Section: New Results

Model-based Verification

We have investigated extensions of regular model-checking to new classes of rewrite relations on terms. We have studied specification and proof of modular imperative programs, as well as of modal workflows.

Tree Automata with Constraints

Participants : Pierre-Cyrille Héam, Olga Kouchnarenko.

Tree automata with constraints are widely used to tackle data base algorithmic problems, particularly to analyse queries over XML documents. The model of Tree Automata with Global Constraints (TAGED) is a model introduced in 2009 for these purposes. The membership problem for TAGED is known to be NP-complete. The emptiness problem for TAGED is known to be decidable and the best known algorithm in the general case is non elementary. In collaboration with Vincent Hugot, we show that if there is at least one negative constraint, the problem is already NP-hard [64] . In the future, we plan to investigate upper bounds for the emptiness problem with a unique negative constraint. We also plan to study the complexity of the universality problem with a single constraint.

Random Generation of Finite Automata

Participant : Pierre-Cyrille Héam.

Developing new algorithms and heuristics raises crucial evaluation issues, as improved worst-case complexity upper-bounds do not always transcribe into clear practical gains. A suite for software performance evaluation can usually gather three types of entries: benchmarks, hard instance and random inputs, that deliver average complexity estimations, for which the catch resides in obtaining a meaningful random distribution (for instance a uniform random distribution).

In collaboration with Jean-Luc Joly, we investigate the problem of randomly and uniformly generating deterministic pushdown automata [65] . Based on a recursive counting approach, we propose a polynomial time algorithm for this purpose. The influence of the accepting condition on the generated automata is also experimentally studied.

Partially ordered automata are finite automata where simple loops have length one. They appear in several verification techniques, such as computing closures under semi-commutation relations or studying FIFO systems. In [68] , we use a Markov chain based approach to randomly - and uniformly - generate deterministic partially ordered automata. The advantage of such a technique is its flexibility, allowing for instance to easily bound the number of loops. Experiments show that the mixing time seems to be polynomial, providing a tractable approach.

Verification of Linear Temporal Patterns over Finite and Infinite Traces

Participants : Pierre-Cyrille Héam, 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 collaboration with Vincent Hugot, 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.

Machine-Learning Techniques for Regular Model-Checking

Participants : Maxime Bride, Pierre-Cyrille Héam.

Using a machine-learning approach, we address the general problem of regular model-checking of computing $R^{*}\left(L\right)$, when $L$ is a regular language and $R$ a relation. Rather than developing specific algorithms to compute $R^{*}\left(L\right)$, it consists in using Angluin style's algorithms. In [58] , we focus on the generation of examples, counter-examples and on the design of an oracle for the specific case of semi-commutation relations. Experiments are promising, particularly for the sizes of the obtained automata, which are quite smaller than with dedicated algorithms.

Constraint Solving for Verifying Modal Workflow Specifications

Participants : Hadrien Bride, Olga Kouchnarenko.

Workflow Petri nets are well suited for modelling and analysing discrete event systems exhibiting behaviours such as concurrency, conflict, and causal dependency between events. They represent finite or infinite-state processes, and several important verification problems, like reachability or soundness, are known to be decidable. Modal specifications introduced in [84] allow loose or partial specifications in a framework based on process algebras.

Our work in [34] focuses on the verification of modal workflow specifications using constraint solving as a computational tool. Its main contribution consists of a formal framework based on constraint systems to model executions of workflow Petri nets and their structural properties, as well as to verify their modal specifications. An implementation and promising experimental results obtained within the proposed approach constitute a practical contribution. In particular, a business process example from the IT domain enables to successfully assess the reliability of our contributions.

Rewriting-based Mathematical Model Transformations

Participants : Walid Belkhir, Alain Giorgetti.

Since 2011 we collaborate 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). The goal is to design a software, called MEMSALab, for the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. We have designed a transformation language facilitating the design of MEMSALab [17] . It is proposed as a Maple${}^{\mathrm{𝖳𝖬}}$ package for rule-based programming, rewriting strategies and their combination with standard Maple${}^{\mathrm{𝖳𝖬}}$ code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation. A more general framework for the derivation of the multi-scale models was established in [29] .