AC-Complete Unification and its Application to Theorem Proving

The inefficiency of AC-completion is mainly due to the doubly exponential number of AC-unifiers and thereby of critical pairs generated. We present AC-complete E-unification, a new technique whose goal is to reduce the number of AC-critical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [24, 25]. The idea is to represent complete sets of AC-unifiers by (smaller) sets of E-unifiers. Not only do the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of E-unifiers those solutions which have no E- instance which is an AC-unifier.

[1]  Alexandre Boudet.  Unification dans les melanges de theories equationnelles , 1990 .

[2]  Jean-Pierre Jouannaud,et al.  Termination and Completion Modulo Associativity, Commutativity and Identity , 1992, Theor. Comput. Sci..

[3]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[4]  François Fages.  Associative-Commutative Unification , 1987, J. Symb. Comput..

[5]  Albert Rubio,et al.  AC-Superposition with Constraints: No AC-Unifiers Needed , 1994, CADE.

[6]  Franz Baader,et al.  Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures , 1992, CADE.

[7]  Rüdiger Loos,et al.  Term Reduction Systems and Algebraic Algorithms , 1981, GWAI.

[8]  Hans-Jürgen Bürckert.  Solving Disequations in Equational Theories , 1988, CADE.

[9]  Eric Domenjoud.  AC unification through order-sorted AC1 unification , 1992 .

[10]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[11]  J. Davenport.  Editor , 1960 .

[12]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[13]  Evelyne Contejean,et al.  A new AC unification algorithm with an algorithm for solving systems of diophantine equations , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[14]  Donald E. Knuth,et al.  Simple Word Problems in Universal Algebras††The work reported in this paper was supported in part by the U.S. Office of Naval Research. , 1970 .

[15]  Laurent Vigneron,et al.  Associative-Commutative Deduction with Constraints , 1994, CADE.

[16]  Claude Marché,et al.  Normalized Rewriting: An Alternative to Rewriting Modulo a Set of Equations , 1996, J. Symb. Comput..

[17]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[18]  M. Schmidt-Schauβ.  Unification in a combination of arbitrary disjoint equational theories , 1989 .

[19]  Claude Kirchner,et al.  Solving Equations in Abstract Algebras: A Rule-Based Survey of Unification , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[20]  Wayne Snyder,et al.  Basic Paramodulation and Superposition , 1992, CADE.

[21]  Claude Marché,et al.  Normalised rewriting and normalised completion , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[22]  Paliath Narendran,et al.  Only Prime Superpositions Need be Considered in the Knuth-Bendix Completion Procedure , 1988, J. Symb. Comput..

[23]  Ralph W. Wilkerson,et al.  Complete Sets of Reductions Modulo Associativity, Commutativity and Identity , 1989, RTA.

[24]  Mark E. Stickel,et al.  A Unification Algorithm for Associative-Commutative Functions , 1981, JACM.

[25]  Albert Rubio,et al.  Basic Superposition is Complete , 1992, ESOP.

[26]  Alexandre Boudet.  Combining Unification Algorithms , 1993, J. Symb. Comput..

[27]  Nachum Dershowitz,et al.  Critical Pair Criteria for Completion , 1988, J. Symb. Comput..

[28]  Nachum Dershowitz,et al.  Completion for Rewriting Modulo a Congruence , 1987, Theor. Comput. Sci..

[29]  D. McIlroy.  Algebraic Simplification , 1966, CACM.