Volume and entropy of regular timed languages

Eugene Asarin (1) , Aldric Degorre (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 VERIMAG - IMAG - VERIMAG

For timed languages, we define measures of their size: volume for 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 comparison of languages, and the entropy can be viewed as information contents of a timed language. In case of languages of deterministic timed automata, we give exact formulas for volumes. Next we characterize the entropy, using methods of functional analysis, as a logarithm of the leading eigenvalue (spectral radius) of a positive integral operator. We devise several methods to compute the entropy: a symbolical one for so-called “1,5 -clock” automata, and two numerical ones: one using techniques of functional analysis, another based on discretization.

Mots clés: timed automata; formal language; entropy; growth rate; Kolmogorov complexity

Domaines: Informatique et langage [cs.CL]; Théorie de l'information [cs.IT]; Théorie de l'information et codage [math.IT]

Eugene Asarin, Aldric Degorre. Volume and entropy of regular timed languages. 2009. ⟨hal-00369812v3⟩