Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals

Abstract Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as first-class citizens. Their expressive power and computational behavior mainly depend on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we consider all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the rationals, and we provide a complete classification of them in terms of their expressiveness and computational complexity by solving the last few open problems.

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

[2]  Dario Della Monica,et al.  The Importance of the Past in Interval Temporal Logics: The Case of Propositional Neighborhood Logic , 2012, Logic Programs, Norms and Action.

[3]  Rosella Gennari,et al.  An AI-Based Process for Generating Games from Flat Stories , 2013, SGAI Conf..

[4]  Gabriele Puppis,et al.  Maximal Decidable Fragments of Halpern and Shoham's Modal Logic of Intervals , 2010, ICALP.

[5]  P. S. Thiagarajan,et al.  A Logical Study of Distributed Transition Systems , 1995, Inf. Comput..

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

[7]  Zhou Chaochen,et al.  Duration Calculus: A Formal Approach to Real-Time Systems , 2004 .

[8]  Peter Roper,et al.  Intervals and tenses , 1980, J. Philos. Log..

[9]  Shaban Laban,et al.  RISMA: A Rule-based Interval State Machine Algorithm for Alerts Generation, Performance Analysis and Monitoring Real-Time Data Processing , 2013 .

[10]  Gabriele Puppis,et al.  A decidable weakening of Compass Logic based on cone-shaped cardinal directions , 2015, Log. Methods Comput. Sci..

[11]  Valentin Goranko,et al.  A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases , 2016, Acta Informatica.

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

[13]  Angelo Montanari,et al.  Interval vs. Point Temporal Logic Model Checking: an Expressiveness Comparison , 2016, FSTTCS.

[14]  Mark Reynolds,et al.  THE TEMPORAL LOGIC OF TWO DIMENSIONAL MINKOWSKI SPACETIME IS DECIDABLE , 2015, The Journal of Symbolic Logic.

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

[16]  Guido Sciavicco,et al.  On the Expressiveness of the Interval Logic of Allen's Relations Over Finite and Discrete Linear Orders , 2014, JELIA.

[17]  Ilya Shapirovsky.  On PSPACE-decidability in Transitive Modal Logic , 2004, Advances in Modal Logic.

[18]  Davide Bresolin,et al.  Interval temporal logics over strongly discrete linear orders: Expressiveness and complexity , 2014, Theor. Comput. Sci..

[19]  Davide Bresolin,et al.  The dark side of interval temporal logic: marking the undecidability border , 2013, Annals of Mathematics and Artificial Intelligence.

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

[21]  Angelo Montanari,et al.  An Optimal Tableau System for the Logic of Temporal Neighborhood over the Reals , 2012, 2012 19th International Symposium on Temporal Representation and Reasoning.

[22]  Benjamin Charles Moszkowski.  Reasoning about Digital Circuits , 1983 .

[23]  Alessio Lomuscio,et al.  Model Checking Multi-Agent Systems against Epistemic HS Specifications with Regular Expressions , 2016, KR.

[24]  Gabriele Puppis,et al.  Decidability of the Interval Temporal Logic $\mathsf{A\bar{A}B\bar{B}}$ over the Rationals , 2014, MFCS.

[25]  Angelo Montanari,et al.  Checking interval properties of computations , 2014, Acta Informatica.

[26]  Davide Bresolin,et al.  What's Decidable about Halpern and Shoham's Interval Logic? The Maximal Fragment ABBL , 2011, 2011 IEEE 26th Annual Symposium on Logic in Computer Science.

[27]  Davide Bresolin,et al.  Optimal Tableau Systems for Propositional Neighborhood Logic over All, Dense, and Discrete Linear Orders , 2011, TABLEAUX.

[28]  Davide Bresolin,et al.  On the Complexity of Fragments of the Modal Logic of Allen's Relations over Dense Structures , 2015, LATA.

[29]  Richard T. Snodgrass,et al.  Reconciling Point-based and Interval-based Semantics in Temporal Relational Databases : A Proper Treatment of the Telic / Atelic Distinction , 2001 .

[30]  Peter van Emde Boas,et al.  The Convenience of Tilings , 2019, complexity, logic, and recursion theory.

[31]  Philippe Schnoebelen,et al.  Lossy Counter Machines Decidability Cheat Sheet , 2010, RP.