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28
Completion Without Failure
, 1989
"... We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational the ..."
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Cited by 141 (21 self)
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We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational theories. The method can also be applied to Horn clauses with equality, in which case it corresponds to positive unit resolution plus oriented paramodulation, with unrestricted simplification.
Equational Reasoning In SaturationBased Theorem Proving
, 1998
"... INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with ..."
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Cited by 38 (2 self)
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INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with equality. We mainly concentrate on refinements of paramodulation, such as the superposition calculus, that have yielded the most promising results to date in automated equational reasoning. We begin with some preliminary material in section 2 and then explain, in section 3, why resolution with the congruence axioms is an impractical theorem proving method for equational logic. In section 4 we outline the main results about paramodulationa more direct equational inference rule. This section also contains a description of the modification method, which can be used to demonstrate that the functional reflexivity axioms are redundant in the context of paramodulation. The modification
Single axioms for groups and Abelian groups with various operations
 Preprint MCSP2701091, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 1991
"... This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In al ..."
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Cited by 34 (11 self)
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This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In all but two of the twelve corresponding theories, we present either the rst single axioms known to us or single axioms shorter than those previously known to us. The automated theoremproving program Otter was used extensively to construct sets of candidate axioms and to search for and nd proofs that given candidate axioms are in fact single axioms. 1
Equational Inference, Canonical Proofs, And Proof Orderings
 Journal of the ACM
, 1992
"... We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a ..."
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Cited by 30 (10 self)
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We describe the application of proof orderingsa technique for reasoning about inference systemsto various rewritebased theoremproving methods, including re#nements of the standard KnuthBendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," coauthored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR8901322, CCR9007195, and CCR9024271. 1 ...
33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies
, 1996
"... Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort &qu ..."
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Cited by 23 (5 self)
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Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easytofind proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ
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 23 (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.
On Using Ground Joinable Equations in Equational Theorem Proving
 PROC. OF THE 3RD FTP, ST. ANDREWS, SCOTTLAND, FACHBERICHTE INFORMATIK. UNIVERSITAT KOBLENZLANDAU
, 2000
"... When rewriting and completion techniques are used for equational theorem proving, the axiom set is saturated with the aim to get a rewrite system that is terminating and confluent on ground terms. To reduce the computational effort it should (1) be powerful for rewriting and (2) create not too ma ..."
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Cited by 12 (2 self)
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When rewriting and completion techniques are used for equational theorem proving, the axiom set is saturated with the aim to get a rewrite system that is terminating and confluent on ground terms. To reduce the computational effort it should (1) be powerful for rewriting and (2) create not too many critical pairs. These problems become especially important if some operators are associative and commutative (AC ). We show in this paper how these two goals can be reached to some extent by using ground joinable equations for simplification purposes and omitting them from the generation of new facts. For the special case of AC operators we present a simple redundancy criterion which is easy to implement, efficient, and effective in practice, leading to significant speedups.
The Decidability of Simultaneous Rigid EUnification with One Variable
 REWRITING TECHNIQUES AND APPLICATIONS
, 1997
"... We show that simultaneous rigid Eunification, or SREU for short, is decidable and in fact EXPTIMEcomplete in the case of one variable. This result implies that the ... fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the ..."
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Cited by 10 (10 self)
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We show that simultaneous rigid Eunification, or SREU for short, is decidable and in fact EXPTIMEcomplete in the case of one variable. This result implies that the ... fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the 99fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is Pcomplete.