A Tableau-Based System for Spatial Reasoning about Directional Relations

  • Davide Bresolin
  • Angelo Montanari
  • Pietro Sala
  • Guido Sciavicco
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5607)


The management of qualitative spatial information is an important research area in computer science and AI. Modal logic provides a natural framework for the formalization and implementation of qualitative spatial reasoning. Unfortunately, when directional relations are considered, modal logic systems for spatial reasoning usually turn out to be undecidable (often even not recursively enumerable). In this paper, we give a first example of a decidable modal logic for spatial reasoning with directional relations, called Weak Spatial Propositional Neighborhood Logic (WSpPNL for short). WSpPNL features two modalities, respectively an east modality and a north modality, to deal with non-empty rectangles over ℕ ×ℕ. We first show the NEXPTIME-completeness of WSpPNL, then we develop an optimal tableau method for it.


Modal Logic Directional Relation Propositional Variable Topological Relation Spatial Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aiello, M., van Benthem, J.: A modal walk through space. Journal of Applied Non-Classical Logic 12(3-4), 319–363 (2002)CrossRefGoogle Scholar
  2. 2.
    Balbiani, P., Condotta, J.F., Fariñas del Cerro, L.: A model for reasoning about bidimensional temporal relations. In: Proc. of the Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR 1998), pp. 124–130 (1998)Google Scholar
  3. 3.
    Balbiani, P., Condotta, J.F., Fariñas del Cerro, L.: A new tractable subclass of the rectangle algebra. In: Proc. of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-1999), pp. 442–447 (1999)Google Scholar
  4. 4.
    Bennett, B.: Spatial reasoning with propositional logics. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) Proc. of the Fourth International Conference on Principles of Knowledge Representation and Reasoning (KR 1994), pp. 51–62. Morgan Kaufmann, San Francisco (1994)Google Scholar
  5. 5.
    Bennett, B.: Modal logics for qualitative spatial reasoning. Journal of the Interest Group in Pure and Applied Logic (IGPL) 4(1), 23–45 (1996)zbMATHGoogle Scholar
  6. 6.
    Bennett, B., Cohn, A.G., Wolter, F., Zakharyaschev, M.: Multi-dimensional modal logic as a framework for spatio-temporal reasoning. Applied Intelligence 17(3), 239–251 (2002)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bresolin, D., Montanari, A., Sciavicco, G.: An optimal decision procedure for right propositional neighborhood logic. Journal of Automated Reasoning 4(3), 305–330 (2007)MathSciNetGoogle Scholar
  8. 8.
    Chittaro, L., Montanari, A.: Temporal representation and reasoning in artificial intelligence: Issues and approaches. Annals of Mathematics and Artificial Intelligence 28(1-4), 47–106 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cohn, A.G., Hazarika, S.M.: Qualitative spatial representation and reasoning: An overview. Fundamenta Informaticae 46(1-2), 1–29 (2001)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood temporal logics. Journal of Universal Computer Science 9(9), 1137–1167 (2003)MathSciNetGoogle Scholar
  11. 11.
    Güsgen, H.: Spatial reasoning based on Allen’s temporal logic. Technical Report ICSI TR89-049, International Computer Science Institute (1989)Google Scholar
  12. 12.
    Kontchakov, R., Pratt-Hartmann, I., Wolter, F., Zakharyaschev, M.: On the computational complexity of spatial logics with connectedness constraints. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS, vol. 5330, pp. 574–589. Springer, Heidelberg (2008)Google Scholar
  13. 13.
    Lutz, C., Wolter, F.: Modal logics of topological relations. Logical Methods in Computer Science 2(2) (2006)Google Scholar
  14. 14.
    Marx, M., Reynolds, M.: Undecidability of compass logic. Journal of Logic and Computation 9(6), 897–914 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Morales, A., Navarrete, I., Sciavicco, G.: A new modal logic for reasoning about space: spatial propositional neighborhood logic. Annals of Mathematics and Artificial Intelligence 51(1), 1–25 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mukerjee, A., Joe, G.: A qualitative model for space. In: Proc. of the of the Eighth National Conference on Artificial Intelligence (AAAI-1990), pp. 721–727 (1990)Google Scholar
  17. 17.
    Nutt, W.: On the translation of qualitative spatial reasoning problems into modal logics. In: Burgard, W., Christaller, T., Cremers, A.B. (eds.) KI 1999. LNCS, vol. 1701, pp. 113–124. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame Journal of Formal Logic 31(4), 529–547 (1990)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Davide Bresolin
    • 1
  • Angelo Montanari
    • 2
  • Pietro Sala
    • 2
  • Guido Sciavicco
    • 3
  1. 1.Department of Computer ScienceUniversity of VeronaVeronaItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of UdineUdineItaly
  3. 3.Department of Information, Engineering and CommunicationsUniversity of MurciaMurciaSpain

Personalised recommendations