Eugene Asarin

Scheduling with timed automata

In this work, we present timed automata as a natural tool for posing and solving scheduling problems. We show how efficient shortest path algorithms for timed automata can find optimal schedules for the classical job-shop problem. We then extend these results to synthesize adaptive scheduling strategies for problems with uncertainty in task durations.

Hybridization methods for the analysis of nonlinear systems

In this article, we describe some recent results on the hybridization methods for the analysis of nonlinear systems. The main idea of our hybridization approach is to apply the hybrid systems methodology as a systematic approximation method. More concretely, we partition the state space of a complex system into regions that only intersect on their boundaries, and then approximate its dynamics in each region by a simpler one. Then, the resulting hybrid system, which we call a hybridization, is used to yield approximate analysis results for the original system. We also prove important properties of the hybridization, and propose two effective hybridization construction methods, which allow approximating the original nonlinear system with a good convergence rate.

Recent progress in continuous and hybrid reachability analysis

Set-based reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachability analysis has been limited so far to simple systems, both in the continuous and the hybrid domain. In this paper we present recent advances that, in combination, significantly improve this applicability, and allow us to find better balance between computational cost and accuracy. The presentation covers, in a unified manner, a variety of methods handling increasingly complex types of continuous dynamics (constant derivative, linear, nonlinear). The improvements include new geometrical objects for representing sets, new approximation schemes, and more flexible combinations of graph-search algorithm and partition refinement. We report briefly some preliminary experiments that have enabled the analysis of systems previously beyond reach.

On the Decidability of the Reachability Problem for Planar Differential Inclusions

In this paper we develop an algorithm for solving the reachability problem of two-dimensional piece-wise rectangular differential inclusions. Our procedure is not based on the computation of the reach-set but rather on the computation of the limit of individual trajectories. A key idea is the use of one-dimensional affine PoincarE maps for which we can easily compute the fixpoints. As a first step, we show that between any two points linked by an arbitrary trajectory there always exists a trajectory without self-crossings. Thus, solving the reachability problem requires considering only those. We prove that, indeed, there are only finitely many qualitative types of those trajectories. The last step consists in giving a decision procedure for each of them. These procedures are essentially based on the analysis of the limits of extreme trajectories. We illustrate our algorithm on a simple model of a swimmer spinning around a whirlpool.

Abstraction by Projection and Application to Multi-affine Systems

In this paper we present an abstraction method for nonlinear continuous systems. The main idea of our method is to project out some continuous variables, say z, and treat them in the dynamics of the remaining variables x as uncertain input. Therefore, the dynamics of x is then described by a differential inclusion. In addition, in order to avoid excessively conservative abstractions, the domains of the projected variables are divided into smaller regions corresponding to different differential inclusions. The final result of our abstraction procedure is a hybrid system of lower dimension with some important properties that guarantee convergence results. The applicability of this abstraction approach depends on the ability to deal with differential inclusions. We then focus on uncertain bilinear systems, a simple yet useful class of nonlinear differential inclusions, and develop a reachability technique using optimal control. The combination of the abstraction method and the reachability analysis technique for bilinear systems allows to treat multi-affine systems, which is illustrated with a biological system.

Some Progress in Satisfiability Checking for Difference Logic

In this paper we report a new SAT solver for difference logic, a propositional logic enriched with timing constraints. The main novelty of our solver is a tighter integration of the incremental analysis of numerical conflicts with the process of Boolean conflict analysis. This and other improvements lead to significant performance gains for some classes of problems.

Balanced timed regular expressions

Several classes of regular expressions for timed languages accepted by timed automata have been suggested in the literature. In this article we introduce balanced timed regular expressions with colored parentheses which are equivalent to timed automata, and, differently from existing definitions, do not refer to clock values, and do not use additional operations such as intersection and renaming.

Noisy Turing Machines

Turing machines exposed to a small stochastic noise are considered. An exact characterisation of their (≈({it Pi}) (_{rm 2}^{rm 0})) computational power (as noise level tends to 0) is obtained. From a probabilistic standpoint this is a theory of large deviations for Turing machines.

Low dimensional hybrid systems - decidable, undecidable, don't know

Even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems, the affair is far from being resolved. More and more low dimensional systems are being shown to be undecidable with respect to reachability, and many open problems in between are being discovered. In this paper, we present various two-dimensional hybrid systems for which the reachability problem is undecidable. We show their undecidability by simulating Minsky machines. Their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable. We also show that for other two-dimensional systems, the reachability question remains unanswered, by proving that it is as hard as the reachability problem for piecewise affine maps on the real line, which is a well known open problem.

Entropy of regular timed languages

To study the size of regular timed languages, we generalize a classical approach introduced by Chomsky and Miller for discrete automata: count words having n symbols, and compute the exponential growth rate of their number (entropy). For timed automata, we replace cardinality by volume and define (volumetric) entropy similarly. It represents the average quantity of information per event in a timed word of the language. We exhibit a criterion for telling apart thick timed automata with non-vanishing entropy, for which typical runs are non-Zeno and discretizable, from thin automata for which all runs behave in a Zeno-like way, implying a quick volume collapse. We associate to every timed automaton a positive integral operator; the entropy equals the logarithm of its spectral radius. This operator has a spectral gap, thus allowing for fast converging numerical procedures to approximate entropy. In a special case, entropy is even characterized symbolically.