Decidability of the Interval Temporal Logic ABB over the Natural Numbers

In this paper, we focus our attention on the interval temporal logic of the Allen's relations ``meets'', ``begins'', and ``begun by'' ($\ABB$ for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties, such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for $\ABB$ over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACE-completeness of the problem.

[1]  Yde Venema,et al.  A Modal Logic for Chopping Intervals , 1991, J. Log. Comput..

[2]  Valentin Goranko,et al.  A Road Map of Interval Temporal Logics and Duration Calculi , 2004, J. Appl. Non Class. Logics.

[3]  Davide Bresolin,et al.  On Decidability and Expressiveness of Propositional Interval Neighborhood Logics , 2007, LFCS.

[4]  Davide Bresolin,et al.  Tableau-based decision procedures for the logics of subinterval structures over dense orderings , 2008 .

[5]  Kamal Lodaya,et al.  Sharpening the Undecidability of Interval Temporal Logic , 2000, ASIAN.

[6]  James F. Allen.  Maintaining knowledge about temporal intervals , 1983, CACM.

[7]  Davide Bresolin,et al.  Decidable and Undecidable Fragments of Halpern and Shoham's Interval Temporal Logic: Towards a Complete Classification , 2008, LPAR.

[8]  Davide Bresolin,et al.  Right Propositional Neighborhood Logic over Natural Numbers with Integer Constraints for Interval Lengths , 2009, 2009 Seventh IEEE International Conference on Software Engineering and Formal Methods.

[9]  Valentin Goranko,et al.  Propositional Interval Neighborhood Temporal Logics , 2003, J. Univers. Comput. Sci..

[10]  D. Gabbay,et al.  Temporal Logic Mathematical Foundations and Computational Aspects , 1994 .

[11]  Guido Sciavicco,et al.  Non-finite Axiomatizability and Undecidability of Interval Temporal Logics with C, D, and T , 2008, CSL.

[12]  Davide Bresolin,et al.  An Optimal Decision Procedure for Right Propositional Neighborhood Logic , 2006, Journal of Automated Reasoning.

[13]  Davide Bresolin,et al.  Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions , 2009, Ann. Pure Appl. Log..

[14]  Yoav Shoham,et al.  A propositional modal logic of time intervals , 1991, JACM.

[15]  Davide Bresolin,et al.  Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders , 2008, JELIA.

[16]  Martin Otto,et al.  Two variable first-order logic over ordered domains , 2001, Journal of Symbolic Logic.

[17]  Carsten Lutz,et al.  Modal Logics of Topological Relations , 2006, Log. Methods Comput. Sci..

[18]  Davide Bresolin,et al.  Tableaux for Logics of Subinterval Structures over Dense Orderings , 2010, J. Log. Comput..

[19]  Gabriele Puppis,et al.  A Decidable Spatial Logic with Cone-Shaped Cardinal Directions , 2009, CSL.