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%
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% This is the encoding vector for Type1 and TrueType fonts to be used
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% with TeX. This file is part of the PSNFSS bundle, version 9
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%
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% Authors: S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry, W. Schmidt
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%
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% Idea is to have all the characters normally included in Type 1 fonts
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% available for typesetting. This is effectively the characters in Adobe
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% Standard Encoding + ISO Latin 1 + extra characters from Lucida + Euro.
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%
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% Character code assignments were made as follows:
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%
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% (1) the Windows ANSI characters are almost all in their Windows ANSI
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% positions, because some Windows users cannot easily reencode the
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% fonts, and it makes no difference on other systems. The only Windows
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% ANSI characters not available are those that make no sense for
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% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
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% (173). quotesingle and grave are moved just because it's such an
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% irritation not having them in TeX positions.
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%
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% (2) Remaining characters are assigned arbitrarily to the lower part
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% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
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%
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% (3) Y&Y Lucida Bright includes some extra text characters; in the
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% hopes that other PostScript fonts, perhaps created for public
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% consumption, will include them, they are included starting at 0x12.
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%
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% (4) Remaining positions left undefined are for use in (hopefully)
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% upward-compatible revisions, if someday more characters are generally
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% available.
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%
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% (5) hyphen appears twice for compatibility with both ASCII and Windows.
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%
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% (6) /Euro is assigned to 128, as in Windows ANSI
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%
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/ogonek /ring /.notdef
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/breve /minus /.notdef
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% These are the only two remaining unencoded characters, so may as
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% well include them.
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/Zcaron /zcaron
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% 0x10
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/caron /dotlessi
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% (unusual TeX characters available in, e.g., Lucida Bright)
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/dotlessj /ff /ffi /ffl
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% very contentious; it's so painful not having quoteleft and quoteright
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% at 96 and 145 that we move the things normally found there down to here.
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/grave /quotesingle
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% 0x20 (ASCII begins)
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/space /exclam /quotedbl /numbersign
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/dollar /percent /ampersand /quoteright
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/parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
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% 0x30
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/zero /one /two /three /four /five /six /seven
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/eight /nine /colon /semicolon /less /equal /greater /question
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% 0x40
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% 0x50
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% 0x60
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/quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
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% 0x70
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/p /q /r /s /t /u /v /w
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/.notdef % rubout; ASCII ends
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% 0x80
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/Euro /.notdef /quotesinglbase /florin
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/quotedblbase /ellipsis /dagger /daggerdbl
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/circumflex /perthousand /Scaron /guilsinglleft
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/OE /.notdef /.notdef /.notdef
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% 0x90
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/quotedblright /bullet /endash /emdash
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/oe /.notdef /.notdef /Ydieresis
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% 0xA0
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/exclamdown /cent /sterling
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/currency /yen /brokenbar /section
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/dieresis /copyright /ordfeminine /guillemotleft
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/logicalnot
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/hyphen % Y&Y (also at 45); Windows' softhyphen
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/registered
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/macron
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% 0xD0
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/degree /plusminus /twosuperior /threesuperior
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/cedilla /onesuperior /ordmasculine /guillemotright
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/onequarter /onehalf /threequarters /questiondown
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% 0xC0
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/Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
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/Egrave /Eacute /Ecircumflex /Edieresis
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/Igrave /Iacute /Icircumflex /Idieresis
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% 0xD0
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/Eth /Ntilde /Ograve /Oacute
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/Ocircumflex /Otilde /Odieresis /multiply
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/Oslash /Ugrave /Uacute /Ucircumflex
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/Udieresis /Yacute /Thorn /germandbls
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% 0xE0
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/agrave /aacute /acircumflex /atilde
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/adieresis /aring /ae /ccedilla
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/egrave /eacute /ecircumflex /edieresis
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/igrave /iacute /icircumflex /idieresis
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% 0xF0
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/eth /ntilde /ograve /oacute
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/ocircumflex /otilde /odieresis /divide
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/oslash /ugrave /uacute /ucircumflex
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/udieresis /yacute /thorn /ydieresis
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% Thomas Esser, Dec 2002. public domain
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%
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% Encoding for:
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% Thomas Esser, Dec 2002. public domain
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%
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% Encoding for:
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% cmsy10 cmsy5 cmsy6 cmsy7 cmsy8 cmsy9
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%
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/TeXbbad153fEncoding [
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/minus /periodcentered /multiply /asteriskmath /divide /diamondmath
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/plusminus /minusplus /circleplus /circleminus /circlemultiply
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/circledivide /circledot /circlecopyrt /openbullet /bullet
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/equivasymptotic /equivalence /reflexsubset /reflexsuperset /lessequal
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/greaterequal /precedesequal /followsequal /similar /approxequal
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/propersubset /propersuperset /lessmuch /greatermuch /precedes /follows
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/arrowleft /arrowright /arrowup /arrowdown /arrowboth /arrownortheast
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/arrowsoutheast /similarequal /arrowdblleft /arrowdblright /arrowdblup
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/arrowdbldown /arrowdblboth /arrownorthwest /arrowsouthwest /proportional
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/prime /infinity /element /owner /triangle /triangleinv /negationslash
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/mapsto /universal /existential /logicalnot /emptyset /Rfractur /Ifractur
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/latticetop /perpendicular /aleph /A /B /C /D /E /F /G /H /I /J /K
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11 dict begin
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end readonly def
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dup 85 /U put
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readonly def
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%%CreationDate: 1993 Sep 17 11:10:37
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968 |
% Math Symbol fonts were designed by the American Mathematical Society.
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969 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
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970 |
11 dict begin
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|
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971 |
/FontInfo 7 dict dup begin
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978 |
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end readonly def
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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1042 |
11 dict begin
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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1197 |
11 dict begin
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1198 |
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/FullName (CMMI6) readonly def
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end readonly def
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readonly def
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1273 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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1274 |
11 dict begin
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1275 |
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1618 |
cleartomark
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1619 |
%%EndFont
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1620 |
%%BeginFont: CMSY8
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1621 |
%!PS-AdobeFont-1.1: CMSY8 1.0
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1622 |
%%CreationDate: 1991 Aug 15 07:22:10
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1623 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
1624 |
11 dict begin
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|
|
1625 |
/FontInfo 7 dict dup begin
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|
1626 |
/version (1.0) readonly def
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|
1627 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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|
|
1628 |
/FullName (CMSY8) readonly def
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|
|
1629 |
/FamilyName (Computer Modern) readonly def
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|
1630 |
/Weight (Medium) readonly def
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1631 |
/ItalicAngle -14.035 def
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1632 |
/isFixedPitch false def
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1633 |
end readonly def
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1634 |
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1635 |
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1636 |
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1637 |
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1638 |
/Encoding 256 array
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1639 |
0 1 255 {1 index exch /.notdef put} for
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1640 |
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readonly def
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/FontBBox{-30 -955 1185 779}readonly def
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1643 |
/UniqueID 5000818 def
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1644 |
currentdict end
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2097 |
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|
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0 1 255 {1 index exch /.notdef put} for
|
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2107 |
dup 0 /.notdef put
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2108 |
readonly def
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2109 |
/FontBBox{-30 -958 1146 777}readonly def
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/UniqueID 5000819 def
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2111 |
currentdict end
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2112 |
currentfile eexec
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2202 |
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2203 |
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2204 |
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2205 |
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2206 |
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2207 |
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2208 |
0000000000000000000000000000000000000000000000000000000000000000
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2209 |
0000000000000000000000000000000000000000000000000000000000000000
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2210 |
0000000000000000000000000000000000000000000000000000000000000000
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2211 |
0000000000000000000000000000000000000000000000000000000000000000
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2212 |
0000000000000000000000000000000000000000000000000000000000000000
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2213 |
0000000000000000000000000000000000000000000000000000000000000000
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2214 |
0000000000000000000000000000000000000000000000000000000000000000
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2215 |
cleartomark
|
|
|
2216 |
%%EndFont
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|
|
2217 |
%%BeginFont: CMTT9
|
|
|
2218 |
%!PS-AdobeFont-1.1: CMTT9 1.0
|
|
|
2219 |
%%CreationDate: 1991 Aug 20 16:46:24
|
|
|
2220 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
2221 |
11 dict begin
|
|
|
2222 |
/FontInfo 7 dict dup begin
|
|
|
2223 |
/version (1.0) readonly def
|
|
|
2224 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
|
|
|
2225 |
/FullName (CMTT9) readonly def
|
|
|
2226 |
/FamilyName (Computer Modern) readonly def
|
|
|
2227 |
/Weight (Medium) readonly def
|
|
|
2228 |
/ItalicAngle 0 def
|
|
|
2229 |
/isFixedPitch true def
|
|
|
2230 |
end readonly def
|
|
|
2231 |
/FontName /CMTT9 def
|
|
|
2232 |
/PaintType 0 def
|
|
|
2233 |
/FontType 1 def
|
|
|
2234 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
|
|
|
2235 |
/Encoding 256 array
|
|
|
2236 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
2237 |
dup 0 /.notdef put
|
|
|
2238 |
readonly def
|
|
|
2239 |
/FontBBox{-6 -233 542 698}readonly def
|
|
|
2240 |
/UniqueID 5000831 def
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|
|
2241 |
currentdict end
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|
|
2242 |
currentfile eexec
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2243 |
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2249 |
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2250 |
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2251 |
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2252 |
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2253 |
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2254 |
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2255 |
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2256 |
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2257 |
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2258 |
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2259 |
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2260 |
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2261 |
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2262 |
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2263 |
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2264 |
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2265 |
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2266 |
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2267 |
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2268 |
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2269 |
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2270 |
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2271 |
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2272 |
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2273 |
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2274 |
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2275 |
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2276 |
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|
|
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2353 |
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2354 |
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2355 |
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2356 |
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2359 |
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2360 |
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2361 |
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2362 |
cleartomark
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2363 |
%%EndFont
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2364 |
%%BeginFont: CMTI9
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2365 |
%!PS-AdobeFont-1.1: CMTI9 1.0
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2366 |
%%CreationDate: 1991 Aug 18 21:08:07
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2367 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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2368 |
11 dict begin
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2369 |
/FontInfo 7 dict dup begin
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2370 |
/version (1.0) readonly def
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2371 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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2372 |
/FullName (CMTI9) readonly def
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2373 |
/FamilyName (Computer Modern) readonly def
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2374 |
/Weight (Medium) readonly def
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2375 |
/ItalicAngle -14.04 def
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2376 |
/isFixedPitch false def
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2377 |
end readonly def
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2378 |
/FontName /CMTI9 def
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2379 |
/PaintType 0 def
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2380 |
/FontType 1 def
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2381 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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2382 |
/Encoding 256 array
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2383 |
0 1 255 {1 index exch /.notdef put} for
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2384 |
dup 0 /.notdef put
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2385 |
readonly def
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2386 |
/FontBBox{-35 -250 1148 750}readonly def
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2387 |
/UniqueID 5000827 def
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2388 |
currentdict end
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2389 |
currentfile eexec
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%%CreationDate: 1991 Aug 20 16:39:02
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
2584 |
11 dict begin
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|
|
2585 |
/FontInfo 7 dict dup begin
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cleartomark
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%!PS-AdobeFont-1.1: CMMI7 1.100
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2688 |
%%CreationDate: 1996 Jul 23 07:53:53
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2689 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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2690 |
11 dict begin
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|
|
2691 |
/FontInfo 7 dict dup begin
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2692 |
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2693 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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2694 |
/FullName (CMMI7) readonly def
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2695 |
/FamilyName (Computer Modern) readonly def
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2696 |
/Weight (Medium) readonly def
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2697 |
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2698 |
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2699 |
end readonly def
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/FontName /CMMI7 def
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2703 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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2704 |
/Encoding 256 array
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2705 |
0 1 255 {1 index exch /.notdef put} for
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2706 |
dup 0 /.notdef put
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readonly def
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/FontBBox{0 -250 1171 750}readonly def
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2710 |
currentdict end
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2733 |
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2734 |
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2735 |
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2736 |
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2737 |
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2738 |
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2739 |
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2740 |
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2741 |
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2742 |
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2743 |
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2744 |
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2745 |
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2747 |
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2749 |
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2750 |
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2751 |
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2752 |
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2753 |
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2754 |
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2755 |
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2756 |
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2757 |
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2758 |
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2760 |
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2761 |
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2762 |
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2763 |
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2779 |
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2780 |
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2783 |
0000000000000000000000000000000000000000000000000000000000000000
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2784 |
0000000000000000000000000000000000000000000000000000000000000000
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2785 |
0000000000000000000000000000000000000000000000000000000000000000
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2786 |
0000000000000000000000000000000000000000000000000000000000000000
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2787 |
0000000000000000000000000000000000000000000000000000000000000000
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2788 |
cleartomark
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|
2789 |
%%EndFont
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|
|
2790 |
%%BeginFont: CMR7
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|
|
2791 |
%!PS-AdobeFont-1.1: CMR7 1.0
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|
|
2792 |
%%CreationDate: 1991 Aug 20 16:39:21
|
|
|
2793 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
2794 |
11 dict begin
|
|
|
2795 |
/FontInfo 7 dict dup begin
|
|
|
2796 |
/version (1.0) readonly def
|
|
|
2797 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
|
|
|
2798 |
/FullName (CMR7) readonly def
|
|
|
2799 |
/FamilyName (Computer Modern) readonly def
|
|
|
2800 |
/Weight (Medium) readonly def
|
|
|
2801 |
/ItalicAngle 0 def
|
|
|
2802 |
/isFixedPitch false def
|
|
|
2803 |
end readonly def
|
|
|
2804 |
/FontName /CMR7 def
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|
2805 |
/PaintType 0 def
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|
2806 |
/FontType 1 def
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|
|
2807 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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|
|
2808 |
/Encoding 256 array
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|
|
2809 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
2810 |
dup 0 /.notdef put
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|
|
2811 |
readonly def
|
|
|
2812 |
/FontBBox{-27 -250 1122 750}readonly def
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|
|
2813 |
/UniqueID 5000790 def
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|
|
2814 |
currentdict end
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|
|
2815 |
currentfile eexec
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2816 |
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2817 |
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2818 |
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2819 |
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2820 |
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2821 |
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2822 |
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2823 |
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2824 |
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2825 |
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2826 |
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2827 |
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2828 |
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2829 |
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2830 |
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2831 |
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2832 |
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2892 |
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2894 |
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2895 |
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2896 |
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2899 |
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2900 |
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2911 |
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2912 |
readonly def
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2913 |
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2914 |
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2915 |
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2947 |
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2961 |
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2962 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
|
2963 |
11 dict begin
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|
2964 |
/FontInfo 7 dict dup begin
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2965 |
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2966 |
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|
2967 |
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2968 |
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2969 |
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2970 |
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2971 |
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2972 |
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3009 |
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3019 |
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3020 |
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3021 |
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3022 |
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3023 |
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3024 |
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3025 |
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cleartomark
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%%EndFont
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%%BeginFont: CMTT10
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%!PS-AdobeFont-1.1: CMTT10 1.00B
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3084 |
%%CreationDate: 1992 Apr 26 10:42:42
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3085 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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3086 |
11 dict begin
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3087 |
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3089 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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3090 |
/FullName (CMTT10) readonly def
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3091 |
/FamilyName (Computer Modern) readonly def
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3092 |
/Weight (Medium) readonly def
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3093 |
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3094 |
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3095 |
end readonly def
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3097 |
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3099 |
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3100 |
/Encoding 256 array
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3101 |
0 1 255 {1 index exch /.notdef put} for
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3102 |
dup 0 /.notdef put
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readonly def
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/FontBBox{-4 -235 731 800}readonly def
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3106 |
currentdict end
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currentfile eexec
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3245 |
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3246 |
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3247 |
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3248 |
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3249 |
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3250 |
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3251 |
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3252 |
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3253 |
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3254 |
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3255 |
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3256 |
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3257 |
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3258 |
0000000000000000000000000000000000000000000000000000000000000000
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3259 |
0000000000000000000000000000000000000000000000000000000000000000
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3260 |
0000000000000000000000000000000000000000000000000000000000000000
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3261 |
0000000000000000000000000000000000000000000000000000000000000000
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3262 |
0000000000000000000000000000000000000000000000000000000000000000
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3263 |
0000000000000000000000000000000000000000000000000000000000000000
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3264 |
0000000000000000000000000000000000000000000000000000000000000000
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3265 |
cleartomark
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3266 |
%%EndFont
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3267 |
%%BeginFont: CMSY10
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3268 |
%!PS-AdobeFont-1.1: CMSY10 1.0
|
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|
3269 |
%%CreationDate: 1991 Aug 15 07:20:57
|
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|
3270 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
3271 |
11 dict begin
|
|
|
3272 |
/FontInfo 7 dict dup begin
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|
|
3273 |
/version (1.0) readonly def
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|
|
3274 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
|
|
|
3275 |
/FullName (CMSY10) readonly def
|
|
|
3276 |
/FamilyName (Computer Modern) readonly def
|
|
|
3277 |
/Weight (Medium) readonly def
|
|
|
3278 |
/ItalicAngle -14.035 def
|
|
|
3279 |
/isFixedPitch false def
|
|
|
3280 |
end readonly def
|
|
|
3281 |
/FontName /CMSY10 def
|
|
|
3282 |
/PaintType 0 def
|
|
|
3283 |
/FontType 1 def
|
|
|
3284 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
|
|
|
3285 |
/Encoding 256 array
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|
|
3286 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
3287 |
dup 0 /.notdef put
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|
|
3288 |
readonly def
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|
|
3289 |
/FontBBox{-29 -960 1116 775}readonly def
|
|
|
3290 |
/UniqueID 5000820 def
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|
|
3291 |
currentdict end
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3292 |
currentfile eexec
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3293 |
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3294 |
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3295 |
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3296 |
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3297 |
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3298 |
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3299 |
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3300 |
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3301 |
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3302 |
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3303 |
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3305 |
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3308 |
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3309 |
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3310 |
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3311 |
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3312 |
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3313 |
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3314 |
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3315 |
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3316 |
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3317 |
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3318 |
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3319 |
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3320 |
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3321 |
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3322 |
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3323 |
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3324 |
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3325 |
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3326 |
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3327 |
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3328 |
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3329 |
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3330 |
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3331 |
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3332 |
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3333 |
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3334 |
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3335 |
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3336 |
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3337 |
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3338 |
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3339 |
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3340 |
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3341 |
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3342 |
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3343 |
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3349 |
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3350 |
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3351 |
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3352 |
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3353 |
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3354 |
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3355 |
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3356 |
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3357 |
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3359 |
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3360 |
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3361 |
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3362 |
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3363 |
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3364 |
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3365 |
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3641 |
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3642 |
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3643 |
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3644 |
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3645 |
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3646 |
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3647 |
cleartomark
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3648 |
%%EndFont
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3649 |
%%BeginFont: CMMI9
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3650 |
%!PS-AdobeFont-1.1: CMMI9 1.100
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3651 |
%%CreationDate: 1996 Jul 23 07:53:55
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3652 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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3653 |
11 dict begin
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3654 |
/FontInfo 7 dict dup begin
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3655 |
/version (1.100) readonly def
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3656 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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3657 |
/FullName (CMMI9) readonly def
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|
|
3658 |
/FamilyName (Computer Modern) readonly def
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|
|
3659 |
/Weight (Medium) readonly def
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|
3660 |
/ItalicAngle -14.04 def
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3661 |
/isFixedPitch false def
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3662 |
end readonly def
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3663 |
/FontName /CMMI9 def
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3664 |
/PaintType 0 def
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3665 |
/FontType 1 def
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3666 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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3667 |
/Encoding 256 array
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3668 |
0 1 255 {1 index exch /.notdef put} for
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|
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3669 |
dup 0 /.notdef put
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3670 |
readonly def
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3671 |
/FontBBox{-29 -250 1075 750}readonly def
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3672 |
/UniqueID 5087384 def
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3673 |
currentdict end
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3674 |
currentfile eexec
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3701 |
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3702 |
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3703 |
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3704 |
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3705 |
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3706 |
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3709 |
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3935 |
3[78 1[60 51 46 55 60 46 60 60 74 51 2[28 60 60 46 51
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3936 |
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3940 |
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3941 |
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3942 |
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3943 |
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3944 |
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3945 |
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3946 |
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3947 |
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3948 |
%%Page: 1 1
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|
|
3949 |
TeXDict begin 1 0 bop 1448 448 a FX(Nominal)30 b(Uni\002cation)1002
|
|
|
3950 |
737 y FW(Christian)20 b(Urban)186 b(Andre)n(w)19 b(Pitts)187
|
|
|
3951 |
b(Murdoch)18 b(Gabbay)1325 911 y FV(Uni)n(v)o(ersity)h(of)g(Cambridge,)
|
|
|
3952 |
h(Cambridge,)f(UK.)759 1221 y FU(Abstract.)41 b FV(W)-6
|
|
|
3953 |
b(e)18 b(present)h(a)e(generalisation)i(of)f(\002rst-order)g
|
|
|
3954 |
(uni\002cation)g(to)g(the)g(practically)759 1313 y(important)i(case)h
|
|
|
3955 |
(of)f(equations)h(between)f(terms)g(in)m(v)o(olving)h
|
|
|
3956 |
FT(binding)g(oper)o(ations)p FV(.)f(A)g(sub-)759 1404
|
|
|
3957 |
y(stitution)26 b(of)f(terms)h(for)f(v)n(ariables)h(solv)o(es)g(such)h
|
|
|
3958 |
(an)f(equation)g(if)f(it)g(mak)o(es)i(the)e(equated)759
|
|
|
3959 |
1495 y(terms)20 b FS(\013)p FT(-equivalent)p FV(,)h(i.e.)f(equal)g(up)h
|
|
|
3960 |
(to)f(renaming)h(bound)h(names.)f(F)o(or)e(the)i(applications)759
|
|
|
3961 |
1587 y(we)27 b(ha)o(v)o(e)f(in)h(mind,)f(we)h(must)g(consider)g(the)g
|
|
|
3962 |
(simple,)f(te)o(xtual)g(form)h(of)f(substitution)i(in)759
|
|
|
3963 |
1678 y(which)h(names)g(occurring)g(in)f(terms)g(may)g(be)g(captured)i
|
|
|
3964 |
(within)d(the)i(scope)g(of)f(binders)759 1769 y(upon)22
|
|
|
3965 |
b(substitution.)f(W)-6 b(e)21 b(are)g(able)g(to)g(tak)o(e)g(a)g
|
|
|
3966 |
(`nominal')g(approach)i(to)d(binding)i(in)f(which)759
|
|
|
3967 |
1861 y(bound)k(entities)d(are)h(e)o(xplicitly)f(named)i(\(rather)f
|
|
|
3968 |
(than)g(using)h(nameless,)f(de)g(Bruijn-style)759 1952
|
|
|
3969 |
y(representations\))d(and)f(yet)f(get)h(a)f(v)o(ersion)h(of)f(this)h
|
|
|
3970 |
(form)f(of)g(substitution)h(that)g(respects)g FS(\013)p
|
|
|
3971 |
FV(-)759 2043 y(equi)n(v)n(alence)f(and)f(possesses)g(good)g
|
|
|
3972 |
(algorithmic)g(properties.)f(W)-6 b(e)16 b(achie)n(v)o(e)h(this)e(by)i
|
|
|
3973 |
(adapt-)759 2135 y(ing)25 b(an)f(e)o(xisting)g(idea)h(and)f
|
|
|
3974 |
(introducing)i(a)e(k)o(e)o(y)h(ne)n(w)f(idea.)g(The)g(e)o(xisting)h
|
|
|
3975 |
(idea)f(is)g(terms)759 2226 y(in)m(v)o(olving)c(e)o(xplicit)e
|
|
|
3976 |
(substitutions)i(of)f(names)g(for)g(names,)g(e)o(xcept)g(that)g(here)g
|
|
|
3977 |
(we)g(only)g(use)759 2317 y FT(e)o(xplicit)k(permutations)h
|
|
|
3978 |
FV(\(bijecti)n(v)o(e)f(substitutions\).)h(The)f(k)o(e)o(y)h(ne)n(w)g
|
|
|
3979 |
(idea)f(is)g(that)g(the)g(uni-)759 2409 y(\002cation)e(algorithm)h
|
|
|
3980 |
(should)g(solv)o(e)g(not)f(only)h(equational)g(problems,)g(b)o(ut)f
|
|
|
3981 |
(also)h(problems)759 2500 y(about)k(the)f FT(fr)m(eshness)h
|
|
|
3982 |
FV(of)f(names)h(for)f(terms.)g(There)g(is)f(a)h(simple)g
|
|
|
3983 |
(generalisation)h(of)f(the)759 2591 y(classical)c(\002rst-order)f
|
|
|
3984 |
(uni\002cation)h(algorithm)h(to)e(this)h(setting)g(which)g(retains)f
|
|
|
3985 |
(the)h(latter')l(s)759 2682 y(pleasant)i(properties:)f(uni\002cation)h
|
|
|
3986 |
(problems)f(in)m(v)o(olving)h FS(\013)p FV(-equi)n(v)n(alence)h(and)f
|
|
|
3987 |
(freshness)759 2774 y(are)c(decidable;)h(and)g(solv)n(able)f(problems)h
|
|
|
3988 |
(possess)g(most)f(general)h(solutions.)523 3042 y FR(1)99
|
|
|
3989 |
b(Intr)n(oduction)523 3231 y FW(Decidability)25 b(of)h(uni\002cation)f
|
|
|
3990 |
(for)g(equations)g(between)h(\002rst-order)e(terms)j(and)e(algorithms)g
|
|
|
3991 |
(for)523 3330 y(computing)d(most)i(general)f(uni\002ers)h(form)f(a)h
|
|
|
3992 |
(fundamental)e(tool)i(of)g(computational)e(logic)h(with)523
|
|
|
3993 |
3430 y(man)o(y)17 b(applications)g(to)h(programming)d(languages)i(and)g
|
|
|
3994 |
(computer)n(-aided)e(reasoning.)i(Ho)n(we)n(v)o(er)m(,)523
|
|
|
3995 |
3530 y(v)o(ery)e(man)o(y)f(potential)h(applications)g(f)o(all)h
|
|
|
3996 |
(outside)g(the)f(scope)h(of)f(\002rst-order)g(uni\002cation,)f(because)
|
|
|
3997 |
523 3629 y(the)o(y)21 b(in)m(v)n(olv)o(e)g(term)h(languages)e(with)j
|
|
|
3998 |
(binding)d(operations)h(where)g(at)i(the)f(v)o(ery)f(least)i(we)f(do)g
|
|
|
3999 |
(not)523 3729 y(wish)f(to)f(distinguish)f(terms)h(dif)n(fering)f(up)g
|
|
|
4000 |
(to)i(the)f(renaming)e(of)i(bound)e(names.)648 3828 y(There)23
|
|
|
4001 |
b(is)j(a)f(lar)o(ge)f(body)f(of)h(w)o(ork)g(studying)f(languages)g
|
|
|
4002 |
(with)i(binders)f(through)e(the)j(use)g(of)523 3928 y(v)n(arious)d
|
|
|
4003 |
FQ(\025)p FW(-calculi)g(as)i(term)e(representation)f(languages,)g
|
|
|
4004 |
(leading)g(to)i FP(higher)n(-or)m(der)f(uni\002cation)523
|
|
|
4005 |
4028 y FW(algorithms)d(for)i(solving)e(equations)h(between)g
|
|
|
4006 |
FQ(\025)p FW(-terms)h(modulo)e FQ(\013\014)t(\021)s FW(-equi)n(v)n
|
|
|
4007 |
(alence.)g(Ho)n(we)n(v)o(er)m(,)523 4127 y(higher)n(-order)j
|
|
|
4008 |
(uni\002cation)i(is)i(technically)e(complicated)f(without)h(being)h
|
|
|
4009 |
(completely)e(satisf)o(ac-)523 4227 y(tory)g(from)g(a)i(pragmatic)d
|
|
|
4010 |
(point)i(of)f(vie)n(w)-5 b(.)23 b(The)h(reason)f(lies)i(in)f(the)g(dif)
|
|
|
4011 |
n(ference)e(between)i(substi-)523 4327 y(tution)j(for)g(\002rst-order)g
|
|
|
4012 |
(terms)h(and)f(for)g FQ(\025)p FW(-terms.)h(The)f(former)g(is)h(a)h
|
|
|
4013 |
(simple)f(operation)e(of)h(te)o(x-)523 4426 y(tual)19
|
|
|
4014 |
b(replacement)f(\(sometimes)g(called)h FP(gr)o(afting)f
|
|
|
4015 |
FW([7],)h(or)f FP(conte)n(xt)h(substitution)g FW([12,)f(Sect.)h
|
|
|
4016 |
(2.1]\),)523 4526 y(whereas)g(the)g(latter)h(also)g(in)m(v)n(olv)o(es)e
|
|
|
4017 |
(renamings)g(to)h(a)n(v)n(oid)g(capture.)f(Capture-a)n(v)n(oidance)f
|
|
|
4018 |
(ensures)523 4625 y(that)22 b(substitution)g(respects)g
|
|
|
4019 |
FQ(\013)p FW(-equi)n(v)n(alence,)e(b)n(ut)i(it)h(complicates)e(higher)n
|
|
|
4020 |
(-order)e(uni\002cation)i(al-)523 4725 y(gorithms.)e(Furthermore)f(it)k
|
|
|
4021 |
(is)f(the)g(simple)f(te)o(xtual)g(form)g(of)g(substitution)g(rather)g
|
|
|
4022 |
(than)g(the)g(more)523 4825 y(complicated)15 b(capture-a)n(v)n(oiding)e
|
|
|
4023 |
(form)j(which)g(occurs)f(in)i(man)o(y)e(informal)g(applications)g(of)i
|
|
|
4024 |
(`uni-)523 4924 y(\002cation)30 b(modulo)f FQ(\013)p
|
|
|
4025 |
FW(-equi)n(v)n(alence'.)e(F)o(or)j(e)o(xample,)f(consider)g(the)h
|
|
|
4026 |
(follo)n(wing)f(schematic)h(rule)p eop end
|
|
|
4027 |
%%Page: 2 2
|
|
|
4028 |
TeXDict begin 2 1 bop 523 448 a FW(which)23 b(might)g(form)g(part)g(of)
|
|
|
4029 |
g(the)h(inducti)n(v)o(e)e(de\002nition)h(of)g(a)h(binary)e(e)n(v)n
|
|
|
4030 |
(aluation)g(relation)h FO(+)h FW(for)523 548 y(the)c(e)o(xpressions)f
|
|
|
4031 |
(of)h(an)g(imaginary)f(functional)f(programming)f(language:)1580
|
|
|
4032 |
671 y FN(app)o FM(\()p FN(fn)j FQ(a:Y)5 b(;)14 b(X)7
|
|
|
4033 |
b FM(\))45 b FO(+)h FQ(V)p 1559 712 787 4 v 1559 786
|
|
|
4034 |
a FN(let)20 b FQ(a)j FM(=)f FQ(X)27 b FN(in)20 b FQ(Y)65
|
|
|
4035 |
b FO(+)46 b FQ(V)2357 733 y(:)928 b FW(\(1\))523 897
|
|
|
4036 |
y(Here)22 b FQ(X)7 b FW(,)22 b FQ(Y)41 b FW(and)21 b
|
|
|
4037 |
FQ(V)42 b FW(are)22 b(meta)n(v)n(ariables)f(standing)f(for)i(unkno)n
|
|
|
4038 |
(wn)d(programming)g(language)h(e)o(x-)523 997 y(pressions.)27
|
|
|
4039 |
b(The)g(binders)g FN(fn)20 b FQ(a:)p FM(\()p FO(\000)p
|
|
|
4040 |
FM(\))28 b FW(and)f FN(let)20 b FQ(a)37 b FM(=)f FQ(X)27
|
|
|
4041 |
b FN(in)20 b FM(\()p FO(\000)p FM(\))28 b FW(may)g(v)o(ery)e(well)i
|
|
|
4042 |
(capture)f(free)523 1097 y(occurrences)16 b(of)i FQ(a)h
|
|
|
4043 |
FW(when)f(we)h(instantiate)f(the)g(schematic)g(rule)g(by)g(replacing)f
|
|
|
4044 |
FQ(Y)37 b FW(with)19 b(an)f(e)o(xpres-)523 1196 y(sion.)i(F)o(or)g
|
|
|
4045 |
(instance,)f(using)h(the)g(rule)g(scheme)g(in)g(a)h(bottom-up)c(search)
|
|
|
4046 |
j(for)g(a)g(proof)f(of)1612 1325 y FN(let)g FQ(a)k FM(=)g(1)d
|
|
|
4047 |
FN(in)g FQ(a)46 b FO(+)g FM(1)991 b FW(\(2\))523 1464
|
|
|
4048 |
y(we)27 b(w)o(ould)g(use)g(a)g(substitution)f(that)h(in)m(v)n(olv)o(es)
|
|
|
4049 |
f(capture,)f(namely)h FM([)p FQ(X)16 b FM(:=)9 b(1)o
|
|
|
4050 |
FQ(;)14 b(Y)28 b FM(:=)9 b FQ(a;)14 b(V)28 b FM(:=)9
|
|
|
4051 |
b(1)o(])p FW(,)27 b(to)523 1564 y(unify)20 b(the)h(goal)g(with)g(the)g
|
|
|
4052 |
(conclusion)f(of)h(\(1\)\227generating)d(the)j(ne)n(w)g(goal)f
|
|
|
4053 |
FN(app)o FM(\()p FN(fn)g FQ(a:a;)14 b FM(1\))25 b FO(+)f
|
|
|
4054 |
FM(1)523 1664 y FW(from)g(the)h(hypothesis)e(of)h(the)h(rule.)f(The)h
|
|
|
4055 |
(problem)e(with)i(this)g(is)h(that)f(in)g(informal)e(practice)h(we)523
|
|
|
4056 |
1763 y(usually)k(identify)g(terms)h(up)f(to)h FQ(\013)p
|
|
|
4057 |
FW(-equi)n(v)n(alence,)e(whereas)h(te)o(xtual)g(substitution)g(does)g
|
|
|
4058 |
(not)h(re-)523 1863 y(spect)20 b FQ(\013)p FW(-equi)n(v)n(alence.)e(F)o
|
|
|
4059 |
(or)i(e)o(xample,)f(up)g(to)i FQ(\013)p FW(-equi)n(v)n(alence,)d(the)i
|
|
|
4060 |
(goal)1620 1991 y FN(let)f FQ(b)k FM(=)g(1)d FN(in)g
|
|
|
4061 |
FQ(b)46 b FO(+)g FM(1)999 b FW(\(3\))523 2131 y(is)16
|
|
|
4062 |
b(the)g(same)f(as)i(\(2\))n(.)f(W)-7 b(e)17 b(might)d(think)h
|
|
|
4063 |
(\(erroneously!\))d(that)j(the)h(conclusion)e(of)h(\(1\))f(is)j(the)e
|
|
|
4064 |
(same)h(as)523 2231 y FN(let)k FQ(b)12 b FM(=)g FQ(X)h
|
|
|
4065 |
FN(in)19 b FQ(Y)31 b FO(+)12 b FQ(V)41 b FW(without)21
|
|
|
4066 |
b(changing)e(the)j(rule')-5 b(s)22 b(hypothesis\227after)d(all,)j(if)g
|
|
|
4067 |
(we)g(are)g(trying)523 2330 y(to)17 b(mak)o(e)f FQ(\013)p
|
|
|
4068 |
FW(-equi)n(v)n(alence)f(disappear)h(into)g(the)h(infrastructure,)d
|
|
|
4069 |
(then)j(we)g(must)g(be)g(able)f(to)h(replace)523 2430
|
|
|
4070 |
y(an)o(y)j FP(part)i FW(of)e(what)h(we)g(ha)n(v)o(e)f(with)g(an)h(equi)
|
|
|
4071 |
n(v)n(alent)e(part.)h(So)g(we)h(might)f(be)h(tempted)e(to)i(unify)e
|
|
|
4072 |
(the)523 2530 y(conclusion)i(with)h(\(3\))g(via)g(the)h(te)o(xtual)e
|
|
|
4073 |
(substitution)h FM([)p FQ(X)15 b FM(:=)9 b(1)p FQ(;)14
|
|
|
4074 |
b(Y)27 b FM(:=)9 b FQ(b;)14 b(V)28 b FM(:=)9 b(1)o(])p
|
|
|
4075 |
FW(,)23 b(and)f(then)f(apply)523 2629 y(this)27 b(substitution)f(to)i
|
|
|
4076 |
(the)e(hypothesis)g(to)h(obtain)f(a)h(wrong)f(goal,)g
|
|
|
4077 |
FN(app)o FM(\()p FN(fn)20 b FQ(a:b;)14 b FM(1\))35 b
|
|
|
4078 |
FO(+)g FM(1)p FW(.)27 b(Using)523 2729 y FQ(\025)p FW(-calculus)d(and)g
|
|
|
4079 |
(higher)n(-order)d(uni\002cation)i(sa)n(v)o(es)h(us)h(from)e(such)h
|
|
|
4080 |
(slopp)o(y)f(thinking,)g(b)n(ut)h(at)g(the)523 2829 y(e)o(xpense)17
|
|
|
4081 |
b(of)i(ha)n(ving)e(to)i(mak)o(e)f(e)o(xplicit)g(the)h(dependence)d(of)i
|
|
|
4082 |
(meta)n(v)n(ariables)g(on)g(bindable)f(names)523 2928
|
|
|
4083 |
y(via)h(the)f(use)h(of)f(function)f(v)n(ariables)h(and)g(application.)f
|
|
|
4084 |
(F)o(or)h(e)o(xample,)f(\(1\))h(w)o(ould)g(be)h(replaced)e(by)593
|
|
|
4085 |
3075 y FN(app)d FM(\()p FN(fn)g FQ(\025a:F)26 b(a)p FM(\))14
|
|
|
4086 |
b FQ(X)53 b FO(+)46 b FQ(V)p 593 3115 848 4 v 644 3194
|
|
|
4087 |
a FN(let)12 b FQ(X)21 b FM(\()p FQ(\025a:F)26 b(a)p FM(\))46
|
|
|
4088 |
b FO(+)g FQ(V)1547 3147 y FW(or)m(,)19 b(modulo)g FQ(\021)s
|
|
|
4089 |
FW(-equi)n(v)n(alence,)2520 3079 y FN(app)12 b FM(\()p
|
|
|
4090 |
FN(fn)i FQ(F)e FM(\))i FQ(X)52 b FO(+)46 b FQ(V)p 2520
|
|
|
4091 |
3120 675 4 v 2603 3194 a FN(let)12 b FQ(X)20 b(F)58 b
|
|
|
4092 |
FO(+)46 b FQ(V)3217 3147 y FW(.)3308 3139 y(\(4\))523
|
|
|
4093 |
3330 y(No)n(w)30 b(goal)f(\(3\))h(becomes)f FN(let)12
|
|
|
4094 |
b FM(1)i FQ(\025b:b)41 b FO(+)g FM(1)30 b FW(and)f(there)h(is)h(no)e
|
|
|
4095 |
(problem)g(unifying)f(it)i(with)g(the)523 3430 y(conclusion)16
|
|
|
4096 |
b(of)i(\(4\))g(via)g(a)g(capture-a)n(v)n(oiding)d(substitution)i(of)h
|
|
|
4097 |
FM(1)g FW(for)g FQ(X)7 b FW(,)17 b FQ(\025c:c)i FW(for)f
|
|
|
4098 |
FQ(F)30 b FW(and)18 b FM(1)g FW(for)f FQ(V)i FW(.)648
|
|
|
4099 |
3530 y(This)31 b(is)i(all)f(v)o(ery)e(\002ne,)h(b)n(ut)g(the)h
|
|
|
4100 |
(situation)f(is)h(not)f(as)h(pleasant)f(as)i(for)d(\002rst-order)g
|
|
|
4101 |
(terms:)523 3629 y(higher)n(-order)19 b(uni\002cation)h(problems)g(can)
|
|
|
4102 |
i(be)f(undecidable,)e(decidable)i(b)n(ut)g(lack)h(most)g(general)523
|
|
|
4103 |
3729 y(uni\002ers,)17 b(or)h(ha)n(v)o(e)g(such)f(uni\002ers)h(only)f
|
|
|
4104 |
(by)h(imposing)e(some)i(restrictions)g([20];)f(see)h([6])g(for)f(a)h
|
|
|
4105 |
(sur)n(-)523 3828 y(v)o(e)o(y)e(of)g(higher)n(-order)e(uni\002cation.)h
|
|
|
4106 |
(W)-7 b(e)18 b(started)e(out)h(w)o(anting)f(to)g(compute)g(with)h
|
|
|
4107 |
(binders)e(modulo)523 3928 y FQ(\013)p FW(-equi)n(v)n(alence,)j(and)h
|
|
|
4108 |
(someho)n(w)g(the)h(process)g(of)f(making)g(possibly-capturing)e
|
|
|
4109 |
(substitution)i(re-)523 4028 y(spectable)26 b(has)h(led)g(to)f
|
|
|
4110 |
(function)f(v)n(ariables,)h(application,)f(capture-a)n(v)n(oiding)e
|
|
|
4111 |
(substitution)j(and)523 4127 y FQ(\014)t(\021)s FW(-equi)n(v)n(alence.)
|
|
|
4112 |
18 b(Does)i(it)h(ha)n(v)o(e)f(to)g(be)g(so?)h(No!)648
|
|
|
4113 |
4227 y(F)o(or)14 b(one)h(thing,)f(se)n(v)o(eral)h(authors)f(ha)n(v)o(e)
|
|
|
4114 |
h(already)f(noted)g(that)h(one)g(can)g(mak)o(e)g(sense)g(of)g
|
|
|
4115 |
(possibly-)523 4327 y(capturing)k(substitution)g(modulo)g
|
|
|
4116 |
FQ(\013)p FW(-equi)n(v)n(alence)f(by)i(using)g FP(e)n(xplicit)h
|
|
|
4117 |
(substitutions)f FW(in)g(the)h(term)523 4426 y(representation)28
|
|
|
4118 |
b(language:)h(see)i([7,)12 b(13,)h(15,)g(26].)29 b(Compared)g(with)h
|
|
|
4119 |
(those)g(w)o(orks,)g(we)g(mak)o(e)g(a)523 4526 y(number)20
|
|
|
4120 |
b(of)h(simpli\002cations.)g(First,)i(we)f(\002nd)f(that)h(we)g(do)f
|
|
|
4121 |
FP(not)i FW(need)e(to)h(use)g(function)e(v)n(ariables,)523
|
|
|
4122 |
4625 y(application)d(or)i FQ(\014)t(\021)s FW(-equi)n(v)n(alence)d(in)j
|
|
|
4123 |
(our)f(representation)f(language\227lea)n(ving)e(just)k(binders)f(and)
|
|
|
4124 |
523 4725 y FQ(\013)p FW(-equi)n(v)n(alence.)e(Secondly)-5
|
|
|
4125 |
b(,)16 b(instead)h(of)h(using)f(e)o(xplicit)h(substitutions)f(of)h
|
|
|
4126 |
(names)f(for)h(names,)f(we)523 4825 y(use)g FP(e)n(xplicit)g
|
|
|
4127 |
(permutations)f FW(of)g(names.)h(The)f(idea)h(of)f(using)h
|
|
|
4128 |
(name-permutations,)c(and)j(in)h(partic-)523 4924 y(ular)23
|
|
|
4129 |
b(name-sw)o(appings,)f(when)h(dealing)f(with)i FQ(\013)p
|
|
|
4130 |
FW(-con)m(v)o(ersion)d(dates)j(back)f(to)h([10])e(and)h(there)h(is)p
|
|
|
4131 |
eop end
|
|
|
4132 |
%%Page: 3 3
|
|
|
4133 |
TeXDict begin 3 2 bop 523 448 a FW(gro)n(wing)14 b(e)n(vidence)g(of)h
|
|
|
4134 |
(its)i(usefulness)e(\(see)h([3,)d(4],)i(for)g(e)o(xample\).)e(When)j(a)
|
|
|
4135 |
g(name)f(substitution)g(is)523 548 y(actually)i(a)h(permutation,)d(the)
|
|
|
4136 |
i(function)f(it)i(induces)e(from)h(terms)g(to)h(terms)f(is)h(a)g
|
|
|
4137 |
(bijection;)f(this)h(bi-)523 648 y(jecti)n(vity)g(gi)n(v)o(es)f(the)h
|
|
|
4138 |
(operation)e(of)i(permuting)e(names)i(v)o(ery)f(good)f(logical)i
|
|
|
4139 |
(properties)f(compared)523 747 y(with)j(name)f(substitution.)f
|
|
|
4140 |
(Consider)h(for)g(e)o(xample)f(the)h FQ(\013)p FW(-equi)n(v)n(alent)f
|
|
|
4141 |
(terms)i FN(fn)13 b FQ(a:b)20 b FW(and)f FN(fn)13 b FQ(c:b)p
|
|
|
4142 |
FW(,)523 847 y(where)j FQ(a)p FW(,)h FQ(b)g FW(and)f
|
|
|
4143 |
FQ(c)h FW(are)g(distinct.)f(If)h(we)g(apply)e(the)i(substitution)f
|
|
|
4144 |
FM([)p FQ(b)p FL(7!)p FQ(a)p FM(])h FW(\(renaming)d(all)j(free)g(occur)
|
|
|
4145 |
n(-)523 946 y(rences)h(of)g FQ(b)h FW(to)f(be)h FQ(a)p
|
|
|
4146 |
FW(\))f(to)h(them)f(we)g(get)h FN(fn)13 b FQ(a:a)19 b
|
|
|
4147 |
FW(and)f FN(fn)13 b FQ(c:a)p FW(,)18 b(which)g(are)g(no)g(longer)f
|
|
|
4148 |
FQ(\013)p FW(-equi)n(v)n(alent.)523 1046 y(Thus)23 b(renaming)e
|
|
|
4149 |
(substitutions)i(do)g(not)g(respect)g FQ(\013)p FW(-equi)n(v)n(alence)e
|
|
|
4150 |
(in)j(general,)e(and)h(an)o(y)f(uni\002ca-)523 1146 y(tion)i(algorithm)
|
|
|
4151 |
g(using)g(them)g(needs)g(to)h(tak)o(e)g(e)o(xtra)f(precautions)f(to)h
|
|
|
4152 |
(not)h(inadv)o(ertently)d(change)523 1245 y(the)e(intended)f(meaning)f
|
|
|
4153 |
(of)i(terms.)g(The)g(traditional)f(solution)g(for)h(this)h(problem)d
|
|
|
4154 |
(is)j(to)f(introduce)523 1345 y(a)32 b(more)g(complicated)e(form)h(of)h
|
|
|
4155 |
(renaming)e(substitution)h(that)h(a)n(v)n(oids)g(capture)f(of)g(names)h
|
|
|
4156 |
(by)523 1445 y(binders.)19 b(In)h(contrast,)f(the)h(simple)g(operation)
|
|
|
4157 |
e(of)i(name-permutation)c(respects)k FQ(\013)p FW(-equi)n(v)n(alence;)
|
|
|
4158 |
523 1544 y(for)j(e)o(xample,)e(applying)h(the)h(name-permutation)d
|
|
|
4159 |
FM(\()p FQ(a)14 b(b)p FM(\))24 b FW(that)f(sw)o(aps)h(all)g
|
|
|
4160 |
(occurrences)d(of)i FQ(a)h FW(and)523 1644 y FQ(b)k FW(\(be)f(the)o(y)g
|
|
|
4161 |
(free,)g(bound)e(or)i(binding\))f(to)i(the)f(terms)h(abo)o(v)o(e)e(gi)n
|
|
|
4162 |
(v)o(es)h FN(fn)13 b FQ(b:a)27 b FW(and)g FN(fn)13 b
|
|
|
4163 |
FQ(c:a)p FW(,)28 b(which)523 1743 y(are)22 b(still)i
|
|
|
4164 |
FQ(\013)p FW(-equi)n(v)n(alent.)c(W)-7 b(e)23 b(e)o(xploit)e(such)h
|
|
|
4165 |
(good)f(properties)g(of)h(name-permutations)c(to)23 b(gi)n(v)o(e)e(a)
|
|
|
4166 |
523 1843 y(conceptually)d(simple)i(uni\002cation)f(algorithm.)648
|
|
|
4167 |
1944 y(In)f(addition)g(to)h(the)g(use)g(of)g(e)o(xplicit)f
|
|
|
4168 |
(name-permutations,)e(we)j(also)g(compute)f(symbolically)523
|
|
|
4169 |
2044 y(with)i(predicates)g(e)o(xpressing)e FP(fr)m(eshness)j
|
|
|
4170 |
FW(of)f(names)g(for)g(terms.)g(This)g(seems)h(to)f(be)g(the)g(k)o(e)o
|
|
|
4171 |
(y)g(no)o(v-)523 2143 y(elty)i(of)f(our)g(approach.)f(Although)f(it)k
|
|
|
4172 |
(arises)f(naturally)f(from)f(the)i(w)o(ork)f(reported)f(in)i([11,)12
|
|
|
4173 |
b(23],)21 b(it)523 2243 y(is)i(easy)g(to)f(see)h(directly)e(why)h
|
|
|
4174 |
(there)g(is)h(a)g(need)e(for)h(computing)e(with)i(freshness,)g(gi)n(v)o
|
|
|
4175 |
(en)f(that)h(we)523 2343 y(tak)o(e)i(a)g(`nominal')e(approach)g(to)i
|
|
|
4176 |
(binders.)f(\(In)g(other)g(w)o(ords)h(we)g(stick)h(with)f(concrete)e(v)
|
|
|
4177 |
o(ersions)523 2442 y(of)d(binding)f(and)h FQ(\013)p FW(-equi)n(v)n
|
|
|
4178 |
(alence)e(in)i(which)g(the)h(bound)d(entity)i(is)h(named)f(e)o
|
|
|
4179 |
(xplicitly)-5 b(,)17 b(rather)i(than)523 2542 y(using)f(de)g
|
|
|
4180 |
(Bruijn-style)g(representations,)f(as)i(for)f(e)o(xample)f(in)h([7,)13
|
|
|
4181 |
b(26].\))k(A)i(basic)g(instance)f(of)g(our)523 2641 y(generalised)27
|
|
|
4182 |
b(form)g(of)h FQ(\013)p FW(-equi)n(v)n(alence)e(identi\002es)i
|
|
|
4183 |
FN(fn)20 b FQ(a:X)35 b FW(with)28 b FN(fn)20 b FQ(b:)p
|
|
|
4184 |
FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)o FQ(X)35 b FW(pro)o(vided)25
|
|
|
4185 |
b FQ(b)k FP(is)523 2741 y(fr)m(esh)22 b(for)i FQ(X)7
|
|
|
4186 |
b FW(,)21 b(where)f(the)i(subterm)e FM(\()p FQ(a)14 b(b)p
|
|
|
4187 |
FM(\))p FK(\001)p FQ(X)28 b FW(indicates)21 b(an)g(e)o(xplicit)g
|
|
|
4188 |
(permutation\227namely)c(the)523 2841 y(sw)o(apping)i(of)i
|
|
|
4189 |
FQ(a)g FW(and)f FQ(b)p FW(\227w)o(aiting)g(to)g(be)h(applied)e(to)i
|
|
|
4190 |
FQ(X)7 b FW(.)20 b(W)-7 b(e)21 b(write)g(`)p FQ(b)f FW(is)i(fresh)e
|
|
|
4191 |
(for)g FQ(X)7 b FW(')20 b(symboli-)523 2940 y(cally)e(as)h
|
|
|
4192 |
FQ(b)j FM(#)i FQ(X)7 b FW(;)18 b(the)g(intended)e(meaning)h(of)g(this)i
|
|
|
4193 |
(relation)e(is)i(that)f FQ(b)g FW(does)g(not)g(occur)f(free)g(in)h(an)o
|
|
|
4194 |
(y)523 3040 y(\(ground\))d(term)j(that)h(may)e(be)i(substituted)e(for)h
|
|
|
4195 |
FQ(X)7 b FW(.)18 b(If)g(we)g(kno)n(w)f(more)h(about)f
|
|
|
4196 |
FQ(X)25 b FW(we)19 b(may)e(be)i(able)523 3140 y(to)24
|
|
|
4197 |
b(eliminate)f(the)h(e)o(xplicit)f(permutation)e(in)j
|
|
|
4198 |
FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)p FQ(X)7 b FW(;)23
|
|
|
4199 |
b(for)g(e)o(xample,)g(if)h(we)g(kne)n(w)f(that)g FQ(a)30
|
|
|
4200 |
b FM(#)g FQ(X)523 3239 y FW(holds)21 b(as)i(well)f(as)g
|
|
|
4201 |
FQ(b)k FM(#)f FQ(X)7 b FW(,)22 b(then)f FM(\()p FQ(a)14
|
|
|
4202 |
b(b)p FM(\))p FK(\001)o FQ(X)29 b FW(can)21 b(be)h(replaced)e(by)i
|
|
|
4203 |
FQ(X)7 b FW(.)21 b(It)h(should)e(already)h(be)h(clear)523
|
|
|
4204 |
3339 y(from)g(these)i(simple)f(e)o(xamples)f(that)i(in)f(our)g(setting)
|
|
|
4205 |
g(the)g(appropriate)e(notion)i(of)g(term-equality)523
|
|
|
4206 |
3438 y(is)e(not)f(a)h(bare)e(equation,)g FQ(t)k FO(\031)f
|
|
|
4207 |
FQ(t)1445 3408 y FL(0)1469 3438 y FW(,)e(b)n(ut)g(rather)g(a)g
|
|
|
4208 |
(hypothetical)e(judgement)g(of)i(the)h(form)1784 3571
|
|
|
4209 |
y FO(r)i(`)g FQ(t)g FO(\031)g FQ(t)2121 3537 y FL(0)3308
|
|
|
4210 |
3571 y FW(\(5\))523 3701 y(where)d FO(r)h FW(is)g(a)f
|
|
|
4211 |
FP(fr)m(eshness)h(en)m(vir)l(onment)q FW(\227a)e(\002nite)i(set)g
|
|
|
4212 |
FO(f)p FQ(a)2238 3713 y FJ(1)2284 3701 y FM(#)9 b FQ(X)2431
|
|
|
4213 |
3713 y FJ(1)2468 3701 y FQ(;)14 b(:)g(:)g(:)g(;)g(a)2697
|
|
|
4214 |
3713 y FI(n)2751 3701 y FM(#)9 b FQ(X)2898 3713 y FI(n)2944
|
|
|
4215 |
3701 y FO(g)20 b FW(of)g(freshness)523 3801 y(assumptions.)h(F)o(or)h
|
|
|
4216 |
(e)o(xample)f FO(f)p FQ(a)13 b FM(#)g FQ(X)r(;)h(b)f
|
|
|
4217 |
FM(#)g FQ(X)7 b FO(g)13 b(`)g FN(fn)19 b FQ(a:X)h FO(\031)13
|
|
|
4218 |
b FN(fn)19 b FQ(b:X)29 b FW(is)24 b(a)e(v)n(alid)g(judgement)f(of)523
|
|
|
4219 |
3900 y(our)e FP(nominal)f(equational)g(lo)o(gic)p FW(.)h(Similarly)-5
|
|
|
4220 |
b(,)19 b(judgements)f(about)h(freshness)g(itself)h(will)h(tak)o(e)e
|
|
|
4221 |
(the)523 4000 y(form)1763 4097 y FO(r)24 b(`)e FQ(a)h
|
|
|
4222 |
FM(#)h FQ(t)f(:)1143 b FW(\(6\))523 4227 y FH(T)-8 b(o)18
|
|
|
4223 |
b(summarise:)f FW(W)-7 b(e)18 b(will)f(represent)f(languages)f(in)m(v)n
|
|
|
4224 |
(olving)g(binders)h(using)g(the)h(usual)f(notion)g(of)523
|
|
|
4225 |
4327 y(\002rst-order)j(terms)i(o)o(v)o(er)f(a)h(man)o(y-sorted)d
|
|
|
4226 |
(signature.)h(These)i(gi)n(v)o(e)f(us)h(terms)g(with:)g(distinguished)
|
|
|
4227 |
523 4426 y(constants)j(naming)f(bindable)g(entities,)h(that)g(we)h
|
|
|
4228 |
(call)g FP(atoms)p FW(;)f(terms)g FQ(a:t)h FW(e)o(xpressing)e(a)i
|
|
|
4229 |
(generic)523 4526 y(form)16 b(of)h FP(binding)e FW(of)i(an)g(atom)g
|
|
|
4230 |
FQ(a)h FW(in)f(a)g(term)g FQ(t)p FW(;)h(and)f(terms)g
|
|
|
4231 |
FQ(\031)s FK(\001)p FQ(X)24 b FW(representing)15 b(an)i(e)o(xplicit)f
|
|
|
4232 |
FP(permu-)523 4625 y(tation)21 b FW(of)h(atoms)g FQ(\031)k
|
|
|
4233 |
FW(w)o(aiting)c(to)g(be)g(applied)f(to)h(whate)n(v)o(er)e(term)i(is)h
|
|
|
4234 |
(substituted)e(for)h(the)g(v)n(ariable)523 4725 y FQ(X)7
|
|
|
4235 |
b FW(.)19 b(Section)f(2)i(presents)e(this)i(term-language)c(together)i
|
|
|
4236 |
(with)h(a)g(syntax-directed)e(inducti)n(v)o(e)g(def-)523
|
|
|
4237 |
4825 y(inition)31 b(of)h(the)g(pro)o(v)n(able)e(judgements)g(of)i(the)g
|
|
|
4238 |
(form)f(\(5\))h(and)f(\(6\))h(which)f(for)h FP(gr)l(ound)g
|
|
|
4239 |
FW(terms)523 4924 y(\(i.e.)23 b(ones)g(with)g(no)f(v)n(ariables\))g
|
|
|
4240 |
(agrees)h(with)g(the)g(usual)g(notions)f(of)h FQ(\013)p
|
|
|
4241 |
FW(-equi)n(v)n(alence)e(and)i(`not)f(a)p eop end
|
|
|
4242 |
%%Page: 4 4
|
|
|
4243 |
TeXDict begin 4 3 bop 523 448 a FW(free)27 b(v)n(ariable)f(of)5
|
|
|
4244 |
b('.)26 b(Ho)n(we)n(v)o(er)m(,)f(on)i(open)f(terms)h(our)f(judgements)g
|
|
|
4245 |
FP(dif)o(fer)j FW(from)d(these)i(standard)523 548 y(notions)h(and)g
|
|
|
4246 |
(appear)f(to)i(be)g(an)f(e)o(xtension)f(that)i(has)g(not)f(yet)h(been)f
|
|
|
4247 |
(studied)g(in)h(the)f(literature)523 648 y(\(including)22
|
|
|
4248 |
b([11,)12 b(24,)h(23]\).)22 b(Section)h(3)h(considers)f(uni\002cation)g
|
|
|
4249 |
(in)h(this)g(setting.)g(Solving)e(equali-)523 747 y(ties)g(between)f
|
|
|
4250 |
(abstractions)g FQ(a:t)k FO(\031)o FM(?)h FQ(a)1673 717
|
|
|
4251 |
y FL(0)1696 747 y FQ(:t)1749 717 y FL(0)1794 747 y FW(entails)c
|
|
|
4252 |
(solving)f(both)f(equalities)h FQ(t)26 b FO(\031)o FM(?)g(\()p
|
|
|
4253 |
FQ(a)14 b(a)3128 717 y FL(0)3151 747 y FM(\))p FK(\001)p
|
|
|
4254 |
FQ(t)3240 717 y FL(0)3285 747 y FW(and)523 847 y(freshness)j(problems)f
|
|
|
4255 |
FQ(a)23 b FM(#?)g FQ(t)1405 817 y FL(0)1428 847 y FW(.)18
|
|
|
4256 |
b(Therefore)d(our)i(general)f(form)g(of)h FP(nominal)f(uni\002cation)f
|
|
|
4257 |
(pr)l(oblem)523 946 y FW(is)k(a)g(\002nite)f(collection)f(of)h(indi)n
|
|
|
4258 |
(vidual)e(equality)i(and)f(freshness)h(problems.)e(Such)i(a)h(problem)d
|
|
|
4259 |
FQ(P)31 b FW(is)523 1046 y(solv)o(ed)21 b(by)h(pro)o(viding)d(not)j
|
|
|
4260 |
(only)f(a)i(substitution)e FQ(\033)26 b FW(\(of)c(terms)g(for)f(v)n
|
|
|
4261 |
(ariables\),)g(b)n(ut)h(also)h(a)f(fresh-)523 1146 y(ness)d(en)m
|
|
|
4262 |
(vironment)d FO(r)k FW(\(as)f(abo)o(v)o(e\),)e(which)h(together)g(ha)n
|
|
|
4263 |
(v)o(e)g(the)h(property)e(that)i FO(r)9 b(`)g FQ(\033)s
|
|
|
4264 |
FM(\()p FQ(t)p FM(\))24 b FO(\031)f FQ(\033)s FM(\()p
|
|
|
4265 |
FQ(t)3349 1116 y FL(0)3373 1146 y FM(\))523 1245 y FW(and)16
|
|
|
4266 |
b FO(r)9 b(`)g FQ(a)g FM(#)g FQ(\033)s FM(\()p FQ(t)1041
|
|
|
4267 |
1215 y FL(00)1085 1245 y FM(\))18 b FW(hold)e(for)g(each)g(indi)n
|
|
|
4268 |
(vidual)f(equality)h FQ(t)23 b FO(\031)o FM(?)h FQ(t)2430
|
|
|
4269 |
1215 y FL(0)2470 1245 y FW(and)16 b(freshness)h FQ(a)22
|
|
|
4270 |
b FM(#?)i FQ(t)3162 1215 y FL(00)3222 1245 y FW(in)16
|
|
|
4271 |
b(the)523 1345 y(problem)23 b FQ(P)12 b FW(.)26 b(Our)e(main)h(result)g
|
|
|
4272 |
(with)g(respect)g(to)g(uni\002cation)f(is)i(that)g FP(solvability)e(is)
|
|
|
4273 |
i(decidable)523 1445 y(and)g(that)h(solvable)g(pr)l(oblems)g(possess)h
|
|
|
4274 |
(most)g(g)o(ener)o(al)e(solutions)h FW(\(for)f(a)i(reasonably)d(ob)o
|
|
|
4275 |
(vious)523 1544 y(notion)j(of)g(`most)h(general'\).)e(The)i(proof)e(is)
|
|
|
4276 |
j(via)f(a)g(uni\002cation)f(algorithm)f(that)j(is)f(v)o(ery)f(simi-)523
|
|
|
4277 |
1644 y(lar)f(to)g(the)g(\002rst-order)f(algorithm)f(gi)n(v)o(en)h(in)h
|
|
|
4278 |
(the)g(no)n(w-common)c(transformational)i(style)i([18].)523
|
|
|
4279 |
1743 y(\(See)21 b([17,)f(Sect.)h(2.6])f(or)h([1,)f(Sect.)i(4.6])e(for)g
|
|
|
4280 |
(e)o(xpositions)f(of)i(this.\))g(Section)g(4)g(considers)f(the)h(re-)
|
|
|
4281 |
523 1843 y(lationship)e(of)h(our)f(v)o(ersion)f(of)i(`uni\002cation)e
|
|
|
4282 |
(modulo)g FQ(\013)p FW(-equi)n(v)n(alence')g(to)i(e)o(xisting)f
|
|
|
4283 |
(approaches.)523 1943 y(Section)26 b(5)h(assesses)h(what)f(has)g(been)f
|
|
|
4284 |
(achie)n(v)o(ed)f(and)h(the)g(prospects)g(for)g(applications.)f(T)-7
|
|
|
4285 |
b(o)27 b(ap-)523 2042 y(preciate)c(the)i(kind)e(of)h(problem)e(that)i
|
|
|
4286 |
(nominal)f(uni\002cation)g(solv)o(es,)g(you)h(might)f(lik)o(e)h(to)h
|
|
|
4287 |
(try)e(the)523 2142 y(follo)n(wing)c(quiz)h(about)f(the)i
|
|
|
4288 |
FQ(\025)p FW(-calculus)f([2])f(before)g(we)i(apply)f(our)f(algorithm)g
|
|
|
4289 |
(to)i(solv)o(e)e(it)j(at)e(the)523 2242 y(end)g(of)g(Section)f(3.)p
|
|
|
4290 |
523 2304 2882 4 v 523 2912 4 609 v 551 2381 a FU(Quiz)j
|
|
|
4291 |
FT(Assuming)h(that)f FS(a)g FT(and)h FS(b)f FT(ar)m(e)h(distinct)f
|
|
|
4292 |
(variables)p FV(,)h(is)f(it)f(possible)i(to)f(\002nd)h
|
|
|
4293 |
FS(\025)p FV(-terms)e FS(M)3062 2389 y FG(1)3097 2381
|
|
|
4294 |
y FS(;)13 b(:)h(:)f(:)g(;)g(M)3342 2389 y FG(7)551 2472
|
|
|
4295 |
y FV(that)19 b(mak)o(e)h(the)f(follo)n(wing)g(pairs)g(of)g(terms)g
|
|
|
4296 |
FS(\013)p FV(-equi)n(v)n(alent?)737 2581 y(1.)k FS(\025a:\025b:)p
|
|
|
4297 |
FF(\()p FS(M)1126 2589 y FG(1)1173 2581 y FS(b)p FF(\))c
|
|
|
4298 |
FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)1744 2589
|
|
|
4299 |
y FG(1)1779 2581 y FF(\))283 b FV(3.)24 b FS(\025a:\025b:)p
|
|
|
4300 |
FF(\()p FS(b)12 b(M)2527 2589 y FG(4)2562 2581 y FF(\))19
|
|
|
4301 |
b FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)3100
|
|
|
4302 |
2589 y FG(5)3135 2581 y FF(\))737 2684 y FV(2.)23 b FS(\025a:\025b:)p
|
|
|
4303 |
FF(\()p FS(M)1126 2692 y FG(2)1173 2684 y FS(b)p FF(\))c
|
|
|
4304 |
FV(and)g FS(\025b:\025a:)p FF(\()p FS(a)12 b(M)1744 2692
|
|
|
4305 |
y FG(3)1779 2684 y FF(\))283 b FV(4.)24 b FS(\025a:\025b:)p
|
|
|
4306 |
FF(\()p FS(b)12 b(M)2527 2692 y FG(6)2562 2684 y FF(\))19
|
|
|
4307 |
b FV(and)g FS(\025a:\025a:)p FF(\()p FS(a)12 b(M)3108
|
|
|
4308 |
2692 y FG(7)3143 2684 y FF(\))p FV(.)551 2778 y(If)26
|
|
|
4309 |
b(it)g(is)h(possible)g(to)f(\002nd)h(a)g(solution)g(for)f(an)o(y)i(of)e
|
|
|
4310 |
(these)h(four)g(problems,)g(can)g(you)h(describe)f(what)g(all)551
|
|
|
4311 |
2869 y(possible)20 b(solutions)f(for)g(that)g(problem)h(are)f(lik)o(e?)
|
|
|
4312 |
3377 2836 y
|
|
|
4313 |
gsave currentpoint currentpoint translate 180 neg rotate neg exch
|
|
|
4314 |
neg exch translate
|
|
|
4315 |
3377 2836 a FU(Answers:)f FV(see)h(Example)h(2.)4149
|
|
|
4316 |
2836 y
|
|
|
4317 |
currentpoint grestore moveto
|
|
|
4318 |
4149 2836 a 3402 2912 4 609 v 523 2915 2882 4
|
|
|
4319 |
v 523 3137 a FR(2)99 b(Nominal)25 b(equational)g(logic)523
|
|
|
4320 |
3330 y FW(W)-7 b(e)27 b(tak)o(e)f(a)g(concrete)f(approach)e(to)j(the)g
|
|
|
4321 |
(syntax)f(of)h(binders)f(in)h(which)f(bound)f(entities)i(are)g(e)o(x-)
|
|
|
4322 |
523 3430 y(plicitly)18 b(named.)f(Furthermore)f(we)j(separate)f(the)g
|
|
|
4323 |
(names)g(of)h(bound)d(entities)j(from)e(the)i(names)f(of)523
|
|
|
4324 |
3530 y(v)n(ariables,)h(which)g(is)h(inspired)f(for)g(e)o(xample)f(by)h
|
|
|
4325 |
(the)h FQ(\031)s FW(-calculus)f([21],)f(in)i(which)f(the)g(restriction)
|
|
|
4326 |
523 3629 y(operator)28 b(binds)h(channel)g(names)g(and)g(these)h(are)g
|
|
|
4327 |
(quite)f(dif)n(ferent)f(from)h(names)h(of)f(unkno)n(wn)523
|
|
|
4328 |
3729 y(processes.)e(Names)h(of)g(bound)e(entities)i(will)g(be)g(called)
|
|
|
4329 |
g FP(atoms)p FW(.)f(This)h(is)h(partly)e(for)g(historical)523
|
|
|
4330 |
3828 y(reasons)e(\(stemming)f(from)g(the)h(w)o(ork)g(by)g(the)g(second)
|
|
|
4331 |
f(tw)o(o)i(authors)e([11]\))g(and)g(partly)h(to)g(indi-)523
|
|
|
4332 |
3928 y(cate)j(that)g(the)g(internal)f(structure)g(of)h(such)g(names)f
|
|
|
4333 |
(is)i(irrele)n(v)n(ant)e(to)h(us:)g(all)h(we)f(care)g(about)f(is)523
|
|
|
4334 |
4028 y(their)c(identity)f(\(i.e.)h(whether)f(or)h(not)f(one)h(atom)g
|
|
|
4335 |
(is)h(the)f(same)g(as)h(another\))d(and)h(that)h(the)h(supply)523
|
|
|
4336 |
4127 y(of)c(atoms)g(is)h(ine)o(xhaustible.)648 4227 y(Although)h(there)
|
|
|
4337 |
h(are)h(se)n(v)o(eral)f(general)g(frame)n(w)o(orks)f(in)i(the)g
|
|
|
4338 |
(literature)f(for)g(specifying)g(lan-)523 4327 y(guages)30
|
|
|
4339 |
b(with)h(binders,)e(not)h(all)h(of)g(them)f(meet)g(the)h(requirements)e
|
|
|
4340 |
(mentioned)f(in)j(the)g(pre)n(vi-)523 4426 y(ous)23 b(paragraph.)e(Use)
|
|
|
4341 |
j(of)f(the)g(simply)g(typed)f FQ(\025)p FW(-calculus)h(for)f(this)i
|
|
|
4342 |
(purpose)e(is)i(common;)e(b)n(ut)h(as)523 4526 y(discussed)e(in)h(the)f
|
|
|
4343 |
(Introduction,)d(it)23 b(leads)e(to)h(a)g(problematic)d(uni\002cation)h
|
|
|
4344 |
(theory)-5 b(.)20 b(Among)g FP(\002r)o(st-)523 4625 y(or)m(der)e
|
|
|
4345 |
FW(frame)n(w)o(orks,)c(Plotkin')-5 b(s)16 b(notion)f(of)h
|
|
|
4346 |
FP(binding)f(signatur)m(e)g FW([25,)d(9],)k(being)f(unsorted,)g
|
|
|
4347 |
(equates)523 4725 y(names)k(used)g(in)h(binding)d(with)j(names)f(of)g
|
|
|
4348 |
(v)n(ariables)f(standing)h(for)f(unkno)n(wn)f(terms;)j(so)f(it)i(is)f
|
|
|
4349 |
(not)523 4825 y(suf)n(\002ciently)i(general)g(for)h(us.)g(A)g
|
|
|
4350 |
(\002rst-order)f(frame)n(w)o(ork)f(that)i(does)g(meet)g(our)f
|
|
|
4351 |
(requirements)g(is)523 4924 y(the)17 b(notion)g(of)g
|
|
|
4352 |
FP(nominal)f(alg)o(ebr)o(as)g FW(in)i([16].)e(The)h FP(nominal)f
|
|
|
4353 |
(signatur)m(es)h FW(that)h(we)f(use)h(in)g(this)g(paper)p
|
|
|
4354 |
eop end
|
|
|
4355 |
%%Page: 5 5
|
|
|
4356 |
TeXDict begin 5 4 bop 523 448 a FW(are)26 b(a)h(mild)e(\(b)n(ut)h
|
|
|
4357 |
(practically)f(useful\))g(generalisation)g(of)h(nominal)e(algebras)i
|
|
|
4358 |
(in)g(which)f(name-)523 548 y(abstraction)h(and)g(pairing)g(can)g(be)h
|
|
|
4359 |
(mix)o(ed)f(freely)g(in)h(arities)g(\(rather)f(than)g(insisting)h(as)h
|
|
|
4360 |
(in)f([16])523 648 y(that)20 b(the)h(ar)o(gument)c(sort)k(of)f(a)g
|
|
|
4361 |
(function)f(symbol)g(be)h(normalised)f(to)h(a)h(tuple)e(of)h
|
|
|
4362 |
(abstractions\).)523 796 y FH(De\002nition)g(1.)41 b
|
|
|
4363 |
FP(A)19 b FE(nominal)g(signature)g FP(is)h(speci\002ed)e(by:)g(a)h(set)
|
|
|
4364 |
h(of)f FE(sorts)h(of)f(atoms)f FP(\(typical)g(sym-)523
|
|
|
4365 |
895 y(bol)j FQ(\027)5 b FP(\);)22 b(a)g(disjoint)f(set)i(of)f
|
|
|
4366 |
FE(sorts)h(of)e(data)g FP(\(typical)g(symbol)g FQ(\016)s
|
|
|
4367 |
FP(\);)h(and)f(a)g(set)i(of)f FE(function)g(symbols)523
|
|
|
4368 |
995 y FP(\(typical)g(symbol)h FQ(f)9 b FP(\),)22 b(eac)o(h)g(of)h(whic)
|
|
|
4369 |
o(h)g(has)f(an)h FE(arity)g FP(of)g(the)g(form)g FQ(\034)38
|
|
|
4370 |
b FO(!)28 b FQ(\016)s FP(.)23 b(Her)m(e)h FQ(\034)33
|
|
|
4371 |
b FP(r)o(ang)o(es)22 b(o)o(ver)523 1094 y(\(compound\))d
|
|
|
4372 |
FE(sorts)24 b FP(given)e(by)g(the)h(gr)o(ammar)e FQ(\034)37
|
|
|
4373 |
b FM(::=)27 b FQ(\027)33 b FO(j)27 b FQ(\016)k FO(j)c
|
|
|
4374 |
FM(1)g FO(j)g FQ(\034)j FO(\002)20 b FQ(\034)37 b FO(j)27
|
|
|
4375 |
b(h)p FQ(\027)5 b FO(i)p FQ(\034)38 b FP(.)23 b(T)-8
|
|
|
4376 |
b(erms)24 b(of)f(sort)523 1194 y FO(h)p FQ(\027)5 b FO(i)p
|
|
|
4377 |
FQ(\034)35 b FP(ar)m(e)25 b(binding)d(abstr)o(actions)h(of)h(atoms)g
|
|
|
4378 |
(of)g(sort)h FQ(\027)30 b FP(o)o(ver)25 b(terms)g(of)f(sort)h
|
|
|
4379 |
FQ(\034)9 b FP(.)25 b(W)-8 b(e)25 b(will)g(e)n(xplain)523
|
|
|
4380 |
1294 y(the)20 b(syntax)g(and)f(pr)l(operties)i(of)f(suc)o(h)g(terms)h
|
|
|
4381 |
(in)f(a)h(moment.)523 1423 y(Example)e(1.)41 b FW(Here)20
|
|
|
4382 |
b(is)h(a)g(nominal)e(signature)g(for)g(e)o(xpressions)g(in)i(a)f(small)
|
|
|
4383 |
h(fragment)d(of)i(ML:)706 1562 y FV(sort)f(of)g(atoms:)42
|
|
|
4384 |
b FD(vid)229 b FV(function)20 b(symbols:)h FC(vr)h FF(:)g
|
|
|
4385 |
FD(vid)29 b FB(!)21 b FD(exp)706 1653 y FV(sort)e(of)g(data:)96
|
|
|
4386 |
b FD(exp)776 b FC(app)22 b FF(:)g FD(exp)h FB(\002)16
|
|
|
4387 |
b FD(exp)27 b FB(!)21 b FD(exp)2044 1744 y FC(fn)h FF(:)g
|
|
|
4388 |
FB(h)p FD(vid)7 b FB(i)p FD(exp)27 b FB(!)21 b FD(exp)2044
|
|
|
4389 |
1836 y FC(lv)h FF(:)g FD(exp)g FB(\002)17 b(h)p FD(vid)8
|
|
|
4390 |
b FB(i)p FD(exp)26 b FB(!)c FD(exp)2044 1927 y FC(lf)g
|
|
|
4391 |
FF(:)g FB(h)p FD(vid)7 b FB(i)p FF(\(\()p FB(h)p FD(vid)g
|
|
|
4392 |
FB(i)p FD(exp)f FF(\))17 b FB(\002)f FD(exp)6 b FF(\))21
|
|
|
4393 |
b FB(!)g FD(exp)27 b FV(.)523 2068 y FW(The)34 b(function)f(symbol)g
|
|
|
4394 |
FN(vr)i FW(constructs)e(terms)i(of)f(sort)g FA(exp)41
|
|
|
4395 |
b FW(representing)32 b(v)n(alue)i(identi\002ers)523 2167
|
|
|
4396 |
y(\(named)19 b(by)h(atoms)g(of)h(sort)f FA(vid)10 b FW(\);)21
|
|
|
4397 |
b FN(app)e FW(constructs)h(application)f(e)o(xpressions)g(from)h(pairs)
|
|
|
4398 |
g(of)g(e)o(x-)523 2267 y(pressions;)d FN(fn)o FW(,)h
|
|
|
4399 |
FN(lv)e FW(and)h FN(lf)g FW(construct)f(terms)h(representing)e
|
|
|
4400 |
(respecti)n(v)o(ely)h(function)f(abstractions)523 2366
|
|
|
4401 |
y(\()p FN(fn)20 b(x)i FM(=)p FQ(>)g FN(e)p FW(\),)e(local)h(v)n(alue)e
|
|
|
4402 |
(declarations)g(\()p FN(let)g(val)h(x)i FM(=)h FN(e1)d(in)g(e2)g(end)o
|
|
|
4403 |
FW(\))g(and)g(local)g(recursi)n(v)o(e)523 2466 y(function)k
|
|
|
4404 |
(declarations)g(\()p FN(let)19 b(fun)g(f)i(x)h FM(=)h
|
|
|
4405 |
FN(e1)d(in)g(e2)g(end)n FW(\).)26 b(The)e(arities)i(of)f(the)g
|
|
|
4406 |
(function)f(sym-)523 2566 y(bols)d(specify)g(which)g(are)h(binders)e
|
|
|
4407 |
(and)h(in)h(which)f(w)o(ay)g(their)g(ar)o(guments)f(are)h(bound.)f
|
|
|
4408 |
(This)h(kind)523 2665 y(of)k(speci\002cation)g(of)g(binding)f(scopes)h
|
|
|
4409 |
(is)h(of)f(course)g(a)h(feature)e(of)i FP(higher)n(-or)m(der)e(abstr)o
|
|
|
4410 |
(act)h(syn-)523 2765 y(tax)16 b FW([22],)f(using)h(function)e(types)i
|
|
|
4411 |
FQ(\027)5 b FO(!)p FQ(\034)27 b FW(in)16 b(simply)g(typed)f
|
|
|
4412 |
FQ(\025)p FW(-calculus)g(where)h(we)g(use)h(abstraction)523
|
|
|
4413 |
2865 y(sorts)k FO(h)p FQ(\027)5 b FO(i)p FQ(\034)k FW(.)22
|
|
|
4414 |
b(W)-7 b(e)22 b(shall)e(see)h(that)g(the)f(latter)h(ha)n(v)o(e)f(much)f
|
|
|
4415 |
(more)h(elementary)f(\(indeed,)f(\002rst-order\))523
|
|
|
4416 |
2964 y(properties)27 b(compared)g(with)h(the)h(former)-5
|
|
|
4417 |
b(.)27 b(T)-7 b(o)29 b(mak)o(e)f(this)h(point)f(clear)g(we)h
|
|
|
4418 |
(deliberately)e(use)i(a)523 3064 y(\002rst-order)g(syntax)g(for)g
|
|
|
4419 |
(terms,)h(and)g FP(not)h FW(higher)n(-order)c(abstract)j(syntax,)f
|
|
|
4420 |
(although)f(we)i(often)523 3163 y(refer)19 b(to)i(abstractions,)e
|
|
|
4421 |
(binders)g(and)h(free)f(atoms)i(by)e(analogy)g(with)h(the)h
|
|
|
4422 |
FQ(\025)p FW(-calculus.)523 3326 y FH(De\002nition)f(2.)41
|
|
|
4423 |
b FP(Given)26 b(a)g(nominal)f(signatur)m(e)o(,)g(we)h(assume)g(that)g
|
|
|
4424 |
(ther)m(e)g(ar)m(e)g(countably)e(in\002nite)523 3425
|
|
|
4425 |
y(and)j(pairwise)i(disjoint)f(sets)i(of)e FE(atoms)g
|
|
|
4426 |
FP(\(typical)g(symbol)g FQ(a)p FP(\))g(for)h(eac)o(h)f(sort)h(of)f
|
|
|
4427 |
(atoms)g FQ(\027)5 b FP(,)29 b(and)523 3525 y FE(variables)22
|
|
|
4428 |
b FP(\(typical)g(symbol)g FQ(X)7 b FP(\))22 b(for)h(eac)o(h)f(sort)h
|
|
|
4429 |
FQ(\034)9 b FP(.)24 b(The)f FE(terms)g FP(o)o(ver)g(a)f(nominal)g
|
|
|
4430 |
(signatur)m(e)f(and)523 3624 y(their)27 b(sorts)h(ar)m(e)f(inductively)
|
|
|
4431 |
f(de\002ned)f(as)i(follows,)g(wher)m(e)g(we)h(write)g
|
|
|
4432 |
FQ(t)35 b FM(:)g FQ(\034)i FP(to)27 b(indicate)f(that)g(a)523
|
|
|
4433 |
3724 y(term)21 b FQ(t)g FP(has)f(sort)h FQ(\034)9 b FP(.)523
|
|
|
4434 |
3878 y FH(Unit)21 b(v)o(alue)40 b FO(hi)24 b FM(:)f(1)p
|
|
|
4435 |
FP(.)523 3975 y FH(P)o(airs)41 b FO(h)p FQ(t)810 3987
|
|
|
4436 |
y FJ(1)848 3975 y FQ(;)14 b(t)915 3987 y FJ(2)952 3975
|
|
|
4437 |
y FO(i)23 b FM(:)g FQ(\034)1089 3987 y FJ(1)1145 3975
|
|
|
4438 |
y FO(\002)18 b FQ(\034)1264 3987 y FJ(2)1302 3975 y FP(,)i(if)h
|
|
|
4439 |
FQ(t)1440 3987 y FJ(1)1501 3975 y FM(:)i FQ(\034)1583
|
|
|
4440 |
3987 y FJ(1)1641 3975 y FP(and)c FQ(t)1816 3987 y FJ(2)1877
|
|
|
4441 |
3975 y FM(:)k FQ(\034)1959 3987 y FJ(2)1996 3975 y FP(.)523
|
|
|
4442 |
4072 y FH(Data)40 b FQ(f)23 b(t)g FM(:)g FQ(\016)s FP(,)d(if)h
|
|
|
4443 |
FQ(f)30 b FP(is)21 b(a)f(function)f(symbol)h(of)g(arity)h
|
|
|
4444 |
FQ(\034)32 b FO(!)24 b FQ(\016)f FP(and)d FQ(t)j FM(:)g
|
|
|
4445 |
FQ(\034)9 b FP(.)523 4169 y FH(Atoms)41 b FQ(a)23 b FM(:)g
|
|
|
4446 |
FQ(\027)5 b FP(,)21 b(if)g FQ(a)g FP(is)g(an)e(atom)h(of)h(sort)g
|
|
|
4447 |
FQ(\027)5 b FP(.)523 4267 y FH(Atom-abstraction)39 b
|
|
|
4448 |
FQ(a:t)23 b FM(:)g FO(h)p FQ(\027)5 b FO(i)p FQ(\034)k
|
|
|
4449 |
FP(,)22 b(if)f FQ(a)f FP(is)i(an)d(atom)h(of)g(sort)h
|
|
|
4450 |
FQ(\027)26 b FP(and)20 b FQ(t)j FM(:)g FQ(\034)9 b FP(.)523
|
|
|
4451 |
4364 y FH(Suspension)42 b FQ(\031)s FK(\001)p FQ(X)d
|
|
|
4452 |
FM(:)34 b FQ(\034)9 b FP(,)26 b(if)g FQ(\031)37 b FM(=)32
|
|
|
4453 |
b(\()p FQ(a)1624 4376 y FJ(1)1675 4364 y FQ(b)1711 4376
|
|
|
4454 |
y FJ(1)1748 4364 y FM(\)\()p FQ(a)1856 4376 y FJ(2)1908
|
|
|
4455 |
4364 y FQ(b)1944 4376 y FJ(2)1981 4364 y FM(\))14 b FO(\001)g(\001)g
|
|
|
4456 |
(\001)f FM(\()p FQ(a)2213 4376 y FI(n)2273 4364 y FQ(b)2309
|
|
|
4457 |
4376 y FI(n)2354 4364 y FM(\))26 b FP(is)g(a)g(\002nite)f(list)i(whose)
|
|
|
4458 |
e(elements)664 4463 y FM(\()p FQ(a)740 4475 y FI(i)782
|
|
|
4459 |
4463 y FQ(b)818 4475 y FI(i)845 4463 y FM(\))g FP(ar)m(e)f(pair)o(s)g
|
|
|
4460 |
(of)g(atoms,)g(with)g FQ(a)1772 4475 y FI(i)1824 4463
|
|
|
4461 |
y FP(and)f FQ(b)2009 4475 y FI(i)2061 4463 y FP(of)h(the)f(same)i
|
|
|
4462 |
(sort,)f(and)f FQ(X)31 b FP(is)24 b(a)g(variable)f(of)664
|
|
|
4463 |
4563 y(sort)e FQ(\034)9 b FP(.)21 b(In)f(the)g(case)h(that)f
|
|
|
4464 |
FQ(\031)k FP(is)d(the)f(empty)g(list)h FM([])p FP(,)g(we)g(just)g
|
|
|
4465 |
(write)g FQ(X)27 b FP(for)21 b FQ(\031)s FK(\001)p FQ(X)7
|
|
|
4466 |
b FP(.)523 4725 y FW(Recall)28 b(that)g(e)n(v)o(ery)e(\002nite)i
|
|
|
4467 |
(permutation)d(can)i(be)h(e)o(xpressed)e(as)i(a)g(composition)e(of)h
|
|
|
4468 |
(sw)o(appings)523 4825 y FM(\()p FQ(a)599 4837 y FI(i)641
|
|
|
4469 |
4825 y FQ(b)677 4837 y FI(i)704 4825 y FM(\))p FW(;)f(the)f(list)h
|
|
|
4470 |
FQ(\031)j FW(of)c(pairs)g(of)f(atoms)h(occurring)e(in)i(a)h(suspension)
|
|
|
4471 |
e(term)g FQ(\031)s FK(\001)p FQ(X)32 b FW(speci\002es)26
|
|
|
4472 |
b(a)f(\002-)523 4924 y(nite)c(permutation)d(of)i(atoms)h(w)o(aiting)f
|
|
|
4473 |
(to)h(be)f(applied)g(once)f(we)i(kno)n(w)f(more)f(about)h(the)h(v)n
|
|
|
4474 |
(ariable)p eop end
|
|
|
4475 |
%%Page: 6 6
|
|
|
4476 |
TeXDict begin 6 5 bop 523 369 2882 4 v 523 975 4 607
|
|
|
4477 |
v 1156 532 a FF([])q Fz(\001)p FS(a)1288 489 y Fy(def)1295
|
|
|
4478 |
532 y FF(=)30 b FS(a)784 742 y FF(\(\()p FS(a)885 750
|
|
|
4479 |
y FG(1)932 742 y FS(a)973 750 y FG(2)1007 742 y FF(\))22
|
|
|
4480 |
b(::)g FS(\031)s FF(\))p Fz(\001)o FS(a)1288 699 y Fy(def)1295
|
|
|
4481 |
742 y FF(=)1385 566 y Fx(8)1385 634 y(>)1385 656 y(<)1385
|
|
|
4482 |
791 y(>)1385 813 y(:)1453 636 y FS(a)1494 644 y FG(1)1640
|
|
|
4483 |
636 y FV(if)c FS(\031)s Fz(\001)o FS(a)j FF(=)g FS(a)1959
|
|
|
4484 |
644 y FG(2)1453 745 y FS(a)1494 753 y FG(2)1640 745 y
|
|
|
4485 |
FV(if)d FS(\031)s Fz(\001)o FS(a)j FF(=)g FS(a)1959 753
|
|
|
4486 |
y FG(1)1453 855 y FS(\031)s Fz(\001)o FS(a)75 b FV(otherwise)2520
|
|
|
4487 |
469 y FS(\031)s Fz(\001)p FB(hi)2675 426 y Fy(def)2682
|
|
|
4488 |
469 y FF(=)30 b FB(hi)2362 587 y FS(\031)s Fz(\001)o
|
|
|
4489 |
FB(h)p FS(t)2491 595 y FG(1)2525 587 y FS(;)13 b(t)2587
|
|
|
4490 |
595 y FG(2)2622 587 y FB(i)2675 545 y Fy(def)2682 587
|
|
|
4491 |
y FF(=)30 b FB(h)p FS(\031)s Fz(\001)o FS(t)2901 595
|
|
|
4492 |
y FG(1)2935 587 y FS(;)14 b(\031)s Fz(\001)o FS(t)3069
|
|
|
4493 |
595 y FG(2)3103 587 y FB(i)2435 705 y FS(\031)s Fz(\001)o
|
|
|
4494 |
FF(\()p FS(f)21 b(t)p FF(\))2675 663 y Fy(def)2682 705
|
|
|
4495 |
y FF(=)30 b FS(f)21 b FF(\()p FS(\031)s Fz(\001)o FS(t)p
|
|
|
4496 |
FF(\))2431 824 y FS(\031)s Fz(\001)o FF(\()p FS(a:t)p
|
|
|
4497 |
FF(\))2675 781 y Fy(def)2682 824 y FF(=)30 b(\()p FS(\031)s
|
|
|
4498 |
Fz(\001)o FS(a)p FF(\))p FS(:)p FF(\()p FS(\031)s Fz(\001)o
|
|
|
4499 |
FS(t)p FF(\))2358 942 y FS(\031)s Fz(\001)o FF(\()p FS(\031)2506
|
|
|
4500 |
910 y Fw(0)2528 942 y Fz(\001)o FS(X)6 b FF(\))2675 899
|
|
|
4501 |
y Fy(def)2682 942 y FF(=)30 b(\()p FS(\031)s FF(@)p FS(\031)2956
|
|
|
4502 |
910 y Fw(0)2977 942 y FF(\))p Fz(\001)p FS(X)d(:)p 3402
|
|
|
4503 |
975 V 523 978 2882 4 v 1360 1121 a FU(Fig)o(.)12 b(1.)18
|
|
|
4504 |
b FV(Permutation)h(action)h(on)f(terms,)f FS(\031)s Fz(\001)p
|
|
|
4505 |
FS(t)p FV(.)523 1423 y FQ(X)36 b FW(\(by)28 b(substituting)f(for)h(it,)
|
|
|
4506 |
i(for)e(e)o(xample\).)e(W)-7 b(e)30 b(represent)e(\002nite)h
|
|
|
4507 |
(permutations)d(in)j(this)g(w)o(ay)523 1523 y(because)f(it)g(is)h
|
|
|
4508 |
(really)f(the)g(operation)f(of)g(sw)o(apping)h(which)f(plays)h(a)h
|
|
|
4509 |
(fundamental)c(role)j(in)g(the)523 1622 y(theory)-5 b(.)19
|
|
|
4510 |
b(Since,)i(semantically)g(speaking)f(\(see)h(Remark)g(1)g(belo)n(w)g
|
|
|
4511 |
(about)f(semantics\),)h(sw)o(apping)523 1722 y(commutes)i(with)i(all)g
|
|
|
4512 |
(term-forming)d(operations,)g(we)j(can)f(normalise)g(terms)g(in)m(v)n
|
|
|
4513 |
(olving)f(an)h(e)o(x-)523 1821 y(plicit)j(sw)o(apping)f(operation)g(by)
|
|
|
4514 |
g(pushing)g(the)h(sw)o(ap)g(in)h(as)f(f)o(ar)g(as)h(it)g(will)g(go,)e
|
|
|
4515 |
(until)h(it)h(reaches)523 1921 y(a)d(v)n(ariable)e(\(cf.)g(Fig.)h(1\);)
|
|
|
4516 |
g(the)g(terms)h(in)f(Def.)g(2)g(are)g(all)h(normalised)d(in)j(this)f(w)
|
|
|
4517 |
o(ay)-5 b(,)24 b(with)g(e)o(xplicit)523 2021 y(sw)o(appings)19
|
|
|
4518 |
b(`piled)h(up')f(in)i(front)e(of)h(v)n(ariables)f(gi)n(ving)g(what)h
|
|
|
4519 |
(we)h(ha)n(v)o(e)e(called)h FP(suspensions)p FW(.)648
|
|
|
4520 |
2128 y(W)-7 b(e)16 b(wish)g(to)g(gi)n(v)o(e)f(a)h(de\002nition)e(of)h
|
|
|
4521 |
FQ(\013)p FW(-equi)n(v)n(alence)f(for)h(terms)g(o)o(v)o(er)g(a)h
|
|
|
4522 |
(nominal)e(signature)g(that)523 2227 y(is)26 b(respected)e(by)g
|
|
|
4523 |
(substitution)h(of)f(terms)h(for)g(v)n(ariables,)e(e)n(v)o(en)h(though)
|
|
|
4524 |
g(the)g(latter)i(may)e(in)m(v)n(olv)o(e)523 2327 y(capture)19
|
|
|
4525 |
b(of)g(atoms)h(by)f(binders.)g(T)-7 b(o)20 b(do)f(so)h(we)h(will)f
|
|
|
4526 |
(need)f(to)h(mak)o(e)g(use)g(of)f(an)h(auxiliary)e(relation)523
|
|
|
4527 |
2426 y(of)24 b FP(fr)m(eshness)h FW(between)e(atoms)i(and)e(terms,)h
|
|
|
4528 |
(whose)g(intended)f(meaning)g(is)i(that)g(the)f(atom)g(does)523
|
|
|
4529 |
2526 y(not)j(occur)f(free)h(in)h(an)o(y)e(substitution)h(instance)g(of)
|
|
|
4530 |
g(the)g(term.)g(As)h(discussed)f(in)h(the)f(Introduc-)523
|
|
|
4531 |
2626 y(tion,)21 b(our)f(judgements)f(about)i(term)f(equi)n(v)n(alence)f
|
|
|
4532 |
(\()p FQ(t)25 b FO(\031)f FQ(t)2227 2596 y FL(0)2250
|
|
|
4533 |
2626 y FW(\))e(need)e(to)h(contain)f(hypotheses)f(about)523
|
|
|
4534 |
2725 y(the)26 b(freshness)g(of)f(atoms)h(with)h(respect)e(to)i(v)n
|
|
|
4535 |
(ariables)e(\()p FQ(a)33 b FM(#)i FQ(X)7 b FW(\);)25
|
|
|
4536 |
b(and)h(the)g(same)g(goes)g(for)f(our)523 2825 y(judgements)g(about)g
|
|
|
4537 |
(freshness)h(itself)g(\()p FQ(a)34 b FM(#)g FQ(t)p FW(\).)27
|
|
|
4538 |
b(Figure)e(2)h(gi)n(v)o(es)g(a)h(syntax-directed)c(inducti)n(v)o(e)523
|
|
|
4539 |
2925 y(de\002nition)c(of)h(equi)n(v)n(alence)e(and)i(freshness)f(using)
|
|
|
4540 |
h(judgements)f(of)h(the)g(form)1463 3122 y FO(r)k(`)e
|
|
|
4541 |
FQ(t)i FO(\031)e FQ(t)1800 3087 y FL(0)1906 3122 y FW(and)82
|
|
|
4542 |
b FO(r)24 b(`)e FQ(a)h FM(#)h FQ(t)523 3319 y FW(where)d
|
|
|
4543 |
FQ(t)h FW(and)f FQ(t)972 3289 y FL(0)1018 3319 y FW(are)g(terms)h(of)f
|
|
|
4544 |
(the)h(same)g(sort)g(o)o(v)o(er)e(a)i(gi)n(v)o(en)e(nominal)h
|
|
|
4545 |
(signature,)f FQ(a)i FW(is)h(an)e(atom,)523 3419 y(and)31
|
|
|
4546 |
b(the)g FH(fr)o(eshness)g(en)m(vir)o(onment)f FO(r)i
|
|
|
4547 |
FW(is)g(a)g(\002nite)f(set)h(of)f FH(fr)o(eshness)g(constraints)g
|
|
|
4548 |
FQ(a)43 b FM(#)h FQ(X)7 b FW(,)523 3518 y(each)21 b(speci\002ed)g(by)g
|
|
|
4549 |
(an)h(atom)f(and)f(a)i(v)n(ariable.)e(Rules)i(\()p FO(\031)p
|
|
|
4550 |
FW(-abstraction-2\),)c(\()p FO(\031)p FW(-suspension\))h(and)523
|
|
|
4551 |
3618 y(\()p FM(#)p FW(-suspension\))f(in)j(Fig.)f(2)g(mak)o(e)g(use)g
|
|
|
4552 |
(of)g(the)g(follo)n(wing)f(de\002nitions.)523 3828 y
|
|
|
4553 |
FH(De\002nition)h(3.)41 b FP(Recall)21 b(fr)l(om)h(Def)o(.)g(2)g(that)f
|
|
|
4554 |
(we)h(specify)g FE(\002nite)g(permutations)f(of)g(atoms)g
|
|
|
4555 |
FP(by)h(\002nite)523 3928 y(lists)17 b FM(\()p FQ(a)749
|
|
|
4556 |
3940 y FJ(1)801 3928 y FQ(b)837 3940 y FJ(1)873 3928
|
|
|
4557 |
y FQ(;)d FM(\)\()p FQ(a)1018 3940 y FJ(2)1070 3928 y
|
|
|
4558 |
FQ(b)1106 3940 y FJ(2)1143 3928 y FM(\))g FO(\001)g(\001)g(\001)g
|
|
|
4559 |
FM(\()p FQ(a)1376 3940 y FI(n)1435 3928 y FQ(b)1471 3940
|
|
|
4560 |
y FI(n)1516 3928 y FM(\))i FP(r)m(epr)m(esenting)f(the)h(composition)f
|
|
|
4561 |
(of)h(\002nitely)f(many)g(swappings)523 4028 y FM(\()p
|
|
|
4562 |
FQ(a)599 4040 y FI(i)641 4028 y FQ(b)677 4040 y FI(i)704
|
|
|
4563 |
4028 y FM(\))p FP(,)25 b(with)h FQ(a)995 4040 y FI(i)1048
|
|
|
4564 |
4028 y FP(and)e FQ(b)1234 4040 y FI(i)1287 4028 y FP(of)h(the)g(same)g
|
|
|
4565 |
(sort.)g(Since)g(we)g(will)h(apply)e(permutations)g(to)h(terms)h(on)523
|
|
|
4566 |
4127 y(the)g(left,)g(the)f(or)m(der)h(of)g(the)f(composition)f(is)j(fr)
|
|
|
4567 |
l(om)f(right)g(to)g(left.)g(So)f(with)h(this)g(r)m(epr)m(esentation,)
|
|
|
4568 |
523 4227 y(the)32 b(composition)f(of)i(a)f(permutation)f
|
|
|
4569 |
FQ(\031)37 b FP(followed)32 b(by)g(a)h(swap)f FM(\()p
|
|
|
4570 |
FQ(a)14 b(b)p FM(\))33 b FP(is)h(given)d(by)i(list-cons,)523
|
|
|
4571 |
4327 y(written)28 b FM(\()p FQ(a)14 b(b)p FM(\))36 b(::)g
|
|
|
4572 |
FQ(\031)s FP(;)28 b(the)f(composition)f(of)h FQ(\031)k
|
|
|
4573 |
FP(followed)c(by)g(another)f(permutation)g FQ(\031)3091
|
|
|
4574 |
4296 y FL(0)3142 4327 y FP(is)i(given)523 4426 y(by)22
|
|
|
4575 |
b(list-concatenation,)d(written)j(as)h FQ(\031)1671 4396
|
|
|
4576 |
y FL(0)1694 4426 y FM(@)p FQ(\031)s FP(;)f(the)g FE(identity)g
|
|
|
4577 |
(permutation)g FP(is)h(given)e(by)h(the)g(empty)523 4526
|
|
|
4578 |
y(list)28 b FM([])p FP(;)e(and)g(the)g FE(in)m(v)o(erse)h
|
|
|
4579 |
FP(of)g(a)f(permutation)f(is)j(given)d(by)i(list)g(r)m(e)o(ver)o(sal,)g
|
|
|
4580 |
(written)g(as)g FQ(\031)3143 4496 y FL(\000)p FJ(1)3233
|
|
|
4581 |
4526 y FP(.)f(The)523 4625 y FE(permutation)21 b(action)p
|
|
|
4582 |
FP(,)f FQ(\031)s FK(\001)p FQ(t)p FP(,)i(of)f(a)g(\002nite)g
|
|
|
4583 |
(permutation)f(of)h(atoms)g FQ(\031)k FP(on)c(a)g(term)h
|
|
|
4584 |
FQ(t)g FP(is)g(de\002ned)e(as)h(in)523 4725 y(F)l(ig)o(.)j(1;)g(it)h
|
|
|
4585 |
(pushes)f(the)h(list)g FQ(\031)k FP(into)24 b(the)g(structur)m(e)h(of)g
|
|
|
4586 |
(the)f(term)h FQ(t)g FP(until)g(it)g(`piles)f(up')g(in)h(fr)l(ont)f(of)
|
|
|
4587 |
523 4825 y(suspensions)g(\(applying)f(the)i(actual)f(permutation)f
|
|
|
4588 |
(that)i FQ(\031)k FP(r)m(epr)m(esents)c(to)g(atoms)g(that)f(it)i(meets)
|
|
|
4589 |
523 4924 y(on)31 b(the)g(way\).)f(The)h FE(disagreement)f(set)i
|
|
|
4590 |
FP(of)f(two)h(permutations)d FQ(\031)35 b FP(and)30 b
|
|
|
4591 |
FQ(\031)2757 4894 y FL(0)2812 4924 y FP(\(used)h(in)g(rule)g(\()p
|
|
|
4592 |
FO(\031)p FP(-)p eop end
|
|
|
4593 |
%%Page: 7 7
|
|
|
4594 |
TeXDict begin 7 6 bop 523 369 2882 4 v 523 1702 4 1334
|
|
|
4595 |
v 556 483 376 4 v 556 555 a FB(r)21 b(`)g(hi)h(\031)f(hi)953
|
|
|
4596 |
499 y Fv(\()p Fu(\031)p Fv(-unit\))1220 446 y FB(r)h(`)f
|
|
|
4597 |
FS(t)1402 454 y FG(1)1458 446 y FB(\031)g FS(t)1567 415
|
|
|
4598 |
y Fw(0)1567 459 y FG(1)1676 446 y FB(r)g(`)g FS(t)1857
|
|
|
4599 |
454 y FG(2)1913 446 y FB(\031)g FS(t)2022 415 y Fw(0)2022
|
|
|
4600 |
459 y FG(2)p 1220 479 836 4 v 1292 555 a FB(r)g(`)g(h)p
|
|
|
4601 |
FS(t)1503 563 y FG(1)1538 555 y FS(;)13 b(t)1600 563
|
|
|
4602 |
y FG(2)1634 555 y FB(i)21 b(\031)g(h)p FS(t)1824 524
|
|
|
4603 |
y Fw(0)1824 569 y FG(1)1859 555 y FS(;)13 b(t)1921 524
|
|
|
4604 |
y Fw(0)1921 569 y FG(2)1955 555 y FB(i)2078 495 y Fv(\()p
|
|
|
4605 |
Fu(\031)p Fv(-pair\))2403 459 y FB(r)22 b(`)f FS(t)g
|
|
|
4606 |
FB(\031)g FS(t)2715 428 y Fw(0)p 2345 479 451 4 v 2345
|
|
|
4607 |
555 a FB(r)h(`)f FS(f)g(t)g FB(\031)g FS(f)g(t)2773 524
|
|
|
4608 |
y Fw(0)2817 495 y Fv(\()p Fu(\031)p Fv(-function)e(symbol\))613
|
|
|
4609 |
694 y FB(r)j(`)f FS(t)g FB(\031)g FS(t)925 662 y Fw(0)p
|
|
|
4610 |
551 714 458 4 v 551 790 a FB(r)h(`)f FS(a:t)g FB(\031)g
|
|
|
4611 |
FS(a:t)987 758 y Fw(0)1031 730 y Fv(\()p Fu(\031)p Fv(-abstraction-1\))
|
|
|
4612 |
1648 675 y FS(a)g FB(6)p FF(=)g FS(a)1832 643 y Fw(0)1929
|
|
|
4613 |
675 y FB(r)h(`)f FS(t)g FB(\031)g FF(\()p FS(a)12 b(a)2337
|
|
|
4614 |
643 y Fw(0)2359 675 y FF(\))p Fz(\001)p FS(t)2442 643
|
|
|
4615 |
y Fw(0)2539 675 y FB(r)21 b(`)g FS(a)g FF(#)g FS(t)2867
|
|
|
4616 |
643 y Fw(0)p 1648 714 1242 4 v 2029 790 a FB(r)g(`)g
|
|
|
4617 |
FS(a:t)h FB(\031)f FS(a)2416 758 y Fw(0)2438 790 y FS(:t)2487
|
|
|
4618 |
758 y Fw(0)2911 730 y Fv(\()p Fu(\031)p Fv(-abstraction-2\))p
|
|
|
4619 |
805 937 338 4 v 805 1005 a FB(r)g(`)h FS(a)f FB(\031)g
|
|
|
4620 |
FS(a)1164 953 y Fv(\()p Fu(\031)p Fv(-atom\))1610 891
|
|
|
4621 |
y FF(\()p FS(a)g FF(#)h FS(X)6 b FF(\))21 b FB(2)h(r)f
|
|
|
4622 |
FV(for)e(all)i FS(a)g FB(2)g FD(ds)7 b FF(\()p FS(\031)s(;)13
|
|
|
4623 |
b(\031)2637 859 y Fw(0)2659 891 y FF(\))p 1610 929 1079
|
|
|
4624 |
4 v 1870 1005 a FB(r)21 b(`)g FS(\031)s Fz(\001)p FS(X)27
|
|
|
4625 |
b FB(\031)21 b FS(\031)2313 973 y Fw(0)2335 1005 y Fz(\001)p
|
|
|
4626 |
FS(X)2710 945 y Fv(\()p Fu(\031)p Fv(-suspension\))p
|
|
|
4627 |
562 1133 361 4 v 562 1206 a FB(r)g(`)g FS(a)g FF(#)g
|
|
|
4628 |
FB(hi)944 1150 y Fv(\()p Ft(#)p Fv(-unit\))1260 1099
|
|
|
4629 |
y FB(r)g(`)h FS(a)f FF(#)g FS(t)1589 1107 y FG(1)1698
|
|
|
4630 |
1099 y FB(r)g(`)h FS(a)f FF(#)g FS(t)2027 1107 y FG(2)p
|
|
|
4631 |
1260 1133 802 4 v 1401 1206 a FB(r)g(`)h FS(a)f FF(#)g
|
|
|
4632 |
FB(h)p FS(t)1760 1214 y FG(1)1794 1206 y FS(;)13 b(t)1856
|
|
|
4633 |
1214 y FG(2)1891 1206 y FB(i)2083 1149 y Fv(\()p Ft(#)p
|
|
|
4634 |
Fv(-pair\))2428 1103 y FB(r)21 b(`)h FS(a)f FF(#)g FS(t)p
|
|
|
4635 |
2399 1137 387 4 v 2399 1206 a FB(r)g(`)h FS(a)f FF(#)g
|
|
|
4636 |
FS(f)g(t)2808 1153 y Fv(\()p Ft(#)p Fv(-function)e(symbol\))p
|
|
|
4637 |
874 1367 391 4 v 874 1436 a FB(r)i(`)h FS(a)f FF(#)g
|
|
|
4638 |
FS(a:t)1287 1384 y Fv(\()p Ft(#)p Fv(-abstraction-1\))1951
|
|
|
4639 |
1325 y FS(a)g FB(6)p FF(=)g FS(a)2135 1294 y Fw(0)2231
|
|
|
4640 |
1325 y FB(r)h(`)f FS(a)g FF(#)g FS(t)p 1951 1360 610
|
|
|
4641 |
4 v 2049 1436 a FB(r)g(`)g FS(a)g FF(#)h FS(a)2391 1404
|
|
|
4642 |
y Fw(0)2413 1436 y FS(:t)2582 1376 y Fv(\()p Ft(#)p Fv
|
|
|
4643 |
(-abstraction-2\))1112 1552 y FS(a)f FB(6)p FF(=)g FS(a)1296
|
|
|
4644 |
1520 y Fw(0)p 1033 1587 364 4 v 1033 1663 a FB(r)g(`)g
|
|
|
4645 |
FS(a)g FF(#)h FS(a)1375 1631 y Fw(0)1419 1603 y Fv(\()p
|
|
|
4646 |
Ft(#)p Fv(-atom\))1869 1556 y FF(\()p FS(\031)1946 1524
|
|
|
4647 |
y Fw(\000)p FG(1)2028 1556 y Fz(\001)o FS(a)f FF(#)h
|
|
|
4648 |
FS(X)6 b FF(\))21 b FB(2)h(r)p 1869 1594 588 4 v 1942
|
|
|
4649 |
1663 a(r)f(`)h FS(a)f FF(#)g FS(\031)s Fz(\001)o FS(X)2478
|
|
|
4650 |
1610 y Fv(\()p Ft(#)p Fv(-suspension\))p 3402 1702 4
|
|
|
4651 |
1334 v 523 1705 2882 4 v 1374 1849 a FU(Fig)o(.)12 b(2.)18
|
|
|
4652 |
b FV(Inducti)n(v)o(e)i(de\002nition)f(of)g FB(\031)g
|
|
|
4653 |
FV(and)g FF(#)p FV(.)523 2139 y FP(suspension\))g(in)h(F)l(ig)o(.)g
|
|
|
4654 |
(2\))g(is)h(de\002ned)e(by)1442 2351 y FA(ds)8 b FM(\()p
|
|
|
4655 |
FQ(\031)s(;)14 b(\031)1695 2316 y FL(0)1719 2351 y FM(\))1774
|
|
|
4656 |
2299 y Fs(def)1777 2351 y FM(=)26 b FO(f)p FQ(a)c FO(j)i
|
|
|
4657 |
FQ(\031)s FK(\001)o FQ(a)f FO(6)p FM(=)g FQ(\031)2304
|
|
|
4658 |
2316 y FL(0)2328 2351 y FK(\001)o FQ(a)p FO(g)g FQ(:)822
|
|
|
4659 |
b FW(\(7\))523 2538 y(Note)17 b(that)g(e)n(v)o(ery)f(disagreement)f
|
|
|
4660 |
(set)j(of)f(the)g(lists)i FQ(\031)i FW(and)16 b FQ(\031)2226
|
|
|
4661 |
2508 y FL(0)2267 2538 y FW(is)i(a)g(subset)f(of)g(the)g
|
|
|
4662 |
FP(\002nite)g FW(set)h(of)f(atoms)523 2638 y(occurring)e(in)i(either)f
|
|
|
4663 |
(of)h(the)f(lists,)i(because)f(if)g FQ(a)g FW(does)f(not)h(occur)f(in)h
|
|
|
4664 |
(those)f(lists,)i(then)f(from)f(Fig.)g(1)523 2737 y(we)21
|
|
|
4665 |
b(get)f FQ(\031)s FK(\001)p FQ(a)j FM(=)f FQ(a)h FM(=)g
|
|
|
4666 |
FQ(\031)1199 2707 y FL(0)1222 2737 y FK(\001)p FQ(a)p
|
|
|
4667 |
FW(.)d(T)-7 b(o)21 b(illustrate)f(the)g(use)h(of)f(disagreement)e
|
|
|
4668 |
(sets,)j(consider)1235 2923 y FO(f)p FQ(a)h FM(#)i FQ(X)6
|
|
|
4669 |
b(;)14 b(c)23 b FM(#)g FQ(X)6 b FO(g)23 b(`)g FM(\()p
|
|
|
4670 |
FQ(a)14 b(c)p FM(\)\()p FQ(a)g(b)p FM(\))p FK(\001)o
|
|
|
4671 |
FQ(X)30 b FO(\031)22 b FM(\()p FQ(b)14 b(c)p FM(\))p
|
|
|
4672 |
FK(\001)p FQ(X)523 3109 y FW(which)h(holds)f(by)h(\()p
|
|
|
4673 |
FO(\031)p FW(-suspension\),)e(since)i(the)g(disagreement)f(set)i(of)8
|
|
|
4674 |
b FM(\()p FQ(a)14 b(c)p FM(\))-8 b(\()p FQ(a)14 b(b)p
|
|
|
4675 |
FM(\))8 b FW(and)g FM(\()p FQ(b)14 b(c)p FM(\))8 b FW(is)g
|
|
|
4676 |
FO(f)p FQ(a;c)p FO(g)p FW(.)523 3292 y FH(Lemma)21 b(1.)41
|
|
|
4677 |
b FO(r)30 b(`)f(\000)g(\031)h(\000)24 b FP(is)g(an)g(equivalence)e(r)m
|
|
|
4678 |
(elation;)h(it)h(is)h(pr)m(eserved)f(by)f(all)h(of)g(the)g(term-)523
|
|
|
4679 |
3391 y(forming)g(oper)o(ations)f(in)h(Def)o(.)h(2;)f(and)f(it)i(r)m
|
|
|
4680 |
(espects)g(the)g(fr)m(eshness)g(r)m(elation)f(\(i.e)o(.)f(if)j
|
|
|
4681 |
FO(r)31 b(`)f FQ(a)h FM(#)g FQ(t)523 3491 y FP(and)24
|
|
|
4682 |
b FO(r)32 b(`)f FQ(t)h FO(\031)f FQ(t)1044 3461 y FL(0)1067
|
|
|
4683 |
3491 y FP(,)26 b(then)e FO(r)32 b(`)f FQ(a)h FM(#)g FQ(t)1672
|
|
|
4684 |
3461 y FL(0)1695 3491 y FP(\).)25 b(Both)f FO(\031)h
|
|
|
4685 |
FP(and)f FM(#)i FP(ar)m(e)f(pr)m(eserved)f(by)h(the)g(permutation)523
|
|
|
4686 |
3591 y(action)c(given)h(in)g(F)l(ig)o(.)g(1)g(in)g(the)g(following)g
|
|
|
4687 |
(sense:)g(if)35 b FO(r)26 b(`)h FQ(t)f FO(\031)g FQ(t)2470
|
|
|
4688 |
3560 y FL(0)2494 3591 y FP(,)c(then)g FO(r)k(`)h FQ(\031)s
|
|
|
4689 |
FK(\001)o FQ(t)g FO(\031)f FQ(\031)s FK(\001)p FQ(t)3207
|
|
|
4690 |
3560 y FL(0)3230 3591 y FP(;)c(and)523 3690 y(if)33 b
|
|
|
4691 |
FO(r)23 b(`)g FQ(a)g FM(#)g FQ(t)p FP(,)e(then)f FO(r)j(`)g
|
|
|
4692 |
FQ(\031)s FK(\001)o FQ(a)g FM(#)h FQ(\031)s FK(\001)o
|
|
|
4693 |
FQ(t)p FP(.)523 3790 y(Pr)l(oof)o(.)e FW(Although)i(reasoning)h(about)g
|
|
|
4694 |
FO(\031)h FW(and)g FM(#)h FW(is)g(rather)e(pleasant)h(once)f(the)h(abo)
|
|
|
4695 |
o(v)o(e)f(f)o(acts)i(are)523 3889 y(pro)o(v)o(ed,)g(establishing)i
|
|
|
4696 |
(them)h(\002rst)g(is)h(rather)e(trick)o(y\227mainly)f(because)h(of)h
|
|
|
4697 |
(the)f(lar)o(ge)g(number)523 3989 y(of)21 b(cases,)i(b)n(ut)e(also)h
|
|
|
4698 |
(because)f(the)h(f)o(acts)g(in)g(the)f(lemma)g(are)h(inter)n
|
|
|
4699 |
(-dependent;)d(in)i(addition)g(some)523 4089 y(further)14
|
|
|
4700 |
b(properties)g(of)h(the)h(permutation)d(action)i(and)g(disagreement)f
|
|
|
4701 |
(sets)j(need)e(to)g(be)h(established)523 4188 y(\002rst)21
|
|
|
4702 |
b(\(statements)f(omitted\).)1376 4158 y Fy(1)3350 4188
|
|
|
4703 |
y FO(u)-55 b(t)648 4369 y FW(The)22 b(main)g(reason)f(for)h(using)g
|
|
|
4704 |
(suspensions)g(in)h(the)f(syntax)g(of)g(terms)h(is)g(to)g(enable)e(a)i
|
|
|
4705 |
(de\002ni-)523 4469 y(tion)e(of)g FP(substitution)f(of)i(terms)g(for)g
|
|
|
4706 |
(variables)f FW(which)g(allo)n(ws)g(capture)f(of)h(free)g(atoms)g(by)g
|
|
|
4707 |
(atom-)523 4568 y(abstractions)c(while)h(still)h(respecting)d
|
|
|
4708 |
FQ(\013)p FW(-equi)n(v)n(alence.)f(The)j(follo)n(wing)e(lemma)h
|
|
|
4709 |
(establishes)h(this.)523 4668 y(First)j(we)g(gi)n(v)o(e)e(some)h
|
|
|
4710 |
(terminology)e(and)h(notation)g(for)h(term-substitution.)p
|
|
|
4711 |
523 4746 473 4 v 558 4801 a Fr(1)606 4833 y FV(A)h(machine-check)o(ed)j
|
|
|
4712 |
(proof)d(of)g(all)g(the)g(results)g(using)h(the)f(theorem)g(pro)o(v)o
|
|
|
4713 |
(er)h(Isabelle)f(can)h(be)f(found)h(at)606 4924 y FC(http)g
|
|
|
4714 |
FF(:)g FS(==)p FC(www)p FS(:)p FC(cl)p FS(:)p FC(cam)p
|
|
|
4715 |
FS(:)p FC(ac)p FS(:)p FC(uk)p FS(=)1480 4916 y Fw(\030)1538
|
|
|
4716 |
4924 y FC(cu200)p FS(=)p FC(Unification)t FV(.)p eop
|
|
|
4717 |
end
|
|
|
4718 |
%%Page: 8 8
|
|
|
4719 |
TeXDict begin 8 7 bop 523 448 a FH(De\002nition)20 b(4.)41
|
|
|
4720 |
b FP(A)32 b FE(substitution)h FQ(\033)j FP(is)d(a)f(sort-r)m(especting)
|
|
|
4721 |
f(function)f(fr)l(om)j(variables)e(to)h(terms)523 548
|
|
|
4722 |
y(with)22 b(the)g(pr)l(operty)g(that)f FQ(\033)s FM(\()p
|
|
|
4723 |
FQ(X)7 b FM(\))27 b(=)e FQ(X)k FP(for)22 b(all)g(b)n(ut)g(\002nitely)g
|
|
|
4724 |
(many)f(variables)g FQ(X)7 b FP(.)22 b(W)-8 b(e)22 b(shall)g(write)523
|
|
|
4725 |
648 y Fq(dom)8 b Fp(\()p Fo(\033)s Fp(\))23 b FP(for)f(the)f(\002nite)g
|
|
|
4726 |
(set)h(of)g(variables)f FQ(X)28 b FP(satisfying)21 b
|
|
|
4727 |
FQ(\033)s FM(\()p FQ(X)7 b FM(\))26 b FO(6)p FM(=)f FQ(X)7
|
|
|
4728 |
b FP(.)21 b(If)g FA(dom)7 b FM(\()p FQ(\033)s FM(\))23
|
|
|
4729 |
b FP(consists)f(of)523 747 y(variables)e FQ(X)922 759
|
|
|
4730 |
y FJ(1)959 747 y FQ(;)14 b(:)g(:)g(:)f(;)h(X)1212 759
|
|
|
4731 |
y FI(n)1278 747 y FP(and)19 b FQ(\033)s FM(\()p FQ(X)1574
|
|
|
4732 |
759 y FI(i)1602 747 y FM(\))24 b(=)e FQ(t)1775 759 y
|
|
|
4733 |
FI(i)1824 747 y FP(for)e FQ(i)j FM(=)g(1)p FQ(::n)p FP(,)d(we)h(shall)f
|
|
|
4734 |
(sometimes)h(write)g FQ(\033)j FP(as)1442 896 y FQ(\033)i
|
|
|
4735 |
FM(=)d([)p FQ(X)1695 908 y FJ(1)1755 896 y FM(:=)g FQ(t)1896
|
|
|
4736 |
908 y FJ(1)1933 896 y FQ(;)14 b(:)g(:)g(:)f(;)h(X)2186
|
|
|
4737 |
908 y FI(n)2254 896 y FM(:=)23 b FQ(t)2395 908 y FI(n)2440
|
|
|
4738 |
896 y FM(])p FQ(:)822 b FW(\(8\))523 1044 y FP(W)-8 b(e)27
|
|
|
4739 |
b(write)h FQ(\033)s FM(\()p FQ(t)p FM(\))g FP(for)f(the)f(r)m(esult)h
|
|
|
4740 |
(of)g FE(applying)e(a)h(substitution)i FQ(\033)i FP(to)d(a)f(term)h
|
|
|
4741 |
FQ(t)p FP(;)g(this)g(is)g(the)g(term)523 1144 y(obtained)19
|
|
|
4742 |
b(fr)l(om)i FQ(t)h FP(by)f(r)m(eplacing)e(eac)o(h)h(suspension)g
|
|
|
4743 |
FQ(\031)s FK(\001)p FQ(X)27 b FP(in)21 b FQ(t)h FP(\(as)e
|
|
|
4744 |
FQ(X)28 b FP(r)o(ang)o(es)20 b(o)o(ver)h FA(dom)8 b FM(\()p
|
|
|
4745 |
FQ(\033)s FM(\))p FP(\))22 b(by)523 1244 y(the)17 b(term)h
|
|
|
4746 |
FQ(\031)s FK(\001)p FQ(\033)s FM(\()p FQ(X)7 b FM(\))18
|
|
|
4747 |
b FP(got)e(by)h(letting)g FQ(\031)k FP(act)c(on)g(the)g(term)h
|
|
|
4748 |
FQ(\033)s FM(\()p FQ(X)7 b FM(\))17 b FP(using)g(the)g(de\002nition)e
|
|
|
4749 |
(in)j(F)l(ig)o(.)e(1.)h(F)-9 b(or)523 1343 y(e)n(xample)o(,)25
|
|
|
4750 |
b(if)h FQ(\033)35 b FM(=)d([)p FQ(X)25 b FM(:=)18 b FO(h)p
|
|
|
4751 |
FQ(b;)c(Y)19 b FO(i)p FM(])26 b FP(and)e FQ(t)33 b FM(=)f
|
|
|
4752 |
FQ(a:)p FM(\()p FQ(a)14 b(b)p FM(\))p FK(\001)o FQ(X)7
|
|
|
4753 |
b FP(,)25 b(then)f FQ(\033)s FM(\()p FQ(t)p FM(\))34
|
|
|
4754 |
b(=)e FQ(a:)p FO(h)p FQ(a;)14 b FM(\()p FQ(a)g(b)p FM(\))p
|
|
|
4755 |
FK(\001)o FQ(Y)19 b FO(i)p FP(.)26 b(Given)523 1443 y(substitutions)20
|
|
|
4756 |
b FQ(\033)k FP(and)19 b FQ(\033)1230 1413 y FL(0)1254
|
|
|
4757 |
1443 y FP(,)h(and)g(fr)m(eshness)g(en)m(vir)l(onments)f
|
|
|
4758 |
FO(r)i FP(and)f FO(r)2533 1413 y FL(0)2556 1443 y FP(,)h(we)g(write)
|
|
|
4759 |
1161 1592 y FM(\()p FQ(a)p FM(\))84 b FO(r)1422 1557
|
|
|
4760 |
y FL(0)1468 1592 y FO(`)23 b FQ(\033)s FM(\()p FO(r)p
|
|
|
4761 |
FM(\))167 b FP(and)e FM(\()p FQ(b)p FM(\))83 b FO(r)23
|
|
|
4762 |
b(`)g FQ(\033)j FO(\031)d FQ(\033)2743 1557 y FL(0)3308
|
|
|
4763 |
1592 y FW(\(9\))523 1740 y FP(to)32 b(mean)f(that)g(\(for)h(a\))f
|
|
|
4764 |
FO(r)1320 1710 y FL(0)1388 1740 y FO(`)44 b FQ(a)g FM(#)g
|
|
|
4765 |
FQ(\033)s FM(\()p FQ(X)7 b FM(\))33 b FP(holds)e(for)h(eac)o(h)f
|
|
|
4766 |
FM(\()p FQ(a)44 b FM(#)h FQ(X)7 b FM(\))43 b FO(2)i(r)32
|
|
|
4767 |
b FP(and)f(\(for)h(b\))523 1840 y FO(r)23 b(`)g FQ(\033)s
|
|
|
4768 |
FM(\()p FQ(X)7 b FM(\))23 b FO(\031)g FQ(\033)1040 1810
|
|
|
4769 |
y FL(0)1064 1840 y FM(\()p FQ(X)7 b FM(\))20 b FP(holds)g(for)h(all)f
|
|
|
4770 |
FQ(X)29 b FO(2)24 b FA(dom)7 b FM(\()p FQ(\033)s FM(\))19
|
|
|
4771 |
b FO([)g FA(dom)7 b FM(\()p FQ(\033)2435 1810 y FL(0)2459
|
|
|
4772 |
1840 y FM(\))p FP(.)523 1980 y FH(Lemma)21 b(2)41 b(\(Substitution\).)
|
|
|
4773 |
14 b FP(Substitution)g(commutes)h(with)h(the)f(permutation)e(action:)h
|
|
|
4774 |
FQ(\033)s FM(\()p FQ(\031)s FK(\001)q FQ(t)p FM(\))23
|
|
|
4775 |
b(=)523 2080 y FQ(\031)s FK(\001)p FM(\()p FQ(\033)s
|
|
|
4776 |
FM(\()p FQ(t)p FM(\)\))p FP(.)f(Substitution)c(pr)m(eserves)j
|
|
|
4777 |
FO(\031)g FP(and)e FM(#)i FP(in)f(the)g(following)g(sense:)581
|
|
|
4778 |
2196 y FO(\017)41 b FP(if)21 b FO(r)800 2166 y FL(0)847
|
|
|
4779 |
2196 y FO(`)h FQ(\033)s FM(\()p FO(r)p FM(\))g FP(and)e
|
|
|
4780 |
FO(r)j(`)g FQ(t)g FO(\031)f FQ(t)1607 2166 y FL(0)1631
|
|
|
4781 |
2196 y FP(,)e(then)g FO(r)1905 2166 y FL(0)1951 2196
|
|
|
4782 |
y FO(`)j FQ(\033)s FM(\()p FQ(t)p FM(\))h FO(\031)f FQ(\033)s
|
|
|
4783 |
FM(\()p FQ(t)2393 2166 y FL(0)2417 2196 y FM(\))p FP(;)581
|
|
|
4784 |
2289 y FO(\017)41 b FP(if)21 b FO(r)800 2259 y FL(0)847
|
|
|
4785 |
2289 y FO(`)h FQ(\033)s FM(\()p FO(r)p FM(\))g FP(and)e
|
|
|
4786 |
FO(r)j(`)g FQ(a)g FM(#)g FQ(t)p FP(,)d(then)g FO(r)1900
|
|
|
4787 |
2259 y FL(0)1947 2289 y FO(`)i FQ(a)h FM(#)h FQ(\033)s
|
|
|
4788 |
FM(\()p FQ(t)p FM(\))p FP(.)523 2411 y(Pr)l(oof)o(.)e
|
|
|
4789 |
FW(The)d(\002rst)i(sentence)e(follo)n(ws)h(by)f(induction)f(on)h(the)h
|
|
|
4790 |
(structure)f(of)g FQ(t)p FW(.)i(The)e(second)g(follo)n(ws)523
|
|
|
4791 |
2511 y(by)h(induction)f(on)h(the)h(proofs)e(of)h FO(r)k(`)f
|
|
|
4792 |
FQ(t)h FO(\031)f FQ(t)1851 2481 y FL(0)1896 2511 y FW(and)d
|
|
|
4793 |
FO(r)k(`)f FQ(a)h FM(#)g FQ(t)d FW(from)e(the)i(rules)f(in)h(Fig.)g(2,)
|
|
|
4794 |
f(using)523 2611 y(the)g(\002rst)h(sentence)f(and)f(the)i(\(proof)d
|
|
|
4795 |
(of\))h(Lemma)h(1.)1302 b FO(u)-55 b(t)648 2751 y FW(W)-7
|
|
|
4796 |
b(e)27 b(claim)f(that)g(the)h(relation)e FO(\031)h FW(de\002ned)f(in)i
|
|
|
4797 |
(Fig.)f(2)g(gi)n(v)o(es)g(the)g(correct)f(notion)g(of)h
|
|
|
4798 |
FQ(\013)p FW(-equi-)523 2851 y(v)n(alence)19 b(for)h(terms)h(o)o(v)o
|
|
|
4799 |
(er)e(a)i(nominal)e(signature.)g(This)h(is)i(reasonable,)c(gi)n(v)o(en)
|
|
|
4800 |
i(Lemma)f(1)i(and)e(the)523 2950 y(f)o(act)e(that,)g(by)f
|
|
|
4801 |
(de\002nition,)f(it)j(satis\002es)g(rules)f(\()p FO(\031)p
|
|
|
4802 |
FW(-abstraction-1\))c(and)j(\()p FO(\031)p FW(-abstraction-2\).)e
|
|
|
4803 |
(Further)523 3050 y(e)n(vidence)23 b(is)j(pro)o(vided)c(by)i(the)h
|
|
|
4804 |
(follo)n(wing)e(theorem,)g(which)h(sho)n(ws)h(that)f(for)g(ground)f
|
|
|
4805 |
(terms)h FO(\031)523 3149 y FW(agrees)c(with)g(the)g(follo)n(wing)f
|
|
|
4806 |
(more)g(traditional)g(de\002nition)g(of)h FQ(\013)p FW(-equi)n(v)n
|
|
|
4807 |
(alence.)523 3290 y FH(De\002nition)g(5)41 b(\(Na)n(\250)-26
|
|
|
4808 |
b(\021v)o(e)20 b Fo(\013)p FH(-equi)o(v)o(alence\).)f
|
|
|
4809 |
FP(De\002ne)h(the)g(binary)g(r)m(elation)f FQ(t)k FM(=)2797
|
|
|
4810 |
3302 y FI(\013)2854 3290 y FQ(t)2884 3260 y FL(0)2928
|
|
|
4811 |
3290 y FP(between)c(terms)523 3389 y(o)o(ver)j(a)g(nominal)e(signatur)m
|
|
|
4812 |
(e)h(to)h(be)f(the)h(least)g(sort-r)m(especting)f(congruence)f(r)m
|
|
|
4813 |
(elation)h(satisfying)523 3489 y FQ(a:t)j FM(=)709 3501
|
|
|
4814 |
y FI(\013)766 3489 y FQ(b:)p FM([)p FQ(a)p FL(7!)p FQ(b)p
|
|
|
4815 |
FM(])p FQ(t)d FP(whene)o(ver)f FQ(b)h FP(is)g(an)g(atom)f(\(of)h(the)f
|
|
|
4816 |
(same)h(sort)h(as)f FQ(a)p FP(\))f(not)h(occurring)f(at)g(all)h(in)g
|
|
|
4817 |
FQ(t)p FP(.)523 3589 y(Her)m(e)g FM([)p FQ(a)p FL(7!)p
|
|
|
4818 |
FQ(b)p FM(])p FQ(t)f FP(indicates)g(the)g(r)m(esult)h(of)f(r)m
|
|
|
4819 |
(eplacing)f(all)h(fr)m(ee)h(occurr)m(ences)f(of)g FQ(a)h
|
|
|
4820 |
FP(with)f FQ(b)h FP(in)f FQ(t)p FP(.)523 3729 y FH(Theor)o(em)g(1)41
|
|
|
4821 |
b(\(Adequacy\).)17 b FP(If)h FQ(t)g FP(and)f FQ(t)1688
|
|
|
4822 |
3699 y FL(0)1729 3729 y FP(ar)m(e)h FH(gr)o(ound)g(terms)g
|
|
|
4823 |
FW(\(i.e.)f(terms)h(with)g(no)f(v)n(ariables)g(and)523
|
|
|
4824 |
3829 y(hence)e(no)h(suspensions\))f FP(o)o(ver)h(a)g(nominal)f
|
|
|
4825 |
(signatur)m(e)o(,)g(then)h(the)g(r)m(elation)g FQ(t)23
|
|
|
4826 |
b FM(=)2804 3841 y FI(\013)2860 3829 y FQ(t)2890 3799
|
|
|
4827 |
y FL(0)2930 3829 y FP(of)16 b(Def)o(.)g(5)g(holds)523
|
|
|
4828 |
3928 y(if)23 b(and)e(only)h(if)h Fn(?)j FO(`)g FQ(t)h
|
|
|
4829 |
FO(\031)f FQ(t)1320 3898 y FL(0)1366 3928 y FP(is)d(pr)l(o)o(vable)e
|
|
|
4830 |
(fr)l(om)i(the)f(rules)h(in)f(F)l(ig)o(.)g(2.)g(Furthermor)m(e)o(,)f
|
|
|
4831 |
Fn(?)26 b FO(`)h FQ(a)f FM(#)h FQ(t)523 4028 y FP(is)21
|
|
|
4832 |
b(pr)l(o)o(vable)e(if)i(and)f(only)f(if)i FQ(a)g FP(is)g(not)f(in)g
|
|
|
4833 |
(the)h(set)g FA(F)-8 b(A)o FM(\()p FQ(t)p FM(\))21 b
|
|
|
4834 |
FP(of)g(fr)m(ee)f(atoms)g(of)h FQ(t)p FP(.)523 4128 y(Pr)l(oof)o(.)h
|
|
|
4835 |
FW(The)e(proof)e(is)k(similar)e(to)g(the)g(proof)f(of)h([11,)f
|
|
|
4836 |
(Proposition)g(2.2].)713 b FO(u)-55 b(t)648 4268 y FW(F)o(or)28
|
|
|
4837 |
b(non-ground)d(terms,)j(the)h(relations)f FM(=)1953 4280
|
|
|
4838 |
y FI(\013)2029 4268 y FW(and)g FO(\031)h FP(dif)o(fer)p
|
|
|
4839 |
FW(.)f(F)o(or)h(e)o(xample)e FQ(a:X)45 b FM(=)3199 4280
|
|
|
4840 |
y FI(\013)3271 4268 y FQ(b:X)523 4368 y FW(al)o(w)o(ays)26
|
|
|
4841 |
b(holds,)e(whereas)h Fn(?)32 b FO(`)g FQ(a:X)38 b FO(\031)32
|
|
|
4842 |
b FQ(b:X)g FW(is)26 b(not)f(pro)o(v)n(able)e(unless)i
|
|
|
4843 |
FQ(a)32 b FM(=)g FQ(b)p FW(.)25 b(This)h(disagree-)523
|
|
|
4844 |
4467 y(ment)c(is)i(to)f(be)f(e)o(xpected,)f(since)i(we)g(noted)e(in)i
|
|
|
4845 |
(the)g(Introduction)c(that)k FM(=)2698 4479 y FI(\013)2768
|
|
|
4846 |
4467 y FW(is)h FP(not)f FW(preserv)o(ed)e(by)523 4567
|
|
|
4847 |
y(substitution,)e(whereas)h(from)f(Lemma)g(2)i(we)f(kno)n(w)f(that)i
|
|
|
4848 |
FO(\031)f FW(is.)523 4707 y FP(Remark)g(1.)41 b FW(Further)17
|
|
|
4849 |
b(e)n(vidence)g(for)h(the)g(status)i(of)e FO(\031)g FW(and)g
|
|
|
4850 |
FM(#)h FW(is)h(pro)o(vided)c(by)i(a)h(natural)e(interpre-)523
|
|
|
4851 |
4807 y(tation)22 b(of)h(judgements)e(pro)o(v)n(able)f(from)h(the)i
|
|
|
4852 |
(rules)g(in)f(Fig.)h(2)g(in)g(the)f(uni)n(v)o(erse)f(of)i(FM-sets)g
|
|
|
4853 |
([11].)523 4906 y(The)d(details)g(will)h(appear)e(in)i(the)f(full)g(v)o
|
|
|
4854 |
(ersion)f(of)h(this)h(paper)-5 b(.)p eop end
|
|
|
4855 |
%%Page: 9 9
|
|
|
4856 |
TeXDict begin 9 8 bop 523 448 a FR(3)99 b(Uni\002cation)523
|
|
|
4857 |
648 y FW(Gi)n(v)o(en)21 b(terms)g FQ(t)h FW(and)f FQ(t)1175
|
|
|
4858 |
618 y FL(0)1221 648 y FW(of)g(the)h(same)f(sort)h(o)o(v)o(er)e(a)i
|
|
|
4859 |
(nominal)e(signature,)g(can)i(we)g(decide)f(whether)523
|
|
|
4860 |
748 y(or)31 b(not)g(there)g(is)h(a)g(substitution)e(of)h(terms)h(for)e
|
|
|
4861 |
(the)i(v)n(ariables)e(in)i FQ(t)f FW(and)g FQ(t)2784
|
|
|
4862 |
717 y FL(0)2839 748 y FW(that)h(mak)o(es)f(them)523 847
|
|
|
4863 |
y(equal)20 b(in)i(the)f(sense)g(of)g(the)g(relation)f
|
|
|
4864 |
FO(\031)h FW(introduced)d(in)k(the)f(pre)n(vious)e(section?)i(Since)g
|
|
|
4865 |
(instances)523 947 y(of)26 b FO(\031)h FW(in)g(general)e(are)i
|
|
|
4866 |
(established)f(modulo)f(freshness)h(constraints,)g(it)h(mak)o(es)g
|
|
|
4867 |
(more)e(sense)i(to)523 1046 y(ask)i(whether)e(or)i(not)f(there)g(is)h
|
|
|
4868 |
(both)f(a)h(substitution)f FQ(\033)k FW(and)c(a)h(freshness)f(en)m
|
|
|
4869 |
(vironment)e FO(r)j FW(for)523 1146 y(which)21 b FO(r)26
|
|
|
4870 |
b(`)g FQ(\033)s FM(\()p FQ(t)p FM(\))h FO(\031)f FQ(\033)s
|
|
|
4871 |
FM(\()p FQ(t)1294 1116 y FL(0)1318 1146 y FM(\))c FW(holds.)f(As)i(for)
|
|
|
4872 |
e(ordinary)f(\002rst-order)g(uni\002cation,)g(solving)h(such)h(an)523
|
|
|
4873 |
1246 y(equational)f(problem)h(may)g(thro)n(w)h(up)f FP(se)o(ver)o(al)h
|
|
|
4874 |
FW(equational)f(subproblems;)f(b)n(ut)i(an)f(added)g(com-)523
|
|
|
4875 |
1345 y(plication)j(here)g(is)h(that)g(because)f(of)g(rule)g(\()p
|
|
|
4876 |
FO(\031)p FW(-abstraction-2\))d(in)k(Fig.)g(2,)f(equational)f(problems)
|
|
|
4877 |
523 1445 y(may)c(generate)f FP(fr)m(eshness)i FW(problems,)e(i.e.)i
|
|
|
4878 |
(ones)f(in)m(v)n(olving)f(the)h(relation)g FM(#)p FW(.)h(W)-7
|
|
|
4879 |
b(e)22 b(are)e(thus)h(led)f(to)523 1545 y(the)g(follo)n(wing)f
|
|
|
4880 |
(de\002nition)g(of)h(uni\002cation)f(problems)f(for)i(nominal)f
|
|
|
4881 |
(equational)g(logic.)523 1719 y FH(De\002nition)h(6.)41
|
|
|
4882 |
b FP(A)17 b FE(uni\002cation)f(problem)g FQ(P)28 b FP(o)o(ver)17
|
|
|
4883 |
b(a)f(nominal)f(signatur)m(e)h(is)h(a)f(\002nite)g(set)h(of)g(atomic)
|
|
|
4884 |
523 1819 y(pr)l(oblems,)27 b(eac)o(h)f(of)h(whic)o(h)g(is)h(either)f
|
|
|
4885 |
(an)g FE(equational)f(problem)g FQ(t)36 b FO(\031)o FM(?)g
|
|
|
4886 |
FQ(t)2728 1789 y FL(0)2779 1819 y FP(wher)m(e)28 b FQ(t)f
|
|
|
4887 |
FP(and)f FQ(t)3246 1789 y FL(0)3297 1819 y FP(ar)m(e)523
|
|
|
4888 |
1919 y(terms)21 b(of)e(the)h(same)g(sort)g(o)o(ver)g(the)g(signatur)m
|
|
|
4889 |
(e)o(,)e(or)i(a)g FE(freshness)h(problem)e FQ(a)k FM(#?)g
|
|
|
4890 |
FQ(t)d FP(wher)m(e)g FQ(a)g FP(is)h(an)523 2018 y(atom)g(and)f
|
|
|
4891 |
FQ(t)i FP(a)g(term)g(o)o(ver)f(the)h(signatur)m(e)o(.)d(A)j
|
|
|
4892 |
FE(solution)g FP(for)g FQ(P)33 b FP(consists)22 b(of)g(a)f(pair)g
|
|
|
4893 |
FM(\()p FO(r)p FQ(;)14 b(\033)s FM(\))23 b FP(wher)m(e)523
|
|
|
4894 |
2118 y FO(r)e FP(is)g(a)f(fr)m(eshness)h(en)m(vir)l(onment)e(and)g
|
|
|
4895 |
FQ(\033)24 b FP(is)d(a)g(substitution)e(satisfying)581
|
|
|
4896 |
2284 y FO(\017)41 b(r)23 b(`)g FQ(a)g FM(#)g FQ(\033)s
|
|
|
4897 |
FM(\()p FQ(t)p FM(\))45 b FP(for)21 b(eac)o(h)42 b FM(\()p
|
|
|
4898 |
FQ(a)23 b FM(#?)h FQ(t)p FM(\))f FO(2)h FQ(P)55 b FP(and)581
|
|
|
4899 |
2384 y FO(\017)41 b(r)23 b(`)g FQ(\033)s FM(\()p FQ(t)p
|
|
|
4900 |
FM(\))h FO(\031)f FQ(\033)s FM(\()p FQ(t)1198 2354 y
|
|
|
4901 |
FL(0)1222 2384 y FM(\))44 b FP(for)21 b(eac)o(h)42 b
|
|
|
4902 |
FM(\()p FQ(t)23 b FO(\031)p FM(?)g FQ(t)1857 2354 y FL(0)1880
|
|
|
4903 |
2384 y FM(\))h FO(2)f FQ(P)12 b FP(.)523 2550 y(Suc)o(h)21
|
|
|
4904 |
b(a)i(pair)f(is)h(a)g FE(most)f(general)g(solution)h
|
|
|
4905 |
FP(for)g FQ(P)34 b FP(if)23 b(given)f(any)g(other)g(solution)g
|
|
|
4906 |
FM(\()p FO(r)3052 2520 y FL(0)3075 2550 y FQ(;)14 b(\033)3162
|
|
|
4907 |
2520 y FL(0)3186 2550 y FM(\))p FP(,)23 b(then)523 2650
|
|
|
4908 |
y(ther)m(e)d(is)h(a)f(substitution)f FQ(\033)1307 2620
|
|
|
4909 |
y FL(00)1370 2650 y FP(satisfying)h FO(r)1778 2620 y
|
|
|
4910 |
FL(0)1824 2650 y FO(`)j FQ(\033)1948 2620 y FL(00)1991
|
|
|
4911 |
2650 y FM(\()p FO(r)p FM(\))e FP(and)e FO(r)2359 2620
|
|
|
4912 |
y FL(0)2406 2650 y FO(`)k FQ(\033)2530 2620 y FL(00)2590
|
|
|
4913 |
2650 y FO(\016)16 b FQ(\033)27 b FO(\031)c FQ(\033)2860
|
|
|
4914 |
2620 y FL(0)2883 2650 y FP(.)e(\(Her)m(e)f(we)g(have)523
|
|
|
4915 |
2749 y(used)e(the)g(notation)f(of)31 b FW(\(9\))n FP(;)19
|
|
|
4916 |
b(and)e FQ(\033)1542 2719 y FL(00)1597 2749 y FO(\016)11
|
|
|
4917 |
b FQ(\033)22 b FP(denotes)17 b(the)i FE(substitution)g(composition)e
|
|
|
4918 |
FP(of)i FQ(\033)j FP(followed)523 2875 y(by)e FQ(\033)672
|
|
|
4919 |
2844 y FL(00)715 2875 y FP(,)h(given)e(by)h FM(\()p FQ(\033)1138
|
|
|
4920 |
2844 y FL(00)1200 2875 y FO(\016)e FQ(\033)s FM(\)\()p
|
|
|
4921 |
FQ(X)7 b FM(\))1505 2823 y Fs(def)1509 2875 y FM(=)25
|
|
|
4922 |
b FQ(\033)1649 2844 y FL(00)1692 2875 y FM(\()p FQ(\033)s
|
|
|
4923 |
FM(\()p FQ(X)7 b FM(\)\))p FP(.\))523 3116 y FH(Theor)o(em)20
|
|
|
4924 |
b(2)41 b(\(Nominal)27 b(uni\002cation\).)g FP(Ther)m(e)g(is)i(an)e
|
|
|
4925 |
(algorithm)f(whic)o(h,)h(given)g(any)g(nominal)523 3207
|
|
|
4926 |
y(uni\002cation)d(pr)l(oblem,)h(decides)h(whether)g(or)g(not)g(it)g
|
|
|
4927 |
(has)g(a)g(solution)f(and)g(if)i(it)g(does,)e(r)m(eturns)h(a)523
|
|
|
4928 |
3299 y(most)21 b(g)o(ener)o(al)e(solution.)523 3390 y(Pr)l(oof)o(.)j
|
|
|
4929 |
FW(W)-7 b(e)16 b(describe)f(an)g(algorithm)f(using)h(labelled)f
|
|
|
4930 |
(transformations)f(directly)i(generalising)e(the)523
|
|
|
4931 |
3481 y(presentation)19 b(of)h(\002rst-order)f(uni\002cation)g(in)i
|
|
|
4932 |
([17,)e(Sect.)i(2.6],)e(which)h(in)h(turn)f(is)h(based)f(upon)f(the)523
|
|
|
4933 |
3573 y(approach)e(in)i([18].)f(\(See)h(also)g([1,)f(Sect.)i(4.6])e(for)
|
|
|
4934 |
g(a)h(detailed)g(e)o(xposition,)e(b)n(ut)i(not)f(using)h(labels.\))523
|
|
|
4935 |
3664 y(W)-7 b(e)21 b(use)g(tw)o(o)f(types)g(of)g(labelled)g
|
|
|
4936 |
(transformation)d(between)j(uni\002cation)f(problems,)f(namely)1527
|
|
|
4937 |
3799 y FS(P)1651 3757 y Fm(\033)1608 3799 y FF(=)-13
|
|
|
4938 |
b FB(\))21 b FS(P)1813 3764 y Fw(0)1910 3799 y FV(and)76
|
|
|
4939 |
b FS(P)2210 3757 y Fw(r)2174 3799 y FF(=)-13 b FB(\))21
|
|
|
4940 |
b FS(P)2379 3764 y Fw(0)523 3932 y FW(where)i(the)g(substitution)f
|
|
|
4941 |
FQ(\033)27 b FW(is)d(either)f(the)g(identity)f FQ(")p
|
|
|
4942 |
FW(,)h(or)g(a)h(single)f(replacement)e FM([)p FQ(X)35
|
|
|
4943 |
b FM(:=)28 b FQ(t)p FM(])p FW(;)c(and)523 4032 y(where)h(the)i
|
|
|
4944 |
(freshness)e(en)m(vironment)e FO(r)k FW(is)g(either)f(empty)f
|
|
|
4945 |
Fn(?)p FW(,)h(or)g(a)g(singleton)f FO(f)p FQ(a)33 b FM(#)i
|
|
|
4946 |
FQ(X)7 b FO(g)p FW(.)25 b(The)523 4132 y(le)o(gal)17
|
|
|
4947 |
b(transformations)e(are)i(gi)n(v)o(en)f(in)i(Fig.)f(3.)g(This)h
|
|
|
4948 |
(\002gure)e(uses)i(the)f(notation)g FQ(P)i FO(])8 b FQ(P)3018
|
|
|
4949 |
4102 y FL(0)3060 4132 y FW(to)17 b(indicate)523 4231
|
|
|
4950 |
y FP(disjoint)26 b(union)e FW(of)i(problem)e(sets;)k(and)d(the)h
|
|
|
4951 |
(notation)f FQ(\033)s(P)39 b FW(to)26 b(indicate)f(the)h(problem)e
|
|
|
4952 |
(resulting)523 4331 y(from)19 b(applying)g(the)h(substitution)f
|
|
|
4953 |
FQ(\033)24 b FW(to)d(all)f(the)h(terms)f(occurring)e(in)i(the)g
|
|
|
4954 |
(problem)f FQ(P)12 b FW(.)648 4422 y(Gi)n(v)o(en)25 b(a)i
|
|
|
4955 |
(uni\002cation)f(problem)f FQ(P)12 b FW(,)27 b(the)f(algorithm)f
|
|
|
4956 |
(proceeds)h(in)g(tw)o(o)h(phases.)g(In)f(the)h(\002rst)523
|
|
|
4957 |
4529 y(phase)17 b(it)i(applies)e(as)i(man)o(y)1384 4482
|
|
|
4958 |
y FI(\033)1337 4529 y FM(=)-14 b FO(\))18 b FW(transformations)e(as)i
|
|
|
4959 |
(possible)g(\(non-deterministically\).)13 b(If)18 b(this)523
|
|
|
4960 |
4620 y(results)23 b(in)f(a)h(problem)e(containing)g(no)h(equational)f
|
|
|
4961 |
(subproblems)f(then)i(it)h(proceeds)e(to)i(the)f(sec-)523
|
|
|
4962 |
4742 y(ond)i(phase;)g(otherwise)g(it)h(halts)g(with)g(f)o(ailure.)f(In)
|
|
|
4963 |
g(the)h(second)f(phase)g(it)h(applies)f(as)i(man)o(y)3310
|
|
|
4964 |
4695 y FL(r)3271 4742 y FM(=)-14 b FO(\))523 4833 y FW(transformations)
|
|
|
4965 |
19 b(as)j(possible)f(\(non-deterministically\).)d(If)j(this)h(does)f
|
|
|
4966 |
(not)g(result)g(in)h(the)f(empty)523 4924 y(problem,)c(then)i(it)h
|
|
|
4967 |
(halts)f(with)g(f)o(ailure;)g(otherwise)f(o)o(v)o(erall)g(it)i(has)f
|
|
|
4968 |
(constructed)f(a)h(transformation)p eop end
|
|
|
4969 |
%%Page: 10 10
|
|
|
4970 |
TeXDict begin 10 9 bop 523 369 2882 4 v 523 2089 4 1721
|
|
|
4971 |
v 546 456 a Fv(\()p Fu(\031)p Ft(?)q Fv(-unit\))558 b
|
|
|
4972 |
FB(fhi)22 b(\031)p FF(?)f FB(hig)c(])h FS(P)1912 413
|
|
|
4973 |
y Fm(")1865 456 y FF(=)-13 b FB(\))23 b FS(P)546 556
|
|
|
4974 |
y Fv(\()p Fu(\031)p Ft(?)q Fv(-pair\))241 b FB(fh)p FS(t)1140
|
|
|
4975 |
564 y FG(1)1174 556 y FS(;)14 b(t)1237 564 y FG(2)1271
|
|
|
4976 |
556 y FB(i)21 b(\031)p FF(?)g FB(h)p FS(t)1497 524 y
|
|
|
4977 |
Fw(0)1497 569 y FG(1)1532 556 y FS(;)13 b(t)1594 524
|
|
|
4978 |
y Fw(0)1594 569 y FG(2)1628 556 y FB(ig)k(])h FS(P)1912
|
|
|
4979 |
513 y Fm(")1865 556 y FF(=)-13 b FB(\))23 b(f)p FS(t)2078
|
|
|
4980 |
564 y FG(1)2134 556 y FB(\031)o FF(?)f FS(t)2279 524
|
|
|
4981 |
y Fw(0)2279 569 y FG(1)2313 556 y FS(;)13 b(t)2375 564
|
|
|
4982 |
y FG(2)2431 556 y FB(\031)p FF(?)21 b FS(t)2576 524 y
|
|
|
4983 |
Fw(0)2576 569 y FG(2)2611 556 y FB(g)c([)g FS(P)546 656
|
|
|
4984 |
y Fv(\()p Fu(\031)p Ft(?)q Fv(-function)i(symbol\))153
|
|
|
4985 |
b FB(f)p FS(f)21 b(t)g FB(\031)p FF(?)h FS(f)f(t)1636
|
|
|
4986 |
624 y Fw(0)1658 656 y FB(g)c(])h FS(P)1912 613 y Fm(")1865
|
|
|
4987 |
656 y FF(=)-13 b FB(\))23 b(f)p FS(t)e FB(\031)p FF(?)g
|
|
|
4988 |
FS(t)2244 624 y Fw(0)2266 656 y FB(g)d([)f FS(P)546 756
|
|
|
4989 |
y Fv(\()p Fu(\031)p Ft(?)q Fv(-abstraction-1\))235 b
|
|
|
4990 |
FB(f)p FS(a:t)22 b FB(\031)o FF(?)g FS(a:t)1636 724 y
|
|
|
4991 |
Fw(0)1658 756 y FB(g)17 b(])h FS(P)1912 713 y Fm(")1865
|
|
|
4992 |
756 y FF(=)-13 b FB(\))23 b(f)p FS(t)e FB(\031)p FF(?)g
|
|
|
4993 |
FS(t)2244 724 y Fw(0)2266 756 y FB(g)d([)f FS(P)546 855
|
|
|
4994 |
y Fv(\()p Fu(\031)p Ft(?)q Fv(-abstraction-2\))213 b
|
|
|
4995 |
FB(f)p FS(a:t)21 b FB(\031)p FF(?)g FS(a)1564 824 y Fw(0)1587
|
|
|
4996 |
855 y FS(:t)1636 824 y Fw(0)1658 855 y FB(g)c(])h FS(P)1912
|
|
|
4997 |
813 y Fm(")1865 855 y FF(=)-13 b FB(\))23 b(f)p FS(t)e
|
|
|
4998 |
FB(\031)p FF(?)g(\()p FS(a)13 b(a)2341 824 y Fw(0)2363
|
|
|
4999 |
855 y FF(\))p Fz(\001)o FS(t)2445 824 y Fw(0)2467 855
|
|
|
5000 |
y FS(;)g(a)21 b FF(#?)h FS(t)2713 824 y Fw(0)2735 855
|
|
|
5001 |
y FB(g)t([)t FS(P)33 b FV(pro)o(vided)20 b FS(a)i FB(6)p
|
|
|
5002 |
FF(=)f FS(a)3382 824 y Fw(0)546 955 y Fv(\()p Fu(\031)p
|
|
|
5003 |
Ft(?)q Fv(-atom\))567 b FB(f)p FS(a)21 b FB(\031)p FF(?)g
|
|
|
5004 |
FS(a)p FB(g)c(])h FS(P)1912 913 y Fm(")1865 955 y FF(=)-13
|
|
|
5005 |
b FB(\))23 b FS(P)546 1055 y Fv(\()p Fu(\031)p Ft(?)q
|
|
|
5006 |
Fv(-suspension\))187 b FB(f)p FS(\031)s Fz(\001)o FS(X)28
|
|
|
5007 |
b FB(\031)p FF(?)21 b FS(\031)1542 1023 y Fw(0)1564 1055
|
|
|
5008 |
y Fz(\001)p FS(X)6 b FB(g)17 b(])h FS(P)1912 1012 y Fm(")1865
|
|
|
5009 |
1055 y FF(=)-13 b FB(\))23 b(f)p FS(a)e FF(#?)h FS(X)27
|
|
|
5010 |
b FB(j)22 b FS(a)f FB(2)g FD(ds)7 b FF(\()p FS(\031)s(;)13
|
|
|
5011 |
b(\031)2736 1023 y Fw(0)2758 1055 y FF(\))p FB(g)k([)g
|
|
|
5012 |
FS(P)546 1193 y Fv(\()p Fu(\031)p Ft(?)q Fv(-v)n(ariable\))1310
|
|
|
5013 |
1147 y FB(f)p FS(t)22 b FB(\031)o FF(?)g FS(\031)s Fz(\001)o
|
|
|
5014 |
FS(X)6 b FB(g)18 b(])f FS(P)1310 1238 y FB(f)p FS(\031)s
|
|
|
5015 |
Fz(\001)p FS(X)27 b FB(\031)p FF(?)22 b FS(t)p FB(g)17
|
|
|
5016 |
b(])g FS(P)1826 1087 y Fx(\033)1908 1150 y Fm(\033)1865
|
|
|
5017 |
1193 y FF(=)-13 b FB(\))23 b FS(\033)s(P)2204 1169 y
|
|
|
5018 |
FV(with)c FS(\033)24 b FF(=)d([)p FS(X)28 b FF(:=)21
|
|
|
5019 |
b FS(\031)2766 1137 y Fw(\000)p FG(1)2848 1169 y Fz(\001)p
|
|
|
5020 |
FS(t)p FF(])2204 1260 y FV(pro)o(vided)f FS(X)25 b FV(does)20
|
|
|
5021 |
b(not)f(occur)h(in)f FS(t)546 1371 y Fv(\()p Ft(#?)q
|
|
|
5022 |
Fv(-unit\))569 b FB(f)p FS(a)22 b FF(#?)f FB(hig)c(])h
|
|
|
5023 |
FS(P)1903 1328 y Fl(?)1865 1371 y FF(=)-13 b FB(\))23
|
|
|
5024 |
b FS(P)546 1482 y Fv(\()p Ft(#?)q Fv(-pair\))410 b FB(f)p
|
|
|
5025 |
FS(a)22 b FF(#?)f FB(h)p FS(t)1497 1490 y FG(1)1532 1482
|
|
|
5026 |
y FS(;)13 b(t)1594 1490 y FG(2)1628 1482 y FB(ig)k(])h
|
|
|
5027 |
FS(P)1903 1439 y Fl(?)1865 1482 y FF(=)-13 b FB(\))23
|
|
|
5028 |
b(f)p FS(a)e FF(#?)h FS(t)2262 1490 y FG(1)2296 1482
|
|
|
5029 |
y FS(;)13 b(a)21 b FF(#?)h FS(t)2542 1490 y FG(2)2576
|
|
|
5030 |
1482 y FB(g)17 b([)h FS(P)546 1592 y Fv(\()p Ft(#?)q
|
|
|
5031 |
Fv(-function)h(symbol\))212 b FB(f)p FS(a)21 b FF(#?)h
|
|
|
5032 |
FS(f)f(t)p FB(g)c(])h FS(P)1903 1549 y Fl(?)1865 1592
|
|
|
5033 |
y FF(=)-13 b FB(\))23 b(f)p FS(a)e FF(#?)h FS(t)p FB(g)17
|
|
|
5034 |
b([)g FS(P)546 1703 y Fv(\()p Ft(#?)q Fv(-abstraction-1\))298
|
|
|
5035 |
b FB(f)p FS(a)22 b FF(#?)f FS(a:t)p FB(g)c(])h FS(P)1903
|
|
|
5036 |
1660 y Fl(?)1865 1703 y FF(=)-13 b FB(\))23 b FS(P)546
|
|
|
5037 |
1814 y Fv(\()p Ft(#?)q Fv(-abstraction-2\))276 b FB(f)p
|
|
|
5038 |
FS(a)21 b FF(#?)h FS(a)1587 1782 y Fw(0)1609 1814 y FS(:t)p
|
|
|
5039 |
FB(g)17 b(])h FS(P)1903 1771 y Fl(?)1865 1814 y FF(=)-13
|
|
|
5040 |
b FB(\))23 b(f)p FS(a)e FF(#?)h FS(t)p FB(g)17 b([)g
|
|
|
5041 |
FS(P)85 b FV(pro)o(vided)21 b FS(a)g FB(6)p FF(=)g FS(a)2987
|
|
|
5042 |
1782 y Fw(0)546 1924 y Fv(\()p Ft(#?)q Fv(-atom\))536
|
|
|
5043 |
b FB(f)p FS(a)21 b FF(#?)h FS(a)1636 1893 y Fw(0)1658
|
|
|
5044 |
1924 y FB(g)17 b(])h FS(P)1903 1881 y Fl(?)1865 1924
|
|
|
5045 |
y FF(=)-13 b FB(\))23 b FS(P)85 b FV(pro)o(vided)20 b
|
|
|
5046 |
FS(a)h FB(6)p FF(=)g FS(a)2613 1893 y Fw(0)546 2037 y
|
|
|
5047 |
Fv(\()p Ft(#?)q Fv(-suspension\))300 b FB(f)p FS(a)22
|
|
|
5048 |
b FF(#?)f FS(\031)s Fz(\001)p FS(X)6 b FB(g)17 b(])h
|
|
|
5049 |
FS(P)1901 1994 y Fw(r)1865 2037 y FF(=)-13 b FB(\))23
|
|
|
5050 |
b FS(P)85 b FV(with)19 b FB(r)i FF(=)g FB(f)p FS(\031)2549
|
|
|
5051 |
2005 y Fw(\000)p FG(1)2632 2037 y Fz(\001)o FS(a)g FF(#)g
|
|
|
5052 |
FS(X)6 b FB(g)p 3402 2089 V 523 2092 2882 4 v 1476 2235
|
|
|
5053 |
a FU(Fig)o(.)12 b(3.)18 b FV(Labelled)h(transformations.)523
|
|
|
5054 |
2500 y FW(sequence)g(of)h(the)g(form)1441 2636 y FS(P)1550
|
|
|
5055 |
2590 y Fm(\033)1586 2600 y Fk(1)1522 2636 y FF(=)-13
|
|
|
5056 |
b FB(\))21 b(\001)13 b(\001)g(\001)1802 2591 y Fm(\033)1838
|
|
|
5057 |
2599 y Fj(n)1778 2636 y FF(=)-13 b FB(\))21 b FS(P)1983
|
|
|
5058 |
2600 y Fw(0)2046 2590 y(r)2097 2600 y Fk(1)2026 2636
|
|
|
5059 |
y FF(=)-13 b FB(\))21 b(\001)14 b(\001)f(\001)2291 2591
|
|
|
5060 |
y Fw(r)2342 2599 y Fj(m)2282 2636 y FF(=)-13 b FB(\))21
|
|
|
5061 |
b Fi(?)793 b FV(\(10\))523 2758 y FW(\(where)21 b FQ(P)841
|
|
|
5062 |
2727 y FL(0)886 2758 y FW(does)h(not)f(contain)g(an)o(y)g(equational)f
|
|
|
5063 |
(subproblems\))f(and)j(the)f(algorithm)g(returns)f(the)523
|
|
|
5064 |
2857 y(solution)f FM(\()p FO(r)912 2869 y FJ(1)969 2857
|
|
|
5065 |
y FO([)f(\001)c(\001)g(\001)19 b([)f(r)1300 2869 y FI(m)1364
|
|
|
5066 |
2857 y FQ(;)27 b(\033)1461 2869 y FI(n)1526 2857 y FO(\016)17
|
|
|
5067 |
b(\001)d(\001)g(\001)19 b(\016)f FQ(\033)1808 2869 y
|
|
|
5068 |
FJ(1)1845 2857 y FM(\))p FW(.)648 2957 y(It)j(is)g(not)g(hard)f(to)h
|
|
|
5069 |
(de)n(vise)f(a)i(well-founded)c(ordering)g(on)j(nominal)e
|
|
|
5070 |
(uni\002cation)h(problems)f(to)523 3056 y(sho)n(w)h(that)g(each)g
|
|
|
5071 |
(phase)g(of)g(the)g(algorithm)f(must)h(terminate.)f(So)h(one)g(just)h
|
|
|
5072 |
(has)f(to)h(sho)n(w)e(that)531 3195 y(\(a\))40 b(if)21
|
|
|
5073 |
b(the)f(algorithm)e(f)o(ails)j(on)f FQ(P)12 b FW(,)20
|
|
|
5074 |
b(then)g FQ(P)33 b FW(has)20 b(no)g(solution;)f(and)526
|
|
|
5075 |
3289 y(\(b\))40 b(if)16 b(the)f(algorithm)f(succeeds)i(on)f
|
|
|
5076 |
FQ(P)d FW(,)j(then)g(the)h(result)f(it)i(produces)c(is)k(a)f(most)f
|
|
|
5077 |
(general)g(solution.)523 3425 y(When)22 b(f)o(ailure)g(happens)f(it)i
|
|
|
5078 |
(is)h(because)d(of)h(certain)g(subproblems)f(that)h(manifestly)g(ha)n
|
|
|
5079 |
(v)o(e)f(no)h(so-)523 3516 y(lution)d(\(e.g.)f(in)h(the)g(\002rst)h
|
|
|
5080 |
(phase,)f FQ(a)k FO(\031)o FM(?)h FQ(a)1726 3486 y FL(0)1769
|
|
|
5081 |
3516 y FW(with)19 b FQ(a)k FO(6)p FM(=)g FQ(a)2135 3486
|
|
|
5082 |
y FL(0)2158 3516 y FW(,)c(and)g FQ(\031)s FK(\001)p FQ(X)29
|
|
|
5083 |
b FO(\031)p FM(?)23 b FQ(f)f(t)e FW(or)f FQ(f)k(t)g FO(\031)o
|
|
|
5084 |
FM(?)g FQ(\031)s FK(\001)p FQ(X)j FW(with)523 3607 y
|
|
|
5085 |
FQ(X)j FW(occurring)19 b(in)j FQ(t)p FW(;)g(in)g(the)g(second)f(phase,)
|
|
|
5086 |
g FQ(a)k FM(#?)i FQ(a)p FW(\).)21 b(Therefore)f(part)h(\(a\))h(is)g(a)g
|
|
|
5087 |
(consequence)e(of)523 3699 y(the)g(follo)n(wing)f(tw)o(o)h(properties)f
|
|
|
5088 |
(of)h(transformations,)e(where)i(we)g(write)h Fh(U)p
|
|
|
5089 |
FM(\()p FQ(P)12 b FM(\))21 b FW(for)f(the)g(set)h(of)f(all)523
|
|
|
5090 |
3790 y(solutions)g(for)f(a)i(problem)d FQ(P)12 b FW(:)683
|
|
|
5091 |
3930 y FV(if)18 b FF(\()p FB(r)841 3894 y Fw(0)863 3930
|
|
|
5092 |
y FS(;)13 b(\033)944 3894 y Fw(0)966 3930 y FF(\))21
|
|
|
5093 |
b FB(2)h Fg(U)p FF(\()p FS(P)11 b FF(\))18 b FV(and)h
|
|
|
5094 |
FS(P)1534 3887 y Fm(\033)1491 3930 y FF(=)-13 b FB(\))21
|
|
|
5095 |
b FS(P)1696 3894 y Fw(0)1718 3930 y FS(;)13 b FV(then)20
|
|
|
5096 |
b FF(\()p FB(r)1994 3894 y Fw(0)2016 3930 y FS(;)13 b(\033)2097
|
|
|
5097 |
3894 y Fw(0)2119 3930 y FF(\))21 b FB(2)h Fg(U)p FF(\()p
|
|
|
5098 |
FS(P)2389 3894 y Fw(0)2410 3930 y FF(\))d FV(and)h FB(r)2650
|
|
|
5099 |
3894 y Fw(0)2693 3930 y FB(`)i FS(\033)2809 3894 y Fw(0)2848
|
|
|
5100 |
3930 y FB(\016)17 b FS(\033)24 b FB(\031)d FS(\033)3099
|
|
|
5101 |
3894 y Fw(0)3280 3930 y FV(\(11\))682 4090 y(if)d FF(\()p
|
|
|
5102 |
FB(r)840 4054 y Fw(0)863 4090 y FS(;)13 b(\033)944 4054
|
|
|
5103 |
y Fw(0)966 4090 y FF(\))21 b FB(2)g Fg(U)p FF(\()p FS(P)11
|
|
|
5104 |
b FF(\))18 b FV(and)i FS(P)1527 4048 y Fw(r)1491 4090
|
|
|
5105 |
y FF(=)-13 b FB(\))21 b FS(P)1696 4054 y Fw(0)1718 4090
|
|
|
5106 |
y FS(;)13 b FV(then)19 b FF(\()p FB(r)1993 4054 y Fw(0)2016
|
|
|
5107 |
4090 y FS(;)13 b(\033)2097 4054 y Fw(0)2119 4090 y FF(\))21
|
|
|
5108 |
b FB(2)g Fg(U)p FF(\()p FS(P)2388 4054 y Fw(0)2410 4090
|
|
|
5109 |
y FF(\))d FV(and)i FB(r)2649 4054 y Fw(0)2693 4090 y
|
|
|
5110 |
FB(`)h FS(\033)2808 4054 y Fw(0)2830 4090 y FF(\()p FB(r)p
|
|
|
5111 |
FF(\))p FS(:)305 b FV(\(12\))523 4204 y FW(F)o(or)20
|
|
|
5112 |
b(part)g(\(b\),)f(one)h(\002rst)h(sho)n(ws)680 4319 y
|
|
|
5113 |
FV(if)d FF(\()p FB(r)838 4283 y Fw(0)860 4319 y FS(;)c(\033)942
|
|
|
5114 |
4283 y Fw(0)964 4319 y FF(\))21 b FB(2)g Fg(U)p FF(\()p
|
|
|
5115 |
FS(P)1233 4283 y Fw(0)1255 4319 y FF(\))d FV(and)i FS(P)1554
|
|
|
5116 |
4276 y Fm(\033)1511 4319 y FF(=)-13 b FB(\))21 b FS(P)1716
|
|
|
5117 |
4283 y Fw(0)1738 4319 y FS(;)13 b FV(then)20 b FF(\()p
|
|
|
5118 |
FB(r)2014 4283 y Fw(0)2036 4319 y FS(;)13 b(\033)2117
|
|
|
5119 |
4283 y Fw(0)2156 4319 y FB(\016)k FS(\033)s FF(\))k FB(2)h
|
|
|
5120 |
Fg(U)p FF(\()p FS(P)11 b FF(\))722 b FV(\(13\))678 4479
|
|
|
5121 |
y(if)19 b FF(\()p FB(r)837 4443 y Fw(0)859 4479 y FS(;)13
|
|
|
5122 |
b(\033)940 4443 y Fw(0)962 4479 y FF(\))21 b FB(2)h Fg(U)p
|
|
|
5123 |
FF(\()p FS(P)1232 4443 y Fw(0)1253 4479 y FF(\))p FS(;)35
|
|
|
5124 |
b(P)1456 4436 y Fw(r)1420 4479 y FF(=)-13 b FB(\))21
|
|
|
5125 |
b FS(P)1625 4443 y Fw(0)1665 4479 y FV(and)f FB(r)1856
|
|
|
5126 |
4443 y Fw(00)1918 4479 y FB(`)h FS(\033)2033 4443 y Fw(0)2055
|
|
|
5127 |
4479 y FF(\()p FB(r)p FF(\))p FS(;)13 b FV(then)19 b
|
|
|
5128 |
FF(\()p FB(r)2454 4443 y Fw(0)2494 4479 y FB([)e(r)2626
|
|
|
5129 |
4443 y Fw(00)2667 4479 y FS(;)c(\033)2748 4443 y Fw(0)2770
|
|
|
5130 |
4479 y FF(\))21 b FB(2)g Fg(U)p FF(\()p FS(P)11 b FF(\))p
|
|
|
5131 |
FS(:)190 b FV(\(14\))523 4601 y FW(From)19 b(these)h(and)g(the)f(f)o
|
|
|
5132 |
(act)h(that)g FM(\()p Fn(?)p FQ(;)14 b(")p FM(\))23 b
|
|
|
5133 |
FO(2)h Fh(U)p FM(\()p Fn(?)p FM(\))p FW(,)c(one)f(gets)i(that)e(if)i(a)
|
|
|
5134 |
f(sequence)f(lik)o(e)h(\(10\))e(e)o(xists,)523 4725 y(then)28
|
|
|
5135 |
b FM(\()p FO(r)p FQ(;)14 b(\033)s FM(\))955 4678 y Fy(def)960
|
|
|
5136 |
4725 y FM(=)43 b(\()p FO(r)1169 4737 y FJ(1)1232 4725
|
|
|
5137 |
y FO([)25 b(\001)14 b(\001)g(\001)24 b([)i(r)1583 4737
|
|
|
5138 |
y FI(m)1646 4725 y FQ(;)i(\033)1744 4737 y FI(n)1814
|
|
|
5139 |
4725 y FO(\016)c(\001)14 b(\001)g(\001)25 b(\016)f FQ(\033)2115
|
|
|
5140 |
4737 y FJ(1)2153 4725 y FM(\))29 b FW(is)h(in)f Fh(U)p
|
|
|
5141 |
FM(\()p FQ(P)12 b FM(\))p FW(.)30 b(Furthermore)c(from)i(\(11\))523
|
|
|
5142 |
4825 y(and)g(\(12\))o(,)h(we)g(get)g(that)f(an)o(y)g(other)g(solution)g
|
|
|
5143 |
FM(\()p FO(r)2017 4795 y FL(0)2041 4825 y FQ(;)14 b(\033)2128
|
|
|
5144 |
4795 y FL(0)2152 4825 y FM(\))39 b FO(2)g Fh(U)p FM(\()p
|
|
|
5145 |
FQ(P)12 b FM(\))30 b FW(satis\002es)g FO(r)2896 4795
|
|
|
5146 |
y FL(0)2959 4825 y FO(`)38 b FQ(\033)3098 4795 y FL(0)3122
|
|
|
5147 |
4825 y FM(\()p FO(r)p FM(\))30 b FW(and)523 4924 y FO(r)592
|
|
|
5148 |
4894 y FL(0)639 4924 y FO(`)22 b FQ(\033)762 4894 y FL(0)805
|
|
|
5149 |
4924 y FO(\016)17 b FQ(\033)27 b FO(\031)c FQ(\033)1076
|
|
|
5150 |
4894 y FL(0)1099 4924 y FW(,)e(so)f(that)h FM(\()p FO(r)p
|
|
|
5151 |
FQ(;)14 b(\033)s FM(\))22 b FW(is)f(indeed)e(a)h(most)h(general)e
|
|
|
5152 |
(solution.)614 b FO(u)-55 b(t)p eop end
|
|
|
5153 |
%%Page: 11 11
|
|
|
5154 |
TeXDict begin 11 10 bop 523 369 2882 4 v 523 1962 4 1594
|
|
|
5155 |
v 533 453 a Ff(P)578 462 y FG(1)654 415 y Fm(")612 453
|
|
|
5156 |
y Ft(=)-12 b Fu(\))o(f)p Fe(fn)13 b Ff(b:)p Fe(app)q
|
|
|
5157 |
Fu(h)p Ff(X)1084 462 y FG(1)1119 453 y Ff(;)e Fe(vr)h
|
|
|
5158 |
Ff(b)p Fu(i)21 b(\031)p Ft(?)e Fe(fn)13 b Ff(b:)p Fe(app)p
|
|
|
5159 |
Fu(h)p Fe(vr)g Ff(b;)f Ft(\()p Ff(a)g(b)p Ft(\))p Fd(\001)q
|
|
|
5160 |
Ff(X)2042 462 y FG(1)2076 453 y Fu(i)p Ff(;)20 b(a)g
|
|
|
5161 |
Ft(#?)g Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)f(X)2794
|
|
|
5162 |
462 y FG(1)2828 453 y Fu(ig)q Fv(\()p Fu(\031)p Ft(?)p
|
|
|
5163 |
Fv(-abstraction-2\))654 508 y Fm(")612 546 y Ft(=)-12
|
|
|
5164 |
b Fu(\))o(f)p Fe(app)r Fu(h)p Ff(X)952 555 y FG(1)987
|
|
|
5165 |
546 y Ff(;)11 b Fe(vr)h Ff(b)p Fu(i)21 b(\031)o Ft(?)f
|
|
|
5166 |
Fe(app)q Fu(h)p Fe(vr)13 b Ff(b;)f Ft(\()p Ff(a)g(b)p
|
|
|
5167 |
Ft(\))p Fd(\001)p Ff(X)1777 555 y FG(1)1812 546 y Fu(i)p
|
|
|
5168 |
Ff(;)20 b(a)g Ft(#?)g Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p
|
|
|
5169 |
Fe(vr)h Ff(a;)e(X)2529 555 y FG(1)2564 546 y Fu(ig)265
|
|
|
5170 |
b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-1\))622 615
|
|
|
5171 |
y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083
|
|
|
5172 |
b(\001)11 b(\001)h(\001)654 646 y Fm(")612 685 y Ft(=)-12
|
|
|
5173 |
b Fu(\))o(f)p Ff(X)818 694 y FG(1)873 685 y Fu(\031)p
|
|
|
5174 |
Ft(?)20 b Fe(vr)12 b Ff(b;)31 b Fe(vr)13 b Ff(b)19 b
|
|
|
5175 |
Fu(\031)p Ft(?)h(\()p Ff(a)13 b(b)p Ft(\))p Fd(\001)p
|
|
|
5176 |
Ff(X)1599 694 y FG(1)1634 685 y Ff(;)19 b(a)h Ft(#?)g
|
|
|
5177 |
Fe(fn)12 b Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)f(X)2324
|
|
|
5178 |
694 y FG(1)2358 685 y Fu(ig)471 b Fv(\()p Fu(\031)p Ft(?)p
|
|
|
5179 |
Fv(-pair\))650 739 y Fm(\033)612 778 y Ft(=)-12 b Fu(\))o(f)p
|
|
|
5180 |
Fe(vr)13 b Ff(b)20 b Fu(\031)p Ft(?)g Fe(vr)12 b Ff(a;)31
|
|
|
5181 |
b(a)20 b Ft(#?)g Fe(fn)13 b Ff(a:)p Fe(app)p Fu(h)p Fe(vr)g
|
|
|
5182 |
Ff(a;)f Fe(vr)g Ff(b)p Fu(ig)68 b Fv(with)20 b Ff(\033)j
|
|
|
5183 |
Ft(=)c([)p Ff(X)2358 787 y FG(1)2412 778 y Ft(:=)g Fe(vr)13
|
|
|
5184 |
b Ff(b)p Ft(])252 b Fv(\()p Fu(\031)p Ft(?)p Fv(-v)n(ariable\))654
|
|
|
5185 |
832 y Fm(")612 871 y Ft(=)-12 b Fu(\))o(f)p Ff(b)20 b
|
|
|
5186 |
Fu(\031)p Ft(?)g Ff(a;)32 b(a)20 b Ft(#?)g Fe(fn)12 b
|
|
|
5187 |
Ff(a:)p Fe(app)q Fu(h)p Fe(vr)h Ff(a;)e Fe(vr)i Ff(b)p
|
|
|
5188 |
Fu(ig)1117 b Fv(\()p Fu(\031)p Ft(?)p Fv(-fnctn)19 b(symbol\))725
|
|
|
5189 |
954 y Ft(F)-8 b(AIL)533 1054 y Ff(P)578 1063 y FG(4)654
|
|
|
5190 |
1016 y Fm(")612 1054 y Ft(=)c Fu(\))o(f)p Fe(fn)13 b
|
|
|
5191 |
Ff(b:)p Fe(app)q Fu(h)p Fe(vr)f Ff(b;)g(X)1228 1063 y
|
|
|
5192 |
FG(6)1263 1054 y Fu(i)20 b(\031)p Ft(?)f Fe(fn)13 b Ff(a:)p
|
|
|
5193 |
Fe(app)q Fu(h)p Fe(vr)g Ff(a;)e(X)1899 1063 y FG(7)1934
|
|
|
5194 |
1054 y Fu(ig)895 b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-1\))654
|
|
|
5195 |
1109 y Fm(")612 1147 y Ft(=)-12 b Fu(\))o(f)p Fe(app)r
|
|
|
5196 |
Fu(h)p Fe(vr)12 b Ff(b;)g(X)1096 1156 y FG(6)1130 1147
|
|
|
5197 |
y Fu(i)21 b(\031)o Ft(?)f Fe(app)q Fu(h)p Fe(vr)13 b
|
|
|
5198 |
Ff(b;)f Ft(\()p Ff(b)g(a)p Ft(\))p Fd(\001)p Ff(X)1777
|
|
|
5199 |
1156 y FG(7)1812 1147 y Fu(i)p Ff(;)32 b(b)19 b Ft(#?)h
|
|
|
5200 |
Fe(app)q Fu(h)p Fe(vr)13 b Ff(a;)f(X)2395 1156 y FG(7)2429
|
|
|
5201 |
1147 y Fu(ig)400 b Fv(\()p Fu(\031)p Ft(?)p Fv(-abstraction-2\))622
|
|
|
5202 |
1216 y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083
|
|
|
5203 |
b(\001)11 b(\001)h(\001)654 1247 y Fm(")612 1286 y Ft(=)-12
|
|
|
5204 |
b Fu(\))o(f)p Ff(b)20 b Fu(\031)p Ft(?)g Ff(b;)31 b(X)1057
|
|
|
5205 |
1295 y FG(6)1112 1286 y Fu(\031)p Ft(?)19 b(\()p Ff(b)13
|
|
|
5206 |
b(a)p Ft(\))p Fd(\001)p Ff(X)1434 1295 y FG(7)1469 1286
|
|
|
5207 |
y Ff(;)31 b(b)20 b Ft(#?)g Fe(app)p Fu(h)p Fe(vr)13 b
|
|
|
5208 |
Ff(a;)f(X)2024 1295 y FG(7)2059 1286 y Fu(ig)770 b Fv(\()p
|
|
|
5209 |
Fu(\031)p Ft(?)p Fv(-fnctn)19 b(symbol\))654 1340 y Fm(")612
|
|
|
5210 |
1379 y Ft(=)-12 b Fu(\))o(f)p Ff(X)818 1388 y FG(6)873
|
|
|
5211 |
1379 y Fu(\031)p Ft(?)20 b(\()p Ff(b)12 b(a)p Ft(\))p
|
|
|
5212 |
Fd(\001)q Ff(X)1196 1388 y FG(7)1230 1379 y Ff(;)31 b(b)20
|
|
|
5213 |
b Ft(#?)g Fe(app)q Fu(h)p Fe(vr)12 b Ff(a;)g(X)1785 1388
|
|
|
5214 |
y FG(7)1820 1379 y Fu(ig)1009 b Fv(\()p Fu(\031)p Ft(?)p
|
|
|
5215 |
Fv(-atom\))650 1433 y Fm(\033)612 1472 y Ft(=)-12 b Fu(\))o(f)p
|
|
|
5216 |
Ff(b)20 b Ft(#?)g Fe(app)q Fu(h)p Fe(vr)13 b Ff(a;)f(X)1265
|
|
|
5217 |
1481 y FG(7)1299 1472 y Fu(ig)67 b Fv(with)21 b Ff(\033)h
|
|
|
5218 |
Ft(=)e([)p Ff(X)1781 1481 y FG(6)1835 1472 y Ft(:=)f(\()p
|
|
|
5219 |
Ff(b)12 b(a)p Ft(\))p Fd(\001)q Ff(X)2144 1481 y FG(7)2178
|
|
|
5220 |
1472 y Ft(])693 b Fv(\()p Fu(\031)p Ft(?)p Fv(-v)n(ariable\))645
|
|
|
5221 |
1537 y Fl(?)612 1576 y Ft(=)-12 b Fu(\))o(f)p Ff(b)20
|
|
|
5222 |
b Ft(#?)g Fu(h)p Fe(vr)13 b Ff(a;)f(X)1159 1585 y FG(7)1193
|
|
|
5223 |
1576 y Fu(ig)1636 b Fv(\()p Ft(#?)p Fv(-fnctn)19 b(symbol\))622
|
|
|
5224 |
1645 y Fu(\001)11 b(\001)g(\001)21 b(\001)12 b(\001)f(\001)2083
|
|
|
5225 |
b(\001)11 b(\001)h(\001)645 1687 y Fl(?)612 1726 y Ft(=)-12
|
|
|
5226 |
b Fu(\))o(f)p Ff(b)20 b Ft(#?)g Ff(a;)32 b(b)19 b Ft(#?)h
|
|
|
5227 |
Ff(X)1230 1735 y FG(7)1265 1726 y Fu(g)1591 b Fv(\()p
|
|
|
5228 |
Ft(#?)p Fv(-fnctn)19 b(symbol\))645 1791 y Fl(?)612 1831
|
|
|
5229 |
y Ft(=)-12 b Fu(\))o(f)p Ff(b)20 b Ft(#?)g Ff(X)980 1840
|
|
|
5230 |
y FG(7)1015 1831 y Fu(g)1841 b Fv(\()p Ft(#?)p Fv(-atom\))643
|
|
|
5231 |
1898 y Fw(r)612 1936 y Ft(=)-12 b Fu(\))o Fc(?)67 b Fv(with)20
|
|
|
5232 |
b Fu(r)g Ft(=)f Fu(f)p Ff(b)h Ft(#)g Ff(X)1359 1945 y
|
|
|
5233 |
FG(7)1393 1936 y Fu(g)1463 b Fv(\()p Ft(#?)p Fv(-suspension\))p
|
|
|
5234 |
3402 1962 V 523 1965 2882 4 v 1553 2109 a FU(Fig)o(.)12
|
|
|
5235 |
b(4.)19 b FV(Example)g(deri)n(v)n(ations)523 2377 y FP(Example)g(2.)41
|
|
|
5236 |
b FW(Using)25 b(the)g(\002rst)h(three)e(function)f(symbols)i(of)f(the)h
|
|
|
5237 |
(nominal)f(signature)g(of)h(Exam-)523 2468 y(ple)18 b(1)g(to)h
|
|
|
5238 |
(represent)e FQ(\025)p FW(-terms,)h(the)g(Quiz)g(at)h(the)f(end)f(of)h
|
|
|
5239 |
(the)h(Introduction)c(translates)j(into)g(the)g(fol-)523
|
|
|
5240 |
2559 y(lo)n(wing)k(four)h(uni\002cation)f(problems)f(o)o(v)o(er)h(that)
|
|
|
5241 |
i(signature,)e(where)g FQ(a)i FW(and)f FQ(b)g FW(are)g(distinct)h
|
|
|
5242 |
(atoms)523 2651 y(of)c(sort)g FA(vid)31 b FW(and)19 b
|
|
|
5243 |
FQ(X)1103 2663 y FJ(1)1140 2651 y FQ(;)14 b(:)g(:)g(:)g(;)g(X)1394
|
|
|
5244 |
2663 y FJ(7)1452 2651 y FW(are)20 b(distinct)g(v)n(ariables)f(of)h
|
|
|
5245 |
(sort)h FA(exp)5 b FW(:)1008 2804 y FS(P)1057 2812 y
|
|
|
5246 |
FG(1)1113 2761 y Fy(def)1120 2804 y FF(=)26 b FB(f)p
|
|
|
5247 |
FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p FC(app)q FB(h)p FS(X)1753
|
|
|
5248 |
2812 y FG(1)1788 2804 y FS(;)g FC(vr)h FS(b)p FB(i)21
|
|
|
5249 |
b(\031)p FF(?)g FC(fn)13 b FS(b:)p FC(fn)h FS(a:)p FC(app)q
|
|
|
5250 |
FB(h)p FC(vr)f FS(a;)g(X)2791 2812 y FG(1)2826 2804 y
|
|
|
5251 |
FB(ig)p FV(,)1008 2922 y FS(P)1057 2930 y FG(2)1113 2879
|
|
|
5252 |
y Fy(def)1120 2922 y FF(=)26 b FB(f)p FC(fn)13 b FS(a:)p
|
|
|
5253 |
FC(fn)g FS(b:)p FC(app)q FB(h)p FS(X)1753 2930 y FG(2)1788
|
|
|
5254 |
2922 y FS(;)g FC(vr)h FS(b)p FB(i)21 b(\031)p FF(?)g
|
|
|
5255 |
FC(fn)13 b FS(b:)p FC(fn)h FS(a:)p FC(app)q FB(h)p FC(vr)f
|
|
|
5256 |
FS(a;)g(X)2791 2930 y FG(3)2826 2922 y FB(ig)p FV(,)1008
|
|
|
5257 |
3040 y FS(P)1057 3048 y FG(3)1113 2997 y Fy(def)1120
|
|
|
5258 |
3040 y FF(=)26 b FB(f)p FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p
|
|
|
5259 |
FC(app)q FB(h)p FC(vr)h FS(b;)f(X)1912 3048 y FG(4)1947
|
|
|
5260 |
3040 y FB(i)21 b(\031)p FF(?)g FC(fn)13 b FS(b:)p FC(fn)h
|
|
|
5261 |
FS(a:)p FC(app)q FB(h)p FC(vr)f FS(a;)g(X)2791 3048 y
|
|
|
5262 |
FG(5)2826 3040 y FB(ig)p FV(,)1008 3158 y FS(P)1057 3166
|
|
|
5263 |
y FG(4)1113 3115 y Fy(def)1120 3158 y FF(=)26 b FB(f)p
|
|
|
5264 |
FC(fn)13 b FS(a:)p FC(fn)g FS(b:)p FC(app)q FB(h)p FC(vr)h
|
|
|
5265 |
FS(b;)f(X)1912 3166 y FG(6)1947 3158 y FB(i)21 b(\031)p
|
|
|
5266 |
FF(?)g FC(fn)13 b FS(a:)p FC(fn)h FS(a:)p FC(app)p FB(h)p
|
|
|
5267 |
FC(vr)f FS(a;)g(X)2798 3166 y FG(7)2833 3158 y FB(ig)p
|
|
|
5268 |
FV(.)523 3288 y FW(Applying)18 b(the)j(nominal)e(uni\002cation)f
|
|
|
5269 |
(algorithm)h(described)g(abo)o(v)o(e,)f(we)j(\002nd)f(that)581
|
|
|
5270 |
3429 y FO(\017)41 b FQ(P)717 3441 y FJ(1)775 3429 y FW(has)21
|
|
|
5271 |
b(no)f(solution;)581 3524 y FO(\017)41 b FQ(P)717 3536
|
|
|
5272 |
y FJ(2)775 3524 y FW(has)21 b(a)g(most)f(general)f(solution)g(gi)n(v)o
|
|
|
5273 |
(en)g(by)h FO(r)2075 3536 y FJ(2)2122 3524 y FM(=)9 b
|
|
|
5274 |
Fn(?)20 b FW(and)f FQ(\033)2468 3536 y FJ(2)2515 3524
|
|
|
5275 |
y FM(=)9 b([)p FQ(X)2681 3536 y FJ(2)2727 3524 y FM(:=)g
|
|
|
5276 |
FN(vr)k FQ(b;)h(X)3067 3536 y FJ(3)3113 3524 y FM(:=)9
|
|
|
5277 |
b FN(vr)k FQ(a)p FM(])p FW(;)581 3618 y FO(\017)41 b
|
|
|
5278 |
FQ(P)717 3630 y FJ(3)775 3618 y FW(has)21 b(a)g(most)f(general)f
|
|
|
5279 |
(solution)g(gi)n(v)o(en)g(by)h FO(r)2075 3630 y FJ(3)2122
|
|
|
5280 |
3618 y FM(=)9 b Fn(?)20 b FW(and)f FQ(\033)2468 3630
|
|
|
5281 |
y FJ(3)2515 3618 y FM(=)9 b([)p FQ(X)2681 3630 y FJ(4)2727
|
|
|
5282 |
3618 y FM(:=)g(\()p FQ(a)14 b(b)p FM(\))p FK(\001)p FQ(X)3078
|
|
|
5283 |
3630 y FJ(5)3115 3618 y FM(])p FW(;)581 3713 y FO(\017)41
|
|
|
5284 |
b FQ(P)717 3725 y FJ(4)773 3713 y FW(has)17 b(a)h(most)f(general)f
|
|
|
5285 |
(solution)h(gi)n(v)o(en)f(by)h FO(r)2052 3725 y FJ(4)2098
|
|
|
5286 |
3713 y FM(=)9 b FO(f)p FQ(b)22 b FM(#)i FQ(X)2434 3725
|
|
|
5287 |
y FJ(7)2471 3713 y FO(g)17 b FW(and)g FQ(\033)2715 3725
|
|
|
5288 |
y FJ(3)2762 3713 y FM(=)9 b([)p FQ(X)2928 3725 y FJ(6)2974
|
|
|
5289 |
3713 y FM(:=)g(\()p FQ(b)14 b(a)p FM(\))p FK(\001)o FQ(X)3324
|
|
|
5290 |
3725 y FJ(7)3361 3713 y FM(])p FW(.)523 3859 y(Deri)n(v)n(ations)21
|
|
|
5291 |
b(for)g FQ(P)1101 3871 y FJ(1)1161 3859 y FW(and)g FQ(P)1356
|
|
|
5292 |
3871 y FJ(4)1416 3859 y FW(are)g(sk)o(etched)g(in)h(Fig.)g(4.)f(Using)h
|
|
|
5293 |
(the)g(Adequac)o(y)d(Theorem)h(1,)i(one)523 3958 y(can)30
|
|
|
5294 |
b(interpret)e(these)i(solutions)f(as)i(the)e(follo)n(wing)g(statements)
|
|
|
5295 |
g(about)g(the)h FQ(\025)p FW(-terms)g(from)e(the)523
|
|
|
5296 |
4058 y(quiz.)p 523 4122 V 523 4963 4 841 v 551 4198 a
|
|
|
5297 |
FU(Quiz)19 b(answers)595 4282 y FV(1.)41 b(There)19 b(is)g(no)g
|
|
|
5298 |
FS(\025)p FV(-term)g FS(M)1347 4290 y FG(1)1400 4282
|
|
|
5299 |
y FV(making)h(the)f(\002rst)f(pair)h(of)g(terms)f FS(\013)p
|
|
|
5300 |
FV(-equi)n(v)n(alent.)595 4374 y(2.)41 b(The)19 b(only)h(solution)f
|
|
|
5301 |
(for)g(the)g(second)h(problem)g(is)f(to)f(tak)o(e)i FS(M)2310
|
|
|
5302 |
4382 y FG(2)2366 4374 y FF(=)h FS(b)d FV(and)i FS(M)2699
|
|
|
5303 |
4382 y FG(3)2755 4374 y FF(=)h FS(a)p FV(.)595 4465 y(3.)41
|
|
|
5304 |
b(F)o(or)21 b(the)g(third)g(problem)h(we)e(can)i(tak)o(e)f
|
|
|
5305 |
FS(M)1810 4473 y FG(5)1866 4465 y FV(to)g(be)g(an)o(y)h
|
|
|
5306 |
FS(\025)p FV(-term,)e(so)h(long)g(as)g(we)g(tak)o(e)h
|
|
|
5307 |
FS(M)3060 4473 y FG(4)3115 4465 y FV(to)f(be)g(the)692
|
|
|
5308 |
4556 y(result)e(of)g(sw)o(apping)h(all)f(occurrences)h(of)f
|
|
|
5309 |
FS(a)g FV(and)g FS(b)g FV(throughout)h FS(M)2470 4564
|
|
|
5310 |
y FG(5)2505 4556 y FV(.)595 4647 y(4.)41 b(F)o(or)22
|
|
|
5311 |
b(the)h(last)f(problem,)h(we)f(can)h(tak)o(e)g FS(M)1802
|
|
|
5312 |
4655 y FG(7)1859 4647 y FV(to)f(be)h(an)o(y)g FS(\025)p
|
|
|
5313 |
FV(-term)f(that)g FT(does)h(not)g(contain)h(fr)m(ee)e(occur)o(-)692
|
|
|
5314 |
4739 y(r)m(ences)f(of)f FS(b)p FV(,)f(so)h(long)h(as)f(we)f(tak)o(e)i
|
|
|
5315 |
FS(M)1698 4747 y FG(6)1752 4739 y FV(to)f(be)g(the)g(result)f(of)h(sw)o
|
|
|
5316 |
(apping)h(all)f(occurrences)h(of)f FS(b)g FV(and)g FS(a)692
|
|
|
5317 |
4830 y FV(throughout)k FS(M)1116 4838 y FG(7)1151 4830
|
|
|
5318 |
y FV(,)e(or)g(equi)n(v)n(alently)i(\(since)e FS(b)g FV(is)g(not)g(free)
|
|
|
5319 |
g(in)h FS(M)2401 4838 y FG(7)2435 4830 y FV(\),)f(taking)h
|
|
|
5320 |
FS(M)2784 4838 y FG(6)2840 4830 y FV(to)g(be)f(the)g(result)g(of)692
|
|
|
5321 |
4921 y(replacing)e(all)f(free)f(occurrences)j(of)e FS(a)f
|
|
|
5322 |
FV(in)h FS(M)1884 4929 y FG(7)1937 4921 y FV(with)g FS(b)p
|
|
|
5323 |
FV(.)p 3402 4963 V 523 4966 2882 4 v eop end
|
|
|
5324 |
%%Page: 12 12
|
|
|
5325 |
TeXDict begin 12 11 bop 523 448 a FP(Remark)20 b(2)41
|
|
|
5326 |
b(\(Atoms)22 b(ar)m(e)g(not)f(variables\).)g FW(Nominal)h
|
|
|
5327 |
(uni\002cation)e(uni\002es)i(v)n(ariables,)f(b)n(ut)h(it)h(does)523
|
|
|
5328 |
548 y(not)i(unify)e(atoms.)i(Indeed)e(the)i(operation)e(of)h
|
|
|
5329 |
(identifying)f(tw)o(o)i(atoms)g(by)f(renaming)f(does)i(not)523
|
|
|
5330 |
648 y(necessarily)e(preserv)o(e)e(the)j(v)n(alidity)e(of)h(the)g
|
|
|
5331 |
(judgements)f(in)h(Fig.)g(2.)g(F)o(or)g(e)o(xample,)f
|
|
|
5332 |
Fn(?)28 b FO(`)g FQ(a:b)g FO(\031)523 747 y FQ(c:b)23
|
|
|
5333 |
b FW(holds)g(if)h FQ(b)k FO(6)p FM(=)g FQ(a;)14 b(c)p
|
|
|
5334 |
FW(;)24 b(b)n(ut)f(renaming)e FQ(b)j FW(to)f(be)g FQ(a)h
|
|
|
5335 |
FW(in)f(this)h(judgement)d(we)j(get)f Fn(?)28 b FO(`)h
|
|
|
5336 |
FQ(a:a)f FO(\031)g FQ(c:a)p FW(,)523 847 y(which)18 b(does)f(not)h
|
|
|
5337 |
(hold)f(so)i(long)e(as)i FQ(a)j FO(6)p FM(=)h FQ(c)p
|
|
|
5338 |
FW(.)18 b(Referring)f(to)h(Def.)g(2,)g(you)f(will)i(see)f(that)h(we)f
|
|
|
5339 |
(do)g(allo)n(w)523 946 y(v)n(ariables)k(ranging)g(o)o(v)o(er)f(sorts)j
|
|
|
5340 |
(of)f(atoms;)g(and)f(such)h(v)n(ariables)f(can)h(be)g(uni\002ed)g(lik)o
|
|
|
5341 |
(e)g(an)o(y)f(other)523 1046 y(v)n(ariables.)17 b(Ho)n(we)n(v)o(er)m(,)
|
|
|
5342 |
e(if)j FQ(A)h FW(is)f(such)f(a)i(v)n(ariable,)d(then)h(it)h(cannot)f
|
|
|
5343 |
(appear)f(in)i(abstraction)f(position,)523 1146 y(i.e.)26
|
|
|
5344 |
b(as)h FQ(A:t)p FW(.)g(This)f(is)i(because)d(we)i(speci\002cally)f
|
|
|
5345 |
(restricted)f(abstraction)g(to)i(range)e(o)o(v)o(er)g(atoms,)523
|
|
|
5346 |
1245 y(rather)31 b(than)g(o)o(v)o(er)g(arbitrary)f(terms)h(of)h(atom)f
|
|
|
5347 |
(sort.)h(Such)f(a)h(restriction)f(seems)h(necessary)f(to)523
|
|
|
5348 |
1345 y(obtain)16 b(single,)g(most)g(general,)f(solutions)h(to)h
|
|
|
5349 |
(nominal)e(uni\002cation)g(problems.)g(F)o(or)h(without)g(such)523
|
|
|
5350 |
1445 y(a)j(restriction,)e(because)h(of)g(rule)g(\()p
|
|
|
5351 |
FO(\031)p FW(-abstraction-2\))d(we)j(w)o(ould)g(also)g(ha)n(v)o(e)g(to)
|
|
|
5352 |
h(allo)n(w)f(v)n(ariables)f(to)523 1544 y(appear)24 b(on)g(the)h
|
|
|
5353 |
(left-hand)f(side)h(of)g(freshness)f(relations)g(and)h(in)g(suspended)e
|
|
|
5354 |
(permutations.)g(So)523 1644 y(then)e(we)g(w)o(ould)g(get)g
|
|
|
5355 |
(uni\002cation)f(problems)g(lik)o(e)i FO(f)p FM(\()p
|
|
|
5356 |
FQ(A)14 b(B)t FM(\))p FK(\001)p FQ(C)31 b FO(\031)p FM(?)25
|
|
|
5357 |
b FQ(C)6 b FO(g)p FW(,)21 b(where)g FQ(A)p FW(,)h FQ(B)k
|
|
|
5358 |
FW(and)20 b FQ(C)29 b FW(are)523 1743 y(v)n(ariables)20
|
|
|
5359 |
b(of)g(atom)g(sort;)h(this)g(has)g(tw)o(o)g(incomparable)d(solutions,)i
|
|
|
5360 |
(namely)g FM(\()p Fn(?)p FQ(;)14 b FM([)p FQ(A)24 b FM(:=)f
|
|
|
5361 |
FQ(B)t FM(]\))f FW(and)523 1843 y FM(\()p FO(f)p FQ(A)h
|
|
|
5362 |
FM(#)g FQ(C)q(;)14 b(B)28 b FM(#)23 b FQ(C)6 b FO(g)p
|
|
|
5363 |
FQ(;)14 b(")p FM(\))p FW(.)523 2166 y FR(4)99 b(Related)26
|
|
|
5364 |
b(w)o(ork)523 2422 y FW(Most)31 b(pre)n(vious)e(w)o(ork)i(on)f
|
|
|
5365 |
(uni\002cation)g(for)g(languages)f(with)i(binders)f(is)i(based)e(on)h
|
|
|
5366 |
(forms)f(of)523 2522 y(higher)n(-order)j(uni\002cation,)h(i.e.)i
|
|
|
5367 |
(solving)e(equations)h(between)g FQ(\025)p FW(-terms)h(modulo)e
|
|
|
5368 |
FQ(\013\014)t(\021)s FW(-equi-)523 2622 y(v)n(alence)15
|
|
|
5369 |
b(by)h(capture-a)n(v)n(oiding)d(substitution)i(of)h(terms)g(for)f
|
|
|
5370 |
(function)f(v)n(ariables.)h(Notable)h(among)523 2721
|
|
|
5371 |
y(that)i(w)o(ork)f(is)h(Miller')-5 b(s)18 b FP(higher)n(-or)m(der)f
|
|
|
5372 |
(pattern)g(uni\002cation)e FW(used)j(in)f(his)h FQ(L)2707
|
|
|
5373 |
2733 y FI(\025)2769 2721 y FW(logic)f(programming)523
|
|
|
5374 |
2821 y(language)g([20].)g(This)h(kind)g(of)g(uni\002cation)f(retains)i
|
|
|
5375 |
(the)f(good)f(properties)g(of)h(\002rst-order)f(uni\002ca-)523
|
|
|
5376 |
2920 y(tion:)22 b(a)h(linear)n(-time)e(decision)h(procedure)d(and)j(e)o
|
|
|
5377 |
(xistence)f(of)h(most)g(general)f(uni\002ers.)h(Ho)n(we)n(v)o(er)523
|
|
|
5378 |
3020 y(it)f(imposes)g(a)g(restriction)f(on)g(the)g(form)g(of)g
|
|
|
5379 |
FQ(\025)p FW(-terms)h(to)g(be)f(uni\002ed;)g(namely)g(that)g(function)f
|
|
|
5380 |
(v)n(ari-)523 3120 y(ables)e(may)f(only)g(be)h(applied)f(to)h(distinct)
|
|
|
5381 |
g(bound)e(v)n(ariables.)h(An)h(empirical)f(study)g(by)g(Michaylo)o(v)
|
|
|
5382 |
523 3219 y(and)24 b(Pfenning)g([19])f(suggests)i(that)g(most)g
|
|
|
5383 |
(uni\002cations)f(arising)h(dynamically)e(in)i(higher)n(-order)523
|
|
|
5384 |
3319 y(logic)16 b(programming)c(satisfy)k(Miller')-5
|
|
|
5385 |
b(s)16 b(restriction,)f(b)n(ut)h(that)g(it)h(rules)e(out)h(some)g
|
|
|
5386 |
(useful)f(program-)523 3419 y(ming)22 b(idioms.)g(F)o(or)g(us,)g(the)h
|
|
|
5387 |
(main)f(disadv)n(antage)e(of)i FQ(L)2158 3431 y FI(\025)2224
|
|
|
5388 |
3419 y FW(is)i(one)e(common)e(to)j(most)f(approaches)523
|
|
|
5389 |
3518 y(based)i(on)g(higher)n(-order)d(abstract)j(syntax:)g(one)f
|
|
|
5390 |
(cannot)g FP(dir)m(ectly)i FW(e)o(xpress)e(the)i(common)d(idiom)523
|
|
|
5391 |
3618 y(of)c(possibly-capturing)c(substitution)j(of)h(terms)g(for)f
|
|
|
5392 |
(meta)n(v)n(ariables.)g(Instead)g(one)h(has)g(to)g(replace)523
|
|
|
5393 |
3717 y(meta)n(v)n(ariables,)j FQ(X)7 b FW(,)21 b(with)h(function)f(v)n
|
|
|
5394 |
(ariables)g(applied)g(to)h(distinct)g(lists)h(of)f(\(bound\))e(v)n
|
|
|
5395 |
(ariables,)523 3817 y FQ(F)26 b(x)649 3829 y FJ(1)700
|
|
|
5396 |
3817 y FQ(:)14 b(:)g(:)g(x)858 3829 y FI(n)904 3817 y
|
|
|
5397 |
FW(,)20 b(and)g(use)g(capture-a)n(v)n(oiding)d(substitution.)648
|
|
|
5398 |
3928 y(Do)n(wek)25 b FP(et)i(al)f FW([8])f(present)h(a)h(v)o(ersion)d
|
|
|
5399 |
(of)i(higher)n(-order)d(pattern)j(uni\002cation)f(for)g
|
|
|
5400 |
FQ(\025\033)31 b FW(\(a)26 b FQ(\025)p FW(-)523 4028
|
|
|
5401 |
y(calculus)i(with)h(de-Bruijn)e(indices)i(and)f(e)o(xplicit)g
|
|
|
5402 |
(substitutions\))f(in)i(which)f(uni\002cation)f(prob-)523
|
|
|
5403 |
4127 y(lems)c(are)f(solv)o(ed,)g(lik)o(e)h(in)f(nominal)f
|
|
|
5404 |
(uni\002cation,)h(by)g(te)o(xtual)f(replacements)h(of)g(terms)g(for)g
|
|
|
5405 |
(v)n(ari-)523 4227 y(ables;)i(ho)n(we)n(v)o(er)e(a)j(`pre-cooking')20
|
|
|
5406 |
b(operation)i(ensures)h(that)h(the)g(te)o(xtual)f(replacements)g(can)h
|
|
|
5407 |
(be)523 4327 y(\(f)o(aithfully\))e(related)i(to)g(capture-a)n(v)n
|
|
|
5408 |
(oiding)d(substitutions.)j(It)g(seems)h(that)f(nominal)f(uni\002cation)
|
|
|
5409 |
523 4426 y(problems)i(can)h(be)h(encoded)d(into)i(higher)n(-order)d
|
|
|
5410 |
(pattern)j(uni\002cation)f(problems)g(using)h(a)h FP(non-)523
|
|
|
5411 |
4526 y(trivial)21 b FW(translation)f(\(the)g(details)g(of)h(this)g
|
|
|
5412 |
(encoding)d(still)j(remain)f(to)h(be)f(in)m(v)o(estigated\).)e(But)j(e)
|
|
|
5413 |
n(v)o(en)523 4625 y(if)h(it)h(turns)e(out)h(that)g(it)h(is)f(possible)g
|
|
|
5414 |
(to)g(simulate)g(nominal)f(uni\002cation)f(by)i(higher)n(-order)c
|
|
|
5415 |
(pattern)523 4725 y(uni\002cation,)k(the)h(calculations)g(in)m(v)n(olv)
|
|
|
5416 |
o(ed)e(in)j(translating)f(our)f(terms)i(into)f(higher)n(-order)d
|
|
|
5417 |
(pattern)523 4825 y(terms)30 b(and)g(then)f(using)g(higher)n(-order)e
|
|
|
5418 |
(pattern)i(uni\002cation)g(are)h(f)o(ar)g(more)f(intricate)h(than)f
|
|
|
5419 |
(our)523 4924 y(simple)20 b(algorithm)f(that)h(solv)o(es)g(nominal)f
|
|
|
5420 |
(uni\002cation)g(problems)g(directly)-5 b(.)p eop end
|
|
|
5421 |
%%Page: 13 13
|
|
|
5422 |
TeXDict begin 13 12 bop 648 448 a FW(Hamana)20 b([13,)12
|
|
|
5423 |
b(14])21 b(manages)f(to)i(add)f(possibly-capturing)d(substitution)i(to)
|
|
|
5424 |
i(a)g(language)d(lik)o(e)523 548 y(Miller')-5 b(s)21
|
|
|
5425 |
b FQ(L)864 560 y FI(\025)907 548 y FW(.)f(This)h(is)g(achie)n(v)o(ed)e
|
|
|
5426 |
(by)g(adding)g(syntax)h(for)f(e)o(xplicit)h(renaming)f(operations)f
|
|
|
5427 |
(and)i(by)523 648 y(recording)g(implicit)i(dependencies)e(of)i(v)n
|
|
|
5428 |
(ariables)f(upon)g(bindable)f(names)i(in)h(a)f(typing)f(conte)o(xt.)523
|
|
|
5429 |
747 y(The)k(mathematical)g(foundation)d(for)j(Hamana')-5
|
|
|
5430 |
b(s)25 b(system)h(is)h(the)e(model)g(of)g(binding)f(syntax)h(of)523
|
|
|
5431 |
847 y(Fiore)f FP(et)g(al)g FW([9].)f(The)h(mathematical)f(foundation)e
|
|
|
5432 |
(for)i(our)g(w)o(ork)g(appeared)g(concurrently)e([10])523
|
|
|
5433 |
946 y(and)k(is)g(in)h(a)f(sense)g(complementary)-5 b(.)22
|
|
|
5434 |
b(F)o(or)j(in)g(Hamana')-5 b(s)24 b(system)i(the)f(typing)e(conte)o(xt)
|
|
|
5435 |
h(restricts)523 1046 y(which)d(terms)h(may)g(be)f(substituted)h(for)f
|
|
|
5436 |
(a)h(v)n(ariable)f(by)g(gi)n(ving)g(a)h(\002nite)g(set)h(of)f(names)f
|
|
|
5437 |
(that)h FP(must)523 1146 y(contain)27 b FW(the)i(free)f(names)g(of)g
|
|
|
5438 |
(such)g(a)h(term;)f(whereas)g(we)h(gi)n(v)o(e)f(a)h(\002nite)f(set)i
|
|
|
5439 |
(of)e(names)g(which)523 1245 y(the)c(term')-5 b(s)24
|
|
|
5440 |
b(free)g(v)n(ariables)f FP(must)h(avoid)p FW(.)f(Since)h
|
|
|
5441 |
FQ(\013)p FW(-con)m(v)o(ersion)d(is)k(phrased)d(in)j(terms)e(of)h(a)n
|
|
|
5442 |
(v)n(oid-)523 1345 y(ance,)18 b(i.e.)g(freshness)g(of)g(names,)f(our)h
|
|
|
5443 |
(approach)e(seems)j(more)e(natural)h(if)g(one)g(w)o(ants)h(to)f
|
|
|
5444 |
(compute)523 1445 y FQ(\013)p FW(-equi)n(v)n(alences)25
|
|
|
5445 |
b(concretely)-5 b(.)25 b(On)i(top)g(of)g(that,)g(our)f(use)h(of)g(name)
|
|
|
5446 |
g(permutations,)e(rather)h(than)523 1544 y(arbitrary)18
|
|
|
5447 |
b(renaming)f(functions,)h(leads)i(to)f(technical)g(simpli\002cations.)g
|
|
|
5448 |
(In)g(an)o(y)g(case,)g(the)h(bottom)523 1644 y(line)i(is)h(that)e
|
|
|
5449 |
(Hamana')-5 b(s)22 b(system)g(seems)g(more)f(complicated)f(than)h(the)h
|
|
|
5450 |
(one)f(presented)g(here)g(and)523 1743 y(does)f(not)g(possess)h(most)f
|
|
|
5451 |
(general)f(uni\002ers.)523 2017 y FR(5)99 b(Conclusion)523
|
|
|
5452 |
2225 y FW(In)21 b(this)h(paper)e(we)i(ha)n(v)o(e)e(proposed)g(a)h
|
|
|
5453 |
(solution)g(to)g(the)g(problem)f(of)h(\002nding)f(possibly-capturing)
|
|
|
5454 |
523 2324 y(substitutions)k(that)g(unify)f(terms)i(in)m(v)n(olving)d
|
|
|
5455 |
(binders)h(up)h(to)h FQ(\013)p FW(-con)m(v)o(ersion.)c(T)-7
|
|
|
5456 |
b(o)25 b(do)f(so)h(we)f(con-)523 2424 y(sidered)f(a)h(man)o(y-sorted)d
|
|
|
5457 |
(\002rst-order)h(term)h(language)f(with)h(distinguished)f(collections)h
|
|
|
5458 |
(of)g(con-)523 2523 y(stants)k(called)f FP(atoms)g FW(and)f(with)i
|
|
|
5459 |
FP(atom-abstr)o(action)c FW(operations)i(for)g(binding)f(atoms)i(in)h
|
|
|
5460 |
(terms.)523 2623 y(This)e(pro)o(vides)d(a)j(simple,)f(b)n(ut)g(\003e)o
|
|
|
5461 |
(xible,)g(frame)n(w)o(ork)e(for)i(specifying)f(binding)f(operations)h
|
|
|
5462 |
(and)523 2723 y(their)g(scopes,)g(in)g(which)f(the)h(bound)f(entities)h
|
|
|
5463 |
(are)g(e)o(xplicitly)f(named.)g(By)h(using)g(v)n(ariables)f(pre-)523
|
|
|
5464 |
2822 y(\002x)o(ed)30 b(with)h(suspended)e(permutations,)g(one)h(can)h
|
|
|
5465 |
(ha)n(v)o(e)f(substitution)g(of)g(terms)h(for)f(v)n(ariables)523
|
|
|
5466 |
2922 y(both)21 b(allo)n(w)g(capture)g(of)g(atoms)h(by)f(binders)g(and)g
|
|
|
5467 |
(respect)g FQ(\013)p FW(-equi)n(v)n(alence)f(\(renaming)f(of)i(bound)
|
|
|
5468 |
523 3022 y(atoms\).)c(The)f(de\002nition)g(of)h FQ(\013)p
|
|
|
5469 |
FW(-equi)n(v)n(alence)e(for)i(the)g(term)g(language)e(mak)o(es)i(use)h
|
|
|
5470 |
(of)f(an)g(auxiliary)523 3121 y FP(fr)m(eshness)23 b
|
|
|
5471 |
FW(relation)f(between)g(atoms)h(and)f(terms)h(which)f(generalises)g
|
|
|
5472 |
(the)h(`not)f(a)h(free)f(atom)h(of)5 b(')523 3221 y(relation)15
|
|
|
5473 |
b(from)f(ground)g(terms)h(to)h(terms)f(with)h(v)n(ariables;)f
|
|
|
5474 |
(furthermore,)d(because)j(v)n(ariables)g(stand)523 3320
|
|
|
5475 |
y(for)i(unkno)n(wn)f(terms,)h(hence)h(with)g(unkno)n(wn)d(free)j
|
|
|
5476 |
(atoms,)f(it)i(is)g(necessary)e(to)h(mak)o(e)f(hypotheses)523
|
|
|
5477 |
3420 y(about)26 b(the)h(freshness)f(of)h(atoms)f(for)h(v)n(ariables)f
|
|
|
5478 |
(in)h(judgements)e(about)h(term)g(equi)n(v)n(alence)f(and)523
|
|
|
5479 |
3520 y(freshness.)20 b(This)i(reliance)e(on)h(`freshness')f(is)i(the)f
|
|
|
5480 |
(main)g(no)o(v)o(elty\227it)e(arises)j(from)e(the)h(w)o(ork)g(re-)523
|
|
|
5481 |
3619 y(ported)e(in)i([11,)12 b(23].)19 b(It)i(leads)g(to)f(a)h(ne)n(w)g
|
|
|
5482 |
(notion)e(of)h(uni\002cation)f(problem)g(in)h(which)g(instances)h(of)
|
|
|
5483 |
523 3719 y(both)c(equi)n(v)n(alence)e(and)i(freshness)g(ha)n(v)o(e)g
|
|
|
5484 |
(to)h(be)f(solv)o(ed)g(by)g(gi)n(ving)g(term-substitutions)e(and)i
|
|
|
5485 |
(\(pos-)523 3819 y(sibly\))i(freshness)h(conditions)e(on)h(v)n
|
|
|
5486 |
(ariables)g(in)h(the)g(solution.)e(W)-7 b(e)21 b(sho)n(wed)e(that)h
|
|
|
5487 |
(this)g(uni\002cation)523 3918 y(problem)f(is)i(decidable)e(and)g
|
|
|
5488 |
(unitary)-5 b(.)648 4011 y(Chene)o(y)30 b(and)i(Urban)f([5])g(are)h(in)
|
|
|
5489 |
m(v)o(estigating)d(the)j(e)o(xtent)g(to)g(which)f(nominal)g
|
|
|
5490 |
(uni\002cation)523 4102 y(can)c(be)g(used)g(in)g(resolution-based)e
|
|
|
5491 |
(proof)g(search)i(for)f(a)i(form)e(of)h(\002rst-order)e(logic)i
|
|
|
5492 |
(program-)523 4194 y(ming)f(for)h(languages)e(with)i(binders)f(\(with)h
|
|
|
5493 |
(a)g(vie)n(w)g(to)g(pro)o(viding)d(better)j(machine-assistance)523
|
|
|
5494 |
4285 y(for)d(structural)f(operational)f(semantics\).)i(Such)g(a)h
|
|
|
5495 |
(logic)e(programming)e(language)i(should)g(per)n(-)523
|
|
|
5496 |
4376 y(mit)j(a)g(concrete,)e(`nominal')g(approach)g(to)i(bound)e
|
|
|
5497 |
(entities)i(in)g(programs)d(while)j(ensuring)e(that)523
|
|
|
5498 |
4468 y(computation)e(\(which)h(in)h(this)h(case)f(is)h(the)f
|
|
|
5499 |
(computation)e(of)i(answers)g(to)g(queries\))f(respects)h
|
|
|
5500 |
FQ(\013)p FW(-)523 4559 y(equi)n(v)n(alence)f(between)i(terms.)g(This)h
|
|
|
5501 |
(is)g(illustrated)f(with)g(the)h(Prolog-lik)o(e)e(program)f(in)i(Fig.)h
|
|
|
5502 |
(5,)523 4650 y(which)18 b(implements)f(a)h(simple)g(typing)f(algorithm)
|
|
|
5503 |
f(for)i FQ(\025)p FW(-terms.)g(Interesting)e(is)j(the)f(third)g
|
|
|
5504 |
(clause.)523 4742 y(First,)23 b(note)g(the)f(term)h Fb(\(lam)49
|
|
|
5505 |
b(x.M\))p FW(,)22 b(which)g(uni\002es)h(with)g(an)o(y)f
|
|
|
5506 |
FQ(\025)p FW(-abstraction.)g(The)g(binder)f Fb(x)p FW(,)523
|
|
|
5507 |
4833 y(roughly)d(speaking,)g(has)i(in)g(the)f(`nominal')f(approach)f(a)
|
|
|
5508 |
k(v)n(alue)e(which)g(can)g(be)h(used)f(in)h(the)g(body)523
|
|
|
5509 |
4924 y(of)e(the)g(clause,)f(for)g(e)o(xample)g(for)g(adding)g
|
|
|
5510 |
Fb(\(pair)48 b(x)i(A\))18 b FW(to)g(the)g(conte)o(xt)f
|
|
|
5511 |
Fb(Gamma)p FW(.)g(Second,)f(the)p eop end
|
|
|
5512 |
%%Page: 14 14
|
|
|
5513 |
TeXDict begin 14 13 bop 523 369 2882 4 v 523 946 4 578
|
|
|
5514 |
v 551 460 a Fa(type)40 b(Gamma)28 b(\(var)39 b(X\))g(A)29
|
|
|
5515 |
b(:-)f(mem)40 b(\(pair)f(X)h(A\))f(Gamma.)551 562 y(type)h(Gamma)28
|
|
|
5516 |
b(\(app)39 b(M)h(N\))f(B)29 b(:-)f(type)39 b(Gamma)h(M)f(\(arrow)g(A)h
|
|
|
5517 |
(B\),)f(type)h(Gamma)f(N)g(A.)551 664 y(type)h(Gamma)28
|
|
|
5518 |
b(\(lam)39 b(x.M\))g(\(arrow)g(A)h(B\))17 b(/)h(x#Gamma)27
|
|
|
5519 |
b(:-)i(type)39 b(\(pair)g(x)h(A\))8 b Ft(:)19 b(:)k Fa(Gamma)39
|
|
|
5520 |
b(M)h(B.)551 794 y(mem)g(A)f(A)8 b Ft(:)19 b(:)k Fa(Tail.)551
|
|
|
5521 |
896 y(mem)40 b(A)f(B)8 b Ft(:)19 b(:)k Fa(Tail)29 b(:-)f(mem)39
|
|
|
5522 |
b(A)h(Tail.)p 3402 946 V 523 949 2882 4 v 571 1092 a
|
|
|
5523 |
FU(Fig)o(.)11 b(5.)19 b FV(A)g(Prolog-lik)o(e)g(program)h(implementing)
|
|
|
5524 |
g(the)f(typing)g(rules)g(for)g(the)g(simply-typed)h FS(\025)p
|
|
|
5525 |
FV(-calculus.)523 1375 y FW(freshness)k(constraint)f
|
|
|
5526 |
Fb(x#Gamma)g FW(ensures)h(that)g Fb(Gamma)g FW(cannot)f(be)h(replaced)f
|
|
|
5527 |
(by)h(a)g(term)g(that)523 1466 y(contains)g Fb(x)i FW(freely)-5
|
|
|
5528 |
b(.)23 b(Since)i(this)h(clause)f(is)h(intended)e(to)h(implement)f(the)h
|
|
|
5529 |
(usual)g(rule)f(for)h(typing)523 1557 y FQ(\025)p FW(-abstractions)1610
|
|
|
5530 |
1689 y FB(f)p FS(x)c FF(:)h FS(A)p FB(g)17 b([)g FS(\000)49
|
|
|
5531 |
b(.)39 b(M)30 b FF(:)21 b FS(B)p 1610 1728 709 4 v 1639
|
|
|
5532 |
1797 a(\000)49 b(.)39 b(\025x:M)29 b FF(:)21 b FS(A)g
|
|
|
5533 |
FB(\033)g FS(B)523 1903 y FW(its)29 b(operational)c(beha)n(viour)h(is)i
|
|
|
5534 |
(gi)n(v)o(en)e(by:)h(choose)g(fresh)g(names)g(for)g Fb(Gamma)p
|
|
|
5535 |
FW(,)g Fb(x)p FW(,)h Fb(M)p FW(,)f Fb(A)h FW(and)f Fb(B)523
|
|
|
5536 |
2003 y FW(\(this)g(is)g(standard)e(in)h(Prolog-lik)o(e)f(languages\),)f
|
|
|
5537 |
(unify)h(the)i(head)e(of)h(the)h(clause)f(with)h(the)f(goal)523
|
|
|
5538 |
2102 y(formula,)17 b(apply)g(the)h(resulting)g(uni\002er)g(to)g(the)h
|
|
|
5539 |
(body)d(of)j(the)f(clause)g(and)g(mak)o(e)g(sure)g(that)h
|
|
|
5540 |
Fb(Gamma)523 2202 y FW(is)30 b(not)f(replaced)f(by)h(a)h(term)e(that)i
|
|
|
5541 |
(contains)e(freely)h(the)g(fresh)g(name)f(we)i(ha)n(v)o(e)e(chosen)h
|
|
|
5542 |
(for)f Fb(x)p FW(.)523 2301 y(Similar)f(f)o(acilities)g(for)f
|
|
|
5543 |
FP(functional)f(pr)l(o)o(gr)o(amming)g FW(already)h(e)o(xist)h(in)g
|
|
|
5544 |
(the)f(FreshML)g(language,)523 2401 y(b)n(uilt)17 b(upon)f(the)i(same)f
|
|
|
5545 |
(foundations:)e(see)j([27])e(and)h FN(www)p FQ(:)p FN(freshml)p
|
|
|
5546 |
FQ(:)p FN(org)-6 b FW(.)18 b(W)-7 b(e)18 b(are)g(also)f(interested)523
|
|
|
5547 |
2501 y(in)22 b(the)g(special)g(case)g(of)g(`nominal)e(matching')g(and)h
|
|
|
5548 |
(its)i(application)e(to)h(term-re)n(writing)d(modulo)523
|
|
|
5549 |
2600 y FQ(\013)p FW(-equi)n(v)n(alence.)648 2700 y(If)j(these)h
|
|
|
5550 |
(applications)f(sho)n(w)g(that)h(nominal)f(uni\002cation)f(is)j
|
|
|
5551 |
(practically)e(useful,)g(then)g(it)i(be-)523 2800 y(comes)31
|
|
|
5552 |
b(important)f(to)h(study)g(its)h(comple)o(xity)-5 b(.)29
|
|
|
5553 |
b(The)i(presentations)f(of)h(the)g(term)g(language)e(in)523
|
|
|
5554 |
2899 y(Section)f(2)g(and)f(of)h(the)g(algorithm)e(in)j(Section)e(3)h
|
|
|
5555 |
(were)g(chosen)f(for)h(clarity)f(and)h(to)g(mak)o(e)f(the)523
|
|
|
5556 |
2999 y(proof)22 b(of)i(correctness)e(easier)1412 2969
|
|
|
5557 |
y Fy(2)1469 2999 y FW(rather)h(than)h(for)f(ef)n(\002cienc)o(y)-5
|
|
|
5558 |
b(.)21 b(In)j(an)o(y)f(case,)h(it)g(remains)f(to)h(be)g(in-)523
|
|
|
5559 |
3098 y(v)o(estigated)30 b(whether)f(the)i(sw)o(apping)f(and)h
|
|
|
5560 |
(freshness)f(computations)f(that)i(we)g(ha)n(v)o(e)g(added)e(to)523
|
|
|
5561 |
3198 y(ordinary)-5 b(,)17 b(\002rst-order)i(uni\002cation)g(result)h
|
|
|
5562 |
(in)h(greater)e(than)h(linear)n(-time)f(comple)o(xity)-5
|
|
|
5563 |
b(.)523 3326 y FU(Ackno)o(wledgements:)38 b FV(W)-6 b(e)19
|
|
|
5564 |
b(thank)h(Gilles)e(Do)n(wek,)i(Ro)o(y)g(Dyckhof)n(f,)g(Dale)g(Miller)m
|
|
|
5565 |
(,)e(Francois)i(Pottier)e(and)523 3426 y(Helmut)g(Schwichtenber)o(g)g
|
|
|
5566 |
(for)g(comments)g(on)g(this)g(w)o(ork.)g(This)f(research)h(w)o(as)g
|
|
|
5567 |
(supported)h(by)f(UK)g(EPSRC)523 3526 y(grants)h(GR/R29697)h(\(Urban\))
|
|
|
5568 |
f(and)h(GR/R07615)g(\(Pitts)d(and)j(Gabbay\).)523 3786
|
|
|
5569 |
y FR(Refer)n(ences)560 3970 y FV(1.)42 b(F)-6 b(.)18
|
|
|
5570 |
b(Baader)h(and)h(T)-6 b(.)18 b(Nipk)o(o)n(w)-5 b(.)28
|
|
|
5571 |
b FT(T)-7 b(erm)19 b(Re)o(writing)f(and)i(All)e(That)p
|
|
|
5572 |
FV(.)26 b(Cambridge)20 b(Uni)n(v)o(ersity)f(Press,)f(1998.)560
|
|
|
5573 |
4059 y(2.)42 b(H.)18 b(P)-8 b(.)18 b(Barendre)o(gt.)27
|
|
|
5574 |
b FT(The)19 b(Lambda)h(Calculus:)f(Its)f(Syntax)i(and)g(Semantics)p
|
|
|
5575 |
FV(.)28 b(North-Holland,)19 b(1984.)560 4149 y(3.)42
|
|
|
5576 |
b(L.)23 b(Caires)g(and)h(L.)f(Cardelli.)41 b(A)23 b(spatial)g(logic)h
|
|
|
5577 |
(for)f(concurrenc)o(y)j(II.)41 b(In)23 b FT(Pr)m(oc.)g(of)g(CONCUR)g
|
|
|
5578 |
(2002)p FV(,)658 4241 y(v)o(olume)c(2421)h(of)f FT(LNCS)p
|
|
|
5579 |
FV(,)f(pages)i(209\226225.)h(Springer)o(-V)-8 b(erlag,)18
|
|
|
5580 |
b(2002.)560 4331 y(4.)42 b(L.)23 b(Cardelli,)g(P)-8 b(.)23
|
|
|
5581 |
b(Gardner)m(,)h(and)h(G.)e(Ghelli.)43 b(Manipulating)25
|
|
|
5582 |
b(trees)f(with)f(hidden)i(labels.)44 b(In)23 b FT(Pr)m(oc.)h(of)658
|
|
|
5583 |
4422 y(FOSSA)n(CS)p FV(,)18 b(v)o(olume)i(2620)g(of)f
|
|
|
5584 |
FT(LNCS)p FV(,)f(pages)h(216\226232.)i(Springer)o(-V)-8
|
|
|
5585 |
b(erlag,)18 b(2003.)560 4512 y(5.)42 b(J.)31 b(Chene)o(y)i(and)g(C.)e
|
|
|
5586 |
(Urban.)69 b FS(\013)p FV(Prolog,)32 b(a)g(fresh)g(approach)i(to)e
|
|
|
5587 |
(logic)g(programming)i(modulo)f FS(\013)p FV(-)658 4603
|
|
|
5588 |
y(equi)n(v)n(alence.)42 b(In)24 b FT(Pr)m(oc.)e(of)h(UNIF)f(2003)p
|
|
|
5589 |
FV(,)i(number)h(DSIC-II/12/03)d(in)i(Departamento)g(de)f(Sistemas)658
|
|
|
5590 |
4694 y(Inform)t(\264)-29 b(aticos)22 b(y)f(Computaci)6
|
|
|
5591 |
b(\264)-31 b(on)23 b(T)-5 b(echnical)22 b(Report)f(Series.)g(Uni)n(v)o
|
|
|
5592 |
(ersidad)h(Polit)t(\264)-29 b(ecnica)21 b(de)h(V)-8 b(alencia,)658
|
|
|
5593 |
4786 y(2003.)p 523 4837 473 4 v 558 4893 a Fr(2)606 4924
|
|
|
5594 |
y FV(See)19 b FC(http)j FF(:)g FS(==)p FC(www)p FS(:)p
|
|
|
5595 |
FC(cl)p FS(:)p FC(cam)p FS(:)p FC(ac)p FS(:)p FC(uk)p
|
|
|
5596 |
FS(=)1607 4916 y Fw(\030)1664 4924 y FC(cu200)p FS(=)p
|
|
|
5597 |
FC(Unification)h FV(for)c(the)g(Isabelle)g(proof)g(scripts.)p
|
|
|
5598 |
eop end
|
|
|
5599 |
%%Page: 15 15
|
|
|
5600 |
TeXDict begin 15 14 bop 560 448 a FV(6.)42 b(G.)15 b(Do)n(wek.)22
|
|
|
5601 |
b(Higher)o(-order)16 b(uni\002cation)h(and)f(matching.)22
|
|
|
5602 |
b(In)16 b(A.)f(Robinson)j(and)e(A.)g(V)-10 b(oronk)o(o)o(v)-5
|
|
|
5603 |
b(,)18 b(editors,)658 540 y FT(Handbook)j(of)e(A)o(utomated)g
|
|
|
5604 |
(Reasoning)p FV(,)g(chapter)h(16,)f(pages)h(1009\2261062.)i(Else)n
|
|
|
5605 |
(vier)m(,)c(2001.)560 631 y(7.)42 b(G.)17 b(Do)n(wek,)h(T)-6
|
|
|
5606 |
b(.)16 b(Hardin,)i(and)g(C.)f(Kirchner)l(.)23 b(Higher)o(-order)18
|
|
|
5607 |
b(uni\002cation)g(via)f(e)o(xplicit)h(substitutions.)24
|
|
|
5608 |
b(In)658 722 y FT(10th)19 b(Symposium)i(of)d(LICS)p FV(,)g(pages)i
|
|
|
5609 |
(366\226374.)h(IEEE)d(Computer)h(Society)g(Press,)f(1995.)560
|
|
|
5610 |
814 y(8.)42 b(G.)25 b(Do)n(wek,)i(T)-6 b(.)25 b(Hardin,)h(C.)g
|
|
|
5611 |
(Kirchner)m(,)g(and)h(F)-6 b(.)25 b(Pfenning.)50 b(Higher)o(-order)26
|
|
|
5612 |
b(uni\002cation)h(via)f(e)o(xplicit)658 905 y(substitutions:)19
|
|
|
5613 |
b(the)g(case)h(of)e(higher)o(-order)i(patterns.)27 b(In)19
|
|
|
5614 |
b FT(Pr)m(oc.)f(of)h(JICSLP)p FV(,)f(pages)i(259\226273,)h(1996.)560
|
|
|
5615 |
996 y(9.)42 b(M.)29 b(P)-8 b(.)27 b(Fiore,)i(G.)f(D.)g(Plotkin,)h(and)g
|
|
|
5616 |
(D.)g(T)m(uri.)58 b(Abstract)29 b(syntax)h(and)f(v)n(ariable)h
|
|
|
5617 |
(binding.)60 b(In)29 b FT(14th)658 1088 y(Symposium)20
|
|
|
5618 |
b(of)f(LICS)p FV(,)f(pages)i(193\226202.)g(IEEE)e(Computer)i(Society)e
|
|
|
5619 |
(Press,)g(1999.)523 1179 y(10.)42 b(M.)19 b(J.)f(Gabbay)i(and)f(A.)f
|
|
|
5620 |
(M.)h(Pitts.)25 b(A)19 b(ne)n(w)g(approach)h(to)f(abstract)g(syntax)h
|
|
|
5621 |
(in)m(v)o(olving)f(binders.)27 b(In)19 b FT(14th)658
|
|
|
5622 |
1270 y(Symposium)h(of)f(LICS)p FV(,)f(pages)i(214\226224.)g(IEEE)e
|
|
|
5623 |
(Computer)i(Society)e(Press,)g(1999.)523 1362 y(11.)42
|
|
|
5624 |
b(M.)26 b(J.)f(Gabbay)i(and)g(A.)e(M.)h(Pitts.)48 b(A)26
|
|
|
5625 |
b(ne)n(w)g(approach)i(to)e(abstract)g(syntax)h(with)e(v)n(ariable)i
|
|
|
5626 |
(binding.)658 1453 y FT(F)-8 b(ormal)19 b(Aspects)g(of)g(Computing)p
|
|
|
5627 |
FV(,)g(13:341\226363,)j(2002.)523 1544 y(12.)42 b(C.)21
|
|
|
5628 |
b(A.)g(Gunter)l(.)35 b FT(Semantics)23 b(of)e(Pr)m(o)o(gr)o(amming)i
|
|
|
5629 |
(Langua)o(g)o(es:)g(Structur)m(es)g(and)f(T)-7 b(ec)o(hniques)p
|
|
|
5630 |
FV(.)37 b(F)o(ounda-)658 1636 y(tions)19 b(of)g(Computing.)h(MIT)e
|
|
|
5631 |
(Press,)g(1992.)523 1727 y(13.)42 b(M.)22 b(Hamana.)40
|
|
|
5632 |
b(A)22 b(logic)h(programming)h(language)g(based)g(on)f(binding)h
|
|
|
5633 |
(algebras.)39 b(In)23 b FT(Pr)m(oc.)f(of)h(T)l(A)n(CS)658
|
|
|
5634 |
1818 y(2001)p FV(,)d(v)o(olume)f(2215)h(of)f FT(LNCS)p
|
|
|
5635 |
FV(,)f(pages)i(243\226262.)h(Springer)o(-V)-8 b(erlag,)18
|
|
|
5636 |
b(2001.)523 1910 y(14.)42 b(M.)24 b(Hamana.)45 b(Simple)24
|
|
|
5637 |
b FS(\014)1358 1918 y FG(0)1393 1910 y FV(-uni\002cation)h(for)f(terms)
|
|
|
5638 |
h(with)f(conte)o(xt)h(holes.)45 b(In)24 b FT(Pr)m(oc.)g(of)g(UNIF)g
|
|
|
5639 |
(2002)p FV(,)658 2001 y(2002.)k(Unpublished)21 b(proceedings.)523
|
|
|
5640 |
2092 y(15.)42 b(M.)19 b(Hashimoto)g(and)h(A.)e(Ohori.)27
|
|
|
5641 |
b(A)19 b(typed)g(conte)o(xt)h(calculus.)27 b FT(TCS)p
|
|
|
5642 |
FV(,)19 b(266:249\226271,)i(2001.)523 2183 y(16.)42 b(F)-6
|
|
|
5643 |
b(.)17 b(Honsell,)h(M.)g(Miculan,)g(and)h(I.)e(Scagnetto.)25
|
|
|
5644 |
b(An)18 b(axiomatic)h(approach)g(to)f(metareasoning)i(on)e(nom-)658
|
|
|
5645 |
2275 y(inal)25 b(algebras)h(in)f(HO)m(AS.)46 b(In)25
|
|
|
5646 |
b FT(Pr)m(oc.)f(of)h(ICALP)f(2001)p FV(,)i(v)o(olume)f(2076)i(of)e
|
|
|
5647 |
FT(LNCS)p FV(,)f(pages)i(963\226978.)658 2366 y(Springer)o(-V)-8
|
|
|
5648 |
b(erlag,)18 b(2001.)523 2457 y(17.)42 b(J.)19 b(W)-7
|
|
|
5649 |
b(.)18 b(Klop.)27 b(T)-5 b(erm)19 b(re)n(writing)f(systems.)28
|
|
|
5650 |
b(In)19 b(S.)f(Abramsk)o(y)-5 b(,)20 b(D.)e(M.)h(Gabbay)-5
|
|
|
5651 |
b(,)20 b(and)g(T)-6 b(.)19 b(S.)f(E.)g(Maibaum,)658 2549
|
|
|
5652 |
y(editors,)h FT(Handbook)i(of)d(Lo)o(gic)i(in)e(Computer)i(Science)o(,)
|
|
|
5653 |
g(V)-8 b(olume)18 b(2)p FV(,)h(pages)h(1\226116.)g(OUP)-8
|
|
|
5654 |
b(,)18 b(1992.)523 2640 y(18.)42 b(A.)17 b(Martelli)g(and)h(U.)f
|
|
|
5655 |
(Montanari.)24 b(An)18 b(ef)n(\002cient)f(uni\002cation)h(algorithm.)24
|
|
|
5656 |
b FT(A)n(CM)17 b(T)l(r)o(ans.)g(Pr)m(o)o(gr)o(amming)658
|
|
|
5657 |
2731 y(Langua)o(g)o(es)k(and)f(Systems)p FV(,)f(4\(2\):258\226282,)i
|
|
|
5658 |
(1982.)523 2823 y(19.)42 b(S.)18 b(Michaylo)o(v)j(and)g(F)-6
|
|
|
5659 |
b(.)18 b(Pfenning.)30 b(An)19 b(empirical)h(study)g(of)f(the)h(runtime)
|
|
|
5660 |
g(beha)o(viour)g(of)g(higher)o(-order)658 2914 y(logic)d(programs.)22
|
|
|
5661 |
b(In)16 b FT(Pr)m(oc.)g(W)-7 b(orkshop)18 b(on)f(the)g
|
|
|
5662 |
FS(\025)p FT(Pr)m(olo)o(g)f(Pr)m(o)o(gr)o(amming)h(Langua)o(g)o(e)p
|
|
|
5663 |
FV(,)h(pages)g(257\226271,)658 3005 y(1992.)28 b(CIS)18
|
|
|
5664 |
b(T)-5 b(echnical)19 b(Report)g(MS-CIS-92-86.)523 3097
|
|
|
5665 |
y(20.)42 b(D.)21 b(Miller)l(.)34 b(A)21 b(logic)h(programming)h
|
|
|
5666 |
(language)g(with)e(lambda-abstraction,)h(function)g(v)n(ariables,)g
|
|
|
5667 |
(and)658 3188 y(simple)d(uni\002cation.)27 b FT(J)n(ournal)20
|
|
|
5668 |
b(of)f(Lo)o(gic)g(and)h(Computation)p FV(,)g(1:497\226536,)h(1991.)523
|
|
|
5669 |
3279 y(21.)42 b(R.)20 b(Milner)m(,)h(J.)f(P)o(arro)n(w)-5
|
|
|
5670 |
b(,)20 b(and)h(D.)g(W)-6 b(alk)o(er)l(.)32 b(A)21 b(calculus)g(of)g
|
|
|
5671 |
(mobile)g(processes)h(\(parts)e(I)h(and)g(II\).)32 b
|
|
|
5672 |
FT(Infor)o(-)658 3371 y(mation)19 b(and)h(Computation)p
|
|
|
5673 |
FV(,)g(100:1\22677,)g(1992.)523 3462 y(22.)42 b(F)-6
|
|
|
5674 |
b(.)13 b(Pfenning)i(and)h(C.)d(Elliott.)j(Higher)o(-order)f(abstract)f
|
|
|
5675 |
(syntax.)19 b(In)14 b FT(Pr)m(oc.)g(A)n(CM-SIGPLAN)e(Confer)m(ence)658
|
|
|
5676 |
3553 y(on)19 b(Pr)m(o)o(gr)o(amming)h(Langua)o(g)o(e)h(Design)e(and)h
|
|
|
5677 |
(Implementation)p FV(,)g(pages)f(199\226208.)i(A)m(CM)e(Press,)f(1988.)
|
|
|
5678 |
523 3645 y(23.)42 b(A.)24 b(M.)g(Pitts.)44 b(Nominal)25
|
|
|
5679 |
b(logic:)f(A)g(\002rst)g(order)h(theory)g(of)f(names)i(and)f(binding.)
|
|
|
5680 |
46 b(In)24 b FT(Pr)m(oc.)g(of)h(T)l(A)n(CS)658 3736 y(2001)p
|
|
|
5681 |
FV(,)20 b(v)o(olume)f(2215)h(of)f FT(LNCS)p FV(,)f(pages)i(219\226242.)
|
|
|
5682 |
h(Springer)o(-V)-8 b(erlag,)18 b(2001.)523 3827 y(24.)42
|
|
|
5683 |
b(A.)18 b(M.)h(Pitts)e(and)j(M.)e(J.)h(Gabbay)-5 b(.)27
|
|
|
5684 |
b(A)19 b(metalanguage)h(for)f(programming)h(with)f(bound)h(names)g
|
|
|
5685 |
(modulo)658 3919 y(renaming.)40 b(In)22 b FT(Pr)m(oc.)g(of)h(MPC2000)p
|
|
|
5686 |
FV(,)g(v)o(olume)g(1837)h(of)f FT(LNCS)p FV(,)e(pages)j(230\226255.)h
|
|
|
5687 |
(Springer)o(-V)-8 b(erlag,)658 4010 y(2000.)523 4101
|
|
|
5688 |
y(25.)42 b(G.)27 b(D.)h(Plotkin.)56 b(An)28 b(illati)n(v)o(e)f(theory)i
|
|
|
5689 |
(of)f(relations.)57 b(In)28 b FT(Situation)h(Theory)f(and)h(its)e
|
|
|
5690 |
(Applications)p FV(,)658 4193 y(v)o(olume)19 b(22)h(of)f
|
|
|
5691 |
FT(CSLI)f(Lecur)m(e)h(Notes)p FV(,)g(pages)h(133\226146.)h(Stanford)e
|
|
|
5692 |
(Uni)n(v)o(ersity)-5 b(,)19 b(1990.)523 4284 y(26.)42
|
|
|
5693 |
b(M.)27 b(Sato,)f(T)-6 b(.)26 b(Sakurai,)g(and)i(Y)-10
|
|
|
5694 |
b(.)26 b(Kame)o(yama.)53 b(A)26 b(simply)h(typed)h(conte)o(xt)f
|
|
|
5695 |
(calculus)g(with)g(\002rst-class)658 4375 y(en)m(vironments.)h
|
|
|
5696 |
FT(J)n(ournal)21 b(of)d(Functional)i(and)g(Lo)o(gic)f(Pr)m(o)o(gr)o
|
|
|
5697 |
(amming)p FV(,)g(2002\(4\),)h(2002.)523 4467 y(27.)42
|
|
|
5698 |
b(M.)20 b(R.)f(Shinwell,)g(A.)h(M.)g(Pitts,)e(and)j(M.)f(J.)f(Gabbay)-5
|
|
|
5699 |
b(.)32 b(FreshML:)20 b(Programming)h(with)e(binders)i(made)658
|
|
|
5700 |
4558 y(simple.)27 b(In)19 b FT(Pr)m(oc.)f(of)h(ICFP)e(2003)p
|
|
|
5701 |
FV(.)j(A)m(CM)f(Press,)f(2003.)p eop end
|
|
|
5702 |
%%Trailer
|
|
|
5703 |
|
|
|
5704 |
userdict /end-hook known{end-hook}if
|
|
|
5705 |
%%EOF
|