On the Expressiveness of the Interval Logic of Allen's Relations Over Finite and Discrete Linear Orders

Interval temporal logics take time intervals, instead of time instants, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). A complete classification of all HS fragments with respect to their relative expressive power has been recently given for the classes of all linear orders and of all dense linear orders. The cases of discrete and finite linear orders turn out to be much more involved. In this paper, we make a significant step towards solving the classification problem over those classes of linear orders. First, we illustrate various non-trivial temporal properties that can be expressed by HS fragments when interpreted over finite and discrete linear orders; then, we provide a complete set of definabilities for the HS modalities corresponding to the Allen’s relations meets, later, begins, finishes, and during, as well as the ones corresponding to their inverse relations. Given the results presented here, the only missing piece of the expressiveness puzzle is that of the definabilities for the modality corresponding to the Allen relation overlaps (those for the inverse relation overlapped by would immediately follow by symmetry).