Minimizable timed automata

Abstract

State minimization plays a fundamental role in both classical automata theory and in the theory of reactive systems. Many algorithms and results are based on the fact that for each finite automaton there exists an equivalent minimum state automaton that can be effectively computed and that is unique up to isomorphism.

Timed safety automata (TSA's) [5], finite automata with clocks, have been used extensively for the specification and verification of real-time systems. However, there does not always exist a unique minimum state TSA that is equivalent to a given TSA. This problem occurs irrespective of the selected notions of state (including or excluding clock values) and equivalence on states (language equivalence, bisimulation equivalence, etc.).

Henzinger, Kopke and Wong-Toi [4] convincingly showed that if states do not include clock values, state minimization for timed automata is neither useful nor interesting. In this paper, we discuss state minimization for states that do include clock values, i.e., at the semantic level, and work in bisimulation equivalence. In this setting, a timed automaton is minimal when there does not exist a pair of bisimilar but distinct states in the transition system induced by the timed automaton.

We present a new model of minimizable timed automata (MTA's), a variant of the TSA model, and prove that

• 1. The MTA and TSA model are equally expressive in the sense that for each MTA there exists a bisimilar TSA and for each TSA there exists a bisimilar MTA.

• 2. For each MTA there exists a bisimilar minimal MTA that can be effectively computed and that is unique up to isomorphism.

Research supported by the Netherlands Organization for Scientific Research (NWO) under contract SION 612-33-006 and by the HCM network EXPRESS.