The light side of Interval Temporal Logics: the Bernays-Schoenfinkel's fragment of CDT

Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the last years. Even though most interval logics turn out to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing suitable semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the nesting degree of modalities. The decidability of the satisfiability problem for the resulting CDT fragment is proved by embedding it into a well-known decidable prefix quantifier class of first-order logic, namely, the Bernays-Scḧonfinkel's class. In addition, we show that such a fragment is in fact NP-complete (the Bernays-Scḧonfinkel's class is NEXPTIME-complete), and that any natural extension of it is undecidable.