Entropy of regular timed languages

Overview & Synopsis

For timed languages, we define size measures: volume for languages with 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 information contents of a timed language. For languages accepted by deterministic timed automata, we give exact formulas for volumes. We show that automata with non-vanishing entropy ("thick") have a normal (non-Zeno, discretizable etc.) behavior for typical runs. Next, we characterize the entropy, using methods of functional analysis, as the logarithm of the leading eigenvalue (spectral radius) of a positive integral operator. We devise a couple of methods to compute the entropy: a symbolical one for so-called "1 1 ⁄2-clock" automata, and a numerical one (with a guarantee of convergence).