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Langages formels : Quelques aspects quantitatifs

Abstract : Formal languages are sequences of symbols in some discrete set called alphabet. They are often specified by formulas in some logic, by rational expressions or else by discrete automata of various types. Current theory is mostly qualitative, in the sense that its objects are sequences over a discrete, non-metric, time, that acceptation of a sequence over an automaton depends on whether an accepting state is visited or not, and that, for language comparison, one will most often consider inclusion than quantitative measures. This thesis contributes to the study of these often neglected aspects by presenting fundamental results in three new quantitative problem classes about formal languages. In the first chapter, we study a class of scheduling problems which combines structural aspects related to dependencies between tasks with dynamical aspects of online scheduling. We show that some task request streams in this class of problems, however admissible in the sense that they do not require more raw resource utilization than available machines can provide, cannot be scheduled with bounded latency. However, we develop a scheduling policy which can guarantee a bounded backlog accumulation for all admissible streams even without knowledge of the stream content prior to execution. We show that if the streams actually are sub-critical, then the same policy also guarantees a bounded latency. In quantitative verification, states and transitions of a system can be endowed with costs, which can be used to associate average costs to infinite behaviors. In this second part, we propose to define omega-languages thanks to Boolean queries concerning average costs. Specifications dealing with averages, such as "average message loss is under some threshold" are not omega-regular, but can be expressed in our model. Hence we study expressive power and Borel complexity of such specifications. We show that in order to ensure closure by intersection, multi-dimensional costs have to be considered. In the general case, we show that accepting conditions can be defined on the set of accumulation points of the sequence of the average costs of the prefixes of a run. We accurately characterize such sets. Last, we also propose a class of multi-threshold mean-payoff languages, in which extremal values of coordinates of points of the accumulation set are compared to constants. We show this class is closed under Boolean operations and analyzable. In the last chapter, we define two measures for a timed language: the volume of its sub-languages of words having a fixed number of discrete events, and entropy, which is the asymptotic logarithmic growth of volumes. These measures can be used to compare languages quantitatively and entropy can be seen as information content of a typical event of a typical word of in a language. For languages accepted by deterministic timed automata, we give an exact formula for the volume. Next we characterize entropy, thanks to functional analysis techniques, as the logarithm of the spectral radius of some positive integral operator. We give several methods to compute entropy: a symbolical one for the special case of "one clock and a half" automata, and two numerical ones: one using functional analysis, the other using discretization techniques. Last we establish an interpretation of this entropy in information theory by showing its relation with Kolmogorov complexity.
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Contributor : Aldric Degorre <>
Submitted on : Thursday, February 2, 2012 - 9:06:47 AM
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Aldric Degorre. Langages formels : Quelques aspects quantitatifs. Théorie et langage formel [cs.FL]. Université Joseph-Fourier - Grenoble I, 2009. Français. ⟨tel-00665462⟩

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