Weighing Timed Regular Languages: The Final Step (long version)

2026-06-09

AI Summary

This work addresses the previously unresolved problem of computing the bandwidth of normal-class timed automata, for which no effective algorithm existed. By transforming the problem into one of finding an optimal gain-cost ratio path in an associated weighted finite graph, the paper presents the first algorithm capable of determining the bandwidth for this class of automata, thereby completing the computational framework for all three classes of timed regular languages. Leveraging techniques from timed automata theory, asymptotic bandwidth analysis, and optimal ratio path algorithms, the study derives a precise characterization of the bandwidth in the form ≈α log(1/ε), offering an exact quantification of the information density inherent to normal-class timed automata.

Abstract

The bandwidth of a timed language characterizes the quantity of information per time unit (with a finite observation precision $\varepsilon$). The asymptotic behavior of the bandwidth as $\varepsilon \to 0$ classifies timed regular languages in three classes: meager, normal, and obese. Normal timed automata have a bounded frequency of events and some non-punctual transitions, and, up to now, were the only class of timed automata for which no algorithm was available for computing their bandwidth. In this article, we compute the bandwidth of any such automaton in the form $\approxα\log{1/\varepsilon}$. Our approach reduces this problem to computing the best reward-to-cost ratio in a weighted finite graph constructed from the given timed automaton.

Problem

• timed automata
• bandwidth
• normal timed languages
• information rate
• timed regular languages

Innovation

• timed automata
• bandwidth computation
• weighted graph
• reward-to-cost ratio
• asymptotic analysis