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38
Ordered Chaining Calculi for FirstOrder Theories of Transitive Relations
 Journal of the ACM
, 1998
"... this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994). ..."
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Cited by 33 (4 self)
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this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994).
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 30 (8 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
Ordered Chainings for Total Orderings
, 1995
"... We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with ..."
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Cited by 25 (7 self)
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We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with a concept of (global) redundancy for clauses and inferences that covers such widely used simplification techniques as tautology deletion, subsumption, and demodulation. A key to the practicality of chaining techniques is the extent to which socalled variable chainings can be restricted. Syntactic ordering restrictions on terms and the rewrite techniques which account for their completeness considerably restrict variable chaining. We show that variable elimination is an admissible simplification techniques within our redundancy framework, and that consequently for dense total orderings without endpoints no variable chaining is needed at all.
The Saturate System
, 1998
"... The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into satur ..."
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Cited by 21 (11 self)
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The Saturate system is an experimental theorem prover for firstorder logic, primarily based on saturation. Saturate uses techniques of ordered chaining for arbitrary transitive relations, including orderings, equivalence relations and congruences, and integrates CNF transformation lazily into saturation.
From Total Equational to Partial First Order Logic
, 1998
"... The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to pa ..."
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Cited by 20 (8 self)
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The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and ordersortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...
AssociativeCommutative Superposition
, 1993
"... We present an associativecommutative paramodulation calculus that generalizes the associativecommutative completion procedure to firstorder clauses. The calculus is parametrized by a selection function (on negative literals) and a wellfounded ordering on terms. It is compatible with an abstract ..."
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Cited by 14 (5 self)
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We present an associativecommutative paramodulation calculus that generalizes the associativecommutative completion procedure to firstorder clauses. The calculus is parametrized by a selection function (on negative literals) and a wellfounded ordering on terms. It is compatible with an abstract notion of redundancy that covers such simplification techniques as tautology deletion, subsumption, and simplification by (associativecommutative) rewriting. The proof of refutational completeness of the calculus is comparatively simple, and the techniques employed may be of independent interest.
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 14 (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.
Towards Specifying with Inclusions
, 1997
"... In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalizatio ..."
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Cited by 5 (2 self)
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In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalization of the conventional algebraic specification by means of equations. The main aim of this generalization is to facilitate the incremental refinement of specifications. Furthermore, inclusional specifications admit a natural visual syntax which can also be used to visualize the reasoning process. We show that reasoning with inclusions is well captured by birewriting, a rewriting technique introduced by Levy and Agust'i [15]. However, there are still key problems to be solved in order to have executable inclusional specifications, necessary for rapid prototyping purposes. The article mainly points to the potentialities and difficulties of specifying with inclusions.