Sophie Tison

Regular Tree Languages And Rewrite Systems

We present a collection of results on regular tree languages and rewrite systems. Moreover we prove the undecidability of the preservation of regularity by rewrite systems. More precisely we prove that it is undecidable whether or not for a set E of equations the set E(R) congruence closure of set R is a regular tree language whenever R is a regular tree language. It is equally undecidable whether or not for a confluent and terminating rewrite system S the set S(R) of ground S-normal forms of set R is a regular tree language whenever R is a regular tree language. Finally we study fragments of the theory of ground term algebras modulo congruence generated by a set of equations which can be compiled in a terminating, confluent rewrite system which preserves regularity.

Solving Systems of Set Constraints using Tree Automata

A set constraint is of the form exp1(subseteq)exp2 where exp1 and exp2 are set expressions constructed using variables, function symbols, and the set union, intersection and complement symbols. An algorithm for solving such systems of set constraints was proposed by Aiken and Wimmers [1]. We present a new algorithm for solving this problem. Indeed, we define a new class of tree automata called Tree Set Automata. We prove that, given a system of set constraints, we can associate a tree set automaton such that the set of tuples of tree languages recognized by this automaton is the set of tuples of solutions of the system. We also prove the converse property. Furthermore, if the system has a solution, we prove, in a constructive way, that there is a regular solution (i.e. a tuple of regular tree languages) and a minimal solution and a maximal solution which are actually regular.

Decidability of confluence for ground term rewriting systems

It is well known that confluence (which is equivalent to Church-Rosser property) is undecidable for arbitrary term rewriting systems. We prove here decidability of confluence for ground term rewriting systems. To obtain this result, we construct a special class of finite state tree transducers that we code in recognizable tree languages. Our work illustrates how tree language theory is useful in term rewriting systems study and we give easily some other results in the ground case (as decidability of uniform termination).

Set constraints and automata

We define a new class of automata which is an acceptor model for mappings from the set of terms T? over a ranked alphabet ? into a set E of labels. When E is a set of tuples of binary values, an automaton can be viewed as an acceptor model for n-tuples of tree languages. We prove decidability of emptiness and closure properties for this class of automata. As a consequence of these results, we prove decidability of satisfiability of systems of positive and negative set constraints without projection symbols. We prove the decidability of the satisfiability problem for systems of positive and negative set constraints without projection symbols. Moreover we prove that a non-empty set of solutions always contain a regular solution (i.e., a n-tuple of regular tree languages). We also deduce decidability results for properties of sets of solutions of systems of set constraints.

Fair termination is decidable for ground systems

By summing up, we have reduced the problem of fair termination to the emptiness of the intersection of two constructible and recognizable forests. Since the family of recognizable forests is closed under intersection and since emptiness is decidable in this family, fair termination is decidable.

Some new Decidability Results on Positive and Negative Set Constraints

A positive set constraint is of the form exp 1

Réduction de la non-linéarité des morphismes d'arbres Recognizable tree-languages and non-linear morphisms

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About Connections Between Syntactical and Computational Complexity

exp 2, a negative set constraint is of the form exp 1