## Section: New Results

### Automata and logics for data words and data trees

Participants : Stéphane Demri, Diego Figueira, Luc Segoufin.

Dahu aim as providing tools for specifying and verifying systems with data. This means finding a suitable logical framework for specifying such systems. A logical framework is suitable if it is expressible enough for modeling the operations of interest. Of course, for the logical framework to be useful, it must come with techniques and tools for reasoning about it, in particular it should be decidable. This can be achieved by compiling the model into some form of decidable automata manipulating data. In the presence of data, the design of appropriate classes of logic and automata with interesting complexities is an on-going research task.

Most of our new results in this direction concerns data words and data trees. Those are words and trees where each position contains a data value together with the classical label. Data words and data trees can model many systems with data with a focus on one variable flow. Data trees can also model XML data.

We have studied several extensions of the classical model of logic and automata with features that could be used for manipulating data. This is done either by using registers or memory explicitly in the model or by restricting the transitions of the automata with constraints that can involve data comparisons. Several models have been considered.

As query languages such as XPath and XML schema are closely related to the two variable fragment of first-order logic, we have studied this fragment over data trees. In [16] it is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result, decidability of satisfiability and containment for a data-aware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

As another line of investigation, we studied a bottom-up model of computation that can test for data equality of distant nodes on different branches of the tree [26] . This model captures XPath with downward and child axes, and has an incomparable expressive power with respect to the previous mentioned approach. The model is decidable in ExpTime.

We have analyzed the computational complexity of the covering and boundedness problems for branching vector addition systems [25] . Branching vector addition systems (BVAS) form a new computational model that is used for instance in computational linguistics and for the verification of cryptographical protocols. This model has tight relationships with data logics intepreted over data trees. Recently, Verma and Goubault-Larrecq (EPI SECSI, LSV) have shown that the covering and boundedness problems for BVAS are decidable. In this work, we have extended and refined the standard proofs for vector addition systems (equivalent to Petri nets) by Rackoff (TCS, 1978) and Lipton (TR, 1976) in order to establish that the covering and boundedness problems for BVAS are 2EXPTIME-complete.

In the article [17] , we have studied decidability and complexity issues for fragments of LTL with Presburger constraints obtained by restricting the syntactic resources of the formulae while preserving the strength of the logical operators. It is shown that model-checking and satisfiability problems for the fragments of LTL with difference constraints restricted to two variables and distance one and to one variable and distance two are highly undecidable, enlarging significantly the class of known undecidable fragments. On the positive side, we prove that the fragment restricted to one variable and to distance one augmented with propositional variables is PSPACE-complete.

In [34] we illustrate two aspects of automata theory related to linear-time temporal logic LTL used for the verification of computer systems. A translation from LTL formulae to Büchi automata is presented with the aim to design an elementary translation which is reasonably efficient and produces small automata. Secondly, we recall how temporal operators can be defined from regular languages and we show why adding even a single operator definable by a context-free language can lead to undecidability.