Bandwidth of Timed Automata: 3 Classes

Abstract: Timed languages contain sequences of discrete events ("letters'') separated by real-valued delays, they can be recognized by timed automata, and represent behaviors of various real-time systems. The notion of bandwidth of a timed language defined in a previous paper characterizes the amount of information per time unit, encoded in words of the language observed with some precision {\epsilon}. In this paper, we identify three classes of timed automata according to the asymptotics of the bandwidth of their languages with respect to this precision {\epsilon}: automata are either meager, with an O(1) bandwidth, normal, with a {\Theta}(log (1/{\epsilon})) bandwidth, or obese, with {\Theta}(1/{\epsilon}) bandwidth. We define two structural criteria and prove that they partition timed automata into these three classes of bandwidth, implying that there are no intermediate asymptotic classes. The classification problem of a timed automaton is PSPACE-complete. Both criteria are formulated using morphisms from paths of the timed automaton to some finite monoids extending Puri's orbit graphs; the proofs are based on Simon's factorization forest theorem.