Sub-propositional Fragments of the Interval Temporal Logic of Allen's Relations

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. The most influential propositional interval-based logic is probably Halpern and Shoham’s Modal Logic of Time Intervals, a.k.a. HS. While most studies focused on the computational properties of the syntactic fragments that arise by considering only a subset of the set of modalities, the fragments that are obtained by weakening the propositional side have received very scarce attention. Here, we approach this problem by considering various sub-propositional fragments of HS, such as the so-called Horn, Krom, and core fragment. We prove that the Horn fragment of HS is undecidable on every interesting class of linearly ordered sets, and we briefly discuss the difficulties that arise when considering the other fragments.