Which fragments of the interval temporal logic HS are tractable in model checking?

Since the 80s, model checking (MC) has been applied to the automatic verification of hardware/software systems. Point-based temporal logics, such as , , ⁎ , and the like, are commonly used in MC as the specification language; however, there are some inherently interval-based properties of computations, e.g., temporal aggregations and durations, that cannot be properly dealt with by these logics, as they model a state-by-state evolution of systems. Recently, an MC framework for the verification of interval-based properties of computations, based on Halpern and Shoham's interval temporal logic ( , for short) and its fragments, has been proposed and systematically investigated. In this paper, we focus on the boundaries that separate tractable and intractable fragments in MC. We first prove that MC for the logic of Allen's relations started-by and finished-by is provably intractable, being Expspace-hard. Such a lower bound immediately propagates to full . Then, in contrast, we show that other noteworthy fragments, i.e., the logic (resp., ) of Allen's relations meets, met-by, starts (resp., finishes), and started-by (resp., finished-by), are well-behaved, and turn out to have the same complexity as (Pspace-complete). Halfway are the fragments and , whose Expspace membership and Pspace hardness are already known. Here, we give an original proof of Expspace membership, that substantially simplifies the complexity of the constructions previously used for such a result. Contraction techniques—suitably tailored to each fragment—are at the heart of our results, enabling us to prove a pair of remarkable small-model properties.