: We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy. Key-words: Term Rewriting, Program Verification, Normal Forms, Descendants, Tree Automata, Approximation, Sufficient Completeness, Reachability, Termination. (R'esum'e : tsvp) Email: Thomas.Genet@loria.fr, http://www.loria.fr/equipe/protheo.html Unite de recherche INRIA Lorraine Technopole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS L ES NA...

We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semi-linear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.

. In this paper, we propose a method allowing us to compare the result of an execution of a logic program and a specification of the intended semantics. This approach is particularly interesting when the set of answers cannot be computed in finite time with usual prolog interpreters. We compute, using a special operational mechanism, a finite set of rewrite rules synthesizing the whole set of answers w.r.t. a goal. Then, we use some tree tuple grammar based techniques to express the languages of the computed answers. An algorithm allows us to compare this language with the intended semantics language which is extracted from a user's specification. This method can be considered as a partial validation mechanism for logic programs. 1 Introduction Validation, partial validation and verification constitute central issues for full logic programming environment building [5]. By partial validation we mean here the comparison between the operational semantics of the program for a given goal, ...

. We give a decision procedure for the whole existential fragment of one-step rewriting first-order theory, in the case where rewrite systems are linear, non left-left-overlapping (i.e. without critical pairs), and non ffl-left-right-overlapping (i.e. no left-hand-side overlaps on top with the right-hand-side of the same rewrite rule 2 ). The procedure is defined by means of tree-tuple synchronized grammars. 1 Introduction Given a signature \Sigma , the theory of one-step rewriting for a finite rewrite system is the first order theory over the universe of ground \Sigma -terms that uses the only predicate symbol !, where x ! y means x rewrites into y by one step. It has been shown undecidable in [11]. Sharper undecidability results have been obtained for some subclasses of rewrite systems, about the 9 8 -fragment [10, 8] and the 9 8 9 -fragment [12]. It has been shown decidable for the positive existential fragment [9], in the case of unary signatures [3], in the case ...

. This paper gives a procedure for solving disequations modulo equational theories, and to decide existence of solutions. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. The procedure represents the possibly infinite set of solutions thanks to a grammar, and decides existence of solutions thanks to an emptiness test. As a consequence, checking whether a linear equality is an inductive theorem is decidable, if assuming moreover sufficient completeness. 1 Introduction The problem that consists in solving symbolic equations modulo a theory is called equational unification. A lot of work has already studied this subject in a theoretical way to know when the problem can be decided, as well as in a practical way to find efficient algorithms that solve the problem. Another interesting problem consists in solving the negation of equations, called disequations, i.e. in finding ...

We give a set of inference rules for E-unification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore,