Laurent Fribourg

AN INVERSE METHOD FOR PARAMETRIC TIMED AUTOMATA

We consider in this paper systems modeled by timed automata. The timing bounds involved in the action guards and location invariants of our timed automata are not constants, but parameters. Those parametric timed automata allow the modelling of various kinds of timed systems, e.g. communication protocols or asynchronous circuits. We will also assume that we are given an initial tuple π0 of values for the parameters, which corresponds to values for which the system is known to behave properly. Our goal is to compute a constraint K0 on the parameters, satisfied by π0, guaranteeing that, under any parameter valuation satisfying K0, the system behaves in the same manner: for any two parameter valuations satisfying K0, the behaviors of the timed automata are (time-abstract) equivalent, i.e., the traces of execution viewed as alternating sequences of actions and locations are identical. We present an algorithm InverseMethod that terminates in the case of acyclic models, and discuss how to extend it in the cyclic case. We also explain how to combine our method with classical synthesis methods which are based on the avoidance of a given set of bad states. A prototype implementation has been done, and various experiments are described.

Proving Safety Properties of Infinite State Systems by Compilation into Presburger Arithmetic

We present in this paper a method combining path decomposition and bottom-up computation features for characterizing the reachability sets of Petri nets within Presburger arithmetic. An application of our method is the automatic verification of safety properties of Petri nets with infinite reachability sets. Our implementation is made of a decomposition module and an arithmetic module, the latter being built upon Boudet-Comons algorithm for solving the decision problem for Presburger arithmetic. Our approach will be illustrated on three nontrivial examples of Petri nets with unbounded places and parametric initial markings.

Reachability Analysis of (Timed) Petri Nets Using Real Arithmetic

In this paper, we address the issue of reachability analysis for Petri nets, viewed as automata with counters. We show that exact reachability analysis can be achieved by treating Petri nets integer variables (counters) as real-valued variables, and using Fourier-Motzkin procedure instead of Presburger elimination procedure. As a consequence, one can safely analyse Petri nets with performant tools, e.g. HyTech, originally designed for analysing automata with real-valued variables (clocks). We also investigate the use of meta-transitions (iterative application of a transition in a single step) and give sufficient conditions ensuring an exact computation in this case. Experimental results with HyTech show an impressive speed-up with respect to previous experiences performed with a Presburger arithmetic solver. The method extends for analysing Petri nets with inhibitors and with timing constraints, but difficulties arise for the treatment of meta-transitions in the latter case.

IMITATOR 2.5: A Tool for Analyzing Robustness in Scheduling Problems

The tool Imitator implements the Inverse Method (IM) for Timed Automata (TAs). Given a TA \(\mathcal{A}\) and a tuple π 0 of reference valuations for timings, IM synthesizes a constraint around π 0 where \(\mathcal{A}\) behaves in the same discrete manner. This provides us with a quantitative measure of robustness of the behavior of \(\mathcal{A}\) around π 0. The new version Imitator 2.5 integrates the new features of stopwatches (in addition to standard clocks) and updates (in addition to standard clock resets), as well as powerful algorithmic improvements for state space reduction. These new features make the tool well-suited to analyze the robustness of solutions in several classes of preemptive scheduling problems.

Coupling and Self-stabilization

A randomized self-stabilizing algorithm \({\cal A}\) is an algorithm that, whatever the initial configuration is, reaches a set \({\cal L}\) of legal configurations in finite time with probability 1. The proof of convergence towards \({\cal L}\) is generally done by exhibiting a potential function ϕ, which measures the “vertical” distance of any configuration to \({\cal L}\), such that ϕ decreases with non-null probability at each step of \({\cal A}\). We propose here a method, based on the notion of coupling, which makes use of a “horizontal” distance δ between any pair of configurations, such that δ decreases in expectation at each step of \({\cal A}\). In contrast with classical methods, our coupling method does not require the knowledge of \({\cal L}\). In addition to the proof of convergence, the method allows us to assess the convergence rate according to two different measures. Proofs produced by the method are often simpler or give better upper bounds than their classical counterparts, as examplified here on Herman’s mutual exclusion and Iterated Prisoner’s Dilemma algorithms in the case of cyclic graphs.

Constraint Logic Programming Applied to Model Checking

We review and discuss here some of the existing approaches based on CLP (Constraint Logic Programming) for verifying properties of various kinds of state-transition systems.

Reachability sets of parameterized rings as regular languages

Abstract We present here a method for deriving a regular language that characterizes the set of reachable states of a given parameterized ring (made of N identical components). The method basically proceeds in two steps: first one generates a regular language L by inductive inference from a finite sample of reachable states; second one formally checks that L characterizes the whole set of reachable states.

Randomized dining philosophers without fairness assumption

Abstract.We consider Lehmann-Rabin’s randomized solution to the well-known problem of the dining philosophers. Up to now, such an analysis has always required a “fairness” assumption on the scheduling mechanism: if a philosopher is continuously hungry then he must eventually be scheduled. In contrast, we modify here the algorithm in order to get rid of the fairness assumption, and we claim that the spirit of the original algorithm is preserved. We prove that, for any (possibly unfair) scheduling, the modified algorithm converges: every computation reaches with probability 1 a configuration where some philosopher eats. Furthermore, we are now able to evaluate the expected time of convergence in terms of the number of transitions. We show that, for some “malicious” scheduling, this expected time is at least exponential in the number N of philosophers.

A Narrowing Procedure for Theories with Constructors

This paper describes methods to prove equational clauses (disjunctions of equations and inequations) in the initial algebra of an equational theory presentation. First we show that the general problem of validity can be converted into the one of satisfiability. Then we present specific procedures based on the narrowing operation, which apply when the theory is defined by a canonical set of rewrite rules. Complete refutation procedures are described and used as invalidity procedures. Finally, a narrowing procedure incorporating structural induction aspects, is proposed and the simplicity of the automated proofs is illustrated through examples.

Behavioral Cartography of Timed Automata

We aim at finding a set of timing parameters for which a given timed automaton has a “good” behavior. We present here a novel approach based on the decomposition of the parametric space into behavioral tiles, i.e., sets of parameter valuations for which the behavior of the system is uniform. This gives us a behavioral cartography according to the values of the parameters. It is then straightforward to partition the space into a “good” and a “bad” subspace, according to the behavior of the tiles. We extend this method to probabilistic systems, allowing to decompose the parametric space into tiles for which the minimal (resp. maximal) probability of reaching a given location is uniform. An implementation has been made, and experiments successfully conducted.