Abstract not found

In [6] we have proposed a general higher-order unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higher-order patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an efficient implementation of the abstract algorithm and its generalization to constraint simplification, which has yielded good experimental results at the core of a higher-order constraint logic programming language.

Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate finitely infinite sets of terms. Since schematizations provide a different approach to solve the same kind of problems as constraints do, they can be viewed as a new type of constraints. The paper presents a new recurrent schematization called primal grammars. The main idea behind the primal grammars is to use primitive recursion as the generating engine of infinite sets. The evaluation of primal grammars is based on substitution and rewriting, hence no particular semantics for them is necessary. This fact allows also a natural integration of primal grammars into Prolog, into functional languages or into other rewrite-based applications. Primal grammars have a decidable unification problem and ...

Abstract. 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. 1

We present an extended completion procedure with builtin theories defined by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm for constructing Gröbner bases for polynomial ideals in this formalism. The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's original algorithm for polynomial rings over a field.

The paper presents a framework to describe, experiment and study the combination of different computational systems including the constraint solving paradigm. Computational systems are interpreted in a first-order setting thanks to an evaluator that rewrites formulas.

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.

Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in order-sorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...

In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the meta-language needed to manipulate the constraints. This has the advantage to provide descriptions that are very close to the proof theoretical setting used now to describe constraint manipulations like unification or numerical constraint solving. We examplify the approach by presenting examples of constraint solvers descriptions and combinations written in the ELAN language. 1

. 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...