## Section: New Results

### Automata theory

Participants : Florent Jacquemard, Thomas Place, Luc Segoufin, Camille Vacher.

The links between models for XML and regular tree languages has been advocated in many places. Tree automata seem to be playing for semi-structured data and XML the role of the relational algebra for relational databases. As XML is central in our research we also study tree automata and regular tree languages.

A first line of research concerns the expressive power of various subclasses of
regular tree languages. It is usually admitted that a fragment is completely
understood, in term of expressive power, when one has a *decidable
characterization* of it. That is an algorithm that given a regular tree
language, presented say as a tree automata, tests whether it belongs to the
class being investigated or not. This question is an active research topic that
turns out to be quite challenging. A regular tree language L is said to be locally
testable if membership of a tree into L depends only on the presence or absence
of some neighborhoods in the tree. In [32] we have shown that it is decidable
whether a regular tree language is locally testable.

We have also considered superclasses of regular tree languages, described by tree automata with features which extend strictly standard tree automata. This is the case of Rigid Tree Automata (RTA), an extension of standard bottom-up tree automata with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA tests for equality of subtrees reaching the same rigid state. In [30] , we have studied the expressiveness of these automata and properties like determinism, pumping lemma, Boolean closure, and several decision problems. Our main result is the decidability of whether a given tree belongs to the rewrite closure of a RTA language under a restricted family of term rewriting systems, whereas this closure is not a RTA language.

We have obtained some other results concerning the transformation of tree
automata languages under various kind of rewriting systems.
In [28] , we show that the transformation of a tree automata
language obtained by application shallow rewrite rules following an innermost
strategy (such strategy corresponds to the *call by value* computation of
programming languages) can be recognized by a tree automaton with equality and
disequality constraints between brothers. This latter class of automata is
another strict extension of tree automata, with the ability to perform some
tests of isomorphism between subtree during computations.
We have also considered the property of unique normalization (UN),
which states that, starting from any tree and applying arbitrarily
transformations defined by a given set of rewrite rules,
one can reach at most one normal form
(one tree which cannot be transformed).
Using tree automata techniques, we have studied in [29]
the decidability of this property for classes of rewrite rules
defined by syntactic restrictions such as linearity
(variables can occur only once in each side of the rules),
flatness (sides of the rules have depth at most one)
and shallowness (variables occur at depth at most one in the rules).