Interval Temporal Logics Model Checking

Model checking is a successful technique widely used in formal verification. Given a model of a system and a formula specifying a desired property of it, one can verify whether the system satisfies the property by checking the formula against the model. Distinctive features of model checking are: (i) it is a fully automatic process, (ii) it exaustively checks all the possible behaviours of the system, and (iii) it produces a counterexample, in case the property is violated. Systems are usually modeled as (finite) Kripke structures, that is, state-transition systems, and their properties are specified by formulas of point-based temporal logics, such as LTL, CTL, and the like. These logics allow one to express requirements on computation states and their relationships; however, they are not well suited to specify conditions on computation stretches, which come into play when dealing with, for instance, actions with duration, accomplishments, and temporal aggregations. To overcome the limitations of point-based logics, one can resort to interval temporal logics (ITLs), that assume time intervals,instead of time points, as their primitive entities. The most well-known ITL is Halpern and Shoham's modal logic of time intervals HS [4], which features one modality for each possible ordering relation between a pair of intervals, apart from equality. The satisfiability problem for HS has been studied in [4], and it turns out to be highly undecidable forall relevant (classes of) linear orders. The same holds for most fragments of it [2]; luckily, some meaningful exceptions exist, including the logic of temporal neighbourhood and the temporal logic of sub-intervals.