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29
Completion for rewriting modulo a congruence
, 1988
"... Abstract. We present completion methods for rewriting modulo a congruence, generalizing previous methods by Peterson and Stickel (1981) and Jouannaud and Kirchner (1986). We formalize our methods as equational inference systems and describe techniques for reasoning about such systems. 1. ..."
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Cited by 31 (6 self)
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Abstract. We present completion methods for rewriting modulo a congruence, generalizing previous methods by Peterson and Stickel (1981) and Jouannaud and Kirchner (1986). We formalize our methods as equational inference systems and describe techniques for reasoning about such systems. 1.
Categorybased Semantics for Equational and Constraint Logic Programming
, 1994
"... This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equation ..."
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Cited by 24 (10 self)
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This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. This is used as a basis for a model theoretic categorybased approach to a paramodulationbased operational semantics for equational logic programming languages. Categorybased equational logic in conjunction with the theory of institutions is used to give mathematical foundations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports i...
Rewrite Methods for Clausal and Nonclausal Theorem Proving
 in Proceedings of the Tenth International Conference on Automata, Languages and Programming
, 1983
"... Effective theorem provers are essential for automatic verification and generation of programs. The conventional resolution strategies, albeit complete, are inefficient. On the other hand, special purpose methods, such as term rewriting systems for solving word problems, are relatively efficient but ..."
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Cited by 21 (10 self)
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Effective theorem provers are essential for automatic verification and generation of programs. The conventional resolution strategies, albeit complete, are inefficient. On the other hand, special purpose methods, such as term rewriting systems for solving word problems, are relatively efficient but applicable to only limited classes of problems. In this paper, a simple canonical set of rewrite rules for Boolean algebra is presented. Based on this set of rules, the notion of term rewriting systems is generalized to provide complete proof strategies for first order predicate calculus. The methods are conceptually simple and can frequently utilize lemmas in proofs. Moreover, when the variables of the predicates involve some domain that has a canonical system, that system can be incorporated as rewrite rules, with the algebraic simplifications being done simultaneously with the merging of clauses. This feature is particularly useful in program verification, data type specification, and programming language design, where axioms can be expressed as equations (rewrite rules). Preliminary results from our implementation indicate that the methods are spaceefficient with respect to the number of rules generated (as compared to the number of resolvents in resolution provers). 2.
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
Any Ground AssociativeCommutative Theory Has a Finite Canonical System
 Proceedings 4th Conference on Rewriting Techniques and Applications
, 1991
"... We show that theories presented by a set of ground equations with several associativecommutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering which is ACcompatible and total on the set of congruence classes generated ..."
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Cited by 18 (4 self)
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We show that theories presented by a set of ground equations with several associativecommutative (AC) symbols always admit a finite canonical system. This result is obtained through the construction of a reduction ordering which is ACcompatible and total on the set of congruence classes generated by the associativity and commutativity axioms. As far as we know, this is the first ordering with such properties, when several AC function symbols and free function symbols are allowed. Such an ordering is also a fundamental tool for deriving complete theorem proving strategies with builtin associative commutative unification.
Paramodulation with Builtin ACTheories and Symbolic Constraints
 Journal of Symbolic Computation
, 1996
"... this paper we overcome these drawbacks by working with clauses with symbolic constraints (Kirchner et al., 1990; Nieuwenhuis and Rubio, 1992; Rubio, 1994; Nieuwenhuis and Rubio, 1995) . A constrained clause C [[ T ]] is a shorthand for the set of ground instances of the clause part C satisfying the ..."
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Cited by 11 (6 self)
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this paper we overcome these drawbacks by working with clauses with symbolic constraints (Kirchner et al., 1990; Nieuwenhuis and Rubio, 1992; Rubio, 1994; Nieuwenhuis and Rubio, 1995) . A constrained clause C [[ T ]] is a shorthand for the set of ground instances of the clause part C satisfying the constraint T . In a constrained equation
Termination and Completion modulo Associativity, Commutativity and Identity
 Theoretical Computer Science
, 1992
"... Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: b ..."
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Cited by 11 (3 self)
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Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: by giving a complete set of rules for completion modulo these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting modulo associativity, commutativity and identity. 1 Introduction Equations are ubiquitous in mathematics and the sciences. Among the most common equations are associativity, commutativity and identity (existence of a neutral element). Rewriting is an efficient way of reasoning with equations, introduced by Knuth and Bendix [12]. When rewriting, equations are used in one direction chosen once and for all. Unfortunately, orientation alone is not a complete inference rule: given a set of equational axioms E, there may be equal terms...
Automated Deduction with Associative Commutative Operators
, 1993
"... We propose a new inference system for automated deduction with equality and associative commutative operators. This system is an extension of the ordered paramodulation strategy. However, rather than using associativity and commutativity as the other axioms, they are handled by the ACunification a ..."
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Cited by 10 (1 self)
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We propose a new inference system for automated deduction with equality and associative commutative operators. This system is an extension of the ordered paramodulation strategy. However, rather than using associativity and commutativity as the other axioms, they are handled by the ACunification algorithm and the inference rules. Moreover, we prove the refutational completeness of this system without needing the functional reflexive axioms or ACaxioms. Such a result is obtained by semantic tree techniques. We also show that the inference system is compatible with simplification rules.
Citius altius fortius: Lessons learned from the Theorem Prover Waldmeister
 Proceedings of the 4th International Workshop on FirstOrder Theorem Proving, number 86.1 in Electronic Notes in Theoretical Computer Science
, 2003
"... In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements Knuth ..."
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Cited by 8 (0 self)
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In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements KnuthBendix completion in its unfailing variant. The system architecture is based on a strict separation of active and passive facts, and is realized via speci cally tailored representations for each of the central data structures: indexing for the active facts, setbased compression for the passive facts, successor sets for the conjectures. In order to cope with large search spaces, specialized redundancy criteria are employed, and the empirically gained control knowledge is integrated to ease the use of the system. We conclude with a discussion of strengths and weaknesses, and a view of future prospects.