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    61 % File 8r.enc  TeX Base 1 Encoding  Revision 2.0  2002-10-30

    62 %

    63 % @@psencodingfile@{

    64 %   author    = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry,

    65 %                W. Schmidt, P. Lehman",

    66 %   version   = "2.0",

    67 %   date      = "30 October 2002",

    68 %   filename  = "8r.enc",

    69 %   email     = "tex-fonts@@tug.org",

    70 %   docstring = "This is the encoding vector for Type1 and TrueType

    71 %                fonts to be used with TeX.  This file is part of the

    72 %                PSNFSS bundle, version 9"

    73 % @}

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    75 % The idea is to have all the characters normally included in Type 1 fonts

    76 % available for typesetting. This is effectively the characters in Adobe

    77 % Standard encoding, ISO Latin 1, Windows ANSI including the euro symbol,

    78 % MacRoman, and some extra characters from Lucida.

    79 % 

    80 % Character code assignments were made as follows:

    81 % 

    82 % (1) the Windows ANSI characters are almost all in their Windows ANSI

    83 % positions, because some Windows users cannot easily reencode the

    84 % fonts, and it makes no difference on other systems. The only Windows

    85 % ANSI characters not available are those that make no sense for

    86 % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen

    87 % (173). quotesingle and grave are moved just because it's such an

    88 % irritation not having them in TeX positions.

    89 % 

    90 % (2) Remaining characters are assigned arbitrarily to the lower part

    91 % of the range, avoiding 0, 10 and 13 in case we meet dumb software.

    92 % 

    93 % (3) Y&Y Lucida Bright includes some extra text characters; in the

    94 % hopes that other PostScript fonts, perhaps created for public

    95 % consumption, will include them, they are included starting at 0x12.

    96 % These are /dotlessj /ff /ffi /ffl.

    97 % 

    98 % (4) hyphen appears twice for compatibility with both ASCII and Windows.

    99 %

   100 % (5) /Euro was assigned to 128, as in Windows ANSI

   101 %

   102 % (6) Missing characters from MacRoman encoding incorporated as follows:

   103 %

   104 %     PostScript      MacRoman        TeXBase1

   105 %     --------------  --------------  --------------

   106 %     /notequal       173             0x16

   107 %     /infinity       176             0x17

   108 %     /lessequal      178             0x18

   109 %     /greaterequal   179             0x19

   110 %     /partialdiff    182             0x1A

   111 %     /summation      183             0x1B

   112 %     /product        184             0x1C

   113 %     /pi             185             0x1D

   114 %     /integral       186             0x81

   115 %     /Omega          189             0x8D

   116 %     /radical        195             0x8E

   117 %     /approxequal    197             0x8F

   118 %     /Delta          198             0x9D

   119 %     /lozenge        215             0x9E

   120 %

   121 /TeXBase1Encoding [

   122 % 0x00

   123  /.notdef /dotaccent /fi /fl

   124  /fraction /hungarumlaut /Lslash /lslash

   125  /ogonek /ring /.notdef /breve

   126  /minus /.notdef /Zcaron /zcaron

   127 % 0x10

   128  /caron /dotlessi /dotlessj /ff

   129  /ffi /ffl /notequal /infinity

   130  /lessequal /greaterequal /partialdiff /summation

   131  /product /pi /grave /quotesingle

   132 % 0x20

   133  /space /exclam /quotedbl /numbersign

   134  /dollar /percent /ampersand /quoteright

   135  /parenleft /parenright /asterisk /plus

   136  /comma /hyphen /period /slash

   137 % 0x30

   138  /zero /one /two /three

   139  /four /five /six /seven

   140  /eight /nine /colon /semicolon

   141  /less /equal /greater /question

   142 % 0x40

   143  /at /A /B /C

   144  /D /E /F /G

   145  /H /I /J /K

   146  /L /M /N /O

   147 % 0x50

   148  /P /Q /R /S

   149  /T /U /V /W

   150  /X /Y /Z /bracketleft

   151  /backslash /bracketright /asciicircum /underscore

   152 % 0x60

   153  /quoteleft /a /b /c

   154  /d /e /f /g

   155  /h /i /j /k

   156  /l /m /n /o

   157 % 0x70

   158  /p /q /r /s

   159  /t /u /v /w

   160  /x /y /z /braceleft

   161  /bar /braceright /asciitilde /.notdef

   162 % 0x80

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   164  /quotedblbase /ellipsis /dagger /daggerdbl

   165  /circumflex /perthousand /Scaron /guilsinglleft

   166  /OE /Omega /radical /approxequal

   167 % 0x90

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   169  /quotedblright /bullet /endash /emdash

   170  /tilde /trademark /scaron /guilsinglright

   171  /oe /Delta /lozenge /Ydieresis

   172 % 0xA0

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   174  /currency /yen /brokenbar /section

   175  /dieresis /copyright /ordfeminine /guillemotleft

   176  /logicalnot /hyphen /registered /macron

   177 % 0xD0

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   179  /acute /mu /paragraph /periodcentered

   180  /cedilla /onesuperior /ordmasculine /guillemotright

   181  /onequarter /onehalf /threequarters /questiondown

   182 % 0xC0

   183  /Agrave /Aacute /Acircumflex /Atilde

   184  /Adieresis /Aring /AE /Ccedilla

   185  /Egrave /Eacute /Ecircumflex /Edieresis

   186  /Igrave /Iacute /Icircumflex /Idieresis

   187 % 0xD0

   188  /Eth /Ntilde /Ograve /Oacute

   189  /Ocircumflex /Otilde /Odieresis /multiply

   190  /Oslash /Ugrave /Uacute /Ucircumflex

   191  /Udieresis /Yacute /Thorn /germandbls

   192 % 0xE0

   193  /agrave /aacute /acircumflex /atilde

   194  /adieresis /aring /ae /ccedilla

   195  /egrave /eacute /ecircumflex /edieresis

   196  /igrave /iacute /icircumflex /idieresis

   197 % 0xF0

   198  /eth /ntilde /ograve /oacute

   199  /ocircumflex /otilde /odieresis /divide

   200  /oslash /ugrave /uacute /ucircumflex

   201  /udieresis /yacute /thorn /ydieresis

   202 ] def

   203 

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   860 TeXDict begin 1 0 bop 705 448 a FB(Classical)30 b(Logic)f(is)h(better)g

   861 (than)h(Intuitionistic)f(Logic:)681 598 y(A)g(Conjectur)n(e)h(about)f

   862 (Double-Negation)h(T)-9 b(ranslations)1417 886 y FA(Christian)21

   863 b(Urban)e(and)h(Diana)g(Ratiu)1410 1060 y Fz(f)p Fy(urban,ratiu)p

   864 Fz(g)p Fy(@math.lmu.de)1642 1152 y Fx(Uni)n(v)o(ersity)f(of)g(Munich)

   865 759 1462 y Fw(Abstract.)41 b Fx(It)20 b(is)h(well-kno)n(wn)g(that)f(in)

   866 h(terms)f(of)g(consistenc)o(y)i(classical)f(logic)f(and)i(intu-)759

   867 1553 y(itionistic)f(logic)h(ha)o(v)o(e)g(equal)g(strength:)g(e)n(v)o

   868 (ery)h(intuitionistic)e(proof)h(is)g(a)f(classical)h(proof)759

   869 1645 y(and)29 b(e)n(v)o(ery)f(classical)g(proof)h(can)f(be)g(embedded)i

   870 (into)e(intuitionistic)f(logic)h(via)g(double)759 1736

   871 y(ne)o(gation)21 b(translations.)f(It)f(is)h(also)g(well-kno)n(wn)h

   872 (that)f(intuitionistic)g(proofs)h(contain)f(wit-)759

   873 1827 y(nesses)33 b(for)f(e)o(xistential)g(statements,)g(which)h(is)f

   874 (not)g(al)o(w)o(ays)h(the)f(case)h(with)f(classical)759

   875 1919 y(proofs.)f(Ho)n(we)n(v)o(er)m(,)g(here)h(we)e(study)i(classical)e

   876 (and)i(intuitionistic)e(logic)h(as)f(reduction)759 2010

   877 y(systems.)24 b(From)g(this)f(perspecti)n(v)o(e,)i(we)f(conjecture)h

   878 (that)e(classical)h(logic)g(is)g(more)g(po)n(w-)759 2101

   879 y(erful)g(than)h(intuitionistic)f(logic.)g(The)g(conjecture)i(links)e

   880 (double-ne)o(gation)i(translations)759 2193 y(to)21 b(the)g(colour)o

   881 (-protocol)i(introduced)f(by)g(Danos)g(et)e(al.)h(for)g

   882 (cut-elimination)h(in)f(classical)759 2284 y(logic.)27

   883 b(If)g(the)g(conjecture)i(turns)e(out)h(to)f(be)g(true,)g(then)h(we)f

   884 (can)h(conclude)g(that)f Fv(not)h(all)759 2375 y Fx(cut-reductions)20

   885 b(can)g(be)f(simulated)g(by)h(double-ne)o(gation)g(translations.)523

   886 2644 y Fu(1)99 b(Intr)n(oduction)523 2832 y FA(Since)25

   887 b(the)f(w)o(orks)g(on)h(double-ne)o(gation)20 b(translations)k(by)g

   888 (Gentzen,)f(G)7 b(\250)-35 b(odel)24 b(and)g(K)m(olmogoro)o(v)-5

   889 b(,)523 2932 y(one)26 b(kno)n(ws)g(that)g(classical)h(logic)f(and)g

   890 (intuitionistic)g(logic)g(ha)n(v)o(e)g(equal)f(strength)h(in)g(terms)h

   891 (of)523 3031 y(consistenc)o(y:)33 b(intuitionistic)h(sequent-proofs)e

   892 (can)i(be)h(seen)g(as)g(classical)g(proofs)f(where)f(the)523

   893 3131 y(right-hand)27 b(side)i(of)g(the)g(sequents)g(is)h(restricted)e

   894 (to)i(maximal)e(one)g(formula)g(and)g(e)n(v)o(ery)g(clas-)523

   895 3231 y(sical)19 b(sequent-proof)c(can)k(be)f(embedded)e(into)j

   896 (intuitionistic)e(logic)h(via)h(double-ne)o(gation)14

   897 b(trans-)523 3330 y(lations.)27 b(As)h(a)f(result,)g(consistenc)o(y)f

   898 (of)h(one)f(logic)h(implies)g(consistenc)o(y)f(of)h(the)g(other)-5

   899 b(.)26 b(In)h(this)523 3430 y(paper)m(,)i(ho)n(we)n(v)o(er)m(,)f(we)j

   900 (focus)f(on)h(the)f(correspondence)e(of)i(intuitionistic)g(and)g

   901 (classical)i(logic)523 3530 y(with)20 b(respect)g(to)h(term-re)n

   902 (writing,)c(proof-normalisation)f(and)k(cut-elimination.)648

   903 3629 y(According)c(to)i(the)g(Curry-Ho)n(w)o(ard)d(correspondence,)f

   904 (the)k(simply-typed)e(lambda-calculus)523 3729 y(can)28

   905 b(be)f(vie)n(wed)g(as)h(a)h(term-assignment)c(for)i(intuitionistic)g

   906 (proofs)g(formalised)f(in)i(Gentzen')-5 b(s)523 3828

   907 y(natural)24 b(deduction)f(calculus)i(NJ.)h(T)-6 b(erm-re)n(writing)22

   908 b(in)k(the)f(simply-typed)e(lambda-calculus)g(is)523

   909 3928 y(a)e(form)f(of)h(computation)e(that)i(con)m(v)o(erts)e(a)i(term)g

   910 (to)g(its)h(simplest)g(form,)d(analogous)g(to)i(symbolic)523

   911 4028 y(e)n(v)n(aluation.)29 b(On)h(the)h(other)e(hand,)g(normalisation)

   912 g(is)j(a)f(method)e(for)h(eliminating)f(certain)h(re-)523

   913 4127 y(dundancies)23 b(in)j(proofs.)e(Applied)g(iterati)n(v)o(ely)-5

   914 b(,)24 b(it)i(transforms)d(a)j(proof)e(to)h(one)g(in)h(normalform.)523

   915 4227 y(Using)19 b(the)f(Curry-Ho)n(w)o(ard)e(correspondence)f(we)k(see)

   916 g(that)g(the)g(tw)o(o)f(notions)g(coincide)f(and)h(con-)523

   917 4327 y(sequently)h(we)i(can)f(talk)g(of)g(a)g(computational)e

   918 (interpretation)g(of)i(intuitionistic)g(proofs.)648 4426

   919 y(T)-7 b(w)o(o)20 b(properties)e(hold)h(for)g(term-re)n(writing)e(in)j

   920 (the)g(simply-typed)d(lambda-calculus)h(and)h(by)523

   921 4526 y(the)k(Curry-Ho)n(w)o(ard)d(correspondence)f(also)k(for)f

   922 (normalisation)e(in)j(NJ:)h(the)o(y)d(are)i(strongly)e(nor)n(-)523

   923 4625 y(malising)k(and)h(Church-Rosser)-5 b(.)24 b(W)m(ith)i(some)g

   924 (limitations)g([8,)12 b(15,)h(18,)g(23],)24 b(the)i(Curry-Ho)n(w)o(ard)

   925 523 4725 y(correspondence)18 b(applies)i(also)h(to)h(the)f

   926 (sequent-calculus)d(LJ)k(and)e(to)h(the)g(process)g(of)f(cut-elimi-)523

   927 4825 y(nation.)28 b(F)o(or)g(sak)o(e)h(of)g(simplicity)-5

   928 b(,)28 b(we)h(shall)g(ignore)f(these)h(limitations)f(here)h(and)f(re)o

   929 (gard)f(cut-)523 4924 y(elimination)k(in)h(intuitionistic)f(logic)g(as)

   930 i(strongly)d(normalising)g(and)h(as)i(ha)n(ving)e(\(morally)f(at)p

   931 eop end

   932 %%Page: 2 2

   933 TeXDict begin 2 1 bop 523 448 a FA(least\))24 b(the)f(Church-Rosser)e

   934 (property)-5 b(.)21 b(This)i(ignorance)e(can)i(be)g(partially)f

   935 (justi\002ed)i(if)f(one)g(sees)523 548 y(behind)c(e)n(v)o(ery)f

   936 (LJ-proof)g(an)i(NJ-proof)f(where)g(the)h(inessential)g(dif)n(ferences)

   937 f(present)g(in)h(sequent)523 648 y(proofs)31 b(\223disappear\224,)g

   938 (and)h(sees)h(cut-elimination)d(as)k(an)e(approximation)d(of)k

   939 (proof-normali-)523 747 y(sation.)648 850 y(Although)19

   940 b(it)i(has)h(been)e(sho)n(wn)g(that)h(some)g(cut-elimination)e

   941 (procdures)g(for)h(classical)i(logic)523 950 y(are)i(strongly)f

   942 (normalising)g(as)i(well,)f(cut-elimination)e(in)j(classical)g(logic)f

   943 (is,)h(gi)n(v)o(en)e(a)h(sensible)523 1049 y(notion)i(of)h

   944 (cut-reductions,)d Ft(not)k FA(Church-Rosser)n(\227not)d(e)n(v)o(en)h

   945 (morally)-5 b(.)25 b(The)i(lack)g(of)g(Church-)523 1149

   946 y(Rosser)18 b(in)f(classical)h(logic)f(is)h(a)g(main)f(theme)f(running)

   947 f(through)g(the)j(w)o(orks)e([2,)d(3,)g(19,)g(22],)j(which)523

   948 1249 y(analyse)j(what)g(this)h(means)f(from)f(a)i(computational)d

   949 (point)h(of)h(vie)n(w)-5 b(.)19 b(F)o(or)g(e)o(xample)f(in)h([19])f(it)

   950 i(has)523 1348 y(been)31 b(sho)n(wn)g(that)g(a)h(lambda-calculus)d

   951 (with)j(a)g(non-deterministic)c(choice-operator)g(can)k(be)523

   952 1448 y(embedded)19 b(into)i(a)g(fragment)f(of)h(classical)h(logic.)e(A)

   953 i(simple)f(classical)h(proof)d(tak)o(en)i(from)f([6,)13

   954 b(9])523 1547 y(shall)21 b(illustrate)f(the)g(lack)g(of)g

   955 (Church-Rosser:)1158 1725 y Fs(A)p 1238 1713 10 38 v

   956 1248 1696 42 4 v 88 w(A)83 b(A)p 1534 1713 10 38 v 1543

   957 1696 42 4 v 88 w(A)p 1158 1745 509 4 v 1178 1818 a(A)19

   958 b Fr(_)g Fs(A)p 1414 1806 10 38 v 1423 1790 42 4 v 88

   959 w(A;)14 b(A)1707 1762 y Fr(_)1762 1774 y Fq(L)p 1178

   960 1854 467 4 v 1228 1928 a Fs(A)19 b Fr(_)g Fs(A)p 1463

   961 1916 10 38 v 1473 1899 42 4 v 88 w(A)1686 1874 y(contr)1879

   962 1886 y Fq(R)2017 1725 y Fs(A)p 2098 1713 10 38 v 2107

   963 1696 42 4 v 88 w(A)84 b(A)p 2393 1713 10 38 v 2403 1696

   964 42 4 v 88 w(A)p 2017 1745 509 4 v 2056 1818 a(A;)14 b(A)p

   965 2236 1806 10 38 v 2246 1790 42 4 v 89 w(A)p Fr(^)p Fs(A)2567

   966 1762 y Fr(^)2622 1774 y Fq(R)p 2056 1854 430 4 v 2106

   967 1928 a Fs(A)p 2187 1916 10 38 v 2196 1899 42 4 v 88 w(A)p

   968 Fr(^)q Fs(A)2527 1874 y(contr)2720 1886 y Fq(L)p 1228

   969 1948 1209 4 v 1608 2021 a Fs(A)p Fr(_)q Fs(A)p 1807 2009

   970 10 38 v 1816 1992 42 4 v 88 w(A)p Fr(^)q Fs(A)2478 1973

   971 y(cut)3308 2021 y FA(\(1\))523 2183 y(The)32 b(cut)g(in)g(this)h(proof)

   972 e(can)h(be)g(eliminated)f(by)h(reducing)e(it)j(to)f(one)g(of)g(the)g

   973 (follo)n(wing)e(tw)o(o)523 2283 y(normalforms)1169 2460

   974 y Fs(A)p 1250 2448 10 38 v 1259 2432 42 4 v 88 w(A)84

   975 b(A)p 1545 2448 10 38 v 1555 2432 42 4 v 88 w(A)p 1169

   976 2480 509 4 v 1209 2554 a(A;)14 b(A)p 1388 2542 10 38

   977 v 1398 2525 42 4 v 88 w(A)p Fr(^)q Fs(A)1719 2497 y Fr(^)1774

   978 2510 y Fq(R)p 1209 2590 430 4 v 1258 2663 a Fs(A)p 1339

   979 2651 10 38 v 1348 2635 42 4 v 88 w(A)p Fr(^)q Fs(A)1679

   980 2609 y(contr)1872 2621 y Fq(L)2005 2460 y Fs(A)p 2086

   981 2448 10 38 v 2096 2432 42 4 v 89 w(A)83 b(A)p 2382 2448

   982 10 38 v 2391 2432 42 4 v 88 w(A)p 2005 2480 509 4 v 2045

   983 2554 a(A;)14 b(A)p 2225 2542 10 38 v 2234 2525 42 4 v

   984 88 w(A)p Fr(^)q Fs(A)2555 2497 y Fr(^)2610 2510 y Fq(R)p

   985 2045 2590 430 4 v 2094 2663 a Fs(A)p 2175 2651 10 38

   986 v 2185 2635 42 4 v 89 w(A)p Fr(^)p Fs(A)2516 2609 y(contr)2709

   987 2621 y Fq(L)p 1258 2683 1167 4 v 1509 2756 a Fs(A)p Fr(_)q

   988 Fs(A)p 1707 2744 10 38 v 1717 2728 42 4 v 88 w(A)p Fr(^)q

   989 Fs(A;)g(A)p Fr(^)p Fs(A)2466 2700 y Fr(_)2521 2712 y

   990 Fq(L)p 1509 2792 665 4 v 1617 2866 a Fs(A)p Fr(_)q Fs(A)p

   991 1816 2854 10 38 v 1825 2837 42 4 v 88 w(A)p Fr(^)q Fs(A)2215

   992 2812 y(contr)2408 2824 y Fq(R)3308 2866 y FA(\(2\))1146

   993 3103 y Fs(A)p 1227 3091 10 38 v 1236 3074 42 4 v 88 w(A)83

   994 b(A)p 1522 3091 10 38 v 1532 3074 42 4 v 89 w(A)p 1146

   995 3123 509 4 v 1167 3196 a(A)19 b Fr(_)f Fs(A)p 1402 3184

   996 10 38 v 1412 3167 42 4 v 89 w(A;)c(A)1695 3140 y Fr(_)1751

   997 3152 y Fq(L)p 1167 3232 467 4 v 1216 3306 a Fs(A)19 b

   998 Fr(_)g Fs(A)p 1452 3294 10 38 v 1461 3277 42 4 v 88 w(A)1675

   999 3251 y(contr)1868 3263 y Fq(R)2005 3103 y Fs(A)p 2086

  1000 3091 10 38 v 2096 3074 42 4 v 89 w(A)83 b(A)p 2382 3091

  1001 10 38 v 2391 3074 42 4 v 88 w(A)p 2005 3123 509 4 v 2026

  1002 3196 a(A)19 b Fr(_)g Fs(A)p 2262 3184 10 38 v 2271 3167

  1003 42 4 v 88 w(A;)14 b(A)2555 3140 y Fr(_)2610 3152 y Fq(L)p

  1004 2026 3232 467 4 v 2076 3306 a Fs(A)19 b Fr(_)g Fs(A)p

  1005 2311 3294 10 38 v 2321 3277 42 4 v 88 w(A)2534 3251 y(contr)2727

  1006 3263 y Fq(R)p 1216 3325 1227 4 v 1497 3399 a Fs(A)p Fr(_)q

  1007 Fs(A;)14 b(A)p Fr(_)q Fs(A)p 1913 3387 10 38 v 1922 3370

  1008 42 4 v 88 w(A)p Fr(^)q Fs(A)2485 3342 y Fr(^)2540 3355

  1009 y Fq(R)p 1497 3435 665 4 v 1606 3508 a Fs(A)p Fr(_)p

  1010 Fs(A)p 1804 3496 10 38 v 1814 3480 42 4 v 89 w(A)p Fr(^)p

  1011 Fs(A)2203 3454 y(contr)2396 3466 y Fq(L)3308 3508 y FA(\(3\))523

  1012 3671 y(which)i(are)h(obtained,)e(respecti)n(v)o(ely)-5

  1013 b(,)15 b(by)h(either)g(permuting)f(the)i(cut)f(to)h(the)g(left)g(o)o(v)

  1014 o(er)f(the)g Fs(contr)3322 3683 y Fq(R)3377 3671 y FA(-)523

  1015 3770 y(rule)f(or)h(to)f(the)h(right)f(o)o(v)o(er)f(the)i

  1016 Fs(contr)1597 3782 y Fq(L)1647 3770 y FA(-rule.)e(Another)h(e)o(xample)

  1017 f(sho)n(wing)g(that)i(in)f(classical)i(logic)523 3870

  1018 y(one)j(can,)f(in)i(general,)e(reach)g(more)g(than)h(one)g(normalform)d

  1019 (is)k(Lafont')-5 b(s)20 b(proof)e([10,)i(P)o(age)f(151].)648

  1020 3973 y(In)26 b(light)h(of)f(the)h(absence)f(of)g(the)h(Church-Rosser)e

  1021 (property)g(for)h(cut-elimination)e(in)j(clas-)523 4072

  1022 y(sical)h(logic)e(and)g(in)h(light)g(of)f(the)h(w)o(ork)f(on)g

  1023 (double-ne)o(gation)d(translations,)i(there)i(seem)g(to)g(be)523

  1024 4172 y(ob)o(vious)k(questions:)g(What)i(is)g(the)g(correspondence)28

  1025 b(between)k(cut-elimination)e(in)j(classical)523 4272

  1026 y(logic)15 b(and)g(the)h(embeddings)d(of)i(classical)h(proofs)e(into)i

  1027 (intuitionistic)e(logic)i(via)f(double-ne)o(gation)523

  1028 4371 y(translations?)23 b(Since)h(cut-elimination)e(in)i

  1029 (intuitionistic)g(logic)f(is)i(Church-Rosser)m(,)3030

  1030 4341 y Fp(1)3085 4371 y FA(which)f(re-)523 4471 y(striction)e(is)h

  1031 (tacitly)f(enforced)f(by)g(a)i(double-ne)o(gation)18

  1032 b(translation)j(so)i(that)f(eliminating)f(cuts)h(in)523

  1033 4570 y(the)i(double-ne)o(gated)c(v)o(ersion)j(of)h(a)g(classical)h

  1034 (proof)e(leads)h(to)g(only)g(a)g(single)g(normalform?)d(Or)p

  1035 523 4654 473 4 v 558 4710 a Fo(1)606 4742 y Fx(As)j(mentioned)i

  1036 (earlier)f(we)f(ignore)h(what)g(we)g(belie)n(v)o(e)g(to)f(be)h

  1037 (super\002cial)g(v)n(ariations)g(between)g(dif)n(fer)o(-)606

  1038 4833 y(ent)c(normalforms)g(reachable)h(from)f(an)g(intuitionistic)f

  1039 (sequent-proof.)i(Ho)n(we)n(v)o(er)f(see)g([18])g(for)g(a)f(more)606

  1040 4924 y(thorough)h(analysis)e(of)g(this)g(aspect.)p eop

  1041 end

  1042 %%Page: 3 3

  1043 TeXDict begin 3 2 bop 523 448 a FA(more)24 b(concisely)f(ask)o(ed,)i

  1044 (do)f(double-ne)o(gation)c(translations)k(correspond)e(to)i(particular)

  1045 f(strate-)523 548 y(gies)e(of)g(ho)n(w)f(to)h(eliminate)g(cuts?)g(Does)

  1046 g(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(lead)g(to)h(the)g

  1047 (same)523 648 y(normalform)i(\(by)j Ft(same)g FA(we)h(mean)e

  1048 (corresponding)e(to)j(one)g(particular)f(normalform)e(obtained)523

  1049 747 y(by)j(cut-elimination)e(in)j(classical)g(logic\)?)f(If)g(not,)g

  1050 (then)g(can)g(one)g(\002nd)g(for)g(e)n(v)o(ery)f(normalform)523

  1051 847 y(of)h(a)g(classical)h(proof)d(a)j(corresponding)22

  1052 b(double-ne)o(gation)g(translation)j(that)h(will)h(produce)c(the)523

  1053 946 y(double-ne)o(gated)g(v)o(ersion)j(of)i(this)g(normalform\227that)c

  1054 (is,)k(can)f(e)n(v)o(ery)f(reduction)g(sequence)g(in)523

  1055 1046 y(classical)16 b(logic)f(be)h(simulated)f(by)g(a)h(\(probably)d

  1056 (carefully)h(chosen\))g(double)g(ne)o(gation)f(translation)523

  1057 1146 y(and)26 b(performing)e(cut-elimination)h(in)i(intuitionistic)f

  1058 (logic?)h(Are)g(there)f(an)o(y)g(double-ne)o(gation)523

  1059 1245 y(translations)20 b(that)g(lead)g(to)g(normalforms)e(that)i(ha)n

  1060 (v)o(e)g(no)f(equi)n(v)n(alent)g(amongst)g(the)h(normalforms)523

  1061 1345 y(reachable)i(by)h(cut-elimination)f(in)h(classical)h(logic?)f

  1062 (Can)h(one)f(characterise)f(someho)n(w)-5 b(,)22 b(which)523

  1063 1445 y(normalforms)16 b(can)i(be)g(reached)f(by)h(double-ne)o(gation)c

  1064 (translations)k(and)g(which)g(can)g(not?)g(In)g(this)523

  1065 1544 y(paper)h(we)i(conjecture)d(answers)i(for)g(all)h(these)f

  1066 (questions.)648 1654 y(Although)k(some)i(special)h(cases)g(seem)g(to)f

  1067 (be)h(answered)e(by)h(e)o(xisting)g(w)o(ork,)f(for)h(e)o(xample)523

  1068 1753 y([5,)13 b(6,)g(14],)21 b(we)i(are)g(una)o(w)o(are)e(of)h(an)o(y)g

  1069 (w)o(ork)g(that)g(treat)h(these)g(questions)e(in)i(full)f(generality)-5

  1070 b(.)21 b(The)523 1853 y(answers)27 b(we)g(shall)g(gi)n(v)o(e)f(to)h

  1071 (these)f(questions)g(are)h(a)g(lot)g(inspired)f(by)g(the)h(comments)e

  1072 (made)h(in)523 1952 y([6,)d(Sec.)g(7].)g(Ho)n(we)n(v)o(er)m(,)e(there)i

  1073 (only)g(one)g(half)g(of)g(the)g(correspondence)d(is)k(considered,)e

  1074 (namely)523 2052 y(ho)n(w)31 b(their)h(v)o(ersion)f(of)h(classical)g

  1075 (logic)g(and)f(cut-elimination)f(can)i(be)g(embedded)e(via)i(some)523

  1076 2152 y(speci\002c)21 b(double-ne)o(gation)c(translations)j(into)g

  1077 (intuitionistic)h(logic.)f(W)-7 b(e)22 b(conjecture)d(also)j(a)f(cor)n

  1078 (-)523 2251 y(respondence)i(in)i(the)g(other)f(direction,)g(namely)g

  1079 (that)h(e)n(v)o(ery)e(double-ne)o(gation)e(translation)j(and)523

  1080 2351 y(corresponding)12 b(reduction)i(sequences)h(can)g(be)g(simulated)

  1081 g(by)g(their)h(cut-elimination)d(procedure.)523 2451

  1082 y(Since)19 b(we)g(shall)h(use)f(as)h(\223point)e(of)g(reference\224)f

  1083 (a)j(more)e(general)g(cut-elimination)f(procedure)f(for)523

  1084 2550 y(cut-elimination)h(in)j(classical)g(logic)f(than)g(the)g(one)g

  1085 (described)f(in)h([6],)g(we)h(are)f(also)h(able)f(to)g(dra)o(w)523

  1086 2650 y(the)26 b(conclusion)e(that)h(gi)n(v)o(en)g(our)g(conjecture)f

  1087 (is)i(true,)f(then)g(double-ne)o(gation)d(translations)j(are)523

  1088 2749 y(not)20 b(enough)e(to)i(describe)g(the)g Ft(full)h

  1089 FA(computational)c(meaning)i(of)h(a)h(classical)g(proof.)648

  1090 2859 y(As)h(can)f(be)h(seen)g(to)f(answer)h(the)f(correspondence)d

  1091 (questions,)j(we)h(\002rst)g(ha)n(v)o(e)f(to)h(mak)o(e)f(pre-)523

  1092 2959 y(cise)28 b(what)e(we)i(mean)e(by)g(cut-elimination)f(in)i

  1093 (classical)h(logic.)e(Most)h(cut-elimination)e(proce-)523

  1094 3058 y(dures,)d(including)f(Gentzen')-5 b(s)23 b(original)f(one,)g

  1095 (only)g(terminate)g(if)h(a)h(particular)d(strate)o(gy)h(for)h(cut-)523

  1096 3158 y(elimination)e(is)i(emplo)o(yed.)c(Common)i(e)o(xamples)g(being)g

  1097 (an)h(innermost)e(reduction)g(strate)o(gy)-5 b(,)21 b(or)523

  1098 3257 y(the)d(elimination)g(of)g(the)g(cut)h(with)f(the)h(highest)f

  1099 (rank.)f(Using)h(those)g(cut-elimination)f(procedures)523

  1100 3357 y(we)29 b(cannot)f(characterise)f(what)i(the)f(set)i(of)e

  1101 Ft(all)h FA(normalforms)d(of)i(a)h(classical)h(proof)d(is\227the)o(y)

  1102 523 3457 y(w)o(ould)c(produce)e(only)i(one)g(or)h(a)g(limited)f(number)

  1103 f(of)h(normalforms.)e(W)-7 b(e)25 b(shall)f(therefore)d(base)523

  1104 3556 y(our)f(ar)o(guments)f(on)i(the)g(cut-elimination)e(procedure)f

  1105 (de)n(v)o(eloped)h(by)h(Urban)h(and)f(Bierman)h([20,)523

  1106 3656 y(21],)h(which)g(is)i(lik)o(e)g(Gentzen')-5 b(s)22

  1107 b(procedure)f(e)o(xcept)g(it)j(imposes)f(one)f(slight)h(restriction)f

  1108 (on)h(ho)n(w)523 3756 y(commuting)g(cuts)j(need)f(to)h(be)g(analysed.)e

  1109 (Since)i(this)g(cut-elimination)d(procedure)g(is)k(strongly)523

  1110 3855 y(normalising,)g(we)i(can)g(calculate)f(all)h(cut-free)f

  1111 (normalforms)e(of)j(a)g(classical)g(proof.)e(Because)523

  1112 3955 y(this)19 b(procedure)d(is)k(not)f(Church-Rosser)m(,)d(the)j

  1113 (collection)f(of)g(normalforms)e(for)i(a)i(classical)f(proof)523

  1114 4054 y(contains)f(in)h(general)e(more)h(than)g(one)g(element\227as)g

  1115 (can)h(be)f(seen)h(for)f(e)o(xample)f(with)i(the)g(proofs)523

  1116 4154 y(\(2\))24 b(and)g(\(3\).)f(As)j(this)f(cut-elimination)d

  1117 (procedure)g(puts)j(only)e(v)o(ery)h(slight)g(restrictions)g(on)g(the)

  1118 523 4254 y(process)i(of)h(cut-elimination)d(we)j(belie)n(v)o(e)f(a)h

  1119 (good)e(case)i(can)g(be)g(made)f(that)g(the)h(collection)f(of)523

  1120 4353 y(normalforms)18 b(calculated)i(by)h(this)g(procedure)d(includes)i

  1121 (all)i(\223essential\224)f(normalforms.)3167 4323 y Fp(2)3218

  1122 4353 y FA(Ho)n(w-)523 4453 y(e)n(v)o(er)e(this)i(is)g(a)g(point)e(we)i

  1123 (shall)f(not)g(be)g(concerned)e(with)j(in)f(this)h(paper)-5

  1124 b(.)p 523 4563 473 4 v 558 4619 a Fo(2)606 4650 y Fx(Making)31

  1125 b(such)e(a)h(case)f(is)g(hopeless)h(for)f(other)h(strongly-normalising)

  1126 h(cut-elimination)e(procedures,)606 4742 y(lik)o(e)f(the)g(one)h(by)f

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