Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

Authors:
Angelo Montanari;Gabriele Puppis;Pietro Sala
Affiliations:
Department of Mathematics and Computer Science, Udine University, Italy;Computing Laboratory, Oxford University, England;Department of Mathematics and Computer Science, Udine University, Italy
Venue:
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
Year:
2010

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Abstract

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 (AABB 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 flows N, Z, Q, and R.