1/* Part of SWI-Prolog 2 3 Author: Jan Wielemaker and Jon Jagger 4 E-mail: J.Wielemaker@vu.nl 5 WWW: http://www.swi-prolog.org 6 Copyright (c) 2001-2021, University of Amsterdam 7 VU University Amsterdam 8 SWI-Prolog Solutions b.v. 9 All rights reserved. 10 11 Redistribution and use in source and binary forms, with or without 12 modification, are permitted provided that the following conditions 13 are met: 14 15 1. Redistributions of source code must retain the above copyright 16 notice, this list of conditions and the following disclaimer. 17 18 2. Redistributions in binary form must reproduce the above copyright 19 notice, this list of conditions and the following disclaimer in 20 the documentation and/or other materials provided with the 21 distribution. 22 23 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS 24 "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT 25 LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS 26 FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE 27 COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, 28 INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, 29 BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 30 LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER 31 CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 32 LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN 33 ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 34 POSSIBILITY OF SUCH DAMAGE. 35*/ 36 37:- module(ordsets, 38 [ is_ordset/1, % @Term 39 list_to_ord_set/2, % +List, -OrdSet 40 ord_add_element/3, % +Set, +Element, -NewSet 41 ord_del_element/3, % +Set, +Element, -NewSet 42 ord_selectchk/3, % +Item, ?Set1, ?Set2 43 ord_intersect/2, % +Set1, +Set2 (test non-empty) 44 ord_intersect/3, % +Set1, +Set2, -Intersection 45 ord_intersection/3, % +Set1, +Set2, -Intersection 46 ord_intersection/4, % +Set1, +Set2, -Intersection, -Diff 47 ord_disjoint/2, % +Set1, +Set2 48 ord_subtract/3, % +Set, +Delete, -Remaining 49 ord_union/2, % +SetOfOrdSets, -Set 50 ord_union/3, % +Set1, +Set2, -Union 51 ord_union/4, % +Set1, +Set2, -Union, -New 52 ord_subset/2, % +Sub, +Super (test Sub is in Super) 53 % Non-Quintus extensions 54 ord_empty/1, % ?Set 55 ord_memberchk/2, % +Element, +Set, 56 ord_symdiff/3, % +Set1, +Set2, ?Diff 57 % SICSTus extensions 58 ord_seteq/2, % +Set1, +Set2 59 ord_intersection/2 % +PowerSet, -Intersection 60 ]). 61:- use_module(library(error)). 62 63:- set_prolog_flag(generate_debug_info, false). 64 65/** <module> Ordered set manipulation 66 67Ordered sets are lists with unique elements sorted to the standard order 68of terms (see sort/2). Exploiting ordering, many of the set operations 69can be expressed in order N rather than N^2 when dealing with unordered 70sets that may contain duplicates. The library(ordsets) is available in a 71number of Prolog implementations. Our predicates are designed to be 72compatible with common practice in the Prolog community. The 73implementation is incomplete and relies partly on library(oset), an 74older ordered set library distributed with SWI-Prolog. New applications 75are advised to use library(ordsets). 76 77Some of these predicates match directly to corresponding list 78operations. It is advised to use the versions from this library to make 79clear you are operating on ordered sets. An exception is member/2. See 80ord_memberchk/2. 81 82The ordsets library is based on the standard order of terms. This 83implies it can handle all Prolog terms, including variables. Note 84however, that the ordering is not stable if a term inside the set is 85further instantiated. Also note that variable ordering changes if 86variables in the set are unified with each other or a variable in the 87set is unified with a variable that is `older' than the newest variable 88in the set. In practice, this implies that it is allowed to use 89member(X, OrdSet) on an ordered set that holds variables only if X is a 90fresh variable. In other cases one should cease using it as an ordset 91because the order it relies on may have been changed. 92*/ 93 94%! is_ordset(@Term) is semidet. 95% 96% True if Term is an ordered set. All predicates in this library 97% expect ordered sets as input arguments. Failing to fullfil this 98% assumption results in undefined behaviour. Typically, ordered 99% sets are created by predicates from this library, sort/2 or 100% setof/3. 101 102is_ordset(Term) :- 103 is_list(Term), 104 is_ordset2(Term). 105 106is_ordset2([]). 107is_ordset2([H|T]) :- 108 is_ordset3(T, H). 109 110is_ordset3([], _). 111is_ordset3([H2|T], H) :- 112 H2 @> H, 113 is_ordset3(T, H2). 114 115 116%! ord_empty(?List) is semidet. 117% 118% True when List is the empty ordered set. Simply unifies list 119% with the empty list. Not part of Quintus. 120 121ord_empty([]). 122 123 124%! ord_seteq(+Set1, +Set2) is semidet. 125% 126% True if Set1 and Set2 have the same elements. As both are 127% canonical sorted lists, this is the same as ==/2. 128% 129% @compat sicstus 130 131ord_seteq(Set1, Set2) :- 132 Set1 == Set2. 133 134 135%! list_to_ord_set(+List, -OrdSet) is det. 136% 137% Transform a list into an ordered set. This is the same as 138% sorting the list. 139 140list_to_ord_set(List, Set) :- 141 sort(List, Set). 142 143 144%! ord_intersect(+Set1, +Set2) is semidet. 145% 146% True if both ordered sets have a non-empty intersection. 147 148ord_intersect([H1|T1], L2) :- 149 ord_intersect_(L2, H1, T1). 150 151ord_intersect_([H2|T2], H1, T1) :- 152 compare(Order, H1, H2), 153 ord_intersect__(Order, H1, T1, H2, T2). 154 155ord_intersect__(<, _H1, T1, H2, T2) :- 156 ord_intersect_(T1, H2, T2). 157ord_intersect__(=, _H1, _T1, _H2, _T2). 158ord_intersect__(>, H1, T1, _H2, T2) :- 159 ord_intersect_(T2, H1, T1). 160 161 162%! ord_disjoint(+Set1, +Set2) is semidet. 163% 164% True if Set1 and Set2 have no common elements. This is the 165% negation of ord_intersect/2. 166 167ord_disjoint(Set1, Set2) :- 168 \+ ord_intersect(Set1, Set2). 169 170 171%! ord_intersect(+Set1, +Set2, -Intersection) 172% 173% Intersection holds the common elements of Set1 and Set2. 174% 175% @deprecated Use ord_intersection/3 176 177ord_intersect(Set1, Set2, Intersection) :- 178 ord_intersection(Set1, Set2, Intersection). 179 180 181%! ord_intersection(+PowerSet, -Intersection) 182% 183% Intersection of a powerset. True when Intersection is an ordered 184% set holding all elements common to all sets in PowerSet. 185% 186% @compat sicstus 187 188ord_intersection(PowerSet, Intersection) :- 189 must_be(list, PowerSet), 190 key_by_length(PowerSet, Pairs), 191 keysort(Pairs, [_-S|Sorted]), 192 l_int(Sorted, S, Intersection). 193 194key_by_length([], []). 195key_by_length([H|T0], [L-H|T]) :- 196 '$skip_list'(L, H, Tail), 197 ( Tail == [] 198 -> key_by_length(T0, T) 199 ; type_error(list, H) 200 ). 201 202l_int(_, [], I) => 203 I = []. 204l_int([], S, I) => 205 I = S. 206l_int([_-H|T], S0, S) => 207 ord_intersection(S0, H, S1), 208 l_int(T, S1, S). 209 210 211%! ord_intersection(+Set1, +Set2, -Intersection) is det. 212% 213% Intersection holds the common elements of Set1 and Set2. Uses 214% ord_disjoint/2 if Intersection is bound to `[]` on entry. 215 216ord_intersection(Set1, Set2, Intersection) :- 217 ( Intersection == [] 218 -> ord_disjoint(Set1, Set2) 219 ; ord_intersection_(Set1, Set2, Intersection) 220 ). 221 222ord_intersection_([], _Int, []). 223ord_intersection_([H1|T1], L2, Int) :- 224 isect2(L2, H1, T1, Int). 225 226isect2([], _H1, _T1, []). 227isect2([H2|T2], H1, T1, Int) :- 228 compare(Order, H1, H2), 229 isect3(Order, H1, T1, H2, T2, Int). 230 231isect3(<, _H1, T1, H2, T2, Int) :- 232 isect2(T1, H2, T2, Int). 233isect3(=, H1, T1, _H2, T2, [H1|Int]) :- 234 ord_intersection_(T1, T2, Int). 235isect3(>, H1, T1, _H2, T2, Int) :- 236 isect2(T2, H1, T1, Int). 237 238 239%! ord_intersection(+Set1, +Set2, ?Intersection, ?Difference) is det. 240% 241% Intersection and difference between two ordered sets. 242% Intersection is the intersection between Set1 and Set2, while 243% Difference is defined by ord_subtract(Set2, Set1, Difference). 244% 245% @see ord_intersection/3 and ord_subtract/3. 246 247ord_intersection([], L, [], L) :- !. 248ord_intersection([_|_], [], [], []) :- !. 249ord_intersection([H1|T1], [H2|T2], Intersection, Difference) :- 250 compare(Diff, H1, H2), 251 ord_intersection2(Diff, H1, T1, H2, T2, Intersection, Difference). 252 253ord_intersection2(=, H1, T1, _H2, T2, [H1|T], Difference) :- 254 ord_intersection(T1, T2, T, Difference). 255ord_intersection2(<, _, T1, H2, T2, Intersection, Difference) :- 256 ord_intersection(T1, [H2|T2], Intersection, Difference). 257ord_intersection2(>, H1, T1, H2, T2, Intersection, [H2|HDiff]) :- 258 ord_intersection([H1|T1], T2, Intersection, HDiff). 259 260 261%! ord_add_element(+Set1, +Element, ?Set2) is det. 262% 263% Insert an element into the set. This is the same as 264% ord_union(Set1, [Element], Set2). 265 266ord_add_element([], El, [El]). 267ord_add_element([H|T], El, Add) :- 268 compare(Order, H, El), 269 addel(Order, H, T, El, Add). 270 271addel(<, H, T, El, [H|Add]) :- 272 ord_add_element(T, El, Add). 273addel(=, H, T, _El, [H|T]). 274addel(>, H, T, El, [El,H|T]). 275 276 277 278%! ord_del_element(+Set, +Element, -NewSet) is det. 279% 280% Delete an element from an ordered set. This is the same as 281% ord_subtract(Set, [Element], NewSet). 282 283ord_del_element([], _El, []). 284ord_del_element([H|T], El, Del) :- 285 compare(Order, H, El), 286 delel(Order, H, T, El, Del). 287 288delel(<, H, T, El, [H|Del]) :- 289 ord_del_element(T, El, Del). 290delel(=, _H, T, _El, T). 291delel(>, H, T, _El, [H|T]). 292 293 294%! ord_selectchk(+Item, ?Set1, ?Set2) is semidet. 295% 296% Selectchk/3, specialised for ordered sets. Is true when 297% select(Item, Set1, Set2) and Set1, Set2 are both sorted lists 298% without duplicates. This implementation is only expected to work 299% for Item ground and either Set1 or Set2 ground. The "chk" suffix 300% is meant to remind you of memberchk/2, which also expects its 301% first argument to be ground. ord_selectchk(X, S, T) => 302% ord_memberchk(X, S) & \+ ord_memberchk(X, T). 303% 304% @author Richard O'Keefe 305 306ord_selectchk(Item, [X|Set1], [X|Set2]) :- 307 X @< Item, 308 !, 309 ord_selectchk(Item, Set1, Set2). 310ord_selectchk(Item, [Item|Set1], Set1) :- 311 ( Set1 == [] 312 -> true 313 ; Set1 = [Y|_] 314 -> Item @< Y 315 ). 316 317 318%! ord_memberchk(+Element, +OrdSet) is semidet. 319% 320% True if Element is a member of OrdSet, compared using ==. Note 321% that _enumerating_ elements of an ordered set can be done using 322% member/2. 323% 324% Some Prolog implementations also provide ord_member/2, with the 325% same semantics as ord_memberchk/2. We believe that having a 326% semidet ord_member/2 is unacceptably inconsistent with the *_chk 327% convention. Portable code should use ord_memberchk/2 or 328% member/2. 329% 330% @author Richard O'Keefe 331 332ord_memberchk(Item, [X1,X2,X3,X4|Xs]) :- 333 !, 334 compare(R4, Item, X4), 335 ( R4 = (>) -> ord_memberchk(Item, Xs) 336 ; R4 = (<) -> 337 compare(R2, Item, X2), 338 ( R2 = (>) -> Item == X3 339 ; R2 = (<) -> Item == X1 340 ;/* R2 = (=), Item == X2 */ true 341 ) 342 ;/* R4 = (=) */ true 343 ). 344ord_memberchk(Item, [X1,X2|Xs]) :- 345 !, 346 compare(R2, Item, X2), 347 ( R2 = (>) -> ord_memberchk(Item, Xs) 348 ; R2 = (<) -> Item == X1 349 ;/* R2 = (=) */ true 350 ). 351ord_memberchk(Item, [X1]) :- 352 Item == X1. 353 354 355%! ord_subset(+Sub, +Super) is semidet. 356% 357% Is true if all elements of Sub are in Super 358 359ord_subset([], _). 360ord_subset([H1|T1], [H2|T2]) :- 361 compare(Order, H1, H2), 362 ord_subset_(Order, H1, T1, T2). 363 364ord_subset_(>, H1, T1, [H2|T2]) :- 365 compare(Order, H1, H2), 366 ord_subset_(Order, H1, T1, T2). 367ord_subset_(=, _, T1, T2) :- 368 ord_subset(T1, T2). 369 370 371%! ord_subtract(+InOSet, +NotInOSet, -Diff) is det. 372% 373% Diff is the set holding all elements of InOSet that are not in 374% NotInOSet. 375 376ord_subtract([], _Not, Diff) => 377 Diff = []. 378ord_subtract(List, [], Diff) => 379 Diff = List. 380ord_subtract([H1|T1], L2, Diff) => 381 diff21(L2, H1, T1, Diff). 382 383diff21([], H1, T1, [H1|T1]). 384diff21([H2|T2], H1, T1, Diff) :- 385 compare(Order, H1, H2), 386 diff3(Order, H1, T1, H2, T2, Diff). 387 388diff12([], _H2, _T2, []). 389diff12([H1|T1], H2, T2, Diff) :- 390 compare(Order, H1, H2), 391 diff3(Order, H1, T1, H2, T2, Diff). 392 393diff3(<, H1, T1, H2, T2, [H1|Diff]) :- 394 diff12(T1, H2, T2, Diff). 395diff3(=, _H1, T1, _H2, T2, Diff) :- 396 ord_subtract(T1, T2, Diff). 397diff3(>, H1, T1, _H2, T2, Diff) :- 398 diff21(T2, H1, T1, Diff). 399 400 401%! ord_union(+SetOfSets, -Union) is det. 402% 403% True if Union is the union of all elements in the superset 404% SetOfSets. Each member of SetOfSets must be an ordered set, the 405% sets need not be ordered in any way. 406% 407% @author Copied from YAP, probably originally by Richard O'Keefe. 408 409ord_union([], Union) => 410 Union = []. 411ord_union([Set|Sets], Union) => 412 length([Set|Sets], NumberOfSets), 413 ord_union_all(NumberOfSets, [Set|Sets], Union, []). 414 415ord_union_all(N, Sets0, Union, Sets) => 416 ( N =:= 1 417 -> Sets0 = [Union|Sets] 418 ; N =:= 2 419 -> Sets0 = [Set1,Set2|Sets], 420 ord_union(Set1,Set2,Union) 421 ; A is N>>1, 422 Z is N-A, 423 ord_union_all(A, Sets0, X, Sets1), 424 ord_union_all(Z, Sets1, Y, Sets), 425 ord_union(X, Y, Union) 426 ). 427 428 429%! ord_union(+Set1, +Set2, -Union) is det. 430% 431% Union is the union of Set1 and Set2 432 433ord_union([], Set2, Union) => 434 Union = Set2. 435ord_union([H1|T1], L2, Union) => 436 union2(L2, H1, T1, Union). 437 438union2([], H1, T1, Union) => 439 Union = [H1|T1]. 440union2([H2|T2], H1, T1, Union) => 441 compare(Order, H1, H2), 442 union3(Order, H1, T1, H2, T2, Union). 443 444union3(<, H1, T1, H2, T2, Union) => 445 Union = [H1|Union0], 446 union2(T1, H2, T2, Union0). 447union3(=, H1, T1, _H2, T2, Union) => 448 Union = [H1|Union0], 449 ord_union(T1, T2, Union0). 450union3(>, H1, T1, H2, T2, Union) => 451 Union = [H2|Union0], 452 union2(T2, H1, T1, Union0). 453 454%! ord_union(+Set1, +Set2, -Union, -New) is det. 455% 456% True iff ord_union(Set1, Set2, Union) and 457% ord_subtract(Set2, Set1, New). 458 459ord_union([], Set2, Set2, Set2). 460ord_union([H|T], Set2, Union, New) :- 461 ord_union_1(Set2, H, T, Union, New). 462 463ord_union_1([], H, T, [H|T], []). 464ord_union_1([H2|T2], H, T, Union, New) :- 465 compare(Order, H, H2), 466 ord_union(Order, H, T, H2, T2, Union, New). 467 468ord_union(<, H, T, H2, T2, [H|Union], New) :- 469 ord_union_2(T, H2, T2, Union, New). 470ord_union(>, H, T, H2, T2, [H2|Union], [H2|New]) :- 471 ord_union_1(T2, H, T, Union, New). 472ord_union(=, H, T, _, T2, [H|Union], New) :- 473 ord_union(T, T2, Union, New). 474 475ord_union_2([], H2, T2, [H2|T2], [H2|T2]). 476ord_union_2([H|T], H2, T2, Union, New) :- 477 compare(Order, H, H2), 478 ord_union(Order, H, T, H2, T2, Union, New). 479 480 481%! ord_symdiff(+Set1, +Set2, ?Difference) is det. 482% 483% Is true when Difference is the symmetric difference of Set1 and 484% Set2. I.e., Difference contains all elements that are not in the 485% intersection of Set1 and Set2. The semantics is the same as the 486% sequence below (but the actual implementation requires only a 487% single scan). 488% 489% == 490% ord_union(Set1, Set2, Union), 491% ord_intersection(Set1, Set2, Intersection), 492% ord_subtract(Union, Intersection, Difference). 493% == 494% 495% For example: 496% 497% == 498% ?- ord_symdiff([1,2], [2,3], X). 499% X = [1,3]. 500% == 501 502ord_symdiff([], Set2, Set2). 503ord_symdiff([H1|T1], Set2, Difference) :- 504 ord_symdiff(Set2, H1, T1, Difference). 505 506ord_symdiff([], H1, T1, [H1|T1]). 507ord_symdiff([H2|T2], H1, T1, Difference) :- 508 compare(Order, H1, H2), 509 ord_symdiff(Order, H1, T1, H2, T2, Difference). 510 511ord_symdiff(<, H1, Set1, H2, T2, [H1|Difference]) :- 512 ord_symdiff(Set1, H2, T2, Difference). 513ord_symdiff(=, _, T1, _, T2, Difference) :- 514 ord_symdiff(T1, T2, Difference). 515ord_symdiff(>, H1, T1, H2, Set2, [H2|Difference]) :- 516 ord_symdiff(Set2, H1, T1, Difference)