Bandwidth of Timed Automata: 3 Classes

TL;DR

This paper classifies timed automata into three classes based on the asymptotic behavior of their language bandwidth with respect to observation precision, using structural criteria and complexity analysis.

Contribution

It introduces a novel classification of timed automata into three bandwidth classes using structural criteria and proves the classification is complete and computationally complex.

Findings

Automata are either meager, normal, or obese in bandwidth.

Structural criteria based on morphisms partition automata into these classes.

Classification problem is PSPACE-complete.

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.