Model checking for fragments of Halpern and Shoham's interval temporal logic based on track representatives

Model checking allows one to automatically verify a specification of the expected properties ofa system against a formal model of its behaviour (generally, a Kripke structure). Point-basedtemporal logics, such as LTL, CTL, and CTL∗, that describe how the system evolves state-by-state,are commonly used as specification languages. They proved themselves quite successful in a varietyof application domains. However, properties constraining the temporal ordering of temporallyextended events as well as properties involving temporal aggregations, which are inherently interval-based, can not be properly dealt with by them. Interval temporal logics (ITLs), that take intervalsas their primitive temporal entities, turn out to be well-suited for the specification and verification ofinterval properties of computations (we interpret all the tracks of a Kripke structure as computationintervals).In this paper, we study the model checking problem for some fragments of Halpern andShoham’s modal logic of time intervals (HS). HS features one modality for each possible ordering re-lation between pairs of intervals (the so-called Allen’s relations). First, we describe an EXPSPACEmodel checking algorithm for the HS fragment of Allen’s relations meets, met-by, starts, started-by,and finishes, which exploits the possibility of finding, for each track (of unbounded length), anequivalent bounded-length track representative. While checking a property, it only needs to con-sider tracks whose length does not exceed the given bound. Then, we prove the model checkingproblem for such a fragment to be PSPACE-hard. Finally, we identify other well-behaved HSfragments which are expressive enough to capture meaningful interval properties of systems, suchas mutual exclusion, state reachability, and non-starvation, and whose computational complexityis less than or equal to that of LTL.