Section: New Results

Model-based Verification

We have investigated extensions of regular model-checking to new classes of rewrite relations on trees. We have studied specification and proof of modular imperative programs.

Algorithms for Tree Walking Automata

Participants : Pierre-Cyrille Héam, Vincent Hugot, Olga Kouchnarenko.

Tree walking automata are widely used to tackle data base algorithmic problems, particularly to analyse queries over XML documents. The emptiness problem for tree walking automata is known to be EXPTIME-complete. The general algorithm to solve this problem consists in transforming the tree walking automaton into a classical top-down tree automaton. The best known in the literature algorithm works in time $O\left(s{2}^{n^2}\right)$ where $n$ is the number of states of the tree walking automaton, and $s$ is the size of the alphabet. In [24] we have proposed a new algorithm based on an overloop concept and working in time $O\left(2^{n^2}\right)$. Then our approach has been improved for deterministic tree walking automata to have in this case a $O\left(2^{nlogn}\right)$ time complexity. Finally, we have also proposed a polynomial-time approximation based semi-algorithm for the emptiness problem. The algorithms have been implemented and experimental results confirm the relevance of the approach.

Algorithms for Tree Automata with Global Constraints

Participants : Pierre-Cyrille Héam, Vincent Hugot, Olga Kouchnarenko.

Extending tree automata models to be able to compare different tree branches is an important and challening issue for systems' modeling and for verifying their properties. Several exetensions have been proposed in the litterature. Among them we are interested in the model of Tree Automata with Global Constraints (TAGED) introduced in 2009. The membership problem for this new model is known to be NP-complete, and the emptyness problem is known to be EXPTIME-complete. In [47] we have investigated some complexity results for tree automata with a bounded number of equality constraints. We have proved that with a unique constraint the emptyness problem is in PTIME and that it is EXPTIME-complete with only two constraints. For a bounded number of constraints, the membership problem is in PTIME.

Verification of Linear Temporal Patterns over Finite and Infinite Traces

Participants : Pierre-Cyrille Héam, Vincent Hugot, Olga Kouchnarenko.

In the regular model-checking framework, reachability analysis can be guided by temporal logic properties, for instance to achieve the counter example guided abstraction refinement (CEGAR) objectives. A way to perform this analysis is to translate a temporal logic formula expressed on maximal rewriting words into a “rewrite proposition” – a propositional formula whose atoms are language comparisons, and then to generate semi-decision procedures based on (approximations of) the rewrite proposition. In [46] we have investigated suitable semantics for LTL on maximal rewriting words and their influence on the feasibility of a translation, and we have proposed a general scheme providing exact results for a fragment of LTL corresponding mainly to safety formulæ, and approximations for a larger fragment.

Rewriting-based Mathematical Model Transformations

Participants : Walid Belkhir, Alain Giorgetti.

We have pursued our collaboration with the Department “Temps-Fréquence” of the FEMTO-ST institute (Franche-Comté Electronique Mécanique Thermique et Optique - Sciences et Technologies, CNRS UMR 6174) on the formalization of asymptotic methods (based on two-scale convergence) to automatically generate asymptotic models of large arrays of micro- and nanosystems. The goal is to provide engineers with an implementation of this mathematical tool inside a modeling software. We follow therefore a multidisciplinary approach which combines a generalization and formalization effort of mathematical asymptotic methods, together with rewriting-based formal transformation techniques. This approach is described in [53] , together with an example and a presentation of the architecture of the software under design. A second contribution [34] is a detailed formal specification and analysis of lazy pattern-matching mechanism modulo associativity and commutativity, and its integration into a strategy language. The pattern-matching solutions are stored in a lazy list composed of a first substitution at the head and a non-evaluated object that encodes the remaining computations. Rule and strategy applications also produce a lazy list of terms. This contribution has been published in EPTCS as the proceedings of the 10th International Workshop on Reduction Strategies in Rewriting and Programming, where a lighter version was presented in 2011  [69] .