Summary: Realtime Logics:
Complexity and Expressiveness \Lambday
Rajeev Alur Thomas A. Henzinger
AT&T Bell Laboratories Computer Science Department
600 Mountain Avenue Cornell University
Murray Hill, NJ 07974 Ithaca, NY 14853
Abstract. The theory of the natural numbers with linear order and monadic predicates
underlies propositional linear temporal logic. To study temporal logics that are suitable
for reasoning about realtime systems, we combine this classical theory of infinite state
sequences with a theory of discrete time, via a monotonic function that maps every state
to its time. The resulting theory of timed state sequences is shown to be decidable, albeit
nonelementary, and its expressive power is characterized by !regular sets. Several more
expressive variants are proved to be highly undecidable.
This framework allows us to classify a wide variety of realtime logics according to
their complexity and expressiveness. Indeed, it follows that most formalisms proposed
in the literature cannot be decided. We are, however, able to identify two elementary
realtime temporal logics as expressively complete fragments of the theory of timed
state sequences, and we present tableaubased decision procedures for checking validity.
Consequently, these two formalisms are wellsuited for the specification and verification
of realtime systems.