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Associative-commutative unification

François Fages
1987 Journal of symbolic computation  
Major results in term rewriting systems, as in (Peterson & Stickel, 1981; Hsiang, 1982) , depend on unification in presence of associative-commutative functions.  ...  Sticke [ (1975; 1981) gave an associative-commutative unification algorithm, but its termination in the general case was still questioned The first part of this paper is an introduction to unification  ...  Unification of terms built over only one function symbol which is associative-commutative-with a unit (ACU), or idempotent in addition (ACUI), is studied in (Livesey & Siekmann, 1976) .  ... 
doi:10.1016/s0747-7171(87)80004-4 fatcat:ck5vohfbdnbxzhlu7op3pythfe

Associative-Commutative Unification [chapter]

François Fages
1984 Lecture Notes in Computer Science  
Associative-commutative unification Francois Fages To cite this version: Francois Fages. Associative-commutative unification. [Research Report] RR-0287, INRIA. 1984. <inria-00076271>  ... 
doi:10.1007/978-0-387-34768-4_12 fatcat:sjyhpgldmrfath7saiw46tf5mu

Adventures in associative-commutative unification

Patrick Lincoln, Jim Christian
1989 Journal of symbolic computation  
In most c~es of AC unification our method obviates the need for solving diophantine equations, and tlms avoids one of the bottlenecks of or.her associative-commutative unification techniques.  ...  We have discovered an efficient a.lgorithm for matching and unification in associative-c~mmutative (AC) equational theories.  ...  AC unifical.ion is unification modulo the theory of associativity and commutativity.  ... 
doi:10.1016/s0747-7171(89)80026-4 fatcat:djot6ulbdnbgjiyxoyqqqdjerm

A Unification Algorithm for Associative-Commutative Functions

Mark E. Stickel
1981 Journal of the ACM  
The algorithm is an efficient alternative to other methods of handling associative-commutative functions KEY WORDS AND PHRASES; automated theorem proving, unification CR CATEGORIES: 3 64, 3 66, 5.21  ...  are associative and commutative (such as the arithmetic addition and multiphcatton functions) are often the subject of automated theorem proving An algorithm which unifies terms whose function is associative  ...  unification (ACI unification). 5 .1 ASSOCIATIVE-COMMUTATIVE UNIFICATION WITH IDENTITY.  ... 
doi:10.1145/322261.322262 fatcat:5ksjwloawvfjpntld6he3vz6gi

On the Parameterized Complexity of Associative and Commutative Unification [article]

Tatsuya Akutsu, Takeyuki Tamura, Atsuhiro Takasu
2013 arXiv   pre-print
It is shown that both associative and associative-commutative unification problems are W[1]-hard.  ...  This paper studies the unification problem with associative, commutative, and associative-commutative functions mainly from a viewpoint of the parameterized complexity on the number of variables.  ...  Associative-Commutative Unification Associative-commutative unification is a variant of unification in which some functions can be both associative and commutative.  ... 
arXiv:1310.0919v1 fatcat:4htswqfykfgwtcoh5xxuvyo2ym

On the Parameterized Complexity of Associative and Commutative Unification [chapter]

Tatsuya Akutsu, Jesper Jansson, Atsuhiro Takasu, Takeyuki Tamura
2014 Lecture Notes in Computer Science  
It is shown that both associative and associative-commutative unification problems are W [1]-hard.  ...  This paper studies the unification problem with associative, commutative, and associative-commutative functions mainly from a viewpoint of the parameterized complexity on the number of variables.  ...  Associative-Commutative Unification Associative-commutative unification is a variant of unification in which some functions can be both associative and commutative.  ... 
doi:10.1007/978-3-319-13524-3_2 fatcat:ixcc6hrujfg4vbnqy36oxby72y

An algebraic approach to unification under associativity and commutativity

Albrecht Fortenbacher
1987 Journal of symbolic computation  
From the work of Siekmann & Livesey, and Stickel it is known how to unify two terms in an associative and commutative theory: transfer the terms into Abelian strings, look for mappings which solve the  ...  One possible solution to this problem is to build associativity and commutativity into the matching and unification algorithms.  ...  Introduction Term rewriting systems are unable to handle mathematical theories with associativity and commutativity in the ordinary way.  ... 
doi:10.1016/s0747-7171(87)80001-9 fatcat:amr4ubrmxfgc5f2q3vbzf3aiqq

Associative-commutative unification [chapter]

François Fages
7th International Conference on Automated Deduction  
Associative-commutative unification Francois Fages To cite this version: Francois Fages. Associative-commutative unification. [Research Report] RR-0287, INRIA. 1984. <inria-00076271>  ... 
doi:10.1007/bfb0047122 fatcat:dns4eq7qx5cqnifforheomzwqm

Expectations for associative-commutative unification speedups in a multicomputer environment

R.W. Wilkerson, B.M. McMillin
[1989] Proceedings of the Thirteenth Annual International Computer Software & Applications Conference  
Unlike standard unification, the associative-commutative (AC) unification of two expressions can lead to the formation of many most general unifiers.  ...  One way around these problems is to remove the troublesome axioms from the system and to employ a unification algorithm which unifies modulo the axioms of associativity and commutativity.  ...  Consider the associative and commutative function f.  ... 
doi:10.1109/cmpsac.1989.65077 dblp:conf/compsac/WilkersonM89 fatcat:osslnciolzeqfkc6l7bx2clyia

Matching, unification and complexity

Deepak Kapur, Paliath Narendran
1987 ACM SIGSAM Bulletin  
The associated axiom(s) with each of the symbols is given below. For example, the symbol A implies that some of the function symbols in the terms under consideration are associative.  ...  When function symbols have properties such as associativity, idempotency, etc., both the problems turn out to much harder (in fact, intractable in most cases).  ...  Showing that these problems can be done in NP has been quite easy in general except in the case of associative-commutative (and associative-commutative-unity as well as associative-commutative-idempotent  ... 
doi:10.1145/36330.36332 fatcat:3wwmsylxyfc5xj6ieiynj5o2ja

Page 77 of Computational Linguistics Vol. 25, Issue 1 [page]

1999 Computational Linguistics  
Default unification is order independent if [ is a commutative and associative binary operation.  ...  by [ is commutative and associative.  ... 

Adding homomorphisms to commutative/monoidal theories or how algebra can help in equational unification [chapter]

Franz Baader, Werner Nutt
1991 Lecture Notes in Computer Science  
We identify a large subclass of commutative/monoidal theories that are of unification type zero by studying equations over the corresponding semiring.  ...  As a second result, we show with methods from linear algebra that unitary and finitary commutative/monoidal theories do not change their unification type when they are augmented by a finite monoid of homomorphisms  ...  Since the implicit operation "*" is associative and commutative, the same is true for "+".  ... 
doi:10.1007/3-540-53904-2_91 fatcat:6jzlkhtkxnccfmbmwzr3xccnra

Editorial

Claude Kirchner
1989 Journal of symbolic computation  
It contains important contributions concerning the classification of theories, general combination of unification algorithms, higher-order unification, the logic of unification and the associative-commutative  ...  unification problem.  ...  This allows to use more etl]ciently Stickel's associative-commutative unification algorithm.  ... 
doi:10.1016/s0747-7171(89)80020-3 fatcat:64aknioac5bebphsc3vvcel6me

Complexity of matching problems

Dan Benanav, Deepak Kapur, Paliath Narendran
1987 Journal of symbolic computation  
It turns out that the similar problems of associative-matching and commutative-matching are also N'P-complete.  ...  The associative-commutative matching problem is shown to be NPcomplete; more precisely, the matching problem for terms in which some function symbols are uninterpreted and others are both associative and  ...  matching is a special case of associative-commutative unification as pointed out by Peterson and Stickel (1981) , associative-commutative unification is NP-hard.  ... 
doi:10.1016/s0747-7171(87)80027-5 fatcat:5xoj5vdsnfg37hcyftwc3vec7m

The unification type of the Pythagorean equation in varieties of nilpotent rings

G. Belitskii, R. Lipyanski
2006 Journal of symbolic computation  
We show also that the variety of associative or commutative-associative nilpotent rings of characteristic 2 has nullary unification type.  ...  The unification type of the Pythagorean equation x 2 1 + x 2 2 = x 2 3 in relatively free rings of varieties of n-nilpotent associative or commutative-associative rings is described (n > 2).  ...  Lipyanski (2002) proved that the unification type of 3-(or 4-) nilpotent associative or associative-commutative rings is nullary.  ... 
doi:10.1016/j.jsc.2005.09.002 fatcat:7zzfbwbhzrhm5aepk4l647zwia
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