A Maximal Entropy Stochastic Process for a Timed Automaton,

Abstract

To cite this version:Nicolas Basset. A maximal entropy stochastic process for a timed automaton. ICALP 2013, Jul 2013, Riga, Latvia. 7966, pp.61-73, 2013 Abstract. Several ways of assigning probabilities to runs of timed automata (TA) have been proposed recently. When only the TA is given, a relevant question is to design a probability distribution which represents in the best possible way the runs of the TA. This question does not seem to have been studied yet. We give an answer to it using a maximal entropy approach. We introduce our variant of stochastic model, the stochastic process over runs which permits to simulate random runs of any given length with a linear number of atomic operations. We adapt the notion of Shannon (continuous) entropy to such processes. Our main contribution is an explicit formula defining a process Y * which maximizes the entropy. This formula is an adaptation of the so-called Shannon-Parry measure to the timed automata setting. The process Y * has the nice property to be ergodic. As a consequence it has the asymptotic equipartition property and thus the random sampling w.r.t. Y * is quasi uniform.

Generation of Signals Under Temporal Constraints for CPS Testing

“…Such a uniform sampling is done by adding probability distributions on time delays along a run of the automaton. In [9] we treat the case of generating infinite timed words, based on the notion of maximal entropy. Let us now explain the essence of these ideas.…”

Uniform Sampling for Timed Automata with Application to Language Inclusion Measurement

“…Thus, volume functions are defined via iteration of the operator Ψ on the constant function 1: v n = Ψ n (1). In [3,7], the state space was decomposed into regions, which guaranteed algebraic properties such as polynomial volume functions at the price of an explosion of the number of locations of the TA. A TA before such a decomposition into regions has volume functions that are complicated (piecewise defined), and hence difficult to handle in practice.…”

“…A natural answer is given by the maximal entropy principle: "without knowledge a priori on the distribution of probability to be taken, the one with maximal entropy should be preferred" [15]. A maximal entropy stochastic process for timed automata was recently proposed in [7]. Essentially, this is the stochastic process that yields the most uniform sampling when the length of the timed words tends to infinity.…”

“…The accuracy of statistical estimation depends on the ability to uniformly sample the executions. The method provided in [7], where the transitions of a TA were annotated with probability functions so that the resulting stochastic process enables random simulation in the most uniform way possible, is based on spectral attributes of a functional operator Ψ (an analogue in the TA context of the adjacency matrix of a graph) [3]. Unfortunately, it is not practical, as it relies on the region graph abstraction and the computation of eigenfunctions.…”

“…Unfortunately, it is not practical, as it relies on the region graph abstraction and the computation of eigenfunctions. In this paper, we overcome this problem by adopting a zone-based approach and approximating the probability functions of [7] with quotients of the volume functions.…”