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Combination Techniques for NonDisjoint Equational Theories
 Proceedings 12th International Conference on Automated Deduction
, 1994
"... ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each ..."
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Cited by 24 (4 self)
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ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each theory may recognize initial, abstraction and introduced variables and never assigns an introduced variable to a nonintroduced one or an abstraction variable to an initial one. With this assumption, our combination algorithm will always make an introduced variable appear in at most one \Gamma i . We may thus also suppose that the domain of each solution does not contain an introduced variable. This does not compromise the soundness of our algorithm. The combination algorithm is described by the two rules given in figure 2. In the rule UnifSolve i , ae SF is obtained by abstracting aliens in the range of ae by fresh variables. ae F i is the substitution such that xae = xae SF ae F i for al...
Unification in a combination of equational theories with shared constants and its application to Primal Algebras
 In Proceedings of the 1st International Conference on Logic Programming and Automated Reasoning, St. Petersburg (Russia), volume 624 of Lecture Notes in Artificial Intelligence
, 1992
"... . We extend the results on combination of disjoint equational theories to combination of equational theories where the only function symbols shared are constants. This is possible because there exist finitely many proper shared terms (the constants) which can be assumed irreducible in any equational ..."
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. We extend the results on combination of disjoint equational theories to combination of equational theories where the only function symbols shared are constants. This is possible because there exist finitely many proper shared terms (the constants) which can be assumed irreducible in any equational proof of the combined theory. We establish a connection between the equational combination framework and a more algebraic one. A unification algorithm provides a symbolic constraint solver in the combination of algebraic structures whose finite domains of values are non disjoint and correspond to constants. Primal algebras are particular finite algebras of practical relevance for manipulating hardware descriptions. 1 Introduction The combination problem for unification can be stated as follows: given two unification algorithms in two (consistent) equational theories E 1 on T (F 1 ; X) and E 2 on T (F 2 ; X), how to design a unification algorithm for E 1 [ E 2 on T (F 1 [ F 2 ; X)? The ge...
A Constraint Solver in Finite Algebras and Its Combination With Unification Algorithms
, 1992
"... In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. A constraint solver in finite algebras is presented for a constraint language including equations, disequations and inequations o ..."
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Cited by 7 (2 self)
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In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. A constraint solver in finite algebras is presented for a constraint language including equations, disequations and inequations on finite domains. The method takes advantage of the embedding of a finite algebra in a primal algebra that can be presented, up to an isomorphism, by an equational presentation. We also show how to combine this constraint solver in finite algebras with other unification algorithms, by extending the techniques used for the combination of unification. 1 Introduction Finite algebras provide valuable domains for constraint logic programming. Unification in this context has attracted considerable interest for its applications: it is of practical relevance for manipulating hardware descriptions and solving formulas of propositional calculus; its implementation in constraint logic programming lan...
ACcomplete Unification and its Application to Theorem Proving
 In Proceedings of the 7th International Conference on Rewriting Techniques and Applications
, 1996
"... . The inefficiency of ACcompletion is mainly due to the doubly exponential number of ACunifiers and thereby of critical pairs generated. We present ACcomplete Eunification, a new technique whose goal is to reduce the number of ACcritical pairs inferred by performing unification in a extension E ..."
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Cited by 5 (0 self)
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. The inefficiency of ACcompletion is mainly due to the doubly exponential number of ACunifiers and thereby of critical pairs generated. We present ACcomplete Eunification, a new technique whose goal is to reduce the number of ACcritical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [21]. The idea is to represent complete sets of ACunifiers by (smaller) sets of Eunifiers. Not only the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of Eunifiers those solutions which have no Einstance which is an ACunifier. First, we define ACcomplete Eunification and describe its fundamental properties. We show how ACcomplete Eunification can be done in the elementary case, and how the known combination techniques for unification algorithms can be reused for our purposes. Finally, we give some evidence of t...
ACUnification of Higherorder Patterns
 IN G. SMOLKA (ED), PROC. PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING  CP'97, LECTURE
, 1997
"... We present a complete algorithm for the unification of higherorder patterns modulo the associativecommutative theory of some constants +1 ; : : : ; +n . Given an ACunification problem over higherorder patterns, the output of the algorithm is a finite set DAG solved forms [9], constrained by some ..."
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We present a complete algorithm for the unification of higherorder patterns modulo the associativecommutative theory of some constants +1 ; : : : ; +n . Given an ACunification problem over higherorder patterns, the output of the algorithm is a finite set DAG solved forms [9], constrained by some flexibleflexible equations with the same head on both sides. Indeed, in the presence of AC constants, such equations are always solvable, but they have no minimal complete set of unifiers [13]. We prove that the algorithm terminates, is sound, and that any solution of the original unification problem is an instance of one of the computed solutions which satisfies the constraints.
Term Rewriting In Associative Commutative Theories With Identities
, 1991
"... of the Thesis Term Rewriting in Associative Commutative Theories with Identities by Martin Joachim Henz Master of Science in Computer Science State University of New York at Stony Brook 1991 Versions of constraint rewriting for completion of rewrite systems in the presence of associative commutative ..."
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of the Thesis Term Rewriting in Associative Commutative Theories with Identities by Martin Joachim Henz Master of Science in Computer Science State University of New York at Stony Brook 1991 Versions of constraint rewriting for completion of rewrite systems in the presence of associative commutative operators with identities have been proposed, in which constraints are used to limit the applicability of rewrite rules. We extend these approaches such that the initially given equations can contain constraints, and such that a suitable version of unification modulo associativity, commutativity and identity can be interleaved with the process of completion. iii To my parents Albert and Klara Henz and my wife Kelly Reedy. Contents Abstract iii Acknowledgements ix 1 Introduction 1 2 Preliminaries 4 2.1 Terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.3 The Associative Commutative Theory wi...