Section: New Results

Automated Deduction

We develop general techniques which allow us to re-use available tools in order to build a new generation of solvers offering a good trade-off between expressiveness, flexibility, and scalability. We focus on the careful integration of combination techniques and rewriting techniques to design decision procedures for a wide range of verification problems.

Building and Verifying decision procedures

Participants : Alain Giorgetti, Olga Kouchnarenko, Christophe Ringeissen.

In the context of the PhD thesis by Elena Tushkanova (defended in 2013), we have developed a methodology to build decision procedures specified by using a superposition calculus [20] which is at the core of all equational theorem provers. This calculus is refutation complete: it provides a semi-decision procedure that halts on unsatisfiable inputs but may diverge on satisfiable ones. Fortunately, it may also terminate for some theories of interest in verification, and thus it becomes a decision procedure. To reason on the superposition calculus, a schematic superposition calculus has been developed to build the schematic form of the saturations allowing to automatically prove decidability of single theories and of their combinations. We have proposed a rule-based logical framework and a tool implementing a complete many-sorted schematic superposition calculus for arbitrary theories. By providing results for unit theories, arbitrary theories, and also for theories with counting operators, we show that this tool is very useful to derive decidability and combinability of theories of practical interest in verification.

Combination of Satisfiability Procedures

Participant : Christophe Ringeissen.

We have continued our work started with Paula Chocron (IIIA-CSIC, U. Barcelona) and Pascal Fontaine (project-team Veridis) on the extension of the Nelson-Oppen combination method to non-disjoint unions of theories. We investigate the case of theories connected via bridging functions [28] . The motivation is, e.g., to solve verification problems expressed in a combination of data structures connected to arithmetic with bridging functions such as the length of lists and the size of trees. We present a sound and complete combination procedure à la Nelson-Oppen for the theory of absolutely free data structures, including lists and trees. This combination procedure is then refined for standard interpretations. The resulting theory has a nice politeness property, enabling combinations with arbitrary decidable theories of elements.

To go beyond the case of absolutely free data structures, we study in [29] the problem of determining the data structure theories for which this combination method is sound and complete. Our completeness proof is based on a rewriting approach where the bridging function is defined as a term rewrite system, and the data structure theory is given by a basic congruence relation. Our contribution is to introduce a class of data structure theories that are combinable with a disjoint target theory via an inductively defined bridging function. This class includes the theory of equality, the theory of absolutely free data structures, and all the theories in between. Hence, our non-disjoint combination method applies to many classical data structure theories admitting a rewrite-based satisfiability procedure.

Unification Modulo Equational Theories

Participant : Christophe Ringeissen.

We investigate a hierarchical combination approach to the unification problem in non-disjoint unions of equational theories. In this approach, the idea is to extend a base theory with some additional axioms given by rewrite rules in such way that the unification algorithm known for the base theory can be reused without loss of completeness. Additional techniques are required to solve a combined problem by reducing it to a problem in the base theory. In [33] we show that the hierarchical combination approach applies successfully to some classes of syntactic theories, such as shallow theories since the required unification algorithms needed for the combination algorithm can always be obtained. We also consider the matching problem in syntactic extensions of a base theory. Due to the more restricted nature of the matching problem, we obtain several improvements over the unification problem.

Enumeration of Planar Proof Terms

Participant : Alain Giorgetti.

By the Curry-Howard isomorphism, simply typed lambda terms correspond to natural deduction proofs in minimal logic. Noam Zeilberger and Alain Giorgetti proved that normal planar lambda terms are in size-preserving bijection with rooted planar maps [21] . Although the formal aspect is not emphasized in the paper, the use of formal representations of both normal planar lambda terms and rooted planar maps, of logic programming and a proof assistant software helped much in more quickly finding the bijection.

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 different physical features and geometries e.g. thin structures, periodic structures, multiple nested scales etc. In [24] , we propose a method called "by-extension-combination", in which the asymptotic models are constructed incrementally so that model features can be included step by step. More precisely, a model derivation is formalised as a rewriting strategy, and its extension is formalised as a second-order rewriting strategy. Thus, our method amounts to defining an operation of combination over a class of second-order rewriting strategies. We illustrate the method by an example of an asymptotic model for the stationary heat equation in a Micro-Mirror Array developed for astrophysics.