An Optimal Decision Procedure for MPNL over the Integers

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reasoning over interval structures, where the truth of formulae is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability p roblem for Metric Propositional Neighborhood Logic (MPNL). MPNL features two modalities to access intervals “to the left” and “to the right” of the current one, respectively, plus an infinite set of length constraints. MPNL, interpreted over the naturals, has been recently shown to be decidable by a doubly exponential procedure. We improve such a result by proving that MPNL is actually EXPSPACE-complete (even when length constraints are encoded in binary), when interpreted over fi nite structures, the naturals, and the integers, by developing an EXPSPACE decision procedure for MPNL over the integers, which can be easily tailored to finite linear orders and the naturals (EXP SPACE-hardness was already known).

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

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

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

[4]  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.

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

[6]  Yde Venema,et al.  Expressiveness and Completeness of an Interval Tense Logic , 1990, Notre Dame J. Formal Log..

[7]  Davide Bresolin,et al.  An Optimal Tableau-Based Decision Algorithm for Propositional Neighborhood Logic , 2007, STACS.

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

[9]  Dario Della Monica.  Expressiveness, decidability, and undecidability of interval temporal logic , 2011 .

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

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

[12]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[13]  Davide Bresolin,et al.  Metric propositional neighborhood logics on natural numbers , 2011, Software & Systems Modeling.

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