Decidability of the Logics of the Reflexive Sub-interval and Super-interval Relations over Finite Linear Orders

An interval temporal logic is a propositional, multi-modal logic interpreted over interval structures of partial orders. The semantics of each modal operator are given in the standard way with respect to one of the natural accessibility relations defined on such interval structures. In this paper, we consider the modal operators based on the (reflexive) sub-interval relation and the (reflexive) super-interval relation. We show that the satisfiability problems for the interval temporal logics featuring either or both of these modalities, interpreted over interval structures of finite linear orders, are all PSPACE-complete. These results fill a gap in the known complexity results for interval temporal logics.