Specifying timed state sequences in powerful decidable logics and timed automata

Abstract

A monadic second-order language, denoted by $\mathcal{L}d$, is introduced for the specification of sets of timed state sequences. A fragment of $\mathcal{L}d$, denoted by

, is proved to be expressively complete for timed automata (Alur and Dill), i.e., every timed regular language is definable by a

-formula and every

-formula defines a timed regular language. As a consequence the satisfiability problem for

is decidable.

Timed temporal logics are shown to be effectively embeddable into

and hence turn out to have a decidable theory. This applies to TL_Г (Manna and Pnueli) and EMITL_p, which is obtained by extending the logic MITL_p (Alur and Henzinger) by automata operators (Sistla, Vardi, and Wolper).

For every positive natural number k the full monadic second-order logic $\mathcal{L}d$ and

are equally expressive modulo the set of timed state sequences of variability ≤ k. Therefore the $\mathcal{L}d$-theory of the set of timed state sequences of variability ≤ k is decidable.

This paper was written during the author's stay at Laboratoire Bordelais de Recherche en Informatique (LaBRI). The presented results are part of the author's dissertation [24].