Aldric Degorre

Volume and Entropy of Regular Timed Languages: Discretization Approach

For timed languages, we define size measures: volume for languages with words having a fixed finite number of events, and entropy (growth rate) as asymptotic measure for an unbounded number of events. These measures can be used for quantitative comparison of languages, and the entropy can be viewed as the information contents of a timed language. For languages accepted by deterministic timed automata, we give exact formulas for volumes. Next, for a large class of timed languages accepted by non-Zeno timed automata, we devise a method to approximate the volumes and the entropy based on discretization. We give an information-theoretic interpretation of the entropy in terms of Kolmogorov complexity.

Two Size Measures for Timed Languages

Quantitative properties of timed regular languages, such as information content (growth rate, entropy) are explored. The approach suggested by the same authors is extended to languages of timed automata with punctual (equalities) and non-punctual (non-equalities) transition guards. Two size measures for such languages are identified: mean dimension and volumetric entropy. The former is the linear growth rate of the dimension of the language; it is characterized as the spectral radius of a max-plus matrix associated to the automaton. The latter is the exponential growth rate of the volume of the language; it is characterized as the logarithm of the spectral radius of a matrix integral operator on some Banach space associated to the automaton. Relation of the two size measures to classical information-theoretic concepts is explored.

Generating functions of timed languages

In order to study precisely the growth of timed languages, we associate to such a language a generating function. These functions (tightly related to volume and entropy of timed languages) satisfy compositionality properties and, for deterministic timed regular languages, can be characterized by integral equations. We provide procedures for closed-form computation of generating functions for some classes of timed automata and regular expressions.

Toward a timed theory of channel coding

The classical theory of constrained-channel coding deals with the following questions: given two languages representing a source and a channel, is it possible to encode source messages to channel messages, and how to realize encoding and decoding by simple algorithms, most often transducers. The answers to this kind of questions are based on the notion of entropy. In the current paper, the questions and the results of the classical theory are lifted to timed languages. Using the notion of entropy of timed languages introduced by Asarin and Degorre, the question of timed coding is stated and solved in several settings.

Spectral gap in timed automata

Various problems about probabilistic and non-probabilistic timed automata (computing probability density, language volume or entropy) can be naturally phrased as iteration of linear operators in Banach spaces. Convergence of such iterations is guaranteed whenever the operators spectrum has a gap. In this article, for operators used in entropy computation, we use the theory of positive operators to establish the existence of such a gap. This allows to devise simple numeric algorithms for computing the entropy and prove their exponential convergence.

Entropy Games and Matrix Multiplication Games

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible Peoples behaviors as small as possible, while Tribune wants to make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP\&coNP.

On Scheduling Policies for Streams of Structured Jobs

We study a class of scheduling problems which combines the structural aspects associated with task dependencies, with the dynamic aspects associated with ongoing streams of requests that arrive during execution. For this class of problems we develop a scheduling policy which can guarantee bounded accumulation of backlog for all admissible request streams. We show, nevertheless, that no such policy can guarantee bounded latency for all admissible request patterns, unless they admit some laxity.

Distance on Timed Words and Applications

We introduce and study a new (pseudo) metric on timed words having several advantages: it is global: it applies to words having different number of events; it is realistic and takes into account imprecise observation of timed events; thus it reflects the fact that the order of events cannot be observed whenever they are very close to each other; it is suitable for quantitative verification of timed systems: we formulate and solve quantitative model-checking and quantitative monitoring in terms of the new distance, with reasonable complexity; it is suitable for information-theoretical analysis of timed systems: due to its pre-compactness the quantity of information in bits per time unit can be correctly defined and computed.