Time Granularities and Ultimately Periodic Automata

The relevance of the problem of managing periodic phenomena is widely recognized in the area of knowledge representation and reasoning. One of the most effective attempts at dealing with this problem has been the addition of a notion of time granularity to knowledge representation systems. Different formalizations of such a notion have been proposed in the literature, following algebraic, logical, string-based, and automaton-based approaches. In this paper, we focus our attention on the automaton-based one, which allows one to represent a large class of granularities in a compact and suitable to algorithmic manipulation form. We further develop such an approach to make it possible to deal with (possibly infinite) sets of granularities instead of single ones. We define a new class of automata, called Ultimately Periodic Automata, we give a characterization of their expressiveness, and we show how they can be used to encode and to solve a number of fundamental problems, such as the membership problem, the equivalence problem, and the problem of granularity comparison. Moreover, we give an example of their application to a concrete problem taken from clinical medicine.