%!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: classif2.dvi %%CreationDate: Tue Oct 7 15:06:50 1997 %%Pages: 52 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips classif2.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1997.10.07:1506 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 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o(y)m(.)300 1972 y Fk(Definition.)27 b Ft(W)m(e)13 b(de\014ne)g(the)h (information)9 b(system)k Fm(C)1213 1978 y Fr(\032)1244 1972 y Ft(=)f(\()p Fs(C)1334 1978 y Fr(\032)1353 1972 y Fs(;)7 b Fk(Con)1456 1978 y Fr(\032)1475 1972 y Fs(;)g Fp(`)1519 1978 y Fr(\032)1538 1972 y Ft(\))13 b(b)o(y)g(induction)300 2022 y(on)h(the)g(t)o(yp)q(e)h Fs(\032)p Ft(:)657 2118 y Fm(C)691 2124 y Fr(\023)717 2118 y Ft(:=)d Fg(N)39 b Ft(\(view)o(ed)14 b(as)g(a)g(\015at)g(information)d(system\))657 2180 y Fm(C)691 2186 y Fr(\032)p Fh(!)p Fr(\033)776 2180 y Ft(:=)g Fm(C)865 2186 y Fr(\032)896 2180 y Fp(!)g Fm(C)983 2186 y Fr(\033)657 2243 y Fm(C)691 2249 y Fr(\032)p Fh(\002)p Fr(\033)768 2243 y Ft(:=)h Fm(C)858 2249 y Fr(\032)887 2243 y Fp(\002)d Fm(C)962 2249 y Fr(\033)985 2243 y Fs(:)300 2338 y Ft(The)21 b Fn(p)n(artial)e(c)n(ontinuous)j(functionals)i Ft(\(o)o(v)o(er)c Fg(N)p Ft(\))f(of)g(t)o(yp)q(e)i Fs(\032)h Fp(!)g Fs(\033)f Ft(are)g(the)f(con)o(tin)o(uous)300 2388 y(functions)14 b(from)e Fp(j)p 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Fe(1)475 366 y Fp(j)p Fs(;)c(:)g(:)g(:)t(;)g(u)603 372 y Fr(n)636 366 y Fp(2)k(j)p Fm(C)721 372 y Fr(\032)738 376 y Ff(n)761 366 y Fp(j)599 454 y Ft([)-7 b([)p Fs(x)640 460 y Fr(i)653 454 y Ft(])g(]\()p Fs(u)710 460 y Fq(1)728 454 y Fs(;)7 b(:)g(:)g(:)e(;)i(u)845 460 y Fr(n)866 454 y Ft(\))42 b(=)g Fs(u)1022 460 y Fr(i)554 516 y Ft([)-7 b([)p Fs(M)5 b(N)g Ft(])-7 b(]\()p Fs(u)711 522 y Fq(1)728 516 y Fs(;)7 b(:)g(:)g(:)e(;)i(u)845 522 y Fr(n)866 516 y Ft(\))42 b(=)g([)-7 b([)p Fs(M)5 b Ft(])-7 b(]\()p Fs(u)1117 522 y Fq(1)1134 516 y Fs(;)7 b(:)g(:)g(:)t(;)g(u)1250 522 y Fr(n)1272 516 y Ft(\)\([)-7 b([)p Fs(N)5 b Ft(])-7 b(]\()p Fs(u)1416 522 y Fq(1)1433 516 y Fs(;)7 b(:)g(:)g(:)e(;)i(u)1550 522 y Fr(n)1572 516 y Ft(\)\))490 579 y([)-7 b([)p Fs(\025xM)5 b Ft(])-7 b(]\()p Fs(u)657 585 y Fq(1)674 579 y Fs(;)7 b(:)g(:)g(:)e(;)i(u)791 585 y Fr(n)813 579 y Ft(\)\()p Fs(v)q Ft(\))42 b(=)g([)-7 b([)p Fs(M)5 b Ft(])-7 b(]\()p Fs(u)1117 585 y Fq(1)1134 579 y Fs(;)7 b(:)g(:)g(:)t(;)g(u)1250 585 y Fr(n)1272 579 y Fs(;)g(v)q Ft(\))p Fs(:)300 667 y Fk(Pr)o(oof)p Ft(.)34 b(By)19 b(induction)f(on)h(the)g(term)f Fs(M)5 b Ft(.)33 b(In)19 b(case)h Fs(M)j Ft(is)c(a)f(v)n(ariable)g Fs(x)1551 673 y Fr(i)1565 667 y Ft(,)h(let)g([)-7 b([)p Fs(x)1702 673 y Fr(i)1715 667 y Ft(])g(])19 b(:=)300 716 y Fs(\031)325 701 y Fr(n)324 727 y(i)348 716 y Ft(,)d(where)i Fs(\031)524 701 y Fr(n)523 727 y(i)563 716 y Ft(is)e(the)h(pro)r (jection)f(af)g(an)g Fs(n)p Ft(-tuple)g(to)g(its)h Fs(i)p Ft(-th)f(comp)q(onen)o(t.)25 b(F)m(ormally)13 b(w)o(e)300 766 y(de\014ne)21 b Fs(\031)451 751 y Fr(n)450 777 y(i)493 766 y Ft(from)d(the)j(pro)r(jections)f Fs(\031)918 772 y Fq(1)937 766 y Fs(;)7 b(\031)980 772 y Fq(2)1017 766 y Ft(\(in)o(tro)q(duced)21 b(ab)q(o)o(v)o(e\))e(as)h(follo)o(ws)e (\(recall)i(that)300 816 y Fs(A)9 b Fp(\002)h Fs(B)i Fp(\002)d Fs(C)14 b Ft(:=)e(\()p Fs(A)d Fp(\002)h Fs(B)r Ft(\))g Fp(\002)f Fs(C)s Ft(\):)18 b Fs(\031)868 801 y Fq(1)867 826 y(1)898 816 y Ft(=)12 b(id,)h Fs(\031)1027 798 y Fr(n)p Fq(+1)1026 827 y Fr(n)p Fq(+1)1103 816 y Ft(=)f Fs(\031)1171 822 y Fq(2)1189 816 y Ft(,)i Fs(\031)1240 798 y Fr(n)p Fq(+1)1239 828 y Fr(i)1316 816 y Ft(=)e Fs(\031)1385 801 y Fr(n)1384 827 y(i)1417 816 y Fp(\016)d Fs(\031)1471 822 y Fq(1)1503 816 y Ft(for)14 b Fs(i)d Ft(=)h(1)7 b Fs(:)g(:)g(:)e(n)p Ft(.)362 866 y(F)m(or)10 b(an)g(application)e(term,)i(de\014ne)h([)-7 b([)p Fs(M)5 b(N)g Ft(])-7 b(])10 b(:=)h(ev)n(al)q Fp(\016)r Ft(\([)-7 b([)p Fs(M)5 b Ft(])-7 b(])r Fp(\002)r Ft([)f([)p Fs(N)5 b Ft(])-8 b(]\).)14 b(This)c(is)g(a)f(con)o(tin)o(uous)300 916 y(function)14 b(b)o(y)f(what)h(w)o(as)g(pro)o(v)o(ed)g(ab)q(o)o(v)o (e,)f(and)478 1004 y([)-7 b([)p Fs(M)5 b(N)g Ft(])-7 b(]\()p Fs(u)635 1010 y Fq(1)651 1004 y Fs(;)7 b(:)g(:)g(:)e(;)i(u)768 1010 y Fr(n)790 1004 y Ft(\))42 b(=)g(ev)n(al)o(\([)-7 b([)p Fs(M)5 b Ft(])-7 b(])7 b Fp(\002)j Ft([)-7 b([)p Fs(N)5 b Ft(])-7 b(]\)\()p Fs(u)1264 1010 y Fq(1)1281 1004 y Fs(;)7 b(:)g(:)g(:)t(;)g(u)1397 1010 y Fr(n)1419 1004 y Ft(\))848 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y Fs(X)i Fp(2)11 b Fk(Con)h Fp(\021)g(8)p Fs(a;)7 b(b)p Fp(2)o Fs(X)s Ft(\()p Fp(f)p Fs(a;)g(b)p Fp(g)k(2)g Fk(Con)p Ft(\))p Fs(:)300 1787 y Ft(Clearly)i(an)o(y)h(\015at)f(information)e(system)j (is)g(coheren)o(t,)h(and)e(moreo)o(v)o(er)g(w)o(e)h(ha)o(v)o(e)300 1894 y Fk(Lemma.)28 b Fl(Let)14 b Fm(A)g Fl(and)g Fm(B)g Fl(b)q(e)g(information)d(systems.)351 1974 y(1.)20 b(If)13 b Fm(B)h Fl(is)g(coheren)o(t,)h(then)f(so)g(is)g Fm(A)e Fp(!)f Fm(B)p Fl(.)351 2056 y(2.)20 b(If)13 b Fm(A)h Fl(and)g Fm(B)g Fl(are)g(coheren)o(t,)h(then)f(so)g(is)g Fm(A)9 b Fp(\002)h Fm(B)p Fl(.)300 2137 y Fk(Pr)o(oof)p Ft(.)19 b(Let)14 b Fm(A)e Ft(=)g(\()p Fs(A;)7 b Fk(Con)776 2143 y Fr(A)803 2137 y Fs(;)g Fp(`)847 2143 y Fr(A)874 2137 y Ft(\))14 b(and)f Fm(B)f Ft(=)g(\()p Fs(B)r(;)7 b Fk(Con)1226 2143 y Fr(B)1255 2137 y Fs(;)g Fp(`)1299 2143 y Fr(B)1327 2137 y Ft(\).)362 2186 y(1.)17 b(Let)11 b Fm(A)h Fp(!)f Fm(B)g Ft(=)h(\()p Fk(Con)774 2192 y Fr(A)803 2186 y Fp(\002)s Fs(B)r(;)7 b Fk(Con)q Fs(;)g Fp(`)p 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Ft(and)f(therefore)j(in)d(particular)h(the)h (functions)f Fs(F)1588 1894 y Fr(\013)1628 1888 y Ft(=)g Fs(F)1710 1873 y Fq(1)1704 1898 y Fr(\013)1745 1888 y Ft(for)300 1938 y Fs(\013)11 b(<)h(")401 1944 y Fq(0)430 1938 y Ft(can)e(b)q(e)h(built)e(from)g(the)h(simple)f(functionals)g Fs(F)1178 1920 y Fr(n)p Fq(+1)1172 1949 y(0)1242 1938 y Ft(,)h(whic)o(h)g(are)h(essen)o(tially)f(iteration)300 1988 y(functionals,)k(b)o(y)h(application)f(alone.)22 b(Note)15 b(that)h(in)e(the)i(resulting)f(represen)o(tation)i(of)e(the) 300 2037 y(functions)f Fs(F)506 2043 y Fr(\013)529 2037 y Ft(,)g Fs(\013)d(<)h(")656 2043 y Fq(0)675 2037 y Ft(,)h(w)o(e)h(do)g (not)g(need)h(the)f(fundamen)o(tal)e(sequences)k Fs(\025)p Ft([)p Fs(x)p Ft(])d(an)o(y)g(more.)300 2148 y Fm(5.2.)18 b Fl(Abstract)13 b(computabilit)o(y)l(.)24 b Ft(With)11 b(the)h(preparations)f(done)h(it)f(is)g(no)o(w)g(rather)i(straigh)o(t-) 300 2198 y(forw)o(ard)18 b(to)g(de\014ne)h(computabilit)o(y)d(in)i(our) g(iterated)h(function)f(spaces)h Fm(C)1528 2204 y Fr(\032)1566 2198 y Ft(\(based)g(on)f Fg(N)300 2248 y Ft(view)o(ed)i(as)g(a)f (\015at)h(information)d(system\).)36 b(The)20 b(tok)o(ens)h(and)e (\014nite)h(sets)h(of)f(tok)o(ens)g(are)300 2298 y(enco)q(dable)g(b)o (y)f(in)o(tegers)g(using)g(sequence-co)q(ding.)35 b(It)19 b(is)g(easy)h(to)f(see)h(that)f(the)h(notions)300 2348 y Fs(X)15 b Fp(2)c Fk(Con)473 2354 y Fr(\032)504 2348 y Ft(of)h(consistency)h(and)f Fs(X)j Fp(`)919 2354 y Fr(\032)951 2348 y Fs(a)c Ft(of)h(en)o(tailmen)o(t)e(corresp)q(ond)k (to)e(recursiv)o(e)i(\(in)d(fact)300 2397 y(elemen)o(tary\))i (relations.)300 2508 y Fk(Definition.)27 b Ft(A)14 b(partial)g(con)o (tin)o(uous)g(functional)f Fs(')h Ft(of)f(t)o(yp)q(e)i Fs(\032)g Ft(is)e(said)h(to)g(b)q(e)h Fn(c)n(omputable)300 2558 y Ft(if)e(-)h(when)g(view)o(ed)g(as)g(a)g(set)h(of)e(\(co)q(des)i (of)s(\))f(tok)o(ens)g(-)g(it)f(is)h(\006)1273 2543 y Fq(0)1273 2568 y(1)1292 2558 y Ft(-de\014nable.)p eop %%Page: 27 27 27 26 bop 300 275 a Fo(5.)17 b(Computability)11 b(in)i(higher)f(typ)n (es)913 b Fv(27)300 366 y Fk(Lemma.)28 b Fl(F)m(or)14 b(all)e(t)o(yp)q(es)j Fs(\032;)7 b(\033)o(;)g(\034)18 b Fl(the)c(functionals)624 456 y Ft(ev)n(al)695 462 y Fr(\032;\033)744 456 y Ft(:)7 b(\()p Fm(C)813 462 y Fr(\032)844 456 y Fp(!)k Fm(C)931 462 y Fr(\033)954 456 y Ft(\))e Fp(\002)h Fm(C)1055 462 y Fr(\032)1086 456 y Fp(!)h Fm(C)1173 462 y Fr(\033)624 518 y Ft(curry)721 529 y Fr(\032;\033)o(;\034)797 518 y Ft(:)c(\()p Fm(C)866 524 y Fr(\032)895 518 y Fp(\002)i Fm(C)970 524 y Fr(\033)1005 518 y Fp(!)i Fm(C)1092 524 y Fr(\034)1113 518 y Ft(\))g Fp(!)h Ft(\()p Fm(C)1244 524 y Fr(\032)1275 518 y Fp(!)f Ft(\()p Fm(C)1378 524 y Fr(\033)1412 518 y Fp(!)h Fm(C)1500 524 y Fr(\034)1521 518 y Ft(\)\))300 608 y Fl(are)i(computable.)300 683 y Fk(Pr)o(oof)p Ft(.)23 b(The)15 b(tok)o(ens)h(of)f(ev)n(al)800 689 y Fr(\032;\033)864 683 y Ft(are)h(of)f(the)g(form)f(\()p Fs(W)o(;)7 b(X)q(;)g(a)p Ft(\))14 b(with)h Fs(W)20 b Fp(2)13 b Fk(Con)1616 689 y Fr(\032)p Fh(!)p Fr(\033)1688 683 y Ft(,)i Fs(X)j Fp(2)300 732 y Fk(Con)384 738 y Fr(\032)418 732 y Ft(and)13 b Fs(a)f Fp(2)f Fs(C)601 738 y Fr(\033)623 732 y Ft(,)i(and)h(w)o(e)g(ha)o(v)o(e)568 823 y(\()p Fs(W)o(;)7 b(X)q(;)g(a)p Ft(\))k Fp(2)g Ft(ev)n(al)g Fp(\021)h Fs(a)f Fp(2)p 981 789 45 2 v 11 w Fs(W)6 b Ft(\()p 1042 789 38 2 v Fs(X)t Ft(\))12 b Fp(\021)g Fs(a)f Fp(2)p 1224 789 83 2 v 11 w Fs(W)6 b(X)15 b Fp(\021)d Fs(W)6 b(X)15 b Fp(`)d Fs(a:)300 913 y Ft(Here)20 b Fs(W)6 b(X)23 b Ft(is)c(the)g(application)f(of)g(\014nite)i(appro)o(ximations) c(in)o(tro)q(duced)k(in)e(4.4,)h(and)g(w)o(e)300 962 y(ha)o(v)o(e)g(made)f(use)h(of)p 646 929 V 19 w Fs(W)6 b(X)23 b Ft(=)p 801 929 45 2 v 21 w Fs(W)6 b Ft(\()p 862 929 38 2 v Fs(X)s Ft(\),)20 b(whic)o(h)f(can)g(b)q(e)h(pro)o(v)o (ed)f(easily)m(.)32 b(Th)o(us)20 b(w)o(e)f(ha)o(v)o(e)g(a)300 1012 y(\006)330 997 y Fq(0)330 1022 y(1)349 1012 y Ft(-de\014nition)13 b(of)g(ev)n(al)665 1018 y Fr(\032;\033)714 1012 y Ft(.)18 b(F)m(or)13 b(curry)j(w)o(e)e(similarly)d(ha)o(v)o(e)381 1108 y(\()p Fs(W)o(;)c(X)q(;)g(Y)r(;)g(a)p Ft(\))k Fp(2)g Ft(curry)i Fp(\021)f Fs(a)f Fp(2)p 865 1074 45 2 v 11 w Fs(W)6 b Ft(\()p 926 1074 118 2 v Fs(X)13 b Fp([)c Fs(Y)h Ft(\))h Fp(\021)h Fs(a)g Fp(2)p 1188 1072 195 2 v 11 w Fs(W)6 b Ft(\()p Fs(X)13 b Fp([)c Fs(Y)g Ft(\))j Fp(\021)f Fs(W)6 b Ft(\()p Fs(X)13 b Fp([)c Fs(Y)h Ft(\))h Fp(`)h Fs(a;)300 1198 y Ft(whic)o(h)i(again)e(is)i(a)g(\006)636 1183 y Fq(0)636 1208 y(1)654 1198 y Ft(-de\014nition.)k(This)c (completes)f(the)i(pro)q(of.)429 b Fj(2)300 1306 y Fk(Lemma.)351 1388 y Fl(1.)20 b(If)13 b Fs(')p Ft(:)7 b Fm(C)525 1394 y Fr(\032)556 1388 y Fp(!)k Fm(C)643 1394 y Fr(\033)680 1388 y Fl(and)j Fs( )q Ft(:)7 b Fm(C)842 1394 y Fr(\033)876 1388 y Fp(!)k Fm(C)963 1394 y Fr(\034)998 1388 y Fl(are)j(computable,)f (then)h(so)g(is)g Fs( )d Fp(\016)e Fs(')p Fl(.)351 1470 y(2.)20 b(If)13 b Ft(\010:)7 b Fm(C)528 1476 y Fr(\032)559 1470 y Fp(!)k Fm(C)646 1476 y Fr(\033)683 1470 y Fl(and)i Fs(')f Fp(2)f(j)p Fm(C)887 1476 y Fr(\032)907 1470 y Fp(j)i Fl(are)h(computable,)f(then)h(so)g(is)g Ft(\010\()p Fs(')p Ft(\))p Fl(.)351 1553 y(3.)20 b(If)13 b Fs(')p Ft(:)7 b Fm(C)525 1559 y Fr(\032)556 1553 y Fp(!)k Fm(C)643 1559 y Fr(\033)680 1553 y Fl(and)j Fs( )q Ft(:)7 b Fm(C)842 1559 y Fr(\032)873 1553 y Fp(!)k Fm(C)960 1559 y Fr(\034)995 1553 y Fl(are)j(computable,)f(then)h(so)g(is)g Fs(')9 b Fp(\002)h Fs( )q Fl(.)351 1635 y(4.)20 b(The)14 b(pro)r(jections)h Fs(\031)727 1641 y Fq(1)745 1635 y Ft(:)7 b Fm(C)798 1641 y Fr(\032)827 1635 y Fp(\002)i Fm(C)902 1641 y Fr(\033)936 1635 y Fp(!)i Fm(C)1023 1641 y Fr(\032)1057 1635 y Fl(and)j Fs(\031)1162 1641 y Fq(2)1180 1635 y Ft(:)7 b Fm(C)1233 1641 y Fr(\032)1261 1635 y Fp(\002)j Fm(C)1337 1641 y Fr(\033)1371 1635 y Fp(!)h Fm(C)1458 1641 y Fr(\033)1495 1635 y Fl(are)j(computable.)300 1717 y Fk(Pr)o(oof)p Ft(.)19 b(1.)e(The)e(tok)o(ens)f(of)f Fs( )d Fp(\016)f Fs(')14 b Ft(are)g(of)f(the)i(form)d(\()p Fs(X)q(;)7 b(b)p Ft(\))13 b(with)h Fs(X)h Fp(2)c Fk(Con)1550 1723 y Fr(\032)1583 1717 y Ft(and)j Fs(b)d Fp(2)g Fs(C)1762 1723 y Fr(\034)1783 1717 y Ft(,)300 1767 y(and)j(w)o(e)g(ha)o(v)o(e)540 1857 y(\()p Fs(X)q(;)7 b(b)p Ft(\))12 b Fp(2)f Fs( )g Fp(\016)e Fs(')i Fp(\021)h(9)p Fs(Y)e Fp(\022)934 1840 y Fq(\014n)973 1857 y Fs(')p Ft([\()p Fs(Y)r(;)d(b)p Ft(\))k Fp(2)g Fs( )g Fp(^)e(8)p Fs(a)p Fp(2)p Fs(Y)16 b Ft(\()p Fs(X)q(;)7 b(a)p Ft(\))12 b Fp(2)f Fs(')p Ft(])p Fs(:)300 1947 y Ft(This)j(is)g(a)f(\006)501 1932 y Fq(0)501 1957 y(1)520 1947 y Ft(-de\014nition,)g(since)h Fs(')g Ft(and)g Fs( )h Ft(are)g(computable.)362 1997 y(2.)j(Since)c(ev)n(al)g (is)f(con)o(tin)o(uous,)h(w)o(e)g(ha)o(v)o(e)f(from)g(PFS)374 2087 y Fs(a)e Fp(2)g Ft(\010\()p Fs(')p Ft(\))h Fp(\021)g(9)p Fs(W)6 b Fp(\022)691 2070 y Fq(\014n)731 2087 y Ft(\010)p Fp(9)p Fs(X)s Fp(\022)853 2070 y Fq(\014n)893 2087 y Fs(')p Ft(\()p Fs(a)12 b Fp(2)p 1009 2053 45 2 v 11 w Fs(W)6 b Ft(\()p 1070 2053 38 2 v Fs(X)s Ft(\)\))12 b Fp(\021)g(9)p Fs(W)6 b Fp(\022)1295 2070 y Fq(\014n)1335 2087 y Ft(\010)p Fp(9)p Fs(X)s Fp(\022)1457 2070 y Fq(\014n)1497 2087 y Fs(')p Ft(\()p Fs(W)g(X)15 b Fp(`)d Fs(a)p Ft(\))p Fs(:)300 2177 y Ft(This)i(again)e(is)i(a)g(\006)612 2162 y Fq(0)612 2187 y(1)644 2177 y Ft(de\014nition.)362 2227 y(3)g(and)g(4)f(are)h(pro)o(v)o(ed)g(similarly)m(.)879 b Fj(2)300 2334 y Fk(Lemma.)27 b Fl(F)m(or)16 b(ev)o(ery)g Fs(\025)p Fl(-term)g Fs(M)851 2319 y Fr(\032)886 2334 y Fl(and)g(ev)o(ery)g(list)g Fs(x)1176 2314 y Fr(\032)1193 2318 y Fe(1)1176 2345 y Fq(1)1211 2334 y Fs(;)7 b(:)g(:)g(:)e(;)i(x) 1328 2319 y Fr(\032)1345 2323 y Ff(n)1328 2345 y Fr(n)1383 2334 y Fl(of)15 b(v)n(ariables)g(con)o(taining)300 2384 y(all)e(v)n(ariables)g(free)i(in)e Fs(M)19 b Fl(the)c(functional)d Ft([)-7 b([)p Fs(M)5 b Ft(])-7 b(]:)7 b Fm(C)1115 2390 y Fr(\032)1132 2394 y Fe(1)1158 2384 y Fp(\002)j Fs(:)d(:)g(:)h Fp(\002)h Fm(C)1333 2390 y Fr(\032)1350 2394 y Ff(n)1385 2384 y Fp(!)i Fm(C)1472 2390 y Fr(\032)1506 2384 y Fl(de\014ned)k(ab)q (o)o(v)o(e)e(is)300 2434 y(computable.)300 2508 y Fk(Pr)o(oof)p Ft(.)35 b(The)19 b(construction)h(of)f([)-7 b([)p Fs(M)5 b Ft(])-7 b(])17 b(\(b)o(y)i(induction)g(on)g Fs(M)5 b Ft(\))19 b(in)g(4.9)f(is)h(suc)o(h)h(that)f(the)300 2558 y(previous)14 b(lemma)d(implies)h(immedia)o(tely)f(its)j (computabilit)o(y)m(.)466 b Fj(2)p eop %%Page: 28 28 28 27 bop 300 275 a Ft(28)777 b Fo(Se)n(ction)11 b(5.)17 b(Computability)11 b(in)i(higher)f(typ)n(es)300 366 y Fm(5.3.)20 b Fl(Least)15 b(\014xed)g(p)q(oin)o(ts.)26 b Ft(Since)15 b(the)g(least)g(\014xed)g(p)q(oin)o(t)f(op)q(erator)h (mak)o(es)e(sense)j(for)e(arbi-)300 416 y(trary)j(information)c (systems,)k(w)o(e)g(for)f(a)g(momen)o(t)e(re-in)o(tro)q(duce)j(that)g (lev)o(el)f(of)g(generalit)o(y)m(.)300 466 y(So)g(let)h Fm(A)g Ft(b)q(e)g(an)f(information)e(system)i(and)h Fs(f)t Ft(:)7 b Fm(A)16 b Fp(!)g Fm(A)p Ft(.)26 b(An)16 b(ob)r(ject)i Fs(x)e Fp(2)f(j)p Fm(A)p Fp(j)h Ft(is)g(a)h Fn(\014xe)n(d)300 516 y(p)n(oint)g Ft(of)12 b Fs(f)18 b Ft(if)12 b Fs(f)t Ft(\()p Fs(x)p Ft(\))h(=)f Fs(x)p Ft(.)17 b(It)c(is)g(called)g(the)g (least)h(\014xed)f(p)q(oin)o(t)f(of)h Fs(f)18 b Ft(if)12 b(for)g(an)o(y)h(other)h Fs(y)f Fp(2)e(j)p Fm(A)p Fp(j)300 565 y Ft(with)j Fs(f)t Ft(\()p Fs(y)q Ft(\))f(=)f Fs(y)j Ft(w)o(e)g(ha)o(v)o(e)e Fs(x)f Fp(\022)f Fs(y)q Ft(.)300 671 y Fk(Theorem.)28 b Fl(\(Fixed)14 b(P)o(oin)o(t)f(Theorem\).)19 b(Let)14 b Fm(A)g Fl(b)q(e)h(an)f(information)d(system)j(and)g Fs(f)t Ft(:)7 b Fm(A)12 b Fp(!)300 721 y Fm(A)p Fl(.)18 b(Then)c Fs(f)19 b Fl(has)14 b(a)g(least)g(\014xed)g(p)q(oin)o(t)g Fs(Y)9 b Ft(\()p Fs(f)t Ft(\))15 b Fl(giv)o(en)e(b)o(y)864 809 y Fs(Y)d Ft(\()p Fs(f)t Ft(\))i(:=)1030 770 y Fi([)1021 859 y Fr(n)p Fh(2)p Fb(N)1092 809 y Fs(f)1116 792 y Fq(\()p Fr(n)p Fq(\))1166 809 y Ft(\()p 1182 773 21 2 v Fp(;)p Ft(\))p Fs(:)300 928 y Fl(Moreo)o(v)o(er,)i Fs(Y)23 b Fl(is)14 b(con)o(tin)o(uous.)300 998 y Fk(Pr)o(oof)p Ft(.)21 b(F)m(rom)p 572 962 V 13 w Fp(;)12 b(\022)h Fs(f)t Ft(\()p 690 962 V Fp(;)q Ft(\))i(w)o(e)g(get)g Fs(f)899 983 y Fq(\()p Fr(n)p Fq(\))948 998 y Ft(\()p 964 962 V Fp(;)p Ft(\))e Fp(\022)g Fs(f)1083 983 y Fq(\()p Fr(n)p Fq(+1\))1174 998 y Ft(\()p 1190 962 V Fp(;)p Ft(\))i(b)o(y)f (induction,)g(since)i Fs(f)j Ft(is)14 b(mono-)300 1047 y(tone.)k(Hence)13 b Fs(x)e Ft(:=)619 1016 y Fi(S)654 1060 y Fr(n)684 1047 y Fs(f)708 1032 y Fq(\()p Fr(n)p Fq(\))757 1047 y Ft(\()p 773 1011 V Fp(;)p Ft(\))g Fp(2)h(j)p Fm(A)p Fp(j)p Ft(.)k(Moreo)o(v)o(er,)c(b)o(y)g(the)g(c)o (haracterization)g(of)f(con)o(tin)o(uit)o(y)300 1097 y(in)i(part)h(3)g(of)f(the)i(prop)q(osition)e(in)h(4.6)670 1183 y Fs(f)t Ft(\()p Fs(x)p Ft(\))f(=)e Fs(f)t Ft(\()853 1144 y Fi([)867 1231 y Fr(n)907 1183 y Fs(f)931 1166 y Fq(\()p Fr(n)p Fq(\))981 1183 y Ft(\()p 997 1147 V Fp(;)p Ft(\))c(\))k(=)1112 1144 y Fi([)1125 1231 y Fr(n)1165 1183 y Fs(f)1189 1166 y Fq(\()p Fr(n)p Fq(+1\))1280 1183 y Ft(\()p 1296 1147 V Fp(;)p Ft(\))h(=)g Fs(x:)300 1308 y Ft(If)i Fs(y)h Ft(is)f(another)g(\014xed)h(p)q(oin)o(t,)d(from)p 890 1272 V 13 w Fp(;)f(\022)h Fs(y)j Ft(w)o(e)g(get)f Fs(f)1156 1293 y Fq(\()p Fr(n)p Fq(\))1205 1308 y Ft(\()p 1221 1272 V Fp(;)p Ft(\))e Fp(\022)f Fs(y)16 b Ft(b)o(y)e(induction)f (and)h(hence)874 1396 y Fs(x)d Ft(=)953 1357 y Fi([)965 1444 y Fr(n)1006 1396 y Fs(f)1030 1379 y Fq(\()p Fr(n)p Fq(\))1079 1396 y Ft(\()p 1095 1360 V Fp(;)p Ft(\))h Fp(\022)f Fs(y)q(:)362 1511 y Ft(Finally)j(w)o(e)h(sho)o(w)g(that)g Fs(Y)24 b Ft(is)14 b(con)o(tin)o(uous.)22 b(Monotonicit)o(y)13 b(follo)o(ws)h(imm)o(ediately)e(from)300 1561 y(its)k(de\014nition.)23 b(F)m(or)15 b(PFS)h(assume)f Fs(b)f Fp(2)g Fs(Y)c Ft(\()p Fs(f)t Ft(\),)16 b(i.e.)f Fs(b)f Fp(2)h Fs(f)1241 1546 y Fq(\()p Fr(m)p Fq(\))1299 1561 y Ft(\()p 1315 1525 V Fp(;)p Ft(\))g(for)h(some)e Fs(m)p Ft(.)24 b(No)o(w)16 b(since)300 1611 y(ev)n(aluation)h(is)h(con)o(tin)o(uous,)g(for)g (\014xed)g Fs(m)h Ft(the)f(function)g Fs(f)23 b Fp(7!)18 b Fs(f)1370 1596 y Fq(\()p Fr(m)p Fq(\))1428 1611 y Ft(\()p 1444 1575 V Fp(;)p Ft(\))g(is)g(con)o(tin)o(uous)g(as)300 1660 y(w)o(ell.)f(Hence)f(b)o(y)d(PFS,)h(for)g(our)g Fs(b)d Fp(2)g Fs(f)915 1645 y Fq(\()p Fr(m)p Fq(\))973 1660 y Ft(\()p 989 1624 V Fp(;)p Ft(\))j(w)o(e)g(\014nd)g(a)g Fs(W)k Fp(\022)1308 1645 y Fq(\014n)1359 1660 y Fs(f)g Ft(suc)o(h)d(that)846 1753 y Fs(b)d Fp(2)p 915 1720 45 2 v 11 w Fs(W)960 1727 y Fq(\()p Fr(m)p Fq(\))1017 1753 y Ft(\()p 1033 1717 21 2 v Fp(;)p Ft(\))g Fp(\022)g Fs(Y)d Ft(\()p 1175 1720 45 2 v Fs(W)d Ft(\))p Fs(:)515 b Fj(2)362 1833 y Ft(Let)14 b Fs(Y)460 1839 y Fr(\032)492 1833 y Ft(b)q(e)f(the)g(least)g(\014xed)h(p)q(oin)o(t)e(op)q(erator)h(asso)q (ciated)h(with)e(the)i(information)c(system)300 1883 y Fm(C)334 1889 y Fr(\032)354 1883 y Ft(,)j(so)h Fs(Y)454 1889 y Fr(\032)474 1883 y Ft(:)7 b(\()p Fm(C)543 1889 y Fr(\032)573 1883 y Fp(!)k Fm(C)660 1889 y Fr(\032)680 1883 y Ft(\))h Fp(!)f Fm(C)795 1889 y Fr(\032)815 1883 y Ft(.)18 b(W)m(e)13 b(no)o(w)h(sho)o(w)g(that)f Fs(Y)1219 1889 y Fr(\032)1253 1883 y Ft(is)g(computable.)300 1988 y Fk(Lemma.)28 b Fs(Y)503 1994 y Fr(\032)536 1988 y Fl(is)14 b(computable.)300 2059 y Fk(Pr)o(oof)p Ft(.)22 b(W)m(e)14 b(need)i(to)f(sho)o(w)g(that)g Fs(Y)904 2065 y Fr(\032)938 2059 y Ft(is)g(\006)1011 2043 y Fq(0)1011 2069 y(1)1029 2059 y Ft(-de\014nable)g(when)h(view)o(ed)f(as)g(a)f(set)i(of)e(tok)o (ens.)300 2108 y(F)m(rom)f(the)j(de\014nitions)f(of)f(appro)o(ximable)e (maps,)i(their)h(corresp)q(ondence)j(with)c(con)o(tin)o(uous)300 2158 y(functions,)f(the)h(de\014nition)f(of)g Fs(Y)816 2164 y Fr(\032)848 2158 y Ft(ab)q(o)o(v)o(e,)g(and)g(the)h(easily)f(v)o (eri\014ed)h(fact)f(that)p 1557 2125 38 2 v 13 w Fs(X)t Ft(\()p 1611 2125 32 2 v Fs(Z)s Ft(\))f(=)p 1714 2125 69 2 v 12 w Fs(X)s(Z)t Ft(,)300 2208 y(w)o(e)i(obtain)585 2288 y(\()p Fs(X)q(;)7 b(a)p Ft(\))12 b Fp(2)f Fs(Y)768 2294 y Fr(\032)829 2288 y Fp(\021)42 b Ft(\()p Fs(X)956 2270 y Fh(0)968 2288 y Fs(;)7 b(a)p Ft(\))k Fp(2)g Fs(Y)1099 2294 y Fr(\032)1160 2288 y Ft(for)j(some)f Fs(X)1365 2273 y Fh(0)1389 2288 y Fp(\022)1421 2273 y Fq(\014n)p 1472 2254 38 2 v 1472 2288 a Fs(X)829 2350 y Fp(\021)42 b Fs(a)11 b Fp(2)g Fs(Y)999 2356 y Fr(\032)1019 2350 y Ft(\()p 1035 2317 V Fs(X)s Ft(\))829 2427 y Fp(\021)42 b(9)p Fs(n)p Ft(\()p Fs(a)11 b Fp(2)p 1039 2393 V 11 w Fs(X)1077 2400 y Fq(\()p Fr(n)p Fq(\))1126 2427 y Ft(\()p 1142 2390 21 2 v Fp(;)o Ft(\)\))829 2496 y Fp(\021)42 b(9)p Fs(n)p Ft(\()p Fs(a)11 b Fp(2)p 1039 2457 107 2 v 11 w Fs(X)1076 2484 y Fq(\()p Fr(n)p Fq(\))1126 2496 y Fp(;)o Ft(\))829 2558 y Fp(\021)42 b(9)p Fs(n)p Ft(\()p Fs(X)1004 2541 y Fq(\()p Fr(n)p Fq(\))1053 2558 y Fp(;)11 b(`)h Fs(a)p Ft(\))p Fs(:)p eop %%Page: 29 29 29 28 bop 300 275 a Fo(5.)17 b(Computability)11 b(in)i(higher)f(typ)n (es)913 b Fv(29)300 366 y Ft(Th)o(us)14 b Fs(Y)429 372 y Fr(\032)462 366 y Ft(is)g(\006)534 351 y Fq(0)534 376 y(1)553 366 y Ft(-de\014nable)g(if)f(the)h(relation)g Fs(X)1041 351 y Fq(\()p Fr(n)p Fq(\))1090 366 y Fp(;)d(`)h Fs(a)i Ft(is.)k(But)c(note)g(that)360 464 y Fs(X)397 447 y Fq(\()p Fr(n)p Fq(\))446 464 y Fp(;)d(`)h Fs(a)g Fp(\021)g(9)622 454 y Fs(~)616 464 y(Z)s Ft([)665 454 y Fs(~)659 464 y(Z)17 b Ft(has)d(length)g Fs(n)p Ft(+1)9 b Fp(^)g Fs(Z)1057 470 y Fq(0)1087 464 y Ft(=)j Fp(;)d(^)g(8)p Fs(i)1051 1396 y(<)1051 1471 y(>)1051 1483 y(:)1094 1373 y Fs(x)50 b Ft(if)13 b Fs(p)f(>)g Ft(0)1094 1423 y Fs(y)54 b Ft(if)13 b Fs(p)f Ft(=)g(0)1094 1473 y Fs(x)50 b Ft(if)13 b Fs(x)f Ft(=)f Fs(y)1094 1522 y Fp(?)41 b Ft(otherwise.)300 1597 y(F)m(rom)12 b(it)i(w)o(e)g(can)g (de\014ne)h(the)f Fn(se)n(quential)h(c)n(onditional)k Ft(cond)14 b(of)f(the)i(same)e(t)o(yp)q(e)h(b)o(y)543 1689 y(cond\()p Fs(p;)7 b(x;)g(y)q Ft(\))12 b(:=)f(p)q(cond\()p Fs(p;)c Ft(p)q(cond\()p Fs(p;)g(x;)g Fp(?)p Ft(\))p Fs(;)g Ft(p)q(cond)o(\()p Fs(p;)g Fp(?)p Fs(;)g(y)q Ft(\)\))p Fs(:)300 1781 y Ft(Then)770 1865 y(cond)q(\()p Fs(p;)g(x;)g(y)q Ft(\))k(=)1047 1794 y Fi(\()1087 1815 y Fs(x)50 b Ft(if)13 b Fs(p)e(>)h Ft(0)1087 1865 y Fs(y)54 b Ft(if)13 b Fs(p)e Ft(=)h(0)1087 1915 y Fp(?)41 b Ft(if)13 b Fs(p)e Ft(=)h Fp(?)p Ft(.)300 1982 y(W)m(e)h(can)i(also)e(de\014ne)i(from)d(p)q(cond) i(the)h Fn(p)n(ar)n(al)r(lel)f(or)k Ft(of)13 b(t)o(yp)q(e)h Fs(\023)d Fp(!)h Fs(\023)f Fp(!)g Fs(\023)i Ft(giv)o(en)h(b)o(y)828 2074 y Fp(_)p Ft(\()p Fs(p;)7 b(q)q Ft(\))k(:=)g(p)q(cond)q(\()p Fs(p;)c Ft(1)p Fs(;)g(q)q Ft(\))p Fs(:)300 2166 y Ft(Then)736 2271 y Fp(_)p Ft(\()p Fs(p;)g(q)q Ft(\))k(=)910 2173 y Fi(8)910 2210 y(>)910 2223 y(<)910 2298 y(>)910 2310 y(:)954 2200 y Ft(1)53 b(if)13 b Fs(p)e(>)h Ft(0)954 2250 y(1)53 b(if)13 b Fs(q)g(>)e Ft(0)954 2299 y(0)53 b(if)13 b Fs(p)e Ft(=)h(0)i(and)g Fs(q)e Ft(=)g(0)954 2349 y Fp(?)41 b Ft(otherwise.)362 2408 y(Because)18 b(these)f(are)g(at)e(the)i(lo)o(w)o(est)e(t)o(yp)q(e)i(lev)o(el,)e(it)h (is)g(necessary)h(only)e(for)h(them)f(to)h(b)q(e)300 2458 y(monotone)g(in)g(order)i(to)f(b)q(e)h(con)o(tin)o(uous,)g(since)g (PFS)f(is)g(trivial)f(o)o(v)o(er)h(the)h(\015at)f(system)g Fg(N)p Ft(.)300 2508 y(But)e(their)g(monotonicit)o(y)d(is)i(clear)h (from)d(their)j(de\014nitions.)k(It)c(is)f(equally)g(clear)g(that)h (eac)o(h)300 2558 y(of)f(these)h(examples)f(is)g(computable.)k(W)m(e)c (only)g(need)h(to)f(sho)o(w)h(that)f(as)g(relations)h(\(i.e.)e(sets)p eop %%Page: 30 30 30 29 bop 300 275 a Ft(30)777 b Fo(Se)n(ction)11 b(5.)17 b(Computability)11 b(in)i(higher)f(typ)n(es)300 366 y Ft(of)g(tok)o(ens)i(of)e(the)i(appropriate)f(t)o(yp)q(e\))h(they)f(are) h(\006)1115 351 y Fq(0)1115 376 y(1)1134 366 y Ft(-de\014nable.)k(But) 13 b(this)g(is)g(immediate:)i(in)300 416 y(the)f(case)g(of)e(p)q(cond)i (the)f(tok)o(ens)h(are)f(all)f(4-tuples)h(\()p Fs(p;)7 b(x;)g(y)q(;)g(z)r Ft(\))12 b(of)h(n)o(um)o(b)q(ers)g(suc)o(h)g(that)h (either)300 466 y Fs(p)d(>)h Ft(0)i(and)g Fs(z)f Ft(=)f Fs(x)p Ft(,)h(or)h Fs(p)d Ft(=)h(0)i(and)g Fs(z)f Ft(=)f Fs(y)q Ft(,)i(or)g Fs(x)d Ft(=)h Fs(y)h Ft(=)f Fs(z)r Ft(.)362 516 y(The)h Fn(c)n(ontinuous)i(existential)e(quanti\014er)k Fp(9)12 b Ft(of)g(t)o(yp)q(e)h(\()p Fs(\023)e Fp(!)g Fs(\023)p Ft(\))g Fp(!)g Fs(\023)h Ft(is)h(de\014ned)g(as)f(follo)o (ws:)711 643 y Fp(9)p Ft(\()p Fs(')p Ft(\))g(:=)860 572 y Fi(\()900 598 y Ft(0)53 b(if)13 b Fs(')p Ft(\()p Fp(?)p Ft(\))e(=)h(0)900 648 y(1)53 b(if)13 b Fs(')p Ft(\()p Fs(n)p Ft(\))f Fs(>)g Ft(0)h(for)h(some)f Fs(n)900 698 y Fp(?)41 b Ft(otherwise.)300 770 y(Clearly)16 b Fp(9)g Ft(is)g(monotone.)24 b(F)m(urthermore,)16 b(if)f Fp(9)p Ft(\()p Fs(')p Ft(\))i(has)f(a)g(de\014ned)h(v)n(alue)f(b)o(y)g(either) h(of)f(the)300 820 y(\014rst)e(t)o(w)o(o)e(clauses,)i(then)g(it)e(ob)o (viously)g(has)h(that)g(same)f(v)n(alue)h(on)g(a)f(\014nite)i (subfunction)f(of)f Fs(')300 870 y Ft(\(only)g(one)g(bit)g(of)g(data)g (is)g(needed\).)19 b(Therefore)14 b Fp(9)e Ft(satis\014es)h(PFS)g(and)f (hence)i(is)e(con)o(tin)o(uous.)362 920 y(Observ)o(e)18 b(on)d(the)i(other)f(hand,)g(that)g(the)h(full)d(existen)o(tial)i(quan) o(ti\014er,)g(where)h(the)g(\014rst)300 970 y(line)g(is)g(replaced)h(b) o(y)f(`0)f(if)g Fs(')p Ft(\()p Fs(n)p Ft(\))i(=)f(0)g(for)f(ev)o(ery)i Fs(n)p Ft(',)f(is)g Fn(not)k Ft(con)o(tin)o(uous.)28 b(F)m(or)17 b(if)f(it)h(w)o(ere,)300 1019 y(then)g(b)o(y)f(PFS)h(its)f (v)n(alue)g(\(namely)e(0\))j(on)f(the)h(total)e(constan)o(t-0)i (function)f(w)o(ould)f(also)h(b)q(e)300 1069 y(attained)e(on)f(some)g (\014nite)h(subfunction;)g(this)g(is)g(imp)q(ossible.)362 1119 y(The)i(con)o(tin)o(uous)f Fp(9)p Ft(-functional)f(is)h (computable)f(since)h(its)g(tok)o(ens)h(are)g(all)d(pairs)i(\()p Fs(X)q(;)7 b(n)p Ft(\))300 1169 y(suc)o(h)16 b(that)g(either)g(\()p Fp(;)p Fs(;)7 b Ft(0\))13 b Fp(2)h Fs(X)19 b Ft(and)c Fs(n)f Ft(=)g(0)h(or)g(else)h(there)h(exists)f(an)f Fs(m)h Ft(and)f(a)g Fs(k)g(>)g Ft(0)g(suc)o(h)300 1219 y(that)f(\()p Fp(f)p Fs(m)p Fp(g)p Fs(;)7 b(k)q Ft(\))k Fp(2)g Fs(X)18 b Ft(and)13 b Fs(n)f Ft(=)g(1.)17 b(Clearly)d(this)g(is)f(another)i (\006)1305 1204 y Fq(0)1305 1229 y(1)1337 1219 y Ft(condition.)300 1326 y Fk(Definition.)27 b Ft(A)14 b(partial)g(con)o(tin)o(uous)g (functional)f(\010)h(of)f(t)o(yp)q(e)i Fs(\032)1348 1332 y Fq(1)1379 1326 y Fp(!)c Fs(:)c(:)g(:)k Fp(!)g Fs(\032)1567 1332 y Fr(p)1599 1326 y Fp(!)g Fs(\023)j Ft(is)g(said)300 1376 y(to)k(b)q(e)h Fn(r)n(e)n(cursive)f(in)i Ft(p)q(cond)f Fn(and)h Fp(9)e Ft(if)g(it)g(can)h(b)q(e)g(de\014ned)g(explicitly)m(,)f (for)g(all)f(argumen)o(ts)300 1425 y Fs(')327 1431 y Fq(1)346 1425 y Fs(;)7 b(:)g(:)g(:)e(;)i(')466 1431 y Fr(p)484 1425 y Ft(,)14 b(b)o(y)f(an)h(equation)765 1512 y(\010\()p Fs(')838 1518 y Fq(1)857 1512 y Fs(;)7 b(:)g(:)g(:)e(;)i(') 977 1518 y Fr(p)996 1512 y Ft(\))12 b(=)f Fs(M)5 b Ft(\()p Fs(')1155 1518 y Fq(1)1174 1512 y Fs(;)i(:)g(:)g(:)e(;)i(')1294 1518 y Fr(p)1313 1512 y Ft(\))300 1599 y(where)22 b Fs(M)j Ft(is)20 b(a)h Fs(\025)p Ft(-term)f(built)g(up)g(from)f(the)i(v)n (ariables)f Fs(')1287 1605 y Fq(1)1306 1599 y Fs(;)7 b(:)g(:)g(:)e(;)i(')1426 1605 y Fr(p)1465 1599 y Ft(together)22 b(with)e(the)300 1649 y(constan)o(ts)14 b(0,)e(successor,)k (predecessor,)f(the)f(\014xed)g(p)q(oin)o(t)e(op)q(erators)i Fs(Y)1433 1655 y Fr(\032)1453 1649 y Ft(,)e(and)h(p)q(cond)h(and)f Fp(9)p Ft(.)300 1756 y Fk(Theorem.)28 b Fl(\(Plotkin\).)17 b(A)c(partial)f(con)o(tin)o(uous)h(functional)f(is)g(computable)g(if)g (and)h(only)f(if)300 1806 y(it)i(is)f(recursiv)o(e)j(in)d Ft(p)q(cond)h Fl(and)g Fp(9)p Fl(.)300 1879 y Fk(Pr)o(oof)h Ft(The)f(examples)f(ab)q(o)o(v)o(e)h(together)g(with)g(the)g(facts)h (pro)o(v)o(ed)f(in)f(5.2)g(that)362 1958 y Fp(\017)21 b Ft(\(pure\))15 b Fs(\025)p Ft(-terms)e(are)i(computable)d(and)362 2040 y Fp(\017)21 b Ft(computable)12 b(functionals)i(are)g(closed)g (under)h(application)300 2119 y(sho)o(w)9 b(that)h(ev)o(ery)g (functional)e(recursiv)o(e)j(in)e(p)q(cond)h(and)f Fp(9)g Ft(is)g(computable.)16 b(F)m(or)9 b(the)g(con)o(v)o(erse,)300 2169 y(w)o(e)k(restrict)h(atten)o(tion)e(to)g(the)h(case)h(of)d(a)i (comp)q(osed)f(t)o(yp)q(e)h Fs(\032)1266 2175 y Fq(1)1296 2169 y Fp(!)e Fs(:)c(:)g(:)j Fp(!)h Fs(\032)1483 2175 y Fr(p)1514 2169 y Fp(!)g Fs(\023)p Ft(;)i(the)g(details)300 2219 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1928 y Ft(suc)o(h)h(that)g Fs(X)1549 1934 y Fq(0)1580 1928 y Ft(=)f Fp(;)g Ft(and)h(the)300 1978 y(follo)o(wing)e(relations)j(are)g(primitiv)o(e)e(recursiv)o(e:) 601 2074 y Fs(X)635 2080 y Fr(n)670 2074 y Fp(\022)g Fs(X)748 2080 y Fr(m)779 2074 y Fs(;)601 2136 y(X)635 2142 y Fr(n)667 2136 y Fp([)d Fs(X)738 2142 y Fr(m)782 2136 y Fp(2)i Fk(Con)905 2142 y Fr(\032)925 2136 y Fs(;)601 2198 y(X)638 2181 y Fr(\032)p Fh(!)p Fr(\033)635 2208 y(n)712 2198 y Fs(X)749 2181 y Fr(\032)746 2208 y(m)789 2198 y Ft(=)h Fs(X)870 2181 y Fr(\033)867 2208 y(k)893 2198 y Fs(;)601 2260 y(X)635 2266 y Fr(n)667 2260 y Fp([)d Fs(X)738 2266 y Fr(m)782 2260 y Ft(=)j Fs(X)860 2266 y Fr(k)963 2260 y Ft(\(with)i Fs(k)f Ft(=)e(0)j(if)f Fs(X)1259 2266 y Fr(n)1291 2260 y Fp([)c Fs(X)1362 2266 y Fr(m)1406 2260 y Fp(62)i Fk(Con)1529 2266 y Fr(\032)1549 2260 y Ft(\))p Fs(:)300 2356 y Ft(W)m(e)i(also)h(assume)f(an)h(en)o (umeration)f(\()p Fs(b)929 2336 y Fr(\032)929 2368 y(i)948 2356 y Ft(\))964 2362 y Fr(i)p Fh(2)p Fb(N)1035 2356 y Ft(of)g(the)i(set)g(of)e(tok)o(ens)h(of)g(t)o(yp)q(e)g Fs(\032)p Ft(.)362 2408 y(Note)j(that)f(an)o(y)g(primitiv)o(e)e (recursiv)o(e)k(function)e Fs(f)21 b Ft(can)16 b(b)q(e)h(regarded)g(as) g(a)e(con)o(tin)o(uous)300 2458 y(functional)h(of)g(t)o(yp)q(e)i Fs(\023)e Fp(!)h Fs(:)7 b(:)g(:)15 b Fp(!)h Fs(\023)g Fp(!)h Fs(\023)f Ft(if)h(w)o(e)g(iden)o(tify)f(it)h(with)f(its)h (strict)h(extension.)28 b(It)300 2508 y(is)14 b(easy)h(to)g(see)g(that) g(an)o(y)f(primitiv)o(e)e(recursiv)o(e)k(function)e(can)h(b)q(e)g (represen)o(ted)i(in)e(this)f(w)o(a)o(y)300 2558 y(b)o(y)j(a)f(term)g (of)g(the)i(simply)d(t)o(yp)q(ed)i Fs(\025)p Ft(-calculus)g(in)o(v)o (olving)d(0,)j(successor,)j(predecessor,)g(the)p eop %%Page: 33 33 33 32 bop 300 275 a Fo(5.)17 b(Computability)11 b(in)i(higher)f(typ)n (es)913 b Fv(33)300 366 y Ft(least)12 b(\014xed)h(p)q(oin)o(t)f(op)q (erators)h Fs(Y)807 372 y Fr(\023)p Fh(!)p Fr(\023)879 366 y Ft(and)f(the)h(sequen)o(tial)f(conditional)f(cond)1518 372 y 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b(b)q(e)h(de\014ned)h(from)d(an)h(auxiliary)f(functional)h (\011)h(of)f(t)o(yp)q(e)h Fs(\032)1404 768 y Fq(1)1437 762 y Fp(!)e Fs(:)7 b(:)g(:)13 b Fp(!)h Fs(\032)1633 768 y Fr(p)1668 762 y Fp(!)g Fs(\023)g Fp(!)300 812 y Fs(\023)d Fp(!)g Fs(\023)j Ft(giv)o(en)f(b)o(y)620 899 y(\011\()s Fs(~)-24 b(')q(;)7 b(m;)g(d;)g Ft(0\))39 b(:=)j Fs(d)556 961 y Ft(\011\()s Fs(~)-24 b(')q(;)7 b(m;)g(d;)g(l)i Ft(+)g(1\))41 b(:=)h(p)q(cond\()p Fs(p)1119 967 y Fr(l)1132 961 y Fs(;)7 b Ft(\011\()s Fs(~)-24 b(';)7 b(m;)g(d;)g(l)q Ft(\))p Fs(;)g(k)q Ft(\()p Fs(m;)g(l)q Ft(\)\))p Fs(;)300 1049 y Ft(where)15 b Fs(p)441 1055 y Fr(l)467 1049 y Ft(denotes)h(incons)731 1055 y Fr(\032)748 1059 y Fe(1)766 1049 y Ft(\()p Fs(')809 1055 y Fq(1)828 1049 y Fs(;)7 b(j)r Ft(\()p Fs(m;)g Ft(1)p Fs(;)g(l)q Ft(\)\))i Fp(_)g Fs(:)e(:)g(:)h Fp(_)h Ft(incons)1276 1055 y Fr(\032)1293 1059 y Ff(p)1313 1049 y Ft(\()p Fs(')1356 1055 y Fr(p)1375 1049 y Fs(;)e(j)r Ft(\()p Fs(m;)g(p;)g(l)q Ft(\)\).)18 b(Hence)494 1179 y Fs(p)515 1185 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b(=)p 1233 2060 V 12 w Fs(X)1270 2085 y Fr(l)p Fq(+1)1267 2108 y Fr(m)1325 2103 y Ft(\()s Fs(~)-24 b(')p Ft(\))p Fs(:)300 2190 y Ft(In)15 b(case)g Fs(p)462 2196 y Fr(l)487 2190 y Ft(=)e Fp(?)h Ft(\014rst)h(assume)g(that)f(\011\()s Fs(~)-24 b(')q(;)7 b(m;)g Fp(?)p Fs(;)g(l)q Ft(\))k(=)i Fp(?)p Ft(.)19 b(Then)c(b)o(y)g(induction)f(h)o(yp)q(othesis)p 300 2214 66 2 v 300 2251 a Fs(X)337 2239 y Fr(l)334 2261 y(m)366 2251 y Ft(\()s Fs(~)-24 b(')p Ft(\))16 b(=)h Fp(?)p Ft(,)f(hence)i(also)p 755 2208 93 2 v 16 w Fs(X)792 2233 y Fr(l)p Fq(+1)789 2256 y Fr(m)847 2251 y Ft(\()s Fs(~)-24 b(')p Ft(\))16 b(=)h Fp(?)e Ft(\(since)j(w)o(e)f(don't)f(ha)o (v)o(e)g Fs(')1440 2257 y Fr(i)1470 2251 y Fp(\023)g Fs(X)1552 2258 y Fr(j)r Fq(\()p Fr(m;i;l)p Fq(\))1684 2251 y Ft(for)h(all)300 2301 y Fs(i)12 b Ft(=)g(1)7 b Fs(:)g(:)g(:)e(p)p Ft(\),)13 b(and)h(\011\()s Fs(~)-24 b(';)7 b(m;)g Fp(?)p Fs(;)g(l)h Ft(+)i(1\))h(=)h Fp(?)h Ft(b)o(y)h(de\014nition)f(of)g(p)q(cond)q(.)18 b(If)13 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Ft(])g(]\()p Fs(G;)7 b(x)p Ft(\))j(:=)h Fs(G)837 1941 y Fi(\022)867 2000 y Fs(\025y)q(;)c(~)-21 b(y)s(:)973 1941 y Fi(\032)1010 1979 y Ft([)-7 b([Y)1058 1985 y Fr(\032;)p Fh(\036)1112 1979 y Ft(])g(]\()p Fs(G;)7 b(y)q(;)g(~)-21 b(y)q Ft(\))42 b(if)13 b Fs(y)g Fp(\036)f Fs(x)1010 2028 y Fp(;)285 b Ft(otherwise)1499 2000 y Fs(;)7 b(x)1542 1941 y Fi(\023)1578 2000 y Fs(:)300 2110 y Ft(This)19 b(equalit)o(y)m(,)f(when)h(view)o(ed) g(as)f(a)h(con)o(v)o(ersion)f(rule)h(from)e(left)i(to)f(righ)o(t,)h (clearly)g(giv)o(es)300 2159 y(rise)e(to)g(an)f(in\014nite)g(reduction) i(sequence:)25 b(just)17 b(expand)g(the)g(subterm)g(corresp)q(onding)g (to)300 2209 y([)-7 b([Y)348 2215 y Fr(\032;)p Fh(\036)402 2209 y Ft(])g(]\()p Fs(G;)7 b(y)q Ft(\))19 b(b)o(y)h(the)g(same)e(rule) i(again,)g(and)f(carry)h(on)g(doing)e(that.)35 b(Hence)21 b(in)f(order)300 2259 y(to)d(ac)o(hiev)o(e)g(termination)f(w)o(e)h(ha)o (v)o(e)g(to)g(mak)o(e)f(use)i(of)f(the)h(fact)f(that)g([)-7 b([Y)1487 2265 y Fr(\032;)p 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1722 a Fs(c)427 1728 y Fh(\036)455 1722 y Fs(M)16 b Fp(!)11 b Fs(c)582 1728 y Fh(\036)610 1722 y Fs(M)655 1705 y Fh(0)672 1696 y Ft(;)820 1680 y Fs(N)17 b Fp(!)11 b Fs(N)961 1663 y Fh(0)p 730 1687 334 2 v 730 1722 a Fs(c)748 1728 y Fh(\036)776 1722 y Fs(M)5 b(N)16 b Fp(!)11 b Fs(c)941 1728 y Fh(\036)969 1722 y Fs(M)5 b(N)1052 1705 y Fh(0)1068 1696 y Ft(;)41 b(similarly)11 b(for)i Fs(c)1372 1702 y Fq(=)1414 1696 y Ft(and)h(S.)362 1812 y(It)g(migh)o(t)e(seem)i(that)g(the)g(simpler)f(rule)305 1911 y(\(iii\)')19 b(Y)435 1917 y Fr(\032;)p Fh(\036)490 1911 y Fs(N)5 b(K)575 1901 y(~)566 1911 y(N)17 b Fp(!)11 b Fs(N)5 b Ft(\()p Fs(\025y)q(;)i(~)-21 b(y)r(:)p Ft(cond\()p Fs(c)940 1917 y Fh(\036)968 1911 y Fs(y)q(K)s Ft(\)\(Y)1090 1917 y Fr(\032;)p Fh(\036)1147 1911 y Fs(N)5 b(y)q(~)-21 b(y)r Ft(\))p Fp(?)p Ft(\))p Fs(K)1340 1901 y(~)1331 1911 y(N)300 2010 y Ft(for)16 b(the)h(Y)471 2016 y Fr(\032;)p Fh(\036)526 2010 y Ft(-op)q(erator)g(w)o(ould)f(su\016ce.)26 b(Ho)o(w)o(ev)o(er,)17 b(it)f(do)q(es)h(not)f(in)g(case)i Fs(K)h Ft(ev)n(aluates)d(to)300 2060 y Fp(?)p Ft(.)h(F)m(or)c(then)h (the)g(outcome)e(of)h(the)h(simpli\014ed)d(rule)j(\(iii'\))d(dep)q (ends)k(on)e(the)h(form)e(of)g Fs(N)19 b Ft(and)300 2110 y(migh)o(t)12 b(b)q(e)i(de\014ned,)h(whic)o(h)f(w)o(ould)f(b)q(e)h (inconsisten)o(t)h(with)e(our)h(denotational)f(seman)o(tics.)362 2165 y(Note)i(that)f(w)o(e)g(do)f Fn(not)18 b Ft(ha)o(v)o(e)362 2264 y Fp(\017)454 2248 y Fs(N)e Fp(!)11 b Fs(N)594 2231 y Fh(0)p 409 2255 242 2 v 409 2290 a Fs(M)5 b(N)16 b Fp(!)11 b Fs(M)5 b(N)639 2273 y Fh(0)362 2398 y Fp(\017)457 2382 y Fs(M)16 b Fp(!)11 b Fs(M)611 2365 y Fh(0)p 409 2389 262 2 v 409 2424 a Fs(\025xM)16 b Fp(!)11 b Fs(\025xM)659 2406 y Fh(0)675 2398 y Ft(,)300 2508 y(for)j(then)i(follo)o(wing)c(an)i (application)g(of)g(rule)h(\(iii\))e(for)i(the)g(b)q(ounded)g(\014xed)h (p)q(oin)o(t)e(op)q(erator)300 2558 y(w)o(e)g(could)g(easily)f (construct)j(an)d(in\014nite)h(reduction)h(sequence.)p eop %%Page: 38 38 38 37 bop 300 275 a Ft(38)774 b Fo(Se)n(ction)11 b(6.)17 b(Bounde)n(d)11 b(\014xe)n(d)h(p)n(oint)g(op)n(er)n(ators)300 366 y Fm(6.4.)19 b Fl(Normalization)11 b(via)h(computabilit)o(y)f (predicates.)28 b Ft(W)m(e)14 b(no)o(w)f(pro)o(v)o(e)h(that)f(ev)o(ery) i(closed)300 416 y(term)g Fs(M)20 b Ft(of)15 b(ground)h(t)o(yp)q(e)g (can)g(b)q(e)g(computed)f(b)o(y)h(the)g(rules)g(of)f(6.3,)g(i.e.)f (that)i(an)o(y)f(reduc-)300 466 y(tion)g(sequence)i(starting)e(with)g Fs(M)k Ft(terminates.)j(It)15 b(clearly)g(follo)o(ws)e(from)h(the)h (form)f(of)g(our)300 516 y(con)o(v)o(ersion)h(rules)g(that)g(the)g (\014nal)f(term)g(in)g(an)o(y)g(reduction)h(sequence)i(m)o(ust)c(b)q(e) j(a)e(constan)o(t)300 565 y(or)k(a)f(n)o(umeral.)28 b(Note)18 b(that)g(it)f(is)h(only)f(in)g(the)i(range)f(of)f(the)h(constan)o(ts)h (other)f(than)g(the)300 615 y(sequen)o(tial)c(conditional)e(cond)j (that)f(w)o(e)g(ha)o(v)o(e)f(a)h(c)o(hoice)g(of)g(where)h(to)e(reduce.) 362 666 y(F)m(or)i(the)h(pro)q(of)g(w)o(e)f(use)i(W.W.)c(T)m(ait's)i (metho)q(d)f(of)h(computabilit)o(y)e(predicates.)24 b(So)16 b(for)300 716 y(ev)o(ery)f(t)o(yp)q(e)g Fs(\032)g Ft(w)o(e)g(de\014ne)g (what)g(it)f(means)f(for)h(a)h(term)e(of)h(t)o(yp)q(e)h Fs(\032)g Ft(to)f(b)q(e)h(computable.)k(The)300 766 y(de\014nition)13 b(is)h(b)o(y)g(induction)f(on)h(t)o(yp)q(es.)351 853 y(1.)20 b(A)f(closed)g(term)f Fs(M)24 b Ft(of)18 b(ground)h(t)o(yp)q(e) g(is)g(computable)e(if)h(an)o(y)h(reduction)g(sequence)404 903 y(starting)14 b(with)f Fs(M)19 b Ft(terminates.)351 992 y(2.)h(A)15 b(closed)g(term)f Fs(M)20 b Ft(of)14 b(t)o(yp)q(e)h Fs(\032)e Fp(!)g Fs(\033)j Ft(is)e(computable)g(if)g (for)g(ev)o(ery)i(closed)f(term)f Fs(N)20 b Ft(of)404 1041 y(t)o(yp)q(e)14 b Fs(\032)p Ft(,)g(if)f Fs(N)19 b Ft(is)14 b(computable,)e(then)i(so)g(is)g Fs(M)5 b(N)g Ft(.)351 1130 y(3.)20 b(A)c(term)g Fs(M)21 b Ft(con)o(taining)16 b(at)g(least)h(one)g(free)g(v)n(ariable)e(is)i(computable)e(if)h(ev)o (ery)h(t)o(yp)q(e)404 1180 y(resp)q(ecting)h(substitution)g(of)e (closed)i(computable)e(terms)g(for)h(the)h(free)f(v)n(ariables)g(in)404 1229 y Fs(M)h Ft(yields)c(a)g(computable)e(term.)300 1345 y Fk(Theorem.)28 b Fl(Ev)o(ery)14 b(term)g Fs(M)k Fl(is)c(computable.)300 1424 y Fk(Pr)o(oof)p Ft(.)19 b(By)14 b(induction)f(on)h Fs(M)5 b Ft(.)362 1475 y Fn(Case)17 b Fs(x)p Ft(.)h(T)m(rivial.)362 1526 y Fn(Case)f Fs(M)5 b(N)g Ft(.)19 b(Let)656 1515 y Fs(~)654 1526 y(L)c Ft(b)q(e)f(a)g (sequence)i(of)e(closed)g(computable)f(terms.)19 b(W)m(e)14 b(ha)o(v)o(e)f(to)h(sho)o(w)300 1576 y(that)j Fs(M)5 b Ft([)452 1565 y Fs(~)450 1576 y(L)p Ft(])p Fs(N)g Ft([)542 1565 y Fs(~)540 1576 y(L)o Ft(])17 b(is)g(computable.)27 b(But)18 b(this)f(holds,)g(since)h(b)o(y)f(induction)g(h)o(yp)q (othesis)h(w)o(e)300 1630 y(kno)o(w)13 b(that)h Fs(M)5 b Ft([)557 1620 y Fs(~)555 1630 y(L)p Ft(])13 b(as)h(w)o(ell)f(as)h Fs(N)5 b Ft([)846 1620 y Fs(~)844 1630 y(L)p Ft(])13 b(are)i(computable.)362 1686 y Fn(Case)i Fs(\025xM)5 b Ft(.)17 b(Let)664 1676 y Fs(~)662 1686 y(L)d Ft(b)q(e)g(a)f(sequence) i(of)e(closed)h(computable)e(terms.)18 b(W)m(e)13 b(ha)o(v)o(e)g(to)g (sho)o(w)300 1736 y(that)j Fs(\025xM)5 b Ft([)499 1726 y Fs(~)497 1736 y(L)p Ft(])16 b(is)g(computable.)24 b(So)16 b(let)h Fs(N)r(;)1033 1726 y(~)1021 1736 y(M)k Ft(b)q(e)c(closed)g (computable)e(terms.)25 b(W)m(e)16 b(m)o(ust)300 1791 y(sho)o(w)d(that)f(\()p Fs(\025xM)5 b Ft([)613 1780 y Fs(~)611 1791 y(L)p Ft(]\))p Fs(N)717 1780 y(~)705 1791 y(M)17 b Ft(is)12 b(computable,)g(i.e.)f(that)i(ev)o(ery)g(reduction)h (sequence)g(starting)300 1841 y(with)19 b(that)h(term)f(terminates.)36 b(But)20 b(from)e(the)i(rules)h(of)e(our)h(op)q(erational)e(seman)o (tics)i(it)300 1891 y(follo)o(ws)11 b(that)i(the)g(second)h(mem)o(b)q (er)d(of)i(an)o(y)f(suc)o(h)h(sequence)i(m)o(ust)d(b)q(e)h Fs(M)5 b Ft([)p Fs(N)r(;)1541 1880 y(~)1540 1891 y(L)p Ft(])1592 1880 y Fs(~)1580 1891 y(M)t Ft(,)13 b(whic)o(h)f(is)300 1940 y(computable)h(b)o(y)g(induction)h(h)o(yp)q(othesis.)362 1992 y Fn(Case)20 b Ft(Y)500 1998 y Fr(\032;)p Fh(\036)555 1992 y Ft(.)27 b(Let)17 b Fs(N)r(;)7 b(K)q(;)788 1981 y(~)780 1992 y(N)21 b Ft(b)q(e)d(closed)f(computable)f(terms.)26 b(W)m(e)16 b(ha)o(v)o(e)h(to)g(sho)o(w)f(that)300 2048 y(Y)331 2054 y Fr(\032;)p Fh(\036)386 2048 y Fs(N)5 b(K)471 2037 y(~)462 2048 y(N)24 b Ft(is)19 b(computable,)g(i.e.)g(that)g(ev)o (ery)h(reduction)g(sequence)h(starting)e(with)g(that)300 2098 y(term)d(terminates.)25 b(Since)17 b Fs(K)i Ft(is)e(computable)e (and)h(of)g(t)o(yp)q(e)h Fs(\023)p Ft(,)f(w)o(e)g(kno)o(w)g(that)h Fs(K)i Ft(reduces)300 2147 y(to)13 b(a)h(constan)o(t)g(or)f(a)h(n)o (umeral)e Fs(c)p Ft(;)h(w)o(e)h(use)g(an)f(auxiliary)f (\(trans\014nite\))j Fp(\036)p Ft(-induction)e(on)h(\(the)300 2202 y(v)n(alue)d(of)s(\))g Fs(c)p Ft(.)18 b(If)11 b Fs(c)g Ft(=)h Fp(?)p Ft(,)f(then)h(ev)o(ery)h(reduction)f(sequence)i (starting)e(with)f(Y)1513 2208 y Fr(\032;)p Fh(\036)1568 2202 y Fs(N)5 b(K)1653 2192 y(~)1644 2202 y(N)17 b Ft(m)o(ust,)300 2252 y(after)d(its)h(\014rst)g(t)o(w)o(o)e(steps,)i(w)o(ork)f(on)g (reducing)h Fs(c)1084 2258 y Fq(=)1111 2252 y Fs(K)s(K)j Ft(un)o(til)13 b(this)h(term)g(has)g(b)q(een)i(reduced)300 2302 y(to)g(a)f(constan)o(t)h(or)g(a)g(n)o(umeral;)e(this)i(follo)o(ws) e(from)g(the)j(form)d(of)h(the)h(con)o(v)o(ersion)g(rules)h(for)300 2351 y(the)e(sequen)o(tial)g(conditional.)j(No)o(w)c(since)h(b)o(y)g (assumption)e Fs(K)k Ft(reduces)g(to)d Fp(?)g Ft(w)o(e)h(m)o(ust)e(get) 300 2401 y Fp(?)p Ft(,)h(and)h(hence)i(Y)589 2407 y Fr(\032;)p Fh(\036)644 2401 y Fs(N)5 b(K)729 2391 y(~)720 2401 y(N)20 b Ft(reduces)d(to)e Fp(?)p Ft(.)21 b(So)15 b(assume)g Fs(c)f Ft(=)g Fm(n)p Ft(.)21 b(Then)16 b(as)f(b)q(efore)h(w)o(e)f(can) 300 2457 y(see)g(that)f(ev)o(ery)h(reduction)f(sequence)i(starting)e (with)g(Y)1204 2463 y Fr(\032;)p Fh(\036)1259 2457 y Fs(N)5 b(K)1344 2447 y(~)1335 2457 y(N)19 b Ft(m)o(ust)13 b(come)g(across)691 2558 y Fs(N)5 b Ft(\()p Fs(\025y)q(;)i(~)-21 b(y)s(:)p Ft(cond\()p Fs(c)963 2564 y Fh(\036)991 2558 y Fs(y)q(K)s Ft(\)\(Y)1113 2564 y Fr(\032;)p Fh(\036)1169 2558 y Fs(N)5 b(y)q(~)-21 b(y)s Ft(\))p Fp(?)p Ft(\))p Fs(K)1362 2547 y(~)1354 2558 y(N)5 b(:)p eop %%Page: 39 39 39 38 bop 300 275 a Fo(6.)17 b(Bounde)n(d)11 b(\014xe)n(d)i(p)n(oint)f (op)n(er)n(ators)909 b Fv(39)300 366 y Ft(Since)14 b Fs(N)19 b Ft(is)14 b(computable)f(b)o(y)g(assumption,)f(it)i(su\016ces) h(to)f(sho)o(w)g(that)f(the)i(closed)f(term)770 464 y Fs(\025y)q(;)7 b(~)-21 b(y)s(:)p Ft(cond)o(\()p Fs(c)987 470 y Fh(\036)1016 464 y Fs(y)q(K)s Ft(\)\(Y)1138 470 y Fr(\032;)p Fh(\036)1194 464 y Fs(N)5 b(y)q(~)-21 b(y)s Ft(\))p Fp(?)300 568 y Ft(is)14 b(computable.)j(So)c(let)j( )-13 b(L)o Fs(;)745 558 y(~)744 568 y(L)14 b Ft(b)q(e)g(a)g(closed)g (computable)f(term.)k(W)m(e)d(m)o(ust)f(sho)o(w)h(that)810 666 y(cond\()p Fs(c)929 672 y Fh(\036)957 666 y Fs(LK)s Ft(\)\(Y)1086 672 y Fr(\032;)p Fh(\036)1142 666 y Fs(N)5 b(L)1210 655 y(~)1208 666 y(L)p Ft(\))p Fp(?)300 764 y Ft(is)15 b(computable.)21 b(Since)16 b Fs(L)g Ft(is)f(computable)f(b) o(y)h(assumption,)f(w)o(e)h(kno)o(w)g(that)h Fs(L)f Ft(reduces)i(to)300 814 y(a)h(constan)o(t)g(or)g(a)g(n)o(umeral)e Fs(d)p Ft(.)30 b(If)18 b Fs(d)g Ft(=)h Fp(?)p Ft(,)e(then)i Fs(c)1148 820 y Fh(\036)1176 814 y Fs(LK)i Ft(m)o(ust)c(reduce)j(to)e Fp(?)f Ft(and)h(hence)300 863 y(the)e(whole)e(term)h(m)o(ust)f(reduce)j (to)e Fp(?)p Ft(.)20 b(So)15 b(assume)g Fs(d)e Ft(=)h Fm(m)p Ft(.)21 b(Then)15 b(the)h(whole)f(term)f(m)o(ust)300 913 y(reduce)i(to)810 973 y(cond\()p Fs(c)929 979 y Fh(\036)957 973 y Fm(mn)p Ft(\)\(Y)1087 979 y Fr(\032;)p Fh(\036)1142 973 y Fs(N)5 b(L)1210 962 y(~)1208 973 y(L)p Ft(\))p Fp(?)300 1057 y Ft(If)20 b Fs(m)i Fp(\036)h Fs(n)d Ft(holds,)h(then)g Fs(c)753 1063 y Fh(\036)781 1057 y Fm(mn)g Fp(!)h Fm(k)e Ft(for)g(some)f Fs(k)k(>)g Ft(0)d(and)g(hence)h(the)g(whole)f(term)300 1107 y(m)o(ust)f(reduce)j(to)e(Y)635 1113 y Fr(\032;)p Fh(\036)690 1107 y Fs(N)5 b(L)758 1097 y(~)756 1107 y(L)p Ft(,)21 b(whic)o(h)f(b)o(y)g(h)o(yp)q(othesis)h(of)e(our)h(auxiliary)f Fp(\036)p Ft(-induction)g(is)300 1157 y(computable.)j(If)15 b Fs(m)g Fp(\036)f Fs(n)i Ft(do)q(es)g(not)f(hold,)g(then)i Fs(c)1117 1163 y Fh(\036)1145 1157 y Fm(mn)c Fp(!)h Ft(0)h(and)h(hence) h(the)f(whole)f(term)300 1207 y(reduces)h(to)e Fp(?)p Ft(.)362 1260 y Fn(Case)k Ft(cond,)c(p)q(cond.)19 b(Let)c Fs(M)r(;)7 b(N)r(;)g(L)14 b Ft(b)q(e)h(closed)g(computable)e(terms.)18 b(W)m(e)c(ha)o(v)o(e)g(to)g(sho)o(w)300 1310 y(that)c(cond)q Fs(M)5 b(N)g(L)10 b Ft(and)g(p)q(cond)q Fs(M)5 b(N)g(L)10 b Ft(are)h(computable,)e(i.e.)h(that)g(ev)o(ery)h(reduction)g(sequence) 300 1359 y(starting)i(with)f(either)i(of)e(them)g(terminates.)17 b(But)c(this)g(clearly)g(holds,)f(since)h(otherwise)h(one)300 1409 y(of)f(the)i(terms)e Fs(M)r(;)7 b(N)r(;)g(L)14 b Ft(w)o(ould)g(ha)o(v)o(e)f(an)h(in\014nite)g(reduction)g(sequence.)362 1462 y(F)m(or)g(the)g(other)h(constan)o(ts)g(of)e Fk(Pcf)925 1468 y Fh(\036)967 1462 y Ft(the)i(argumen)o(t)d(is)i(similar.)360 b Fj(2)362 1515 y Ft(Note)18 b(that)f(this)h(pro)q(of)f(in)o(v)o(olv)o (es)f(computabilit)o(y)f(predicates)j(of)f(arbitrary)g(t)o(yp)q(es)i (and)300 1565 y(hence)e(is)f(not)g(formalizable)d(in)i(P)o(eano)h (arithmetic)f(in)g(all)g(\014nite)h(t)o(yp)q(es.)25 b(It)16 b(is)g(w)o(ell-kno)o(wn)300 1615 y(that)i(this)f(is)h(necessarily)h(so) e(for)h(ev)o(ery)g(pro)q(of)f(of)h(a)f(result)h(implying)d(the)j (existence)i(of)d(a)300 1665 y(normal)12 b(form)g(for)h(ev)o(ery)i (term)e Fs(M)5 b Ft(.)300 1790 y Fm(6.5.)22 b Fl(Reduction)15 b(of)f Ft(Y)675 1796 y Fr(\032;<)745 1790 y Fl(to)h(higher)g(order)g (primitiv)o(e)e(recursion.)27 b Ft(W)m(e)14 b(\014rst)i(consider)g(the) 300 1840 y(case)e(where)h(the)f(w)o(ell-ordering)f Fp(\036)h Ft(is)f(the)h(standard)g(ordering)f Fs(<)h Ft(of)f(the)h(natural)f(n)o (um)o(b)q(ers.)300 1889 y(Then)f(the)f(b)q(ounded)h(\014xed)g(p)q(oin)o (t)e(op)q(erator)i(is)f(de\014nable)g(b)o(y)g(means)f(of)h(primitiv)o (e)e(recursion,)300 1939 y(as)14 b(follo)o(ws.)362 1992 y(De\014ne)j(a)f(functional)f Fs(H)762 1998 y Fr(\032)797 1992 y Ft(of)h(t)o(yp)q(e)g([\()p Fs(\023)f Fp(!)g Fs(\032)p Ft(\))h Fp(!)f Fs(\023)g Fp(!)g Fs(\032)p Ft(])g Fp(!)g Fs(\023)g Fp(!)g Fs(\023)g Fp(!)g Fs(\032)i Ft(b)o(y)f(primitiv)o(e)300 2042 y(recursion,)f(as)f(in)f(G\177)-21 b(odel's)13 b(system)h Fs(T)20 b Ft([20)o(]:)720 2140 y Fs(H)755 2146 y Fr(\032)773 2140 y Ft(\()p Fs(G;)7 b Fp(?)p Ft(\))75 b(:=)11 b Fs(\025y)q(;)c(~)-21 b(y)9 b Fp(?)p Fs(;)720 2202 y(H)755 2208 y Fr(\032)773 2202 y Ft(\()p Fs(G;)e Ft(0\))87 b(:=)11 b Fs(\025y)q(;)c(~)-21 b(y)9 b Fp(?)p Fs(;)720 2264 y(H)755 2270 y Fr(\032)773 2264 y Ft(\()p Fs(G;)e(n)i Ft(+)g(1\))j(:=)f Fs(K)1055 2270 y Fr(\032)1075 2264 y Ft(\()p Fs(G;)c(n;)g(H)1222 2270 y Fr(\032)1239 2264 y Ft(\()p Fs(G;)g(n)p Ft(\)\))p Fs(;)300 2362 y Ft(where)15 b Fs(K)455 2368 y Fr(\032)488 2362 y Ft(of)f(t)o(yp)q(e)g([\()p Fs(\023)d Fp(!)g Fs(\032)p Ft(\))h Fp(!)f Fs(\023)g Fp(!)g 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y(st)o(yle)20 b(deriv)n(ations)f(in)g Fs(!)q Ft(-arithmetic.)35 b(In)20 b(that)f(pap)q(er,)j(the)e(basic)g (idea)f(is)h(to)g(in)o(tro)q(duce)300 2458 y(a)d(primitiv)o(e)f (recursiv)o(e)j(notation)d(system)i(for)f(w)o(ell-founded)g Fs(!)q Ft(-deriv)n(ations)g(in)g(the)h(same)300 2508 y(w)o(a)o(y)g(as)h(one)g(usually)f(in)o(tro)q(duces)j(an)d(ordinal)g (notation)g(system)h(as)g(a)g(system)f(of)h(terms)300 2558 y(generated)13 b(from)d(constan)o(ts)i(for)f(particular)h (ordinals)f(b)o(y)g(function)g(sym)o(b)q(ols)f(corresp)q(onding)p eop %%Page: 44 44 44 43 bop 300 275 a Ft(44)541 b Fo(Se)n(ction)11 b(7.)17 b(Elimination)11 b(of)i(detours)f(thr)n(ough)g(higher)g(typ)n(es)300 366 y Ft(to)h(certain)h(ordinal)e(functions.)18 b(A)13 b(less-than)h(relation)e(b)q(et)o(w)o(een)j(ordinal)d(notations)h(is)g (then)300 416 y(deriv)o(ed)i(from)d(the)i(prop)q(erties)i(of)d(the)i (denoted)g(ordinals)e(and)h(ordinal)f(functions.)19 b(Instead)300 466 y(of)12 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a Fo(R)n(efer)n(enc)n(es)1278 b Fv(47)300 366 y Ft(T)m(r)343 348 y Fr(\034)358 352 y Ff(i)343 376 y Fr(\034)373 366 y Ft(\(T)m(r)432 348 y Fr(\034)432 376 y(\034)447 380 y Ff(i)463 366 y Ft(\()p Fs(u)p Ft(\)\))11 b(=)h Fs(u)p Ft(\).)18 b(With)12 b(these)j(T)m(r)914 348 y Fr(\034)914 376 y(\034)929 380 y Ff(i)945 366 y Ft(\()p Fs(u)985 372 y Fr(i)999 366 y Ft(\))e(build)f(a)g(function)h Fs(w)h Ft(of)e(t)o(yp)q(e)i Fs(\023)d Fp(!)g Fs(\034)18 b Ft(suc)o(h)13 b(that)300 416 y Fs(w)q Ft(\()p Fs(a)369 422 y Fr(i)383 416 y Ft(\))h(=)g(T)m(r)502 398 y Fr(\034)502 426 y(\034)517 430 y Ff(i)533 416 y Ft(\()p Fs(u)573 422 y Fr(i)586 416 y Ft(\).)23 b(F)m(rom)14 b Fs(w)i Ft(one)g(can)g(then)g(read)g(o\013)f(the)h(functionals)f Fs(u)1510 422 y Fr(i)1539 416 y Ft(assigned)h(to)f Fs(x)1781 422 y Fr(i)300 466 y Ft(b)o(y)h Fs(u)384 472 y Fr(i)413 466 y Ft(=)g(T)m(r)503 448 y Fr(\034)518 452 y Ff(i)503 476 y Fr(\034)534 466 y Ft(\()p Fs(w)q Ft(\()p Fs(a)619 472 y Fr(i)633 466 y Ft(\)\).)25 b(With)15 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