The unification problem in first-order predicate calculus is described in general terms as the solution of a system of equations, and a nondeterministic algorithm is given. A new unification algorithm, characterized by having the acyclicity test efficiently embedded into it, is derived from the nondeterministic one, and a PASCAL implementation is given. A comparison with other well-known unification algorithms shows that the algorithm described here performs well in all cases.

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. In this article we show that the three equations known as commutativity, associativity, and the Robbins equation are a basis for the variety of Boolean algebras. The problem was posed by Herbert Robbins in the 1930s. The proof was found automatically by EQP, a theorem-proving program for equational logic. We present the proof and the search strategies that enabled the program to find the proof. Key words: Associative-commutative unification, Boolean algebra, EQP, equational logic, paramodulation, Robbins algebra, Robbins problem. 1. Introduction This article contains the answer to the Robbins question of whether all Robbins algebras are Boolean. The answer is yes, all Robbins algebras are Boolean. The proof that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra [6, 5]: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativit...

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An informal tutorial is presented for program synthesis, with an emphasis on deductive methods. According to this approach, to construct a program meeting a given specification, we prove the existence of an object meeting the specified conditions. The proof is restricted to be sufficiently constructive, in the sense that, in establishing the existence of the desired output, the proof is forced to indicate a computational method for finding it. That method becomes the basis for a program that can be extracted from the proof. The exposition is based on the deductive-tableau system, a theorem-proving framework particularly suitable for program synthesis. The system includes a nonclausal resolution rule, facilities for reasoning about equality, and a well-founded induction rule. INTRODUCTION This is an introduction to program synthesis, the derivation of a program to meet a given specification. It focuses on the deductive approach, in which the derivation task is regarded as a problem of ...

Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de Paris-Sud, 91405 ORSAY cedex, France. E-mail: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...

This paper presents a new notion of typing for logic programs which generalizes the notion of directional types. The generation of type dependencies for a logic program is fully automatic with respect to a given domain of types. The analysis method is based on a novel combination of program abstraction and ACI-unification which is shown to be correct and optimal. Type dependencies are obtained by abstracting programs, replacing concrete terms by their types, and evaluating the meaning of the abstract programs using a standard semantics for logic programs enhanced by ACI-unification. This approach is generic and can be used with any standard semantics. The method is both theoretically clean and easy to implement using general purpose tools. The proposed domain of types is condensing which means that analyses can be carried out in both top-down or bottom-up frameworks with no loss of precision for goal-independent analyses. The proposed method has been fully implemented within a bottom-up approach and the experimental results are promising.

this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction

A completion procedure for computing a canonical basis for a k-subalgebra is proposed. Using this canonical basis, the membership problem for a k-subalgebra can be solved. The approach follows Buchberger's approach for computing a Gröbner basis for a polynomial ideal and is based on rewriting concepts. A canonical basis produced by the completion procedure shares many properties of a Grobner basis such as reducing an element of a k-subalgebra to 0 and generating unique normal forms for the equivalence classes generated by a k-subalgebra. In contrast to Shannon and Sweedler's approach using tag variables, this approach is direct. One of the limitations of the approach however is that the procedure may not terminate for some term orderings thus giving an infinite canonical basis. The procedure is illustrated using examples.

Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easy-to-find proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ