Hubert Comon

Completion of Rewrite Systems with Membership Constraints

We consider a constrained equational logic where the constraints are membership conditions t ∈ s where s is interpreted as a regular tree language. Our logic includes a fragment of second order equational logic (without λ-expressions) where second order variables range over regular sets of contexts. The problem with constrained equational logics is the failure of the critical pair lemma. This is the reason why we propose new deduction rules for which the critical pair lemma is restored. Computing critical pairs requires however to solve some constraints in a second-order logic with membership constraints. This is the most difficult result of the paper: we give a terminating set of transformation rules for these formulas, which decides the existence of a solution.

On unification of terms with integer exponents

We consider terms in which some patterns can be repeatedn times.n is an integer variable which is part of the syntax of the terms (and hence may occur more than once in them). We show that unification of such terms is decidable and finitary, extending Chen and Hsiangs result onp-term unification. Finally, extending slightly the syntax yields an undecidable unification problem.We consider terms in which some patterns can be repeatedn times.n is an integer variable which is part of the syntax of the terms (and hence may occur more than once in them). We show that unification of such terms is decidable and finitary, extending Chen and Hsiangs result onp-term unification. Finally, extending slightly the syntax yields an undecidable unification problem.

Complete axiomatizations of some quotient term algebras

We show that T(F)/ =E can be completely axiomatized when =E is a quasi-free theory. Quasi-free theories are a wider class of theories than permutative theories of [Mal71] for which Malcev gave decision results. As an example of application, we show that the first order theory of T(F)/ =E is decidable when E is a set of ground equations. Besides, we prove that the ∑1-fragment of the theory of T(F)/ =E is decidable when E is acompact set of axioms. In particular, the existential fragment of the theory of associative-commutative function symbols is decidable.

Pumping, Cleaning and Symbolic Constraints Solving

We define a new class of tree automata which generalizes both the encompassment automata of [3] and the automata with tests between brothers of [2]. We give a pumping lemma for these automata, which implies that the emptiness of the corresponding language is decidable. Then, we show how to decide emptiness by means of a ”cleaning” algorithm, which leads to more effective decision procedures.

Ordering Constraints on Trees

We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature.

The first-order theory of lexicographic path orderings is undecidable

We show, under some assumption on the signature, that the *This formula not viewable on a Text-Browser* fragment of the theory of any lexicographic path ordering is undecidable. This applies to partial and to total precedences. Our result implies in particular that the simplification rule of ordered completion is undecidable.

Ground Reducibility and Automata with Disequality Constraints

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.

Constraints in Term Algebras: An Overview of Constraint Solving Techniques

We will give a very brief overview on three methods for solving constraints over term algebras, namely formula rewriting, automata techniques and combination techniques. For results which illustrate the specific methods, we give literature pointers (which may be indirect ones, i.e., to more extensive surveys).