On the Expressive Power of Temporal Logic for Infinite Words

We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the “restricted” temporal logic (RTL), obtained by discarding the “until” operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads to a polynomial time algorithm to check whether the formal language accepted by an n-state deterministic automaton is RTL-expressible. Temporal logic is a particular case of modal logic. It was introduced by Pnueli [16] in connection with applications to the specification, development and verification of possibly parallel or non-deterministic processes. This logical language admits several variations, one of them being propositional linear temporal logic (PTL). It uses three connectives suggestively called “next”, “eventually” and “until”. In this paper we are interested in the descriptive power of propositional linear temporal logic and of a restriction of temporal logic (RTL) obtained by considering only the operators “next” and “eventually”. In both cases, we interpret temporal logic on finite words only. In this case, a temporal formula defines a set of words (that is, a formal language) and our problem is to determine precisely which formal languages can be specified in this way. In the case of PTL, the solution has been known for some time, as a consequence of a series of deep results. Indeed, Kamp [6] has shown that PTL is expressively equivalent to first-order logic when interpreted on words. Next, McNaughton [10] proved that a formal language is first-order definable if and only if it is star-free. Finally, star-free languages are characterized by a ∗Research on this paper was partially supported by PRC “Mathématiques et Informatique”.