A Decidable Spatial Generalization of Metric Interval Temporal Logic

Temporal reasoning plays an important role in artificial intelligence. Temporal logics provide a natural framework for its formalization and implementation. A standard way of enhancing the expressive power of temporal logics is to replace their unidimensional domain by a multidimensional one. In particular, such a dimensional increase can be exploited to obtain spatial counterparts of temporal logics. Unfortunately, it often involves a blow up in complexity, possibly losing decidability. In this paper, we propose a spatial generalization of the decidable metric interval temporal logic RPNL+INT, called Directional Area Calculus (DAC). DAC features two modalities, that respectively capture (possibly empty) rectangles to the north and to the east of the current one, and metric operators, to constrain the size of the current rectangle. We prove the decidability of the satisfiability problem for DAC, when interpreted over frames built on natural numbers, and we analyze its complexity. In addition, we consider a weakened version of DAC, called WDAC, which is expressive enough to capture meaningful qualitative and quantitative spatial properties and computationally better.

[1]  Brandon Bennett,et al.  Spatial Reasoning with Propositional Logics , 1994, KR.

[2]  Brandon Bennett,et al.  Modal Logics for Qualitative Spatial Reasoning , 1996, Log. J. IGPL.

[3]  Anthony G. Cohn,et al.  Qualitative Spatial Representation and Reasoning: An Overview , 2001, Fundam. Informaticae.

[4]  Anthony G. Cohn,et al.  Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning , 2002, Applied Intelligence.

[5]  Maarten Marx,et al.  Undecidability of Compass Logic , 1999, J. Log. Comput..

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

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

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

[9]  Frank Wolter,et al.  Logics of metric spaces , 2003, TOCL.

[10]  Johan van Benthem,et al.  Handbook of Spatial Logics , 2007 .

[11]  Soumitra Dutta,et al.  Approximate spatial reasoning: Integrating qualitative and quantitative constraints , 1991, Int. J. Approx. Reason..

[12]  Luis Fariñas del Cerro,et al.  A Model for Reasoning about Bidemsional Temporal Relations , 1998, KR.

[13]  Amitabha Mukerjee,et al.  A Qualitative Model for Space , 1990, AAAI.

[14]  Frank Wolter,et al.  From topology to metric: modal logic and quantification in metric spaces , 2006, Advances in Modal Logic.

[15]  Guido Sciavicco,et al.  A new modal logic for reasoning about space: spatial propositional neighborhood logic , 2007, Annals of Mathematics and Artificial Intelligence.

[16]  Johan van Benthem,et al.  A ModalWalk Through Space , 2002, J. Appl. Non Class. Logics.

[17]  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.

[18]  Davide Bresolin,et al.  A Tableau-Based System for Spatial Reasoning about Directional Relations , 2009, TABLEAUX.

[19]  Luis Fariñas del Cerro,et al.  A New Tractable Subclass of the Rectangle Algebra , 1999, IJCAI.

[20]  Frank Wolter,et al.  Automated Reasoning About Metric and Topology , 2006, JELIA.

[21]  Werner Nutt.  On the Translation of Qualitative Spatial Reasoning Problems into Modal Logics , 1999, KI.

[22]  Frank Wolter,et al.  On the Computational Complexity of Spatial Logics with Connectedness Constraints , 2008, LPAR.

[23]  Jean-François Condotta.  The Augmented Interval and Rectangle Networks , 2000, KR.