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%
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626 |
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627 |
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628 |
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629 |
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630 |
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631 |
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632 |
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633 |
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634 |
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635 |
0000000000000000000000000000000000000000000000000000000000000000
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636 |
0000000000000000000000000000000000000000000000000000000000000000
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637 |
0000000000000000000000000000000000000000000000000000000000000000
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638 |
0000000000000000000000000000000000000000000000000000000000000000
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639 |
cleartomark
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640 |
%%EndFont
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641 |
%%BeginFont: CMBX10
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642 |
%!PS-AdobeFont-1.1: CMBX10 1.00B
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643 |
%%CreationDate: 1992 Feb 19 19:54:06
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644 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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645 |
11 dict begin
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646 |
/FontInfo 7 dict dup begin
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647 |
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648 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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649 |
/FullName (CMBX10) readonly def
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650 |
/FamilyName (Computer Modern) readonly def
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651 |
/Weight (Bold) readonly def
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652 |
/ItalicAngle 0 def
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653 |
/isFixedPitch false def
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654 |
end readonly def
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655 |
/FontName /CMBX10 def
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656 |
/PaintType 0 def
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657 |
/FontType 1 def
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658 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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659 |
/Encoding 256 array
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660 |
0 1 255 {1 index exch /.notdef put} for
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661 |
dup 0 /.notdef put
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662 |
readonly def
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663 |
/FontBBox{-301 -250 1164 946}readonly def
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664 |
/UniqueID 5000768 def
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665 |
currentdict end
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666 |
currentfile eexec
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667 |
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690 |
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691 |
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692 |
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693 |
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694 |
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695 |
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696 |
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697 |
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698 |
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699 |
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700 |
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701 |
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703 |
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704 |
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705 |
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706 |
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707 |
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708 |
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709 |
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710 |
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711 |
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712 |
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713 |
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714 |
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715 |
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716 |
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717 |
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718 |
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719 |
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720 |
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724 |
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725 |
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727 |
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805 |
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810 |
/FullName (CMMIB10) readonly def
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811 |
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812 |
/Weight (Bold) readonly def
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813 |
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814 |
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815 |
end readonly def
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816 |
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817 |
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819 |
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820 |
/Encoding 256 array
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821 |
0 1 255 {1 index exch /.notdef put} for
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822 |
dup 0 /.notdef put
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readonly def
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825 |
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826 |
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858 |
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860 |
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861 |
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862 |
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cleartomark
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873 |
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874 |
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875 |
%%CreationDate: 1992 Feb 19 19:56:16
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876 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
877 |
11 dict begin
|
|
|
878 |
/FontInfo 7 dict dup begin
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|
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879 |
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880 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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|
|
881 |
/FullName (CMTI10) readonly def
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|
|
882 |
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|
883 |
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884 |
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|
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885 |
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886 |
end readonly def
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890 |
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0 1 255 {1 index exch /.notdef put} for
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893 |
dup 0 /.notdef put
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readonly def
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896 |
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897 |
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918 |
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920 |
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921 |
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0 1 255 {1 index exch /.notdef put} for
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dup 0 /.notdef put
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1095 |
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1096 |
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1097 |
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1098 |
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1099 |
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1100 |
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1101 |
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1102 |
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1103 |
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1104 |
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1105 |
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1106 |
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1107 |
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1108 |
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1109 |
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1110 |
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1111 |
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1112 |
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1119 |
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1120 |
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1121 |
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1122 |
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1124 |
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1125 |
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1126 |
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1127 |
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1128 |
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1129 |
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1175 |
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1176 |
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1177 |
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1178 |
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1179 |
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1180 |
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1181 |
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1182 |
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1183 |
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1184 |
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1185 |
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1186 |
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1187 |
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1188 |
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1189 |
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1190 |
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1191 |
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1192 |
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1193 |
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1194 |
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1195 |
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1196 |
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1197 |
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1198 |
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1199 |
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1200 |
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1201 |
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1202 |
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1203 |
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1204 |
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1205 |
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1206 |
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1207 |
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1208 |
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1209 |
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1210 |
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1211 |
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1212 |
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1213 |
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1214 |
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1215 |
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1216 |
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1217 |
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1218 |
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1219 |
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1220 |
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1221 |
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1222 |
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1223 |
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1224 |
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1225 |
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1226 |
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1227 |
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1231 |
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1232 |
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1233 |
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1235 |
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1236 |
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1237 |
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cleartomark
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%%EndFont
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%%CreationDate: 1991 Aug 20 16:34:54
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1601 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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1602 |
11 dict begin
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1603 |
/FontInfo 7 dict dup begin
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/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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1606 |
/FullName (CMBX12) readonly def
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1607 |
/FamilyName (Computer Modern) readonly def
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1608 |
/Weight (Bold) readonly def
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1609 |
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/isFixedPitch false def
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1611 |
end readonly def
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1615 |
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/Encoding 256 array
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1617 |
0 1 255 {1 index exch /.notdef put} for
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1618 |
dup 0 /.notdef put
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readonly def
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/FontBBox{-53 -251 1139 750}readonly def
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1621 |
/UniqueID 5000769 def
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currentdict end
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1825 |
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1826 |
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1827 |
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1828 |
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1829 |
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1830 |
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1832 |
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1833 |
/FullName (CMMI9) readonly def
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1834 |
/FamilyName (Computer Modern) readonly def
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1835 |
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1836 |
/ItalicAngle -14.04 def
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1837 |
/isFixedPitch false def
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1838 |
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1839 |
/FontName /CMMI9 def
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1840 |
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1841 |
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1842 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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1843 |
/Encoding 256 array
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1844 |
0 1 255 {1 index exch /.notdef put} for
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1845 |
dup 0 /.notdef put
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1846 |
readonly def
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1847 |
/FontBBox{-29 -250 1075 750}readonly def
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1848 |
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1849 |
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1866 |
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1867 |
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1868 |
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1869 |
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1873 |
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1874 |
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1875 |
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1876 |
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1877 |
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1878 |
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1879 |
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1880 |
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1881 |
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1882 |
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1883 |
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1884 |
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1885 |
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1886 |
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1887 |
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1888 |
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1889 |
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1890 |
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1891 |
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1892 |
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1893 |
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1894 |
cleartomark
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1895 |
%%EndFont
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1896 |
%%BeginFont: CMBX9
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1897 |
%!PS-AdobeFont-1.1: CMBX9 1.0
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|
1898 |
%%CreationDate: 1991 Aug 20 16:36:25
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|
|
1899 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
1900 |
11 dict begin
|
|
|
1901 |
/FontInfo 7 dict dup begin
|
|
|
1902 |
/version (1.0) readonly def
|
|
|
1903 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
|
|
|
1904 |
/FullName (CMBX9) readonly def
|
|
|
1905 |
/FamilyName (Computer Modern) readonly def
|
|
|
1906 |
/Weight (Bold) readonly def
|
|
|
1907 |
/ItalicAngle 0 def
|
|
|
1908 |
/isFixedPitch false def
|
|
|
1909 |
end readonly def
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|
|
1910 |
/FontName /CMBX9 def
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1911 |
/PaintType 0 def
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1912 |
/FontType 1 def
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|
1913 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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1914 |
/Encoding 256 array
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1915 |
0 1 255 {1 index exch /.notdef put} for
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|
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1916 |
dup 0 /.notdef put
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1917 |
readonly def
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1918 |
/FontBBox{-58 -250 1195 750}readonly def
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1919 |
/UniqueID 5000767 def
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1920 |
currentdict end
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1921 |
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1922 |
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1923 |
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1924 |
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1925 |
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1926 |
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1927 |
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1928 |
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1929 |
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1930 |
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1931 |
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1932 |
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1933 |
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1934 |
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1935 |
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1936 |
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1937 |
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1938 |
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1939 |
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1940 |
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1941 |
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1942 |
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1943 |
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1944 |
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1945 |
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1946 |
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1947 |
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1948 |
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1949 |
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1950 |
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|
1951 |
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|
1952 |
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|
1953 |
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1954 |
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1955 |
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1956 |
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1957 |
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1958 |
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1959 |
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1960 |
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1961 |
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1962 |
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1963 |
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1964 |
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1965 |
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1966 |
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1967 |
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1968 |
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1969 |
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|
|
1970 |
6755EA37C4432D04D23E1AF8AAB4B6DA95AED308C181CB041E9F2F0D1B854A3D
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1971 |
1B2CE716E02F7DA94FC207F6B25D1D9A20F9E9CDB8240E30D6FA6419CB17C97A
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|
|
1972 |
DD5E9F7A672E8F2F3802A68D38CC9EA0BF77BCC84D0ACEE0C72CA1495755A074
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|
|
1973 |
4968844B9D04995D4228D7406754AEE641551A506C85161BD700E375B02FD3E4
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|
1974 |
DA6DB524FF19EDD08979735528EC2072A293130E91C3B3BED0AE8B9B0AC300C4
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|
|
1975 |
895DB32998F27390BF847F45A7892629CF30EE986951337C3114BF8B7E7FC778
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|
|
1976 |
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|
|
1977 |
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|
1978 |
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1979 |
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|
1980 |
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|
|
1981 |
E0CCC3FFE9DF19D01C3A28EA35E2B12D84844D4726C8860DE35CF93E62CCE0DC
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|
|
1982 |
D67355FC6EFFB8B97A1D1C0F33D9EEFD3D5191F0667CCCD7463A961791DD52D8
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|
|
1983 |
C570062FD58963C7826415678F9511ED4E4C1D029F2EBF92EA4F3E587505163E
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1984 |
7E77DEFA0180F1115015E6B0524A2529A733CCB4D9642B744B29C1E553AA5A00
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|
1985 |
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1986 |
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1987 |
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|
1988 |
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1989 |
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|
1990 |
B9BA5A6E39641D809FE3DE7038CE61A8C7A79CBBDEFC19E9B608C8CEE46693AF
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1991 |
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1992 |
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1993 |
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1994 |
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1995 |
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1996 |
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1997 |
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1998 |
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1999 |
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2000 |
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2001 |
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2002 |
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2003 |
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2004 |
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2005 |
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2006 |
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2007 |
0000000000000000000000000000000000000000000000000000000000000000
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2008 |
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2009 |
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2010 |
0000000000000000000000000000000000000000000000000000000000000000
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2011 |
cleartomark
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|
2012 |
%%EndFont
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2013 |
%%BeginFont: CMTT9
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|
|
2014 |
%!PS-AdobeFont-1.1: CMTT9 1.0
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|
|
2015 |
%%CreationDate: 1991 Aug 20 16:46:24
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|
2016 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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|
2017 |
11 dict begin
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|
2018 |
/FontInfo 7 dict dup begin
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2019 |
/version (1.0) readonly def
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2020 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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2021 |
/FullName (CMTT9) readonly def
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2022 |
/FamilyName (Computer Modern) readonly def
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2023 |
/Weight (Medium) readonly def
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2024 |
/ItalicAngle 0 def
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2025 |
/isFixedPitch true def
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|
2026 |
end readonly def
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2027 |
/FontName /CMTT9 def
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2028 |
/PaintType 0 def
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2029 |
/FontType 1 def
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2030 |
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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2031 |
/Encoding 256 array
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2032 |
0 1 255 {1 index exch /.notdef put} for
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2033 |
dup 0 /.notdef put
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2034 |
readonly def
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2035 |
/FontBBox{-6 -233 542 698}readonly def
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2036 |
/UniqueID 5000831 def
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2037 |
currentdict end
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2038 |
currentfile eexec
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2039 |
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2040 |
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2041 |
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2042 |
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2043 |
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2044 |
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2045 |
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2046 |
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2047 |
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2048 |
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2049 |
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2051 |
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2052 |
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2053 |
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2054 |
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2055 |
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2056 |
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2057 |
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2058 |
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2060 |
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2064 |
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2065 |
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2066 |
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2067 |
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2068 |
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2069 |
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2070 |
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2071 |
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2072 |
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2073 |
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2074 |
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2075 |
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2076 |
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2077 |
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2078 |
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2079 |
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2080 |
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2081 |
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2082 |
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2083 |
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2084 |
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2085 |
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2086 |
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2087 |
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2088 |
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2089 |
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2090 |
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2091 |
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2092 |
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2093 |
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2094 |
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2095 |
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2096 |
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2097 |
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2098 |
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2099 |
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2100 |
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2101 |
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2102 |
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2103 |
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2105 |
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2106 |
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2107 |
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2108 |
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2109 |
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2110 |
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2111 |
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2112 |
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2113 |
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2114 |
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2115 |
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2116 |
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2117 |
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2118 |
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2119 |
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2120 |
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2121 |
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2122 |
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2123 |
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2126 |
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2127 |
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2128 |
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2129 |
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2130 |
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2131 |
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2132 |
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2135 |
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2136 |
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2138 |
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2139 |
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2140 |
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2141 |
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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2538 |
11 dict begin
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2539 |
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2542 |
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2543 |
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2544 |
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2545 |
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2547 |
end readonly def
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2548 |
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2549 |
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2550 |
/FontType 1 def
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2551 |
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2552 |
/Encoding 256 array
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2553 |
0 1 255 {1 index exch /.notdef put} for
|
|
|
2554 |
dup 0 /.notdef put
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2555 |
readonly def
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2556 |
/FontBBox{-20 -250 1193 750}readonly def
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2557 |
/UniqueID 5000789 def
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2558 |
currentdict end
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2559 |
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2599 |
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2600 |
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2601 |
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2602 |
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2603 |
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2605 |
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2608 |
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2613 |
cleartomark
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2614 |
%%EndFont
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2615 |
%%BeginFont: CMR7
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2616 |
%!PS-AdobeFont-1.1: CMR7 1.0
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|
2617 |
%%CreationDate: 1991 Aug 20 16:39:21
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|
2618 |
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
|
|
|
2619 |
11 dict begin
|
|
|
2620 |
/FontInfo 7 dict dup begin
|
|
|
2621 |
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|
|
|
2622 |
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
|
|
|
2623 |
/FullName (CMR7) readonly def
|
|
|
2624 |
/FamilyName (Computer Modern) readonly def
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|
|
2625 |
/Weight (Medium) readonly def
|
|
|
2626 |
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|
|
2627 |
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2628 |
end readonly def
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2629 |
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2630 |
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2631 |
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2632 |
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2633 |
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2634 |
0 1 255 {1 index exch /.notdef put} for
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2635 |
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2636 |
readonly def
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2637 |
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2638 |
/UniqueID 5000790 def
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2639 |
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2655 |
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cleartomark
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%%EndFont
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%%CreationDate: 1992 Feb 19 19:54:52
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end readonly def
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0 1 255 {1 index exch /.notdef put} for
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dup 0 /.notdef put
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readonly def
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TeXDict begin 39158280 55380996 1000 600 600 (aprolog-colognet5.dvi)
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3164 |
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3165 |
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3166 |
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3167 |
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3168 |
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3169 |
Fl(Christian.Urban@cl.cam.ac.uk)286 1465 y Fk(Abstract.)42
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3170 |
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3171 |
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3172 |
286 1556 y(guage)c(with)f(a)g(built-in)f(notion)h(of)h(names,)g
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3173 |
(binding)e(and)h(uni\014cation)g(mo)r(dulo)g Fj(\013)p
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3174 |
Fm(-equiv)l(alence.)g Fj(\013)p Fm(Prolog)286 1648 y(is)g(based)f(on)g
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3175 |
(a)h(mild)f(extension)g(of)h(\014rst-order)f(Horn)g(form)n(ulae:)i
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3176 |
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3177 |
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3178 |
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3179 |
286 1830 y Fk(Note:)f Fm(This)i(pap)r(er)e(is)i(a)f(revised)f(v)n
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3180 |
(ersion)h(of)h([1].)50 2111 y Fi(1)112 b(In)m(tro)s(duction)50
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3181 |
2310 y Fp(Logic)33 b(programming)f(is)h(particularly)g(suited)h(for)g
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3182 |
(implemen)n(ting)g(inference)g(rules)f(de\014ning)h(relations)f(o)n(v)n
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3183 |
(er)50 2410 y(terms.)26 b(Man)n(y)f(in)n(teresting)h(examples,)f(ho)n
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3184 |
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|
3185 |
b(ariables)24 b(and)i Fh(\013)p Fp(-equiv)-5 b(alence.)50
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|
3186 |
2509 y(F)e(or)27 b(example)g(the)h(t)n(yping)f(relation)g(for)g
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3187 |
Fh(\025)p Fp(-terms)g(is)h(often)g(form)n(ulated)f(with)h(the)g
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3188 |
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|
3189 |
Fg(2)23 b Fh(\000)p 642 2708 341 4 v 653 2782 a(\000)30
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|
3190 |
b(.)18 b(x)24 b Fp(:)f Fh(T)1148 2683 y(\000)30 b(.)18
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3191 |
b(M)g Fp(:)23 b Fh(T)34 b Fg(!)23 b Fh(T)1684 2652 y
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3192 |
Ff(0)1790 2683 y Fh(\000)30 b(.)18 b(N)g Fp(:)23 b Fh(T)p
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3193 |
1148 2703 974 4 v 1412 2782 a(\000)30 b(.)18 b(M)9 b(N)18
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3194 |
b Fp(:)23 b Fh(T)1835 2751 y Ff(0)2287 2662 y Fg(f)p
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3195 |
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3196 |
b(M)g Fp(:)23 b Fh(T)2986 2632 y Ff(0)p 2287 2703 722
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|
3197 |
4 v 2309 2782 a Fh(\000)30 b(.)18 b(\025x:M)h Fp(:)k
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|
3198 |
Fh(T)34 b Fg(!)23 b Fh(T)2964 2751 y Ff(0)50 2948 y Fp(with)29
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|
3199 |
b(the)g(implicit)h(side-condition)e(that)h(in)h Fh(\000)40
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3200 |
b Fp(\(a)29 b(set)g(of)f(t)n(yp)r(e)i(assignmen)n(ts)d(for)h(v)-5
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|
3201 |
b(ariables\))28 b(no)h(v)-5 b(ariable)28 b(has)50 3047
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3202 |
y(more)34 b(than)h(one)f(assignmen)n(t.)g(This)h(side-condition)f(is)h
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3203 |
(violated)f(if)i(the)f(third)g(rule)f(is)h(applied)g(b)r(ottom-up)50
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3204 |
3147 y(and)27 b Fh(\000)40 b Fp(in)28 b(the)g(conclusion)f(already)f
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|
3205 |
(con)n(tains)h(an)g(assignmen)n(t)g(for)g Fh(x)p Fp(.)h(In)g(this)g
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3206 |
(case)f(the)h(b)r(ound)g(v)-5 b(ariable)27 b Fh(x)h Fp(in)50
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|
3207 |
3247 y Fh(\025x:M)37 b Fp(needs)28 b(to)g(b)r(e)g(renamed)f(to)h(a)f
|
|
|
3208 |
(new)h Fh(x)1454 3217 y Ff(0)1506 3247 y Fp(so)f(that)h(a)g(t)n(yp)r(e)
|
|
|
3209 |
g(assignmen)n(t)f(for)g Fh(x)2647 3217 y Ff(0)2699 3247
|
|
|
3210 |
y Fp(can)g(b)r(e)i(safely)e(added)h(to)f Fh(\000)12 b
|
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3211 |
Fp(.)50 3346 y(It)32 b(is)h(understo)r(o)r(d)f(that)g(this)h(renaming)e
|
|
|
3212 |
(do)r(es)h(not)h(a\013ect)f(the)h(`meaning')f(of)g(the)h(term,)f(b)r
|
|
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3213 |
(ecause)g(terms)g(are)50 3446 y(iden)n(ti\014ed)c(mo)r(dulo)f
|
|
|
3214 |
Fh(\013)p Fp(-equiv)-5 b(alence.)174 3546 y(Ho)n(w)n(ev)n(er,)35
|
|
|
3215 |
b(this)i(in)n(tuitiv)n(e)f(accoun)n(t)g(of)h Fh(\013)p
|
|
|
3216 |
Fp(-equiv)-5 b(alence)36 b(do)r(es)g(not)h(carry)d(o)n(v)n(er)h(to)i(a)
|
|
|
3217 |
f(correct)f(declarativ)n(e)50 3645 y(implemen)n(tation)27
|
|
|
3218 |
b(in)h(Prolog.)e(F)-7 b(or)27 b(example,)g(if)h(w)n(e)f(na)-9
|
|
|
3219 |
b(\177)-32 b(\020v)n(ely)26 b(implemen)n(t)i(the)g(t)n(yping)f(rules)g
|
|
|
3220 |
(with)i(the)e(clauses)443 3822 y Fe(member)41 b(X)i(X::L.)443
|
|
|
3221 |
3921 y(member)e(X)i(Y::L)893 b(:-)23 b(member)41 b(X)i(L.)443
|
|
|
3222 |
4068 y(type)f(Gamma)f(\(var)h(X\))h(T)589 b(:-)23 b(member)41
|
|
|
3223 |
b(\(pair)g(X)j(T\))e(Gamma.)443 4168 y(type)g(Gamma)f(\(app)h(M)h(N\))g
|
|
|
3224 |
(T')458 b(:-)23 b(type)42 b(Gamma)f(M)i(\(arrow)e(T)j(T'\),)2015
|
|
|
3225 |
4267 y(type)e(Gamma)f(N)i(T.)443 4367 y(type)f(Gamma)f(\(lam)h(X)h(M\))
|
|
|
3226 |
g(\(arrow)e(T)i(T'\))22 b(:-)h(type)42 b(\(pair)f(X)i(T\)::Gamma)d(M)j
|
|
|
3227 |
(T'.)50 4550 y Fp(where)20 b Fe(X)f Fp(is)i(a)e(\(logical\))h(v)-5
|
|
|
3228 |
b(ariable,)19 b(then)i(w)n(e)f(get)g(incorrect)f(answ)n(ers)f(for)i
|
|
|
3229 |
Fh(\025)p Fp(-terms)g(suc)n(h)g(as)f Fh(\025x:\025x:)p
|
|
|
3230 |
Fp(\()p Fh(x)26 b(x)p Fp(\).)21 b(F)-7 b(or)20 b(this)50
|
|
|
3231 |
4649 y(un)n(t)n(ypable)g(term,)i(the)f(na)-9 b(\177)-32
|
|
|
3232 |
b(\020v)n(e)19 b(implemen)n(tation)i(erroneously)e(returns)i(t)n(w)n(o)
|
|
|
3233 |
f(p)r(ossible)h(t)n(yp)r(es:)g Fh(\034)33 b Fg(!)23 b
|
|
|
3234 |
Fp(\()p Fh(\034)33 b Fg(!)23 b Fh(\033)s Fp(\))h Fg(!)f
|
|
|
3235 |
Fh(\033)50 4749 y Fp(and)g(\()p Fh(\034)33 b Fg(!)23
|
|
|
3236 |
b Fh(\033)s Fp(\))h Fg(!)f Fh(\034)33 b Fg(!)23 b Fh(\033)s
|
|
|
3237 |
Fp(.)h(Although)g(this)g(problem)f(can)g(b)r(e)h(\014xed)g(b)n(y)f
|
|
|
3238 |
(judicious)h(use)f(of)h(cut,)g(\014rst-order)d(terms)50
|
|
|
3239 |
4848 y(are)26 b(un)n(wieldy)i(for)f(implemen)n(ting)h(relations)e(ab)r
|
|
|
3240 |
(out)i(syn)n(tax)f(with)h(binders)f(correctly)-7 b(.)174
|
|
|
3241 |
4948 y(The)30 b(traditional)f(solution)h(for)f(this)h(problem)g(is)g
|
|
|
3242 |
(to)f(use)h(a)g(higher-order)d(logic)i(programming)f(language,)50
|
|
|
3243 |
5048 y(suc)n(h)20 b(as)g Fh(\025)p Fp(Prolog)e(\(see)i(for)g(example)g
|
|
|
3244 |
([2]\),)h(with)g(higher-order)d(terms.)i(Suc)n(h)g(languages)f(pro)n
|
|
|
3245 |
(vide)g(elegan)n(t)h(mec)n(h-)50 5147 y(anisms)26 b(for)f(dealing)h
|
|
|
3246 |
(with)h(binders,)f(but)h(they)f(also)g(in)n(tro)r(duce)g(new)g
|
|
|
3247 |
(problems:)g(uni\014cation)g(for)g(higher-order)50 5247
|
|
|
3248 |
y(terms)i(is)g(in)g(general)f(undecidable)h(and)g(if)h(uni\014ers)f
|
|
|
3249 |
(exist)g(for)g(a)g(problem)f(they)i(ma)n(y)e(not)i(b)r(e)f(subsumed)h
|
|
|
3250 |
(b)n(y)f(a)50 5347 y(single)g(most)g(general)e(uni\014er.)j(F)-7
|
|
|
3251 |
b(urthermore,)27 b(inference)h(rules)g(suc)n(h)g(as)f(the)i(ones)e(giv)
|
|
|
3252 |
n(en)h(ab)r(o)n(v)n(e)f(are)g(sp)r(eci\014ed)p eop end
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|
|
3253 |
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|
3254 |
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|
|
|
3255 |
b(ariables)24 b(\(often)i(also)e(referred)g(to)h(as)g(con)n(text-v)-5
|
|
|
3256 |
b(ariables,)23 b(whic)n(h)j(can)f(b)r(e)h(substituted)g(with)g(an)n(y)
|
|
|
3257 |
50 340 y(term\);)j(using)g(higher-order)e(uni\014cation)j(one)f(needs)g
|
|
|
3258 |
(to)g(replace)f(these)i(meta-v)-5 b(ariables)27 b(with)j(function)g(v)
|
|
|
3259 |
-5 b(ari-)50 440 y(ables)26 b(and)h(moreo)n(v)n(er)e(has)h(to)h
|
|
|
3260 |
(explicitly)g(sp)r(ecify)h(the)f(parameters)f(the)h(function)h(v)-5
|
|
|
3261 |
b(ariables)26 b(dep)r(end)h(on.)g(This)50 540 y(leads)g(to)h(enco)r
|
|
|
3262 |
(dings)f(that)i(are)e(quite)h(di\013eren)n(t)g(in)g(form)g(from)g
|
|
|
3263 |
(their)g(in)n(tuitiv)n(e)g(p)r(en-and-pap)r(er)f(coun)n(terparts)50
|
|
|
3264 |
639 y(since)g(there)h(dep)r(endences)f(are)g(left)h(implicit)h(as)e(m)n
|
|
|
3265 |
(uc)n(h)g(as)g(p)r(ossible.)174 739 y(In)32 b(this)h(pap)r(er)e(w)n(e)h
|
|
|
3266 |
(shall)f(presen)n(t)g(a)h(logic)f(programming)e(language)i(called)g
|
|
|
3267 |
Fh(\013)p Fp(Prolog)f(in)i(whic)n(h)g(one)g(can)50 839
|
|
|
3268 |
y(implemen)n(t)c(the)g(inference)f(rules)g(giv)n(en)g(ab)r(o)n(v)n(e)f
|
|
|
3269 |
(correctly)h(as)f(follo)n(ws:)443 992 y Fe(type)42 b(Gamma)f(\(var)h
|
|
|
3270 |
(X\))h(T)589 b(:-)23 b(member)41 b(\(pair)g(X)j(T\))e(Gamma.)443
|
|
|
3271 |
1092 y(type)g(Gamma)f(\(app)h(M)h(N\))g(T')458 b(:-)23
|
|
|
3272 |
b(type)42 b(Gamma)f(M)i(\(arrow)e(T)j(T'\),)2015 1192
|
|
|
3273 |
y(type)e(Gamma)f(N)i(T.)443 1291 y(type)f(Gamma)f(\(lam)h(x.M\))g
|
|
|
3274 |
(\(arrow)f(T)i(T'\))h(/)h(x#Gamma)1904 1391 y(:-)23 b(type)42
|
|
|
3275 |
b(\(pair)f(x)i(T\)::Gamma)d(M)j(T'.)50 1546 y Fp(P)n(osing)26
|
|
|
3276 |
b(the)i(queries)43 1704 y(\(a\))42 b Fe(type)g(nil)g(\(lam)g(x.\(lam)f
|
|
|
3277 |
(x.)i(\(var)f(x\)\)\))g(A)193 b Fp(and)39 1802 y(\(b\))42
|
|
|
3278 |
b Fe(type)g(nil)g(\(lam)g(x.\(lam)f(x.)i(\(app)f(\(var)g(x\))h(\(var)f
|
|
|
3279 |
(x\)\)\))g(B)50 1961 y Fp(to)29 b Fh(\013)p Fp(Prolog)e(pro)r(duces)h
|
|
|
3280 |
(the)h(unique)g(answ)n(er)f([)p Fe(A)d Fp(:=)g Fe(arrow)41
|
|
|
3281 |
b(T)i(\(arrow)f(T')g(T'\))o Fp(])29 b(for)g(\(a\),)g(and)g(no)f(answ)n
|
|
|
3282 |
(er)g(for)50 2061 y(\(b\).)174 2161 y(Tw)n(o)d(no)n(v)n(el)e(features)i
|
|
|
3283 |
(of)g Fh(\013)p Fp(Prolog)e(are)h(illustrated)h(b)n(y)f(the)i(third)f
|
|
|
3284 |
(clause)f(of)h Fe(type)p Fp(.)f(First,)h Fd(abstr)l(actions)32
|
|
|
3285 |
b Fp(are)50 2260 y(a)e(built-in)h(notion)g(of)g Fh(\013)p
|
|
|
3286 |
Fp(Prolog;)d(th)n(us)j(w)n(e)g(can)f(write)g Fe(\(lam)42
|
|
|
3287 |
b(x.M\))29 b Fp(in)i(whic)n(h)g Fe(x)g Fp(is)f(tak)n(en)g(from)h(the)g
|
|
|
3288 |
(syn)n(tactic)50 2360 y(category)c(of)i(names,)g Fe(M)g
|
|
|
3289 |
Fp(is)g(a)g(v)-5 b(ariable)28 b(and)h Fe(lam)f Fp(a)h(function)h(sym)n
|
|
|
3290 |
(b)r(ol.)f(This)g(term)g(is)g(in)n(tended)h(to)f(unify)h(with)50
|
|
|
3291 |
2459 y(an)n(y)d Fh(\025)p Fp(-abstraction,)f(for)h(example)g
|
|
|
3292 |
Fe(\(lam)42 b(x.var)g(x\))p Fp(,)27 b Fe(\(lam)41 b(y.app)h(\(var)g
|
|
|
3293 |
(x\))h(\(var)f(y\)\))26 b Fp(and)h(so)g(on,)h(where)50
|
|
|
3294 |
2559 y(in)35 b(the)f(\014rst)h(case)e(the)i(uni\014er)g(is)f(the)h
|
|
|
3295 |
(capturing)e(substitution)j([)p Fe(M)e Fp(:=)g Fe(var)42
|
|
|
3296 |
b(x)p Fp(].)34 b(Note)h(that)g(if)g(w)n(e)f(had)g(used)50
|
|
|
3297 |
2659 y(higher-order)j(uni\014cation,)i(then)h Fe(\(lam)i(x.M\))c
|
|
|
3298 |
Fp(do)r(es)h Fd(not)48 b Fp(unify)40 b(with)g Fe(\(lam)i(x.var)f(x\))p
|
|
|
3299 |
Fp(,)e(b)r(ecause)g(there)g(is)50 2758 y(no)30 b(capture-a)n(v)n
|
|
|
3300 |
(oiding)e(substitution)k(that)f(mak)n(es)f(b)r(oth)h(terms)g(equal.)f
|
|
|
3301 |
(Also)h(note)f(that)h(in)h Fh(\013)p Fp(Prolog)c(binders)50
|
|
|
3302 |
2858 y(ha)n(v)n(e)e(concrete)g(names,)h(whic)n(h)g(are)f(not)h
|
|
|
3303 |
(restricted)g(to)g(the)g(scop)r(e)g(of)g(the)h(abstraction.)d(These)i
|
|
|
3304 |
(names)g(can)g(b)r(e)50 2958 y(therefore)h(used)h(in)g(the)g(b)r(o)r
|
|
|
3305 |
(dy)g(of)g(the)g(clause,)f(for)h(example)f(to)h(add)g
|
|
|
3306 |
Fe(\(pair)41 b(x)i(T\))29 b Fp(to)f(the)i(con)n(text)e
|
|
|
3307 |
Fe(Gamma)p Fp(.)f(In)50 3057 y(con)n(trast,)c(in)i(higher-order)d
|
|
|
3308 |
(abstract)i(syn)n(tax)g(binders)g(are)g(anon)n(ymous)f(and)h(cannot)g
|
|
|
3309 |
(escap)r(e)g(their)h(scop)r(e.)f(So,)50 3157 y(in)k Fh(\025)p
|
|
|
3310 |
Fp(Prolog,)d(a)j(binder)f(from)g(the)h(head)g(cannot)f(b)r(e)h(used)f
|
|
|
3311 |
(in)h(the)g(b)r(o)r(dy)g(of)g(a)f(clause.)174 3256 y(Second,)19
|
|
|
3312 |
b Fd(fr)l(eshness)j(c)l(onstr)l(aints)p Fp(,)d(here)f
|
|
|
3313 |
Fe(x#Gamma)p Fp(,)e(can)i(b)r(e)i(attac)n(hed)e(to)g(the)i(heads)e(of)h
|
|
|
3314 |
(clauses;)f(the)h(freshness)50 3356 y(constrain)n(t)33
|
|
|
3315 |
b Fe(x#Gamma)e Fp(ensures)i(that)h(the)g(v)-5 b(ariable)33
|
|
|
3316 |
b Fe(Gamma)f Fp(is)i(not)g(instan)n(tiated)g(with)g(a)g(term)g(con)n
|
|
|
3317 |
(taining)e Fe(x)50 3456 y Fp(freely)-7 b(.)25 b(Since)g(the)h(third)g
|
|
|
3318 |
(clause)e(is)h(in)n(tended)h(to)f(implemen)n(t)h(the)g(t)n(yp)r(e)f
|
|
|
3319 |
(inference)g(rule)g(for)g Fh(\025)p Fp(-abstractions,)f(its)50
|
|
|
3320 |
3555 y(op)r(erational)30 b(b)r(eha)n(viour)h(is)g(giv)n(en)g(b)n(y:)h
|
|
|
3321 |
(c)n(ho)r(ose)e(fresh)i(names)f(for)g Fe(Gamma)p Fp(,)f
|
|
|
3322 |
Fe(x)p Fp(,)i Fe(M)p Fp(,)f Fe(T)h Fp(and)f Fe(T')g Fp(\(this)h(is)g
|
|
|
3323 |
(standard)50 3655 y(in)h(Prolog-lik)n(e)c(languages\),)i(unify)i(the)g
|
|
|
3324 |
(head)f(of)h(the)g(clause)f(with)h(the)g(goal)e(form)n(ula,)h(apply)g
|
|
|
3325 |
(the)h(resulting)50 3755 y(uni\014er)h(to)h(the)g(b)r(o)r(dy)g(of)g
|
|
|
3326 |
(the)g(clause)f(and)h(mak)n(e)f(sure)g(that)h Fe(Gamma)e
|
|
|
3327 |
Fp(is)h(not)h(substituted)h(with)f(a)f(term)h(that)50
|
|
|
3328 |
3854 y(con)n(tains)26 b(free)i(o)r(ccurrences)e(of)h(the)h(fresh)g
|
|
|
3329 |
(name)f(w)n(e)g(ha)n(v)n(e)g(c)n(hosen)f(for)h Fe(x)p
|
|
|
3330 |
Fp(.)50 4113 y Fi(2)112 b(Nominal)38 b(terms)g(and)g(nominal)h
|
|
|
3331 |
(uni\014cation)50 4306 y Fp(T)-7 b(o)27 b(calculate)g(uni\014ers,)g
|
|
|
3332 |
Fh(\013)p Fp(Prolog)e(emplo)n(ys)i(the)h(nominal)f(uni\014cation)g
|
|
|
3333 |
(algorithm)g(of)g(Urban)g(et)h(al.)f([3],)g(whic)n(h)50
|
|
|
3334 |
4405 y(pro)r(duces)36 b(uni\014ers)h(that)g(mak)n(e)g(t)n(w)n(o)f
|
|
|
3335 |
(terms)h(equal)f(mo)r(dulo)h Fh(\013)p Fp(-equiv)-5 b(alence.)37
|
|
|
3336 |
b(Nominal)g(uni\014cation)g(retains)50 4505 y(the)32
|
|
|
3337 |
b(main)h(adv)-5 b(an)n(tages)30 b(of)i(\014rst-order)f(uni\014cation:)h
|
|
|
3338 |
(all)g(uni\014cation)g(problems)g(are)f(decidable)h(and)g(solv)-5
|
|
|
3339 |
b(able)50 4604 y(problems)18 b(p)r(ossess)f(most)i(general)e
|
|
|
3340 |
(uni\014ers.)i(It)g(mak)n(es)f(use)g(of)h(the)g(notion)g(of)f(p)r
|
|
|
3341 |
(ossibly)h(capturing)f(substitutions,)50 4704 y(i.e.)40
|
|
|
3342 |
b(substitutions)g(that)h(allo)n(w)d Fe(M)i Fp(to)g(b)r(e)g(substituted)
|
|
|
3343 |
h(with)g Fe(\(var)h(x\))d Fp(in)h(the)g(term)g Fe(\(lam)i(x.M\))d
|
|
|
3344 |
Fp(without)50 4804 y(in)n(tro)r(ducing)26 b(a)h(new)g(name)f(for)h(the)
|
|
|
3345 |
g(binder)g Fe(x)p Fp(,)g(and)f(the)i(notion)e(of)h(name)g(sw)n(apping.)
|
|
|
3346 |
f(A)h(name)g(sw)n(apping)f(is)h(a)50 4903 y(pair)g(of)g(the)h(form)f
|
|
|
3347 |
(\()p Fe(x)c(y)p Fp(\),)28 b(and)f(if)i(applied)e(for)g(example)g(to)h
|
|
|
3348 |
(the)g(term)950 5103 y Fe(lam)22 b(x)p Fh(:)14 b Fe(app)22
|
|
|
3349 |
b(\()o(lam)g(x)p Fh(:)14 b Fe(var)21 b(x\))o(\(app)h(\()o(var)g(y\))g
|
|
|
3350 |
(\(var)g(z\)\))50 5247 y Fp(giv)n(es)946 5347 y Fe(lam)g(y)p
|
|
|
3351 |
Fh(:)14 b Fe(app)21 b(\()o(lam)h(y)p Fh(:)14 b Fe(var)22
|
|
|
3352 |
b(y\))o(\()o(app)g(\(var)g(x\))g(\()o(var)g(z\)\))g Fh(:)p
|
|
|
3353 |
eop end
|
|
|
3354 |
%%Page: 3 3
|
|
|
3355 |
TeXDict begin 3 2 bop 50 241 a Fp(As)30 b(can)f(b)r(e)h(seen)f(the)h
|
|
|
3356 |
(name)g(sw)n(apping)e(acts)h(on)h(all)f(o)r(ccurrences)f(of)i
|
|
|
3357 |
Fe(x)f Fp(and)g Fe(y)h Fp(regardless)d(whether)i(they)h(are)50
|
|
|
3358 |
340 y(b)r(ound,)f(free)g(or)f(binding.)i(The)f(crucial)f(p)r(oin)n(t)h
|
|
|
3359 |
(ab)r(out)h(name)e(sw)n(apping)g(is)h(that)h(it)f(preserv)n(es)e
|
|
|
3360 |
Fh(\013)p Fp(-equiv)-5 b(alence.)50 440 y(F)e(or)19 b(instance,)g
|
|
|
3361 |
(consider)f Fe(lam)k(y)p Fh(:)p Fe(var)f(x)e Fp(and)h
|
|
|
3362 |
Fe(lam)h(z)p Fh(:)p Fe(var)h(x)o Fp(,)e(whic)n(h)f(are)g
|
|
|
3363 |
Fh(\013)p Fp(-equiv)-5 b(alen)n(t.)19 b(Sw)n(apping)f
|
|
|
3364 |
Fe(x)h Fp(and)h Fe(y)f Fp(results)50 540 y(in)25 b Fe(lam)d(x)p
|
|
|
3365 |
Fh(:)p Fe(var)f(y)k Fp(and)g Fe(lam)d(z)p Fh(:)p Fe(var)f(y)p
|
|
|
3366 |
Fp(,)k(whic)n(h)g(are)g(still)g Fh(\013)p Fp(-equiv)-5
|
|
|
3367 |
b(alen)n(t,)25 b(but)h(na)-9 b(\177)-32 b(\020v)n(ely)23
|
|
|
3368 |
b(substituting)j Fe(y)e Fp(for)h Fe(x)g Fp(results)50
|
|
|
3369 |
639 y(in)j(v)-5 b(ariable)26 b(capture)h(in)h(the)g(\014rst)f(term.)174
|
|
|
3370 |
741 y(Sw)n(appings)c(are)g(emplo)n(y)n(ed)g(in)h(the)g(nominal)f
|
|
|
3371 |
(uni\014cation)h(algorithm)f(when)g(abstractions)g(are)f(uni\014ed.)j
|
|
|
3372 |
(F)-7 b(or)50 840 y(example)32 b(if)i(one)e(uni\014es)h(the)g(term)g
|
|
|
3373 |
Fe(lam)42 b(x.M)32 b Fp(with)h Fe(lam)43 b(y.var)e(y)p
|
|
|
3374 |
Fp(,)33 b(then)g Fe(M)g Fp(will)g(b)r(e)g(instan)n(tiated)f(with)i(the)
|
|
|
3375 |
50 940 y(term)20 b Fe(var)43 b(x)20 b Fp(whic)n(h)h(can)f(b)r(e)h
|
|
|
3376 |
(obtained)f(b)n(y)h(applying)f(the)h(sw)n(apping)e(\()p
|
|
|
3377 |
Fe(x)k(y)p Fp(\))e(to)g Fe(var)42 b(y)20 b Fp(\(written)h(as)f(\()p
|
|
|
3378 |
Fe(x)j(y)p Fp(\))5 b Fg(\001)g Fe(var)21 b(y)p Fp(\).)50
|
|
|
3379 |
1039 y(Ho)n(w)n(ev)n(er)j(care)h(needs)h(to)f(b)r(e)i(tak)n(en)e(if)i
|
|
|
3380 |
Fe(x)e Fp(o)r(ccurs)h(freely)f(in)h(the)h(term)f(to)g(whic)n(h)g(the)g
|
|
|
3381 |
(sw)n(apping)f(is)h(applied.)g(F)-7 b(or)50 1139 y(example)27
|
|
|
3382 |
b(the)h(uni\014cation)f(problem)1148 1345 y Fe(lam)22
|
|
|
3383 |
b(x)p Fh(:)p Fe(M)45 b Fg(\031)p Fp(?)h Fe(lam)22 b(y)p
|
|
|
3384 |
Fh(:)p Fe(app)g Fp(\()p Fe(var)g(y)p Fp(\))h(\()p Fe(var)f(x)p
|
|
|
3385 |
Fp(\))50 1501 y(do)r(es)f(not)g(ha)n(v)n(e)f(an)n(y)g(solution)h
|
|
|
3386 |
(\(just)h(replacing)e Fe(M)h Fp(b)n(y)g(the)g(`sw)n(app)r(ed')g(v)n
|
|
|
3387 |
(ersion)f(of)h Fe(app)h Fp(\()p Fe(var)g(y)p Fp(\))h(\()p
|
|
|
3388 |
Fe(var)f(x)p Fp(\))f(do)r(es)g(not)50 1601 y(yield)f(t)n(w)n(o)f
|
|
|
3389 |
Fh(\013)p Fp(-equiv)-5 b(alen)n(t)20 b(terms\).)h(This)f(problem)g(is)g
|
|
|
3390 |
(solv)n(ed)f(in)h(nominal)g(uni\014cation)g(b)n(y)g(freshness)f
|
|
|
3391 |
(constrain)n(ts,)50 1700 y(whic)n(h)31 b(ensure)g(that)h(p)r(ossible)f
|
|
|
3392 |
(solutions)f(resp)r(ect)h Fh(\013)p Fp(-equiv)-5 b(alence.)31
|
|
|
3393 |
b(So)g(for)g(solving)g(the)g(uni\014cation)h(problem)50
|
|
|
3394 |
1800 y(giv)n(en)27 b(ab)r(o)n(v)n(e)f(one)h(needs)g(to)h(solv)n(e)652
|
|
|
3395 |
2006 y Fe(M)46 b Fg(\031)p Fp(?)g(\()p Fe(x)23 b(y)p
|
|
|
3396 |
Fp(\))18 b Fg(\001)g Fe(app)k Fp(\()p Fe(var)g(y)p Fp(\))h(\()p
|
|
|
3397 |
Fe(var)f(x)p Fp(\))83 b(and)g Fe(x)23 b Fp(#?)g Fe(app)f
|
|
|
3398 |
Fp(\()p Fe(var)g(y)p Fp(\))h(\()p Fe(var)f(x)p Fp(\))50
|
|
|
3399 |
2162 y(where)38 b(the)h(\014rst)g(problem)f(asks)g(whether)h
|
|
|
3400 |
Fe(M)f Fp(and)h Fe(app)22 b Fp(\()p Fe(var)g(x)p Fp(\))h(\()p
|
|
|
3401 |
Fe(var)f(y)p Fp(\))39 b(unify)-7 b(,)39 b(and)g(the)g(second)g(problem)
|
|
|
3402 |
50 2262 y(asks)33 b(whether)h Fe(x)g Fp(is)h(fresh)f(for)f
|
|
|
3403 |
Fe(app)22 b Fp(\()p Fe(var)g(y)p Fp(\))h(\()p Fe(var)f(x)p
|
|
|
3404 |
Fp(\),)35 b(i.e.)f Fe(app)23 b(\(var)g(y\))g(\(var)g(x\))33
|
|
|
3405 |
b Fp(cannot)h(con)n(tain)g(an)n(y)f(free)50 2362 y(o)r(ccurrence)26
|
|
|
3406 |
b(of)i Fe(x)p Fp(.)f(Since)h(the)g(latter)f(do)r(es)g(not)h(hold,)f
|
|
|
3407 |
(the)h(uni\014cation)g(problem)f(fails|as)g(exp)r(ected.)174
|
|
|
3408 |
2463 y(While)22 b(in)e(the)h(examples)f(ab)r(o)n(v)n(e)g(the)h(v)-5
|
|
|
3409 |
b(ariable)19 b Fe(M)i Fp(is)f(uni\014ed)h(with)h(ground)d(terms)i(only)
|
|
|
3410 |
-7 b(,)20 b(nominal)g(uni\014cation)50 2562 y(also)25
|
|
|
3411 |
b(solv)n(es)f(more)h(general)g(uni\014cation)g(problems)h(in)n(v)n
|
|
|
3412 |
(olving)e(op)r(en)i(terms.)g(F)-7 b(or)25 b(this)h(the)g(notion)g(of)g
|
|
|
3413 |
(freshness)50 2662 y(constrain)n(t)d(needs)h(some)g(sligh)n(t)g
|
|
|
3414 |
(extension|for)f(the)i(details)f(w)n(e)f(refer)h(the)h(reader)d(to)j
|
|
|
3415 |
([3])f(whic)n(h)g(also)f(includes)50 2762 y(references)j(ab)r(out)i(a)g
|
|
|
3416 |
(formal)f(v)n(eri\014cation)f(of)i(the)g(correctness)e(of)i(the)g
|
|
|
3417 |
(nominal)f(uni\014cation)h(algorithm)e(using)50 2861
|
|
|
3418 |
y(the)i(pro)r(of)f(assistan)n(t)f(Isab)r(elle.)50 3135
|
|
|
3419 |
y Fi(3)112 b Fc(\013)p Fi(Prolog)37 b(examples)50 3343
|
|
|
3420 |
y Fp(Considering)30 b(the)j(\014rst)e(exp)r(erience)h(w)n(e)f(gained)h
|
|
|
3421 |
(with)g Fh(\013)p Fp(Prolog)e(there)i(is)g(go)r(o)r(d)f(evidence)h
|
|
|
3422 |
(that)g(abstractions)50 3443 y(and)21 b(freshness)f(constrain)n(ts)f
|
|
|
3423 |
(from)i(nominal)f(uni\014cation)h(pro)n(vide)f(useful)h(programming)e
|
|
|
3424 |
(constructs.)h(Consider)50 3542 y(for)27 b(example)g(t)n(w)n(o)g(of)g
|
|
|
3425 |
(the)h(transformations)e(required)h(for)g(calculating)g(a)g
|
|
|
3426 |
(prenex-normal)e(form)j(of)f(a)g(form)n(ula)675 3728
|
|
|
3427 |
y Fg(8)p Fh(x:P)k Fg(^)19 b(8)p Fh(x:Q)k Fp(=)-14 b Fg(\))23
|
|
|
3428 |
b(8)p Fh(x:)p Fp(\()p Fh(P)31 b Fg(^)19 b Fh(Q)p Fp(\))83
|
|
|
3429 |
b(and)g(\()p Fg(8)p Fh(x:P)12 b Fp(\))19 b Fg(^)g Fh(Q)k
|
|
|
3430 |
Fp(=)-14 b Fg(\))23 b(8)p Fh(x:)p Fp(\()p Fh(P)31 b Fg(^)19
|
|
|
3431 |
b Fh(Q)p Fp(\))50 3916 y(where)30 b(in)h(the)h(second)e(transformation)
|
|
|
3432 |
f Fh(Q)i Fp(m)n(ust)g(not)g(con)n(tain)f(an)n(y)g(free)h(o)r(ccurrence)
|
|
|
3433 |
f(of)h Fh(x)p Fp(.)g(These)g(transfor-)50 4016 y(mations)c(can)g(b)r(e)
|
|
|
3434 |
h(implemen)n(ted)g(as)f(the)h(follo)n(wing)f(clauses.)531
|
|
|
3435 |
4183 y Fe(prenex)41 b(\(and)h(\(forall)f(x.P\))h(\(forall)f(x.Q\)\))g
|
|
|
3436 |
(\(forall)g(x.\(and)g(P)i(Q\)\).)531 4283 y(prenex)e(\(and)h(\(forall)f
|
|
|
3437 |
(x.P\))h(Q\))h(\(forall)d(x.\(and)i(P)h(Q\)\))f(/)h(x#Q.)50
|
|
|
3438 |
4450 y Fp(In)29 b(the)h(\014rst)f(clause,)g(the)h(use)f(of)h(nominal)f
|
|
|
3439 |
(uni\014cation)g(allo)n(ws,)f(roughly)g(sp)r(eaking,)h(to)g(sync)n
|
|
|
3440 |
(hronise)f(the)i(t)n(w)n(o)50 4550 y(binders.)25 b(In)g(e\013ect)g
|
|
|
3441 |
(this)g(clause)g(is)f(applicable)h(for)f(an)n(y)g(term)h(of)g(the)h
|
|
|
3442 |
(form)e Fe(\(and)42 b(\(forall)f(y....\))g(\(forall)50
|
|
|
3443 |
4649 y(z....\))p Fp(.)27 b(What)i(is)h(pleasan)n(t)e(ab)r(out)h
|
|
|
3444 |
Fh(\013)p Fp(Prolog)f(is)h(that)h(in)f(this)h(clause)e(no)i(explicit)f
|
|
|
3445 |
(renaming)f(of)i(the)f(binders)50 4749 y(is)34 b(required:)g(it)h(will)
|
|
|
3446 |
g(b)r(e)g(done)f(implicitly)i(b)n(y)e(the)h(sw)n(apping)f(op)r(eration)
|
|
|
3447 |
f(of)i(the)g(uni\014cation)f(algorithm.)g(In)50 4848
|
|
|
3448 |
y(the)27 b(second)g(clause,)g(the)g(freshness)g(constrain)n(t)f
|
|
|
3449 |
Fe(x#Q)g Fp(mak)n(es)g(sure)h(that)g(the)h(side-condition)e(placed)h
|
|
|
3450 |
(up)r(on)h(the)50 4948 y(second)34 b(transformation)g(holds.)h(In)g
|
|
|
3451 |
(con)n(trast,)f(in)i(a)f(higher-order)d(abstract)j(syn)n(tax)f(enco)r
|
|
|
3452 |
(ding,)h(w)n(e)g(cannot)50 5048 y(explicitly)25 b(sa)n(y)f(that)h(a)g
|
|
|
3453 |
(v)-5 b(ariable)24 b(m)n(ust)h(not)g(o)r(ccur)f(in)h(a)g(term;)g
|
|
|
3454 |
(instead)g(w)n(e)f(can)h(only)f(sa)n(y)g(that)h(a)g(v)-5
|
|
|
3455 |
b(ariable)24 b(ma)n(y)50 5147 y(o)r(ccur)30 b(in)h(a)g(term.)g(So,)f
|
|
|
3456 |
(for)h(example,)f(w)n(e)h(w)n(ould)f(ha)n(v)n(e)g(to)g(mak)n(e)h(the)g
|
|
|
3457 |
(p)r(oten)n(tial)g(dep)r(endences)g(on)f Fe(x)h Fp(explicit)50
|
|
|
3458 |
5247 y(ev)n(erywhere)23 b(in)i Fh(\025)p Fp(Prolog.)f(In)h(case)f(of)h
|
|
|
3459 |
(the)h(prenex)e(transformation,)g(this)h(results)g(in)n(to)g(the)g
|
|
|
3460 |
(follo)n(wing)f Fh(\025)p Fp(Prolog)50 5347 y(clauses:)p
|
|
|
3461 |
eop end
|
|
|
3462 |
%%Page: 4 4
|
|
|
3463 |
TeXDict begin 4 3 bop 248 227 a Fe(prenex)41 b(\(and)h(\(forall)f(x\\P)
|
|
|
3464 |
h(x\))h(\(forall)e(x\\Q)h(x\)\))g(\(forall)f(x\\)i(\(and)f(\(P)g(x\))h
|
|
|
3465 |
(\(Q)g(x\)\)\).)248 327 y(prenex)e(\(and)h(\(forall)f(x\\P)h(x\))h(Q\))
|
|
|
3466 |
g(\(forall)d(x\\\(and)h(\(P)i(x\))g(Q\)\).)174 485 y
|
|
|
3467 |
Fp(Another)30 b Fh(\013)p Fp(Prolog)e(example)h(is)g(giv)n(en)g(b)r
|
|
|
3468 |
(elo)n(w.)h(It)g(implemen)n(ts)g(the)g(capture-a)n(v)n(oiding)d
|
|
|
3469 |
(substitution)j(of)g(a)50 584 y(v)-5 b(ariable)26 b(in)i(a)f
|
|
|
3470 |
Fh(\025)p Fp(-term)774 554 y Fo(3)812 584 y Fp(.)421
|
|
|
3471 |
724 y Fe(id)43 b(X)g(X.)421 824 y(subst)e(\(var)h(X\))h(X)g(T)g(T.)421
|
|
|
3472 |
923 y(subst)e(\(var)h(Y\))h(X)g(T)g(\(var)f(Y\))284 b(:-)23
|
|
|
3473 |
b(not\(id)41 b(X)i(Y\).)421 1023 y(subst)e(\(app)h(M)i(N\))e(X)h(T)h
|
|
|
3474 |
(\(app)e(M')g(N'\))22 b(:-)h(subst)41 b(M)i(X)h(T)f(M',)f(subst)g(N)h
|
|
|
3475 |
(X)g(T)g(N'.)421 1122 y(subst)e(\(lam)h(y.M\))g(X)h(T)h(\(lam)e(y.M'\))
|
|
|
3476 |
64 b(/)j(y#T,X)1795 1222 y(:-)23 b(subst)41 b(M)i(X)h(T)f(M'.)50
|
|
|
3477 |
1380 y Fp(Giv)n(en)38 b(this)g(program,)e(p)r(osing)i(the)g(query)h
|
|
|
3478 |
Fe(subst)i(\(lam)h(\(x.)h(var)f(y\)\))g(y)i(\(var)d(x\))i(M)38
|
|
|
3479 |
b Fp(yields)g(the)g(result)50 1479 y([)p Fe(M)23 b Fp(:=)f
|
|
|
3480 |
Fe(lam)43 b(x'.var)e(x)o Fp(])28 b(where)f Fe(x')g Fp(is)g(c)n(hosen)g
|
|
|
3481 |
(freshly)g(during)g(pro)r(of)g(searc)n(h.)174 1579 y(Our)33
|
|
|
3482 |
b(\014nal)h(example)f(is)g(the)h(de\014nition)g(of)f(ev)-5
|
|
|
3483 |
b(aluation)33 b(for)g Fh(\025)p Fp(-terms.)h(W)-7 b(e)34
|
|
|
3484 |
b(\014rst)f(de\014ne)h(a)f(relation)f(whic)n(h)50 1679
|
|
|
3485 |
y(expresses)26 b Fh(\014)t Fp(-reduction:)857 1854 y
|
|
|
3486 |
Fe(beta)42 b(\(app)g(\(lam)g(\(x.E\)\))f(E'\))h(R)23
|
|
|
3487 |
b(:-)f(subst)42 b(E)h(x)g(E')g(R.)50 2032 y Fp(No)n(w)27
|
|
|
3488 |
b(w)n(e)g(de\014ne)h(relations)f Fe(step)f Fp(that)i(express)f
|
|
|
3489 |
(one-step)g(reduction)g(of)h Fh(\025)p Fp(-terms,)g(and)f
|
|
|
3490 |
Fe(steps)p Fp(,)f(the)i(re\015exiv)n(e,)50 2132 y(transitiv)n(e)e
|
|
|
3491 |
(closure)h(of)g Fe(step)p Fp(:)617 2290 y Fe(step)42
|
|
|
3492 |
b(M)h(M')894 b(:-)23 b(beta)42 b(M)h(M'.)617 2389 y(step)f(\(app)g(M)h
|
|
|
3493 |
(N\))g(\(app)f(M')h(N\))196 b(:-)23 b(step)42 b(M)h(M'.)617
|
|
|
3494 |
2489 y(step)f(\(app)g(M)h(N\))g(\(app)f(M)h(N'\))196
|
|
|
3495 |
b(:-)23 b(step)42 b(N)h(N'.)617 2589 y(step)f(\(lam)g(\(x.M\)\))f
|
|
|
3496 |
(\(lam)h(\(x.M'\)\))20 b(:-)j(step)42 b(M)h(M'.)617 2788
|
|
|
3497 |
y(steps)f(M)h(M.)617 2888 y(steps)f(M)h(M'')806 b(:-)23
|
|
|
3498 |
b(step)42 b(M)h(M',)f(steps)g(M')g(M''.)174 3045 y Fp(F)-7
|
|
|
3499 |
b(rom)32 b(the)h(exp)r(erience)f(w)n(e)g(gained)g(so)f(far,)h(all)g
|
|
|
3500 |
(inference)h(rules)e(in)n(v)n(olving)g Fh(\025)p Fp(-terms)i(can)f(in)g
|
|
|
3501 |
Fh(\013)p Fp(Prolog)f(b)r(e)50 3145 y(implemen)n(ted)24
|
|
|
3502 |
b(nearly)e(`one-to-one'.)f(W)-7 b(e)24 b(wish)f(to)g(stress,)f(ho)n(w)n
|
|
|
3503 |
(ev)n(er,)f(that)i Fh(\013)p Fp(Prolog)e(is)i(not)g(limited)h(to)f
|
|
|
3504 |
(de\014ning)50 3244 y(relations)28 b(on)g Fh(\025)p Fp(-terms.)h(W)-7
|
|
|
3505 |
b(e)30 b(ha)n(v)n(e)d(c)n(hosen)i(the)g Fh(\025)p Fp(-calculus)g(as)f
|
|
|
3506 |
(an)h(example)f(b)r(ecause)h(it)g(w)n(ell-kno)n(wn)f(and)h(the)50
|
|
|
3507 |
3344 y(issues)20 b(raised)f(b)n(y)h(binding)g(and)g(capture-a)n(v)n
|
|
|
3508 |
(oiding)e(substitution)j(are)e(widely)h(appreciated.)g(Ho)n(w)n(ev)n
|
|
|
3509 |
(er,)e Fh(\013)p Fp(Prolog)50 3444 y(can)33 b(also)f(b)r(e)h(used)g(to)
|
|
|
3510 |
g(write)g(programs)e(ab)r(out)i(languages)e(with)j(other)e(forms)h(of)g
|
|
|
3511 |
(binding,)g(pro)n(vided)f(these)50 3543 y(forms)d(admit)g
|
|
|
3512 |
Fh(\013)p Fp(-equiv)-5 b(alence.)29 b(Some)h(examples)e(include)i
|
|
|
3513 |
(concurrency)e(calculi)h(suc)n(h)g(as)g(the)g Fh(\031)s
|
|
|
3514 |
Fp(-calculus)g(and)50 3643 y(the)c(Calculus)g(of)g(Mobile)g(Am)n(bien)n
|
|
|
3515 |
(ts,)h(in)f(whic)n(h)g(names)g(can)g(b)r(e)g(b)r(ound)h(but)g(are)e
|
|
|
3516 |
(not)h(sub)5 b(ject)26 b(to)f(substitution)50 3742 y(b)n(y)e(other)g
|
|
|
3517 |
(\(pro)r(cess\))g(terms,)h(only)f(renaming.)g(Another)g(example)h(is)f
|
|
|
3518 |
(mo)r(deling)h(imp)r(erativ)n(e)f(languages)f(lik)n(e)h(C,)50
|
|
|
3519 |
3842 y(in)h(whic)n(h)h(v)-5 b(ariables)22 b(serv)n(e)h(a)h(dual)g(role)
|
|
|
3520 |
g(as)f(references)g(to)i(memory)e(cells)h(and)g(their)g(con)n(ten)n
|
|
|
3521 |
(ts.)g(Also,)g(freshness)50 3942 y(constrain)n(ts)c(and)h(fresh-name)g
|
|
|
3522 |
(generation)f(ma)n(y)g(b)r(e)i(of)g(in)n(terest)f(in)h(their)f(o)n(wn)g
|
|
|
3523 |
(righ)n(t,)g(for)g(example)g(in)g(mo)r(deling)50 4041
|
|
|
3524 |
y(fresh-nonce)26 b(generation)g(in)i(securit)n(y)f(proto)r(cols.)50
|
|
|
3525 |
4302 y Fi(4)112 b(Conclusion)50 4497 y Fp(Curren)n(tly)-7
|
|
|
3526 |
b(,)41 b(w)n(e)g(ha)n(v)n(e)f(implemen)n(ted)i(a)f(protot)n(yp)r(e)g
|
|
|
3527 |
(in)n(terpreter)g(for)g Fh(\013)p Fp(Prolog)e(that)j(p)r(ermits)g
|
|
|
3528 |
(user-de\014ned)50 4596 y(datat)n(yp)r(es)26 b(and)h(p)r
|
|
|
3529 |
(olymorphically)e(t)n(yp)r(ed)i(Horn)g(clauses.)f(W)-7
|
|
|
3530 |
b(ork)26 b(on)g(v)n(erifying)g(the)h(correctness)e(of)i
|
|
|
3531 |
Fh(\013)p Fp(Prolog's)50 4696 y(pro)r(of)j(searc)n(h)f(algorithm)h
|
|
|
3532 |
(relativ)n(e)g(to)h(Nominal)f(Logic)g(and)h(on)f(optimizing)h(the)g
|
|
|
3533 |
(nominal)g(uni\014cation)g(algo-)50 4796 y(rithm)36 b(is)g(in)h
|
|
|
3534 |
(progress.)d(In)i(the)h(future,)f(w)n(e)g(plan)g(to)h(adapt)e(more)h
|
|
|
3535 |
(adv)-5 b(anced)35 b(logic)h(program)e(analyses)h(for)50
|
|
|
3536 |
4895 y Fh(\013)p Fp(Prolog)29 b(programs,)g(suc)n(h)i(as)f(mo)r(de,)h
|
|
|
3537 |
(determinism,)h(and)f(termination)g(analyses.)e(Man)n(y)i(in)n
|
|
|
3538 |
(teresting)f(prop-)50 4995 y(erties)d(of)g(languages)f(can)h(b)r(e)h
|
|
|
3539 |
(framed)f(in)h(terms)f(of)h(suc)n(h)f(prop)r(erties)g(on)g(relations;)f
|
|
|
3540 |
(for)h(example)h(unicit)n(y)f(and)50 5095 y(decidabilit)n(y)32
|
|
|
3541 |
b(of)g(t)n(yp)r(ec)n(hec)n(king)g(can)g(b)r(e)h(expressed)e(in)i(terms)
|
|
|
3542 |
f(of)g(determinism)h(and)f(termination)g(of)h(an)f(ap-)50
|
|
|
3543 |
5194 y(propriate)27 b(mo)r(ding)i(of)g(the)h(t)n(yping)f(relation.)f
|
|
|
3544 |
(Ev)n(en)n(tually)f(w)n(e)i(wish)g(to)g(supp)r(ort)g(general)e
|
|
|
3545 |
(inductiv)n(e)j(theorem)p 50 5261 473 4 v 73 5315 a Fn(3)133
|
|
|
3546 |
5347 y Fl(not)c Fm(is)g(the)g(usual)g(negation-b)n(y-failure.)p
|
|
|
3547 |
eop end
|
|
|
3548 |
%%Page: 5 5
|
|
|
3549 |
TeXDict begin 5 4 bop 50 241 a Fp(pro)n(ving)20 b(ab)r(out)i
|
|
|
3550 |
Fh(\013)p Fp(Prolog)e(programs.)f(W)-7 b(e)23 b(b)r(eliev)n(e)e(that)h
|
|
|
3551 |
(this)h(will)f(lead)f(to)h(b)r(etter)g(to)r(ols)f(for)h
|
|
|
3552 |
(computer-assisted)50 340 y(reasoning)j(ab)r(out)j(prop)r(erties)f(of)g
|
|
|
3553 |
(languages.)50 440 y Fb(Ac)m(kno)m(wledgemen)m(ts:)39
|
|
|
3554 |
b Fp(Dominik)i(W)-7 b(ee)40 b(implemen)n(ted)h(a)f(preliminary)f(v)n
|
|
|
3555 |
(ersion)g(of)h Fh(\013)p Fp(Prolog)e(as)h(a)h(thesis)50
|
|
|
3556 |
540 y(pro)5 b(ject.)21 b(W)-7 b(e)22 b(thank)g(Marinos)f(Georgiades)f
|
|
|
3557 |
(who)i(in)n(vited)g(us)f(to)h(write)g(this)g(pap)r(er)g(for)f(the)i
|
|
|
3558 |
(CoLogNet)d(newslet-)50 639 y(ter.)50 905 y Fi(References)50
|
|
|
3559 |
1096 y Fm(1.)42 b(J.)36 b(Cheney)e(and)h(C.)g(Urban.)62
|
|
|
3560 |
b(System)34 b(description:)i(alpha-Prolog,)h(a)e(fresh)g(approac)n(h)g
|
|
|
3561 |
(to)g(logic)i(programming)151 1187 y(mo)r(dulo)32 b(alpha-equiv)l
|
|
|
3562 |
(alence.)50 b(In)30 b(J.)i(Levy)-6 b(,)30 b(M.)i(Kohlhase,)f(J.)h
|
|
|
3563 |
(Niehren,)f(and)g(M.)g(Villaret,)i(editors,)f Fa(Pr)l(o)l(c)l(e)l(e)l
|
|
|
3564 |
(dings)151 1279 y(of)d(the)h(17th)g(International)g(Workshop)g(on)g
|
|
|
3565 |
(Uni\014c)l(ation,)f(UNIF'03)p Fm(,)e(pages)h(15{19,)i(V)-6
|
|
|
3566 |
b(alencia,)28 b(Spain,)f(June)g(2003.)151 1370 y(Departamen)n(to)d(de)f
|
|
|
3567 |
(Sistemas)h(Informaticos)h(y)e(Computacion,)i(Univ)n(ersidad)e(P)n
|
|
|
3568 |
(olitecnica)i(de)e(V)-6 b(alencia.)31 b(T)-6 b(ec)n(hnical)151
|
|
|
3569 |
1461 y(Rep)r(ort)25 b(DSIC-I)r(I/12/03.)50 1553 y(2.)42
|
|
|
3570 |
b(Gopalan)32 b(Nadath)n(ur)f(and)g(Dale)h(Miller.)52
|
|
|
3571 |
b(Higher-order)32 b(logic)g(programming.)53 b(In)31 b(D.)g(Gabba)n(y)-6
|
|
|
3572 |
b(,)31 b(C.)h(Hogger,)g(and)151 1644 y(A.)26 b(Robinson,)h(editors,)g
|
|
|
3573 |
Fa(Handb)l(o)l(ok)j(of)e(L)l(o)l(gic)h(in)f(AI)g(and)g(L)l(o)l(gic)h
|
|
|
3574 |
(Pr)l(o)l(gr)l(amming,)g(V)-6 b(olume)28 b(5:)g(L)l(o)l(gic)h(Pr)l(o)l
|
|
|
3575 |
(gr)l(amming)p Fm(.)151 1735 y(Oxford,)d(1986.)50 1826
|
|
|
3576 |
y(3.)42 b(C.)28 b(Urban,)f(A.)g(M.)g(Pitts,)h(and)f(M.)h(J.)g(Gabba)n
|
|
|
3577 |
(y)-6 b(.)38 b(Nominal)27 b(uni\014cation.)39 b(In)27
|
|
|
3578 |
b(M.)g(Baaz,)i(editor,)f Fa(Computer)i(Scienc)l(e)151
|
|
|
3579 |
1918 y(L)l(o)l(gic)i(and)f(8th)h(Kurt)g(Go)l(del)g(Col)t(lo)l(quium)e
|
|
|
3580 |
(\(CSL'03)i(&)f(K)n(GC\))p Fm(,)f(Lecture)g(Notes)g(in)f(Computer)h
|
|
|
3581 |
(Science,)g(Vienna,)151 2009 y(Austria,)c(2003.)i(Springer-V)-6
|
|
|
3582 |
b(erlag.)34 b(T)-6 b(o)26 b(app)r(ear.)p eop end
|
|
|
3583 |
%%Trailer
|
|
|
3584 |
|
|
|
3585 |
userdict /end-hook known{end-hook}if
|
|
|
3586 |
%%EOF
|