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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 higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logics---logics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a general framework---the substitutional framework---for integrating logical deduction and sortal deduction to form a deductive system for sorted logic. This paper also presents results that provide the theoretical underpinnings of the framework. A distinguishing characteristic of a deductive system that is structured according to the substitutional framework is that the sort subsystem is invoked only when the logic subsystem performs unification, and thus sort information is used only in determining what substitutions to make for variables. Unlike every other known approach to sorted deduction, the substitutional framework provides for a systematic transformation of unsorted deductive systems ...

Set-Formers In the practice of mathematics, only seldom a set S is denoted extensionally , that is by enumeration of its elements. Much more often, one provides a condition ' j '[x] that is necessary and sufficient for an element x to belong to S. This intensional denotation of a set is achieved by use of an abstract set-former, whose typical syntactic form is f x : '[x] g. A notation of this kind is very useful in programming languages that embody sets (cf., e.g., [45]). The rest of this section is dedicated to the development of this feature in flogg. A major theoretical difficulty related to the use of the notation f x : '[x] g is that the latter does not make much sense unless one can show that 9 S 8x ( x 2 S $ ' ) follows as a theorem from the axioms of the set theory at hand. No theory of sets can make all formulas of this kind provable, without being inconsistent; this is why convenient syntactic restrictions are usually placed on ', to the effect that whenever a set-form...

. 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 apply---namely that unification with so-called linear constant restrictions is decidable in the single theories---is 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 combinability---that 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...

2.1.1 The Snark language................... 6

This paper exploits the point of view of constraint programming as computation in a logical system, namely constraint logic. We define the basic ingredients of constraint logic, such as constraint models and generalised polynomials. We show that constraint logic is an institution, and we internalise the study of constraint logic to the framework of category-based equational logic. By showing that constraint logic is a special case of category-based equational logic, we integrate the constraint logic programming paradigm into equational logic programming. Results include a Herbrand theorem for constraint logic programming characterising Herbrand models as initial models in constraint logic.

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We describe a refined superposition calculus for cancellative abelian monoids. They encompass not only abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose superposition theorem prover, as they create many variants of clauses which contain sums. Our calculus requires neither explicit inferences with the theory clauses for cancellative abelian monoids nor extended equations or clauses. Improved ordering constraints allow us to restrict to inferences that involve the maximal term of the maximal sum in the maximal literal. Furthermore, the search space is reduced drastically by certain variable elimination techniques. Keywords Automated Theorem Proving, First-Order Logic, Superposition, Cancellative Abelian Monoids, Associativity, Commutativity, Variable Elimination, Term Rewriting. 1 Introduction To be useful in applications such as program verification and synthesis, a...

. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of built-in theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with built-in constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higher-order resolution [16]. The idea is to distinguish two levels of deduction and t...