Interval vs. Point Temporal Logic Model Checking: An Expressiveness Comparison

In recent years, model checking with interval temporal logics is emerging as a viable alternative to model checking with standard point-based temporal logics, such as LTL, CTL, CTL∗, and the like. The behavior of the system is modeled by means of (finite) Kripke structures, as usual. However, while temporal logics which are interpreted “point-wise” describe how the system evolves state-by-state, and predicate properties of system states, those which are interpreted “interval-wise” express properties of computation stretches, spanning a sequence of states. A proposition letter is assumed to hold over a computation stretch (interval) if and only if it holds over each component state (homogeneity assumption). A natural question arises: is there any advantage in replacing points by intervals as the primary temporal entities, or is it just a matter of taste? In this article, we study the expressiveness of Halpern and Shoham’s interval temporal logic (HS) in model checking, in comparison with those of LTL, CTL, and CTL∗. To this end, we consider three semantic variants of HS: the state-based one, introduced by Montanari et al. in [30, 34], that allows time to branch both in the past and in the future, the computation-tree-based one, that allows time to branch in the future only, and the trace-based variant, that disallows time to branch. These variants are compared among themselves and to the aforementioned standard logics, getting a complete picture. In particular, we show that HS with trace-based semantics is equivalent to LTL (but at least exponentially more succinct), HS with computation-tree-based semantics is equivalent to finitary CTL∗, and HS with state-based semantics is incomparable with all of them (LTL, CTL, and CTL∗).