Undecidability of the Logic of Overlap Relation over Discrete Linear Orderings

Authors:
Davide Bresolin;Dario Della Monica;Valentin Goranko;Angelo Montanari;Guido Sciavicco
Affiliations:
University of Verona, Verona, Italy;University of Udine, Udine, Italy;Technical University of Denmark, Lyngby, Denmark;University of Udine, Udine, Italy;University of Murcia, Murcia, Spain
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2010

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Abstract

The validity/satisfiability problem for most propositional interval temporal logics is (highly) undecidable, under very weak assumptions on the class of interval structures in which they are interpreted. That, in particular, holds for most fragments of Halpern and Shoham's interval modal logic HS. Still, decidability is the rule for the fragments of HS with only one modal operator, based on an Allen's relation. In this paper, we show that the logic O of the Overlap relation, when interpreted over discrete linear orderings, is an exception. The proof is based on a reduction from the undecidable octant tiling problem. This is one of the sharpest undecidability result for fragments of HS.