Classifying Discrete Temporal Properties

This paper surveys recent results on the classiication of discrete temporal properties, gives an introduction to the methods that have been developed to obtain them, and explains the connections to the theory of nite automata, the theory of nite semigroups, and to rst-order logic. The salient features of temporal logic 1 are its modalities, which allow it to express temporal relationships. So it is only natural to investigate how and how much each individual modality contributes to the expressive power of temporal logic. One would like to be able to answer questions like: Can a given property be expressed without using the modality nextt? What properties can be expressed using formulas where the nesting depth in the modality untill is at most 2? This survey reports on recent progress on answering such questions, presenting results from the papers 3], 2], 14], 11], and 16] and the thesis 17]. The results fall into three categories: (A) characterizations of fragments of future temporal logic, where a fragment is determined by which future modalities (modalities referring to the future and present only) are allowed in building formulas; (B) characterizations of symmetric fragments, where with each modality its symmetric past/future counterpart is allowed; (C) characterization of the levels of the until hierarchy, where the nesting depth in the untill modality required to express a property in future temporal logic determines its level. An almost complete account of the results from category (A) will be given in Sections 2 through 4, including full proofs. These results can be obtained with a reasonable eeort in an automata-theoretic framework and the methods used to obtain them are fundamental to the whole subject, in particular, to the results from categories (B) and (C). The results from these two categories are presented in Sections 5 and 6 without going into details of the proofs, which would require a thorough background in nite semigroup theory. In computer science applications, temporal formulas are interpreted in ((nite or innnite) sequences (colored discrete linear orderings), which are nothing else than words (strings or !-words). Therefore, the set of models of a temporal formulaathe property deened by ittcan be viewed as a formal language, in fact, a regular language. In other words, characterizing a fragment of temporal logic amounts to characterizing a certain class of regular languages. 1 I use temporal logicc as a synonym for propositional linear-time temporal logic..