Undecidable Problems About Timed Automata

TL;DR

This paper proves several fundamental decision problems about timed automata are undecidable, including determinization, language complement, clock minimization, and language shuffle, with some problems being highly undecidable beyond the arithmetical hierarchy.

Contribution

It establishes the undecidability of key problems in timed automata theory, advancing understanding of their computational limits.

Findings

Decidability of automaton determinization is impossible.

Complement of a timed regular language cannot be decided.

Minimizing clocks in a timed automaton is undecidable.

Abstract

We solve some decision problems for timed automata which were recently raised by S. Tripakis in [ Folk Theorems on the Determinization and Minimization of Timed Automata, in the Proceedings of the International Workshop FORMATS'2003, LNCS, Volume 2791, p. 182-188, 2004 ] and by E. Asarin in [ Challenges in Timed Languages, From Applied Theory to Basic Theory, Bulletin of the EATCS, Volume 83, p. 106-120, 2004 ]. In particular, we show that one cannot decide whether a given timed automaton is determinizable or whether the complement of a timed regular language is timed regular. We show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. It is also undecidable whether the shuffle of two timed regular languages is timed regular. We show that in the case of timed B\"uchi automata accepting infinite timed words some of these problems are Pi^1_1-hard, hence highly undecidable (located beyond the arithmetical hierarchy).