Michel Sintzoff

Expressing program developments in a design calculus

The present paper describes a step in the study of means to express software developments. This study is also related to approaches where programs are extracted from proofs, and it is influenced by the spirit and the techniques of constructive logic.

Iterative Synthesis of Control Guards Ensuring Invariance and Inevitability in Discrete-Decision Games

Reactive and hybrid systems are modeled by games where players make strategic decisions in a temporally discrete manner. The dynamics of players use dense or discrete time. In order to guarantee invariance and inevitability properties, the proponent moves are restricted by “winning guards”. The winning strategy determined by these guards does not exclude any initial state from which a winning strategy exists. Sets of such initial states constitute winning regions and are defined by fixed points. The iterates which yield winning regions are structured as unions of iterates which yield winning guards.

Abstract verification of structured dynamical systems

Dynamical systems combining different kinds of time are analyzed with the help of homomorphisms which allow time abstraction besides the usual state abstraction and which do preserve fundamental temporal properties. Dynamical systems are composed by restriction, union, synchronization, concatenation and iteration. Thanks to abstraction and structure, the qualitative analysis of systems which are hard to understand can be reduced to that of simpler, homomorphic systems.

A Semiring Approach to Equivalences, Bisimulations and Control

Equivalences, partitions and (bi)simulations are usually tackled using concrete relations. There are only few treatments by abstract relation algebra or category theory. We give an approach based on the theory of semirings and quantales. This allows applying the results directly to structures such as path and tree algebras which is not as straightforward in the other approaches mentioned. Also, the amount of higher-order formulations used is low and only a one-sorted algebra is used. This makes the theory suitable for fully automated first-order proof systems. As a small application we show how to use the algebra to construct a simple control policy for infinite-state transition systems.

Model refinement using bisimulation quotients

The paper shows how to refine large-scale or even infinite transition systems so as to ensure certain desired properties. First, a given system is reduced into a smallish, finite bisimulation quotient. Second, the reduced system is refined in order to ensure a given property, using any known finite-state method. Third, the refined reduced system is expanded back into an adequate refinement of the system given initially. The proposed method is based on a Galois connection between systems and their quotients. It is applicable to various models and bisimulations and is illustrated with a few qualitative and quantitative properties.

Synthesis of Optimal Control Policies for Some Infinite-State Transition Systems

We develop a symbolic, logic-based technique for constructing optimal control policies in some transition systems where state spaces are large or infinite. These systems are presented as iterations of finite sets of guarded assignments which have costs. The optimality objective is to minimize the total costs of system executions reaching the set characterized by a given target predicate. Guards are predicates and control policies are expressed by tuples of guards. The optimal control policy refines the control policy of the given system. It is generated from the target predicate by an iteration based on backwards induction. This iterative procedure amounts to a variant of the symbolic algorithm generating the reachability precondition; the latter characterizes the states from which some system execution reaches the target set. The main difference is the introduction of greedy and cost-dependent iteration steps.

Algebraic Composition and Refinement of Proofs

We present an algebraic calculus for proof composition and refinement. Fundamentally, proofs are expressed at successive levels of abstraction, with the perhaps unconventional principle that a formula is considered to be its own most abstract proof, which may be refined into increasingly concrete proofs. Consequently, we suggest a new paradigm for expressing proofs, which views theorems and proofs as inhabiting the same semantic domain. This algebraic/model-theoretical view of proofs distinguishes our approach from conventional typetheoretical or sequent-based approaches in which theorems and proofs are different entities. All the logical concepts that make up a formal system — formulas, inference rules, and derivations — are expressible in terms of the calculus itself. Proofs are constructed and structured by means of a composition operator and a consequential rule-forming operator. Their interplay and their relation wrt. the refinement order are expressed as algebraic laws.

On the design of correct and optimal dynamical systems and games

There exist various methods for designing dynamical systems and dynamical games in order to ensure correctness and optimality. In the paper, they are systematically organized as follows. Two variational principles are recalled. Firstly, solutions must be stationary: this leads to necessary conditions and to gradient algorithms. Secondly, solutions, if any, must be optimal or correct; this leads to sufficient conditions and to dynamic-programming algorithms. Methods based on these principles allow to design dynamical systems and games such as control systems, hybrid systems and reactive ones. Time may be discrete or continuous; correctness can be viewed as an abstraction of optimality. The structured presentation of design methods is intended to foster their understanding, integration, cross-fertilization and improvement.