Volume and entropy of regular timed languages

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. 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 1 2 -clock” automata, and two numerical ones: one using techniques of functional analysis, another based on discretization. We give an information-theoretic interpretation of the entropy in terms of Kolmogorov complexity.