Maximal Decidable Fragments of Halpern and Shoham's Modal Logic of Intervals

Angelo Montanari (1) , Gabriele Puppis (2) , Pietro Sala (1)

1 DIMI - Dipartimento di Matematica e Informatica - Universita Udine
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations "meets", met by", "begun by", and "begins" of Allen's interval algebra (AAB B logic). AABB properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BB of Allen's "begun by" / "begins" relations and propositional neighborhood logic AA, in its many variants (including metric ones). We prove that the satisfiability problem for AABB, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AABB turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AABB is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time ows N, Z, and R.

Domaines: Logique en informatique [cs.LO]

Angelo Montanari, Gabriele Puppis, Pietro Sala. Maximal Decidable Fragments of Halpern and Shoham's Modal Logic of Intervals. Proceedings of ICALP 2010, 2010, Bordeaux, France. pp.345-356, ⟨10.1007/978-3-642-14162-1_29⟩. ⟨hal-00717788⟩