Volume and entropy of regular timed languages

27 pages. Two papers based on this work will appear at Concur'09 and Formats'09.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