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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
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 12 (7 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
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 11 (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.
Oriented Equational Logic Programming is Complete
 JOURNAL OF SYMBOLIC COMPUTATION
, 1996
"... ..."
Strict Basic Superposition
, 1998
"... In this paper we solve a longstanding open problem by showing that strict superpositionthat is, superposition without equality factoringis refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality f ..."
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Cited by 7 (0 self)
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In this paper we solve a longstanding open problem by showing that strict superpositionthat is, superposition without equality factoringis refutationally complete. The difficulty of the problem arises from the fact that the strict calculus, in contrast to the standard calculus with equality factoring, is not compatible with arbitrary removal of tautologies, so that the usual techniques for proving the (refutational) completeness of paramodulation calculi are not directly applicable. We deal with the problem by introducing a suitable notion of direct rewrite proof and modifying proof techniques based on candidate models and counterexamples in that we define these concepts in terms of, not semantic truth, but direct provability. We introduce a corresponding concept of redundancy with which strict superposition is compatible and that covers most simplification techniques. We also show that certain superposition inferences from variables are redundanta result that is relevant, ...
Redundancy Criteria for Constrained Completion
 5th International Conference on Rewriting Techniques and Applications (RTA)', LNCS 690
, 1995
"... This paper studies completion in the case of equations with constraints consisting of firstorder formulae over equations, disequations, and an irreducibility predicate. We present several inference systems which show in a very precise way how to take advantage of redundancy notions in this framewor ..."
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Cited by 5 (1 self)
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This paper studies completion in the case of equations with constraints consisting of firstorder formulae over equations, disequations, and an irreducibility predicate. We present several inference systems which show in a very precise way how to take advantage of redundancy notions in this framework. A notable feature of these systems is the variety of tradeooes they present for removing redundant instances of the equations involved in an inference. The irreducibility predicates simulate redundancy criteria based on reducibility (such as prime superposition and Blocking in Basic Completion) and the disequality predicates simulate the notion of subsumed critical pairs; in addition, since constraints are passed along with equations, we can perform hereditary versions of all these redundancy checks. This combines in one consistent framework stronger versions of all practical critical pair criteria. We also provide a rigorous analysis of the problem with completing sets of equation...
Reasoning and Rewriting with SetRelations I: Ground Completeness
, 1994
"... . The paper investigates reasoning with setrelations: intersection, inclusion and identity of 1element sets. A language is introduced which, interpreted in a multialgebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally groundcompl ..."
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Cited by 4 (2 self)
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. The paper investigates reasoning with setrelations: intersection, inclusion and identity of 1element sets. A language is introduced which, interpreted in a multialgebraic semantics, allows one to specify such relations. An inference system is given and shown sound and refutationally groundcomplete for a particular proof strategy which selects only maximal literals from the premise clauses. Each of the introduced setrelations satisfies only two among the three properties of the equivalence relations  we study rewriting with such nonequivalence relations and point out differences from the equational case. As a corollary of the main groundcompleteness theorem we obtain groundcompleteness of the introduced rewriting technique. 1 Introduction Reasoning with sets becomes an important issue in different areas of computer science. Its relevance can be noticed in constraint and logic programming e.g. [SD86, DO92, Jay92, Sto93], in algebraic approach to nondeterminism e.g. [Hus93, He...
Local Simplification
, 1994
"... We present a modification to the paramodulation inference system, where semantic equality and nonequality literals are stored as local simplifiers with each clause. The local simplifiers are created when new clauses are generated and inherited by the descendants of that clause. Then the local simpl ..."
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Cited by 2 (0 self)
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We present a modification to the paramodulation inference system, where semantic equality and nonequality literals are stored as local simplifiers with each clause. The local simplifiers are created when new clauses are generated and inherited by the descendants of that clause. Then the local simplifiers can be used to perform demodulation and unit simplification, if certain conditions are satisfied. This reduces the search space of the theorem proving procedure and the length of the proofs obtained. In fact, we show that for ground SLD resolution with any selection rule, any set of clauses has a polynomial length proof. Without this technique, proofs may be exponential. We show that this process is sound, complete, and compatible with deletion rules (e.g., demodulation, subsumption, unit simplification, and tautology deletion), which do not have to be modified to preserve completeness. We also show the relationship between this technique and model elimination.