Roughly speaking, an equational problem is a first order formula whose only predicate symbol is =. We propose some rules for the transformation of equational problems and study their correctness in various models. Then, we give completeness results with respect to some “simple ” problems called solved forms. Such completeness results still hold when adding some control which moreover ensures termination. The termination proofs are given for a “weak ” control and thus hold for the (large) class of algorithms obtained by restricting the scope of the rules. Finally, it must be noted that a by-product of our method is a decision procedure for the validity in the Herbrand Universe of any

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...

We propose a set of transformation rules for first order formulae whose atoms are either equations between terms or "membership constraints" t 2 i. i can be interpreted as a regular tree language (i is called a sort in the algebraic specification community) or as a tree language in any class of languages which satisfies some adequate closure and decidability properties. This set of rules is proved to be correct, terminating and complete. This provides a quantifier elimination procedure: for every regular tree language L, the first order theory of some structure defining L is decidable. This extends the results of Mal'cev (1971), Maher (1988), Comon and Lescanne (1989). We also show how to apply our results to automatic inductive proofs in equational theories. Introduction To unify two terms s and t means to turn the equation s = t into an equivalent solved form which is either ? (this means that s = t has no solution, or, in other words, that s and t are not unifiable) or else a form...

We present a method for compiling pattern matching on lazy languages based on previous work by Laville and Huet-Levy. It consists of coding ambiguous linear sets of patterns using "Term Decomposition," and producing non ambiguous sets over terms with structural constraints on variables. The method can also be applied to strict languages giving a match algorithm that includes only unavoidable tests when such an algorithm exists.

Abstract. It is quite appealing to base the description of pattern-based searches on positive as well as negative conditions. We would like for example to specify that we search for white cars that are not station wagons. To this end, we define the notion of anti-patterns and their semantics along with some of their properties. We then extend the classical notion of matching between patterns and ground terms to matching between anti-patterns and ground terms. We provide a rule-based algorithm that finds the solutions to such problems and prove its correctness and completeness. Anti-pattern matching is by nature different from disunification and quite interestingly the anti-pattern matching problem is unitary. Therefore the concept is appropriate to ground a powerful extension to pattern-based programming languages and we show how this is used to extend the expressiveness and usability of the Tom language. 1

. Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We also consider whether important properties of rewrite systems such as confluence and termination can be proved in a modular way. Finally, we stress the links between rewriting and tree automata. Previous surveys include [21; 18; 37; 12; 45; 46]. The present one owes much to [21]. Keywords. completion, confluence, critical pair, ground reducibility, inductive completion, local confluence, modularity, narrowing, order-sorted algebras, rewrite rule, rewriting, term algebra, termination, tree automata. 1 Introduction The use of equations is traditional in mathematics. Its use in computer science has culminated with the success of algebraic specifications, a method of specifying software by enc...

Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negation-as-failure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higher-order HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.

Using the automata with constraints, we give an algorithm for the decision of ground reducibility of a term t w.r.t. a rewriting system R. The complexity of the algorithm is doubly exponential in the maximum of the depths of t and R and the cardinal of R. Introduction Ground reducibility of a term t w.r.t. a term rewriting system R is the property that all ground instances (i.e. instances without variables) of t are reducible by R. This property, which is also known as "quasi-reducibility" and "inductive reducibility", has been used by several authors for proving properties of algebraic specifications (the sufficient completeness) as well as in inductive theorem proving (see [3, 6, 9--15] among others). Ground reducibility has been shown to be decidable for an arbitrary rewrite system by D. Plaisted [15]. Further decidability proofs where given by Kapur, Narendran and Zhang [11] and Kounalis [13]. These three proofs are based on a "test set" method: they show that there is a bound B(R...

1.45> ? The main contribution of rewrite theory was first to explicitly give some axiomatizations A for which the above method is indeed correct and for which the consistency of E 0 [A is decidable. Several such sets of axioms A have been successively (and successfully) proposed, each of them assuming some strong properties of E. For example, D. Musser [22] assumes that E contains an equational axiomatization of equality, and A is the set ftrue 6= falseg. G. Huet and J.-M. Hullot [15] assume a constructor theory and A expresses that two pure constructor terms are distinct (see also [21]). J.-P. Jouannaud and E. Kounalis [16] and L. Bachmair [1] assume that E is given by a ground-convergent rewrite system (see also [17]), then A expresses that two equal

In this paper, we develop a new approach for mechanizing induction on complex data structures (like bags, sorted lists, trees, powerlists. . . ) by adapting and generalizing works in tree automata with constraints. The key idea of our approach is to compute a tree grammar with constraints which describes the initial model of the given specification. This grammar