A mathematical framework for the semantics of symbolic languages representing periodic time

In several areas, including Temporal DataBases (TDB), Presburger arithmetic has been chosen as a standard reference for the semantics of languages representing periodic time, and to study their expressiveness. On the other hand, the proposal of most symbolic languages in the AI literature has not been paired with an adequate semantic counterpart, making the task of studying the expressiveness of such languages and of comparing them a very complex one. In this paper, we first define a representation language which enables us to handle each temporal point as a complex object enriched with all the structure it is immersed in, and then we use it in order to provide a Presburger semantics for classes of symbolic languages coping with periodicity. Finally, we use the semantics to compare a few AI and TDB symbolic approaches.