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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 75 (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.
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 46 (18 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
Proving FirstOrder Equality Theorems with HyperLinking
, 1995
"... Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving p ..."
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Cited by 2 (0 self)
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Lee and Plaisted recently developed a new automated theorem proving strategy called hyperlinking. As part of his dissertation, Lee developed a roundbyround implementation of the hyperlinking strategy, which competes well with other automated theorem provers on a wide range of theorem proving problems. However, Lee's roundbyround implementation of hyperlinking is not particularly well suited for the addition of special methods in support of equality. In this dissertation, we describe, as alternative to the roundbyround hyperlinking implementation of Lee, a smallest instance first implementation of hyperlinking which addresses many of the inefficiencies of roundbyround hyperlinking encountered when adding special methods in support of equality. Smallest instance first hyperlinking is based on the formalization of generating smallest clauses first, a heuristic widely used in other automated theorem provers. We prove both the soundness and logical completeness of smallest instance first hyperlinking and show that it always generates smallest clauses first under
Killer Transformations
, 1994
"... This paper deals with methods of faithful transformations between logical systems. Several methods for developing transformations of logical formulae are defined which eliminate unwanted properties from axiom systems without losing theorems. The elementary examples we present are permutation, transi ..."
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Cited by 2 (2 self)
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This paper deals with methods of faithful transformations between logical systems. Several methods for developing transformations of logical formulae are defined which eliminate unwanted properties from axiom systems without losing theorems. The elementary examples we present are permutation, transitivity, equivalence relation properties of predicates and congruence properties of functions. Various translations between logical systems are shown to be instances of Ktransformations, for example the transition from relational to functional translation of modal logic into predicate logic, the transition from axiomatic specifications of logics via unary provability relations to a binary consequence relations, and the development of neighbourhood semantics for nonclassical propositional logics. Furthermore we show how to eliminate self resolving clauses like the condensed detachment clause, resulting in dramatic improvements of the performance of automated theorem provers on extremely hard ...
The Use of Semantics in InstanceBased Proof Procedures
, 1994
"... . We survey our past work in theorem proving, leading up to the semantic hyperlinking method, and clause linking with semantics. We highlight some basic problems in automated deduction, and show that significant progress has been made. We present our belief that these problems are about to be solve ..."
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Cited by 2 (0 self)
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. We survey our past work in theorem proving, leading up to the semantic hyperlinking method, and clause linking with semantics. We highlight some basic problems in automated deduction, and show that significant progress has been made. We present our belief that these problems are about to be solved completely. We present some strategies that we believe will accomplish this. 1 Introduction During CADE12, a tutorial was held entitled "The Use of Semantics in a HerbrandBased Proof Procedure," in which the author, as well as Heng Chu and ShieJue Lee, participated. The author gave a brief survey talk, then ShieJue Lee described his clause linking theorem prover, and finally Heng Chu described clause linking with semantics. Both of these provers are instance (Herbrand) based, that is, they generate a set of ground instances of the input clauses, and then apply propositional methods to detect unsatisfiability. In this they recall some of the very early, preresolution approaches to th...
Approved by:
, 1997
"... ALL RIGHTS RESERVEDEarly refutational theorem proving procedures were direct applications of Herbrand's version of the completeness theorem for firstorder logic. These instancebased theorem provers created propositional instances of the firstorder clauses to be proved unsatisfiable, and tested th ..."
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ALL RIGHTS RESERVEDEarly refutational theorem proving procedures were direct applications of Herbrand's version of the completeness theorem for firstorder logic. These instancebased theorem provers created propositional instances of the firstorder clauses to be proved unsatisfiable, and tested the instances on a propositional calculus prover. This methodology was not pursued for several decades as it was thought to be too slow. Moreover, the invention of the resolution inference rule changed the direction of theorem proving forever. The success of resolution was largely due to unification. Recently, unification has been incorporated in creating instances of firstorder clauses. Furthermore, highperformance propositional calculus provers have been developed in the past few years. As a result, it is possible to realize effective instancebased firstorder methods for several applications. We describe the design of instancebased methods for three different