Decidability of the interval temporal logic AABB over the rationals

The classification of the fragments of Halpern and Shoham’s logic with respect to decidability/undecidability of the satisfiability problem is now very close to the end. We settle one of the few remaining questions concerning the fragment AĀBB̄, which comprises Allen’s interval relations “meets” and “begins” and their symmetric versions. We already proved that AĀBB̄ is decidable over the class of all finite linear orders and undecidable over ordered domains isomorphic to N. In this paper, we first show that AĀBB̄ is undecidable over R and over the class of all Dedekind-complete linear orders. We then prove that the logic is decidable over Q and over the class of all linear orders.