Entropy of regular timed languages

To study the size of regular timed languages, we generalize a classical approach introduced by Chomsky and Miller for discrete automata: count words having n symbols, and compute the exponential growth rate of their number (entropy). For timed automata, we replace cardinality by volume and define (volumetric) entropy similarly. It represents the average quantity of information per event in a timed word of the language. We exhibit a criterion for telling apart “thick” timed automata with non-vanishing entropy, for which typical runs are non-Zeno and discretizable, from “thin” automata for which all runs behave in a Zeno-like way, implying a quick volume collapse. We associate to every timed automaton a positive integral operator; the entropy equals the logarithm of its spectral radius. This operator has a spectral gap, thus allowing for fast converging numerical procedures to approximate entropy. In a special case, entropy is even characterized symbolically.