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Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 83 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
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 34 (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
ProofSearch in Intuitionistic Logic Based on Constraint Satisfaction
 Theorem Proving with Analytic Tableaux and Related Methods. 5th International Workshop, TABLEAUX '96, volume 1071 of Lecture Notes in Artificial Intelligence
, 1996
"... We characterize provability in intuitionistic predicate logic in terms of derivation skeletons and constraints and study the problem of instantiations of a skeleton to valid derivations. We prove that for two different notions of a skeleton the problem is respectively polynomial and NPcomplete. As ..."
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Cited by 20 (7 self)
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We characterize provability in intuitionistic predicate logic in terms of derivation skeletons and constraints and study the problem of instantiations of a skeleton to valid derivations. We prove that for two different notions of a skeleton the problem is respectively polynomial and NPcomplete. As an application of our technique, we demonstrate PSPACEcompleteness of the prenex fragment of intuitionistic logic. We outline some applications of the proposed technique in automated reasoning. y y Copyright c fl 1995, 1996 Andrei Voronkov. This technical report and other technical reports in this series can be obtained at http://www.csd.uu.se/~thomas/reports.html or at ftp.csd.uu.se in the directory pub/papers/reports. Some reports can be updated, check one of these addresses for the latest version. Section 1 Introduction The characterization of provability for classical logic in terms of matrices was given by Kanger [9, 10] and Prawitz [19, 20] and is in a fact a reformulation of the...
AssociativeCommutative Deduction with Constraints
, 1993
"... Associativecommutative equational reasoning is known to be highly complex for theorem proving. Hence, it is very important to focus deduction by adding constraints, such as unification and ordering, and to define efficient strategies, such as the basic requirements `a la Hullot. Constraints are f ..."
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Cited by 18 (3 self)
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Associativecommutative equational reasoning is known to be highly complex for theorem proving. Hence, it is very important to focus deduction by adding constraints, such as unification and ordering, and to define efficient strategies, such as the basic requirements `a la Hullot. Constraints are formulas used for pruning the set of ground instances of clauses deduced by a theorem prover. We propose here an extension of ACparamodulation and ACsuperposition with these constraint mechanisms ; we do not need to compute ACunifiers anymore. The method is proved to be refutationally complete, even with simplification. The power of this approach is exemplified by a very short proof of the equational version of SAM's Lemma using DATAC, our implementation of the strategy.
Prototyping completion with constraints using computational systems
"... We use computational systems to express a completion with constraints procedure that gives priority to simplifications. Computational systems are rewrite theories enriched by strategies. The implementation of completion in ELAN, an interpretor of computational systems, is especially convenient for e ..."
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Cited by 16 (9 self)
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We use computational systems to express a completion with constraints procedure that gives priority to simplifications. Computational systems are rewrite theories enriched by strategies. The implementation of completion in ELAN, an interpretor of computational systems, is especially convenient for experimenting with different simplification strategies, thanks to the powerful strategy language of ELAN.
Combination of Constraint Solving Techniques: An Algebraic Point of View
 In Proceedings of the 6th International Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science
"... . In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is dec ..."
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Cited by 16 (7 self)
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. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is decidable in the single theoriesis equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras called combinabilitythat clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...
Superposition Theorem Proving for Abelian Groups Represented as Integer Modules
 THEORETICAL COMPUTER SCIENCE
, 1996
"... We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equation ..."
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Cited by 13 (4 self)
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We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the lefthand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the ACunification problems which are generated. That is, ACunification is not necessary at the top of a term, only below some nonACsymbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible.