 b(Clearly)27 b F2(R)911 3956 y Gb(3)967 3944 y FV(\033\031)f Ge(\(let)16 b FZ(~)-31 b F0(y)26 b Ge(be)h(the)f(empty)f(string\).)43 b(F)o(or)26 b(the)g(re)n(v)o(erse)g(inclusion,)g(under)f F0(P)3219 3956 y Gb(1)3275 3944 y FV(\031)c F0(P)3404 3956 y Gb(2)450 4052 y Ge(we)29 b(ha)n(v)o(e:)43 b FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\))p F0(P)994 4064 y Gb(1)1028 4052 y FV(f)-10 b FZ(~)-32 b F0(y)o FZ(=)-15 b(~)-27 b F0(z)p FV(g)23 b(\031)g FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)1562 4064 y Gb(1)1596 4052 y FV(j)p F1(P)p FX([)p F0(z)1738 4064 y FT(i)1783 4052 y FV(!)23 b F0(y)1926 4064 y FT(i)1948 4052 y FX(])1971 4022 y Gc(t)1998 4033 y Fn(i)2020 4052 y FX(\))g FV(\031)g FX(\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)2392 4064 y Gb(2)2426 4052 y FV(j)p F1(P)p FX([)p F0(z)2568 4064 y FT(i)2613 4052 y FV(!)24 b F0(y)2757 4064 y FT(i)2778 4052 y FX(])2801 4022 y Gc(t)2828 4033 y Fn(i)2850 4052 y FX(\))f FV(\031)g FX(\()p F1(n)p FZ(~)-32 b F0(y)p FX(\))p F0(P)3190 4064 y Gb(2)3224 4052 y FV(f)-10 b FZ(~)-32 b F0(y)p FZ(=)-15 b(~)-27 b F0(z)p FV(g)p Ge(,)450 4160 y(where)23 b(the)g(\002rst)h(and) f(the)g(last)h(equations)e(are)h(by)g(the)g(cop)o(y-cat)f(la)o(w)h (\(see)h(Proposition)e(5.4\),)g(while)450 4268 y(the)d(second)g (equation)f(is)i(by)g(closure)e(of)i FV(\031)f Ge(under)f(parallel)h (composition)f(and)h(hiding.)k(Thus)c F0(P)p F2(R)3344 4280 y Gb(3)3379 4268 y F0(Q)450 4376 y Ge(implies)h F0(P)e FV(\031)g F0(Q)p Ge(,)j(that)f(is)h F2(R)1250 4388 y Gb(3)1304 4376 y FV(\032\031)p Ge(.)j(This)c(sho)n(ws)h F2(R)f Ge(is)h(indeed)f(a)g(bisimulation,)f(hence)g(as)i(required.)1128 4596 y F3(C.4.)94 b(PR)m(OOF)31 b(OF)h(PR)m(OPOSITION)g(4.5)533 4757 y Ge(Let)20 b F2(R)742 4710 y FQ(def)747 4757 y FX(=)j FV(fh)p F0(P)-11 b FZ(;)9 b F0(Q)p FV(i)20 b(j)g F1(\301)1193 4769 y FT(e)1243 4757 y FV(`)d F0(P)h FX(=)g F0(Q)p FV(g)p Ge(.)25 b(The)19 b(statement)h(says)g F2(R)e FV(\032\031)p Ge(.)25 b(It)20 b(is)h(enough)d(to)i(sho)n(w)f(this)i (in-)450 4865 y(clusion)e(under)f(Con)m(v)o(ention)f(3.1)h(since)i FV(\031)f Ge(is)i(already)d(closed)h(under)f FV(\021)p Ge(.)25 b(By)19 b(Proposition)f(4.3)h(\(3\),)g(it)450 4973 y(suf)n(\002ces)e(to)f(sho)n(w)g F2(R)p FV([)g(\021)1186 4985 y Gc(a)1245 4973 y Ge(is)i(a)e(bisimulation)g(with)g(respect)h(to) f(prime)g(syntactic)g(transition.)23 b(W)-7 b(e)18 b(\002rst)p eop %%Page: 51 51 51 50 bop 1151 -257 a FW(STR)m(ONG)16 b(NORMALISA)-7 b(TION)16 b(IN)h(THE)e FP(p)p FW(-CALCULUS)701 b FX(51)450 -4 y Ge(consider)22 b(the)i(pair)f(from)g FM(\(E1\))p Ge(,)d F0(C)r FX([)p 1483 -50 38 4 v F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)25 b Ge(and)19 b F0(C)r FX([\()p F1(n)o FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])p Ge(.)36 b(Let)24 b F0(R)c FX(=)c F0(C)r FX([)p 2982 -50 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)450 122 y Ge(and)20 b(set)h FV(`)d F0(R)i FZ(.)h F0(A)p Ge(.)k(If)20 b FV(`)e F0(R)1275 76 y FT(l)1218 122 y FV(\000)-14 b(!)18 b F0(R)1421 92 y Gd(0)1442 122 y Ge(,)j(we)f(ha)n(v)o(e)g(the)g(follo)n (wing)f(cases.)522 318 y FX(\()p Ge(1)p FX(\))45 b F0(C)r FX([)p 753 272 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)1285 271 y FT(l)1228 318 y FV(\000)-14 b(!)14 b F0(C)1433 288 y Gd(0)1454 318 y FX([)p 1477 272 V F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)133 b FX(\()p Ge(2)p FX(\))45 b F0(C)r FX([)p 2298 272 V F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)2827 271 y Gc(t)2773 318 y FV(\000)-14 b(!)19 b FX(\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()l F0(C)r FX([)p F0(P)p FX(])p FV(j)p F0(Q)p FX(\))522 448 y(\()p Ge(3)p FX(\))45 b F0(C)r FX([)p 753 402 V F0(x)p 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FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(Q)315 b FX(\()p Ge(6)p FX(\))45 b F0(C)r FX([)p 2298 544 V F0(x)q FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)p 2789 501 28 3 v 2789 535 a FT(x)q FU(\()-8 b FS(~)-23 b FT(y)p FU(\))2773 590 y FV(\000)-14 b(!)15 b F0(C)r FX([)p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)p FZ(:)450 760 y Ge(W)-7 b(e)24 b(shall)f(no)n(w)g(sho)n(w)f(that)h(only)f(the)h (processes)g(in)g(\(1\),)g(\(2\))f(and)g(\(3\))g(are)h(typable.)32 b(T)-7 b(o)23 b(this)h(end)e(we)450 867 y(sho)n(w)i(by)h(induction)d (on)j(the)f(deri)n(v)n(ation)f(of)h FV(`)d F0(R)13 b FZ(.)g F0(A)24 b Ge(that)h FJ(l)p F0(x)c FV(2)h F0(A)p Ge(.)38 b(This)25 b(implies)g(that)20 b F0(C)r FX([)h(])k Ge(cannot)450 975 y(contain)16 b(an)h(input)f(at)h F0(x)p Ge(.)25 b(Hence)16 b(\(4\))g(is)i(not)f(typable.)23 b(Similarly)-5 b(,)16 b(no)h(typable)e(observ)o(er)g(could)h(contain)450 1083 y(an)k(output)f(or)h(an)g(input)g(at)g F0(x)p Ge(,)h(making)e(in)h (\(5\))g(and)f(\(6\))h(untypable.)533 1207 y(The)i(transition)g(\(1\))g (is)i(matched)d(by)i(a)g(transition)18 b F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])2547 1160 y FT(l)2490 1207 y FV(\000)-14 b(!)16 b F0(C)2697 1177 y Gd(0)2718 1207 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])24 b Ge(while)e(the)450 1315 y(empty)14 b(transition)h(sequence)10 b F0(C)r FX([\()p F1(n)o FZ(~)-31 b F0(y)o FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])16 b Ge(matches)f(\(2\))g(because)10 b F0(C)r FX([\()p F1(n)p FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])14 b FV(\021)g FX(\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()l F0(C)r FX([)p F0(P)p FX(])p FV(j)p F0(Q)p FX(\))p Ge(,)450 1442 y(as)16 b(can)f(be)g(sho)n(wn)g(by) f(induction)g(on)h(the)g(deri)n(v)n(ation)e(of)i(\(2\).)23 b(It)15 b(is)h(easy)g(to)f(see)h(that)11 b F0(C)r FX([\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])3375 1395 y FT(l)3318 1442 y FV(\000)-15 b(!)446 1550 y F0(C)r FX([\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)763 1520 y Gd(0)784 1550 y FV(j)p F0(Q)p FX(\)])21 b Ge(is)g(an)f(admissible)g(match)g(for)f(\(4\).)533 1676 y(No)n(w)f(assume)g(that)h F0(R)d FX(=)c F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])1781 1629 y FT(l)1724 1676 y FV(\000)-14 b(!)17 b F0(R)1926 1646 y Gd(0)1946 1676 y Ge(,)i FV(`)e F0(R)10 b FZ(.)g F0(A)17 b Ge(and)h FV(`)e F0(R)2565 1629 y FT(l)2508 1676 y FV(\000)-14 b(!)17 b F0(R)2710 1646 y Gd(0)2730 1676 y Ge(.)25 b(W)-7 b(e)20 b(ha)n(v)o(e)d(the)h (follo)n(w-)450 1784 y(ing)i(causes)g(of)g(the)g(transition.)546 1980 y FX(\()p Ge(1)p FX(\))45 b F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)q FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])1218 1933 y FT(l)1162 1980 y FV(\000)-15 b(!)15 b F0(C)1367 1950 y Gd(0)1388 1980 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])154 b(\()p Ge(2)p FX(\))46 b F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])2600 1933 y Gc(t)2546 1980 y FV(\000)-14 b(!)14 b F0(C)2751 1950 y Gd(0)2772 1980 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)q FX(\)\()p F1(n)-15 b FZ(~)-27 b F0(z)q FX(\)\()p F0(P)3163 1950 y Gd(0)3184 1980 y FV(j)p F0(Q)3267 1950 y Gd(0)3288 1980 y FX(\)])546 2110 y(\()p Ge(3)p FX(\))45 b F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)q FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])1218 2063 y FT(l)1162 2110 y FV(\000)-15 b(!)15 b F0(C)1367 2080 y Gd(0)1388 2110 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)1638 2080 y Gd(0)1659 2110 y FV(j)p F0(Q)p FX(\)])133 b(\()p Ge(4)p FX(\))46 b F0(C)r FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])2603 2063 y FT(l)2546 2110 y FV(\000)-14 b(!)14 b F0(C)2751 2080 y Gd(0)2772 2110 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)q FX(\)\()p F0(P)p FV(j)p F0(Q)3106 2080 y Gd(0)3127 2110 y FX(\)])450 2321 y Ge(\(1\))29 b(is)i(matched)e(by)c F0(C)r FX([)p 1164 2275 38 4 v F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)1701 2274 y FT(l)1644 2321 y FV(\000)-14 b(!)20 b F0(C)1855 2290 y Gd(0)1876 2321 y FX([)p 1899 2275 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)p Ge(.)54 b(F)o(or)30 b(\(2\),)h(we)f(\002rst)h(sho)n(w)-5 b(,)31 b(by)f(in-)450 2429 y(duction)24 b(on)h(the)h(transition,)g (that)21 b F0(C)r FX([)h(])k Ge(must)g(be)f(a)h(reduction)e(conte)o (xt.)40 b(If)25 b(the)h(conte)o(xt)20 b F0(C)r FX([)h(])27 b Ge(in)e(the)450 2536 y(de\002nition)f(of)g(e)o(xtended)f(reduction)h (\(De\002nition)f(3.1\))h(is)i(restricted)f(to)g(a)g(reduction)e(conte) o(xt,)i(then)450 2644 y(the)18 b(resulting)f(relation)g(coincides)g (with)h FV(\000)-14 b(!)p Ge(,)19 b(hence)e(also)h(with)2406 2597 y Gc(t)2353 2644 y FX(=)-14 b FV(\))18 b Ge(by)g(Proposition)e (4.2.)23 b(Thus)18 b(we)450 2768 y(obtain)j F0(C)r FX([)p 760 2722 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(Q)1292 2721 y Gc(t)1239 2768 y FV(\000)-15 b(!)18 b F0(C)r FX([\()p F1(n)o FZ(~)-31 b F0(y)p FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])1923 2721 y FT(l)1867 2768 y FV(\000)-15 b(!)18 b F0(C)2075 2738 y Gd(0)2096 2768 y FX([\()p F1(n)-10 b FZ(~)-32 b F0(y)p FX(\)\()p F1(n)-15 b FZ(~)-27 b F0(z)q FX(\)\()p F0(P)2486 2738 y Gd(0)2508 2768 y FV(j)p F0(Q)2591 2738 y Gd(0)2612 2768 y FX(\)])27 b Ge(as)f(matching)f(transition)450 2876 y(sequence.)f(The)c(remaining) e(cases)j(\(3\))f(and)f(\(4\))h(are)g(dealt)g(with)g(in)h(the)f(same)g (w)o(ay)-5 b(.)533 2984 y(Similarly)17 b(for)h(the)f(pair)h(from)e FM(\(E2\))p Ge(,)e F0(C)r FX([)p 1679 2938 V F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)p FX(])p FV(j)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(Q)19 b Ge(and)13 b F0(C)r FX([\()p F1(n)o FZ(~)-31 b F0(y)o FX(\)\()p F0(P)p FV(j)p F0(Q)p FX(\)])p FV(j)p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)q FX(\))p FZ(:)p F0(Q)p Ge(.)24 b(Finally)18 b(we)450 3092 y(can)j(immediately)f(reason)g (about)g(the)i(pair)e(from)h FM(\(E3\))p Ge(,)g FX(\()p F1(n)9 b F0(x)p FX(\))p Ge(!)p F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p FZ(:)p F0(P)22 b Ge(and)f F2(0)o Ge(,)h(since)f(no)g (transition)f(is)450 3200 y(possible)g(in)g(either)g(process.)p 1391 3200 50 50 v 1281 3410 a F3(C.5.)95 b(PR)m(OOF)31 b(OF)h(LEMMA)f(4.1)533 3541 y Ge(F)o(or)f(\(1\),)h(if)g(a)f Fl(B)p Ge(-normal)e(form)h(contains)g(hiding)g(and/or)g(redundant)e F2(0)p Ge(,)32 b FV(\021)2842 3510 y Gd(0)2893 3541 y Ge(cannot)d(equate)h(it)450 3648 y(with)21 b(the)g(result)g(of)f (stripping)g(them)g(of)n(f;)h(while)f(if)h(it)h(doesn')o(t,)d(since)i FZ(.)g Ge(only)f(strips)h(of)n(f)f(\(rather)g(than)450 3756 y(increases\))f(them,)f(applying)g FZ(.)h Ge(is)h(the)f(same)h (thing)e(as)i(applying)d FV(\021)2421 3726 y Gd(0)2442 3756 y Ge(.)25 b(\(2\))19 b(is)h(immediate)e(by)h(reaching)450 3864 y(an)31 b(ENF)h(by)f(Theorem)e(3.2)i(and)g(then)g(by)g(stripping)f (redundant)f(hiding)h(and)g F2(0)i Ge(by)f FZ(.)g Ge(\(which)f(is)450 3972 y(inside)d FV(\021)p Ge(\).)45 b(F)o(or)26 b(\(3\),)i(by)f (de\002nition)e(the)i(processes)g(generated)e(by)i(the)g(rules)g(in)g (Proposition)e(3.3)450 4080 y(do)d(not)g(contain)f(hiding)g(and)h (redundant)d F2(0)p Ge(.)32 b(F)o(or)21 b(the)i(con)m(v)o(erse)d(we)j (ar)o(gue)d(by)i(strucrural)f(induction)450 4188 y(combined)c(with)i (these)g(tw)o(o)g(conditions)e(to)i(sho)n(w)g FZ(.)p Ge(-normal)d(forms)i(can)h(be)g(generated)e(by)h(the)h(three)450 4296 y(rules)28 b(in)g(Proposition)f(3.3.)48 b(\(4\))27 b(is)i(immediate)e(from)g(\(3\).)48 b(F)o(or)28 b(\(5\),)h(we)f(sho)n (w)g(this)h(for)e F0(P)c FV(2)g FM(NF)3407 4308 y FT(e)450 4404 y Ge(which)g(is)i(enough.)33 b(Suppose)23 b F0(P)d FV(2)h FM(NF)1611 4416 y FT(e)1666 4404 y Ge(and)i F0(P)1935 4357 y Gc(t)1882 4404 y 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Gd(0)2178 1342 y FX([)p F0(P)2244 1354 y Gb(2)2279 1342 y FX(])p Ge(,)19 b(i.e.)h F0(C)r FX([)p F0(P)2587 1354 y Gb(2)2622 1342 y FX(])c FV(+)2712 1312 y FT(i)2712 1363 y(x)2744 1342 y Ge(.)24 b(If)18 b(not,)h(then)e (suppose)446 1450 y F0(C)r FX([)p F0(P)569 1462 y Gb(1)603 1450 y FX(])h FV(!)-37 b(!)13 b F0(C)843 1420 y Gd(0)864 1450 y FX([)p F0(P)930 1462 y Gb(1)965 1450 y F1(s)p FX(])19 b Ge(where)c F0(C)1333 1420 y Gd(0)1354 1450 y FX([)p F0(P)1420 1462 y Gb(1)1454 1450 y F1(s)p FX(])20 b Ge(is)g(the)f(\002rst)g(con\002guration)e(in)i(which)f(the)h(input)g (pre\002x)f(is)i(tak)o(en)450 1558 y(of)n(f.)44 b(Using)26 b(cop)o(ycats,)i(we)f(can)f(represent)g F1(s)h Ge(by)f(parallel)h (composition)d(and)j(hiding,)f(so)i(that)e(the)450 1666 y(former)19 b(condition)f(gi)n(v)o(es)i(us)c F0(C)r FX([)p F0(P)1437 1678 y Gb(2)1472 1666 y FX(])i FV(+)1564 1636 y FT(i)1564 1687 y(x)1595 1666 y Ge(,)j(as)g(required.)533 1774 y(\(2-b\))i(is)j(immediate)e(by)h(performing)d(e)o(xtended)h (reduction)g(at)i(occurrences)e(of)i F0(x)g Ge(in)g F0(P)h Ge(on)e(both)450 1882 y(sides,)d(and)e(noting)g FV(7!\032\031\032)1309 1860 y(\030)1309 1886 y FX(=)1372 1882 y Ge(.)533 2003 y(F)o(or)24 b(\(3\),)h(suppose)f F0(P)1181 1956 y FQ(def)1186 2003 y FX(=)p 1276 1957 V 25 w F0(x)p FX(\()-10 b FZ(~)-32 b F0(y)p FX(\))p F0(P)1465 1973 y Gd(0)1512 2003 y Ge(has)25 b(type)f(?)t F0(A)h Ge(and)f(tak)o(e)h(the)g(conte)o(xt)19 b F0(C)r FX([)25 b(])h Ge(from)d F0(A)i Ge(to)g F0(u)12 b Ge(:)g Fy(B)f Ge(.)45 b(By)450 2111 y(\(2\))21 b(abo)o(v)o(e)f(we)j (can)e(set)e F0(C)r FX([)k(])f Ge(has)g(form)f FX(\()p F1(n)9 b FZ(~)-41 b F0(w)p FX(\)\()p F0(S)q FV(j)p F0(R)p FV(j)p FX([)22 b(]\))h Ge(where)e F0(R)h Ge(is)h(the)f(composition)e (of)h(replicated)450 2219 y(processes)g(compensating)e F0(A)p Ge(.)28 b(Since)22 b F0(u)f Ge(cannot)f(occur)g(in)i F0(R)f Ge(it)h(occurs)f(in)g F0(S)q Ge(,)g(whose)g(beha)n(viour)f(at)h F0(u)450 2327 y Ge(does)d(not)g(depend)f(on)h F0(P)h Ge(in)14 b F0(C)r FX([)p 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FX(\()o FZ(~)-41 b F0(w)q FX(\))p F1(P)p FX([)p F0(w)2808 2915 y FT(i)2852 2903 y FV(!)23 b F0(z)2990 2915 y FT(i)3012 2903 y FX(])p 3035 2835 56 3 v -30 x Gc(r)3069 2884 y Fn(i)3090 2903 y Ge(.)50 b(Let)29 b F0(P)22 b FV(\021)446 3011 y F0(C)r FX([)p 526 2965 37 4 v F0(y)p FX(\()-15 b FZ(~)-27 b F0(z)p FX(\))p F0(R)710 3023 y Gb(1)745 3011 y FX(])p FZ(::)p FX([)p 837 2965 V F0(y)p FX(\()-15 b FZ(~)-27 b F0(z)q FX(\))p F0(R)1022 3023 y FT(n)1056 3011 y FX(])30 b Ge(where)p 1341 2965 V 28 w F0(y)p FX(\()-15 b FZ(~)-27 b F0(z)p FX(\))p F0(R)1534 3023 y FT(j)1586 3011 y Ge(e)o(xhausts)28 b(all)i(prime)e(outputs)g(in)g F0(P)i Ge(\(these)e(conte)o(xts)g(can)g (be)450 3119 y(nested\).)c(Then)c(we)g(ha)n(v)o(e:)506 3316 y FX(\()p F1(n)10 b F0(y)p FX(\)\()p F0(P)p FV(j)p FX([)p F0(y)18 b FV(!)h F0(x)p FX(])1006 3286 y Gc(t)1037 3316 y FX(\))68 b FV(7!)1220 3286 y FT(n)p FU(+)p Gb(1)1398 3316 y F0(C)r FX([\()p F1(n)-5 b FZ(~)-27 b F0(z)p FX(\)\()p F0(R)1710 3328 y Gb(1)1745 3316 y FV(j)p 1768 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1180 y Gb(2)1607 1168 y FX(\))1639 1144 y Gd(\016)1674 1168 y FZ(;)-6 b(~)-27 b F0(z)10 b Ge(:)1763 1164 y FZ(~)1780 1168 y F1(t)p Ge(.)533 1279 y(\(4\))46 b FV(`)20 b Fw(Sum)p FV(h)p F0(m)p FZ(;)-6 b(~)-27 b F0(z)q FZ(;)9 b FV(f)p FX(\()p F0(x)1203 1291 y FT(i)1224 1279 y FX(\))p F0(M)1325 1291 y FT(i)1347 1279 y FV(gi)1421 1249 y FT(T)1489 1279 y FZ(.)21 b F0(m)11 b Ge(:)p 1657 1207 V 11 w FX(\()p F0(T)1729 1291 y Gb(1)1777 1279 y FX(+)h F0(T)1894 1291 y Gb(2)1928 1279 y FX(\))1960 1255 y Gd(\016)1995 1279 y FZ(;)d F0(E)2084 1249 y Gd(\016)2143 1279 y Ge(if)25 b FV(`)20 b FX([)-9 b([)p F0(M)2396 1291 y FT(i)2429 1279 y Ge(:)11 b F0(T)2520 1249 y Gd(0)2503 1302 y FT(i)2541 1279 y FX(])-9 b(])2578 1291 y FT(u)2635 1279 y FZ(.)22 b F0(u)11 b Ge(:)g F0(T)2843 1249 y Gd(0)2826 1302 y FT(i)2864 1239 y Gd(\016)2919 1279 y FV(!)21 b F0(E)3080 1249 y Gd(\016)3139 1279 y Ge(\()p F0(i)g FX(=)f Ge(1)p FZ(;)9 b Ge(2\))450 1387 y(for)20 b(some)g F0(T)820 1357 y Gd(0)804 1411 y Gb(1)p FS(;)p Gb(2)886 1387 y Ge(.)533 1540 y(F)o(or)28 b(proofs,)h(\(1\))f(is)i(immediate)d(from)h (Proposition)f(5.4)h(\(1\).)49 b(The)28 b(remaining)f(statements)h(are) 450 1648 y(direct)18 b(from)g(the)h(de\002nition.)k(F)o(or)18 b(e)o(xample,)f(let)j FV(`)c FX([)-9 b([)p F0(N)23 b Ge(:)17 b F0(T)2158 1618 y Gd(0)2180 1648 y FX(])-9 b(])2217 1660 y FT(u)2270 1648 y FZ(.)19 b F0(u)8 b Ge(:)g F0(T)2468 1618 y Gd(0)2489 1608 y(\016)2524 1648 y FZ(;)h F0(E)2613 1618 y Gd(\016)2648 1648 y Ge(.)25 b(F)o(or)18 b(simplicity)-5 b(,)18 b(assume)450 1756 y F0(T)507 1726 y Gd(\016)565 1756 y FX(=)24 b(\()p F1(t)p FX(\))754 1726 y Gb(!)814 1756 y Ge(and,)31 b(accordingly)-5 b(,)14 b FZ(~)-27 b F0(z)24 b FX(=)f F0(z)p Ge(.)55 b(By)30 b(\(1\))f(we)h(ha)n(v)o(e,)i (noting)c(that)i FM(md)p FX(\()p F1(t)p FX(\))25 b(=)p FV(")k Ge(in)h(this)g(case,)450 1864 y FV(`)18 b F0(c)p FX(\()p F0(w)p FX(\))p FZ(:)p Fw(Msg)q FV(h)p F0(w)-6 b FZ(;)9 b F0(z)p FV(i)1023 1834 y FT(T)1066 1812 y Fk(\016)1120 1864 y FZ(.)20 b F0(c)9 b Ge(:)g FX(\()p 1292 1800 92 4 v F0(T)1350 1840 y Gd(\016)1384 1864 y FX(\))1416 1834 y Gd(#)1470 1864 y FV(!)18 b F0(z)9 b Ge(:)g F1(t)p Ge(.)27 b(T)-7 b(ogether)19 b(with)h(the)h(gi)n(v)o(en)e(assumption,)f(we)j (obtain:)873 2051 y FV(`)p 943 2005 60 4 v 19 w F0(m)p FX(\()p F0(nc)p FX(\)\([)-9 b([)p F0(N)23 b Ge(:)c F0(T)1391 2017 y Gd(0)1412 2051 y FX(])-9 b(])1449 2063 y FT(n)1505 2051 y FV(j)21 b F0(c)p FX(\()p F0(w)p FX(\))p FZ(:)p Fw(Msg)p FV(h)p F0(w)-6 b FZ(;)9 b F0(z)p FV(i)2052 2017 y FT(T)2096 1994 y Fk(\016)2129 2051 y FX(\))21 b FZ(.)f F0(m)9 b Ge(:)g FX(\()p F0(T)2434 2017 y Gd(0)2455 2007 y(\016)2490 2051 y FX(\()p 2522 1987 92 4 v F0(T)2579 2027 y Gd(\016)2614 2051 y FX(\))2646 2017 y Gd(#)2681 2051 y FX(\))2713 2017 y Gb(?)2749 2051 y FZ(;)g F0(z)g Ge(:)g F1(t)p FZ(;)g F0(E)2979 2017 y Gd(\016)450 2238 y Ge(as)21 b(required.)p 920 2238 50 50 v 533 2395 a F0(Remark.)115 b Ge(In)23 b(the)f(abo)o(v)o(e,)g F0(m)h Ge(in)g FM(Arg)o FV(h)p F0(m)p FZ(;)9 b F0(N)c 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