Abstract
It is widely accepted that spatial reasoning plays a central role in artificial intelligence, for it has a wide variety of potential applications, e.g., in robotics, geographical information systems, and medical analysis and diagnosis. While spatial reasoning has been extensively studied at the algebraic level, modal logics for spatial reasoning have received less attention in the literature. In this paper we propose a new modal logic, called spatial propositional neighborhood logic (SpPNL for short) for spatial reasoning through directional relations. We study the expressive power of SpPNL, we show that it is able to express meaningful spatial statements, we prove a representation theorem for abstract spatial frames, and we devise a (non-terminating) sound and complete tableaux-based deduction system for it. Finally, we compare SpPNL with the well-known algebraic spatial reasoning system called rectangle algebra.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)
Aiello, M., van Benthem J.: A modal walk through space. J. Appl. Non-Class. Log. 12(3–4), 319–363 (2002)
Balbiani, P., Condotta, J.F., Fariñas del Cerro L.: A model for reasoning about bidemsional temporal relations. In: Proceedings of the Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR’98), pp. 124–130 (1998)
Balbiani, P., Condotta, J.F., Fariñas del Cerro L.: A new tractable subclass of the rectangle algebra. In: Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence (IJCAI-99), pp. 442–447 (1999)
Bennett, B., Cohn, A.G., Wolter, F., Zakharyaschev, M.: Multi-dimensional modal logic as a framework for spatio-temporal reasoning. Appl. Intell. 17(3), 239–251 (2002)
Bennett, B.: Spatial reasoning with propositional logics. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) KR’94: Principles of Knowledge Representation and Reasoning, pp. 51–62. Morgan Kaufmann, San Francisco, CA (1994)
Bennett, B.: Modal logics for qualitative spatial reasoning. Log. J. IGPL 4(1), 23–45 (1996)
Bresolin, D., Montanari, A., Sciavicco G.: An optimal decision procedure for right propositional neighborhood logic. J. Autom. Reason. 4(3), 305–330 (2006)
Bresolin, D., Montanari, A., Goranko, V., Sciavicco G.: On decidability and expressiveness of propositional neighborhood logics. In: Proceedings of the Symposium on Logical Foundations of Computer Science, LNCS, vol. 4514, pp. 84–99 (2007)
Cohn, A.G., Hazarika, S.M.: Qualitative spatial representation and reasoning: an overview. Fundam. Inform. 46(1–2), 1–29 (2001)
Egenhofer, M.J., Franzosa, R.D.: Point set topological relations. Int. J. Geogr. Inf. Syst. 5, 161–174 (1991)
Freksa, C., Habel, C., Friedrich, Wender K., (eds.): Spatial Cognition, An Interdisciplinary Approach to Representing and Processing Spatial Knowledge, vol. 1404 of LNCS. Springer (1998)
Güesgen, H.: Spatial reasoning based on Allen’s temporal logic. Technical Report ICSI TR89-049, International Computer Science Institute (1989)
Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood temporal logics. J. Univers. Comput. Sci. 9(9), 1137–1167 (2003)
Levinson, S.C.: Frames of reference and molyneux’s question: cross-linguistic evidence. In: Bloom, P., Peterson, M., Nadel, L., Garrett, M. (eds.) Language and Space. MIT Press, MA (1996)
Lutz, C., Wolter, F.: Modal logics of topological relations. In: Proceedings of Advances in Modal Logics 2004 (2004)
Mukerjee, A., Joe, G.: A qualitative model for space. In: Proceedings of the Eighth National Conference on Artificial Intelligence (AAAI-90), pp. 721–727 (1990)
Morales, A., Navarrete, I., Sciavicco, G.: Proof methods for spatial propositional neighborhood logic. In: Guesguen, H.W., Ligozat, G., Rodríguez, R.V. (eds.) Proceedings of the IJCAI-07 Workshop on Spatial and Temporal Reasoning, pp. 80–89 (2007)
Marx, M., Reynolds, M.: Undecidability of compass logic. J. Log. Comput. 9(6), 897–914 (1999)
Montanari, A., Sciavicco, G.: A decidability proof for propositional neighborhood logic. Contributed Talk, Trends in Logics III Conference, Warsaw–Ruciane Nida (Poland) (2005)
Morales, A., Sciavicco, G.: Una comparativa entre el algebra de rectangulos y la logica SpPNL (in Spanish). In: Proceedings of the Conference ‘Campus Multidisciplinar en Perception e Inteligencia’ (CMPI-06), pp. 576–587 (2006)
Morales, A., Sciavicco, G.: Using time for spatial reasoning: spatial propositional neighborhood logic. In: Proceedings of the 13th International Symposium on Temporal Representation and Reasoning (TIME-06), pp. 50–57 (2006)
Montanari, A., Sciavicco, G., Vitacolonna, N.: Decidability of interval temporal logics over split-frames via granularity. In: Proceedings of the European Conference on Logic in Artificial Intelligence 2002, vol. 2424 of LNAI, pp. 259–270. Springer (2002)
Nutt, W.: On the translation of qualitative spatial reasoning problems into modal logics. In: KI ’99: Proceedings of the 23rd Annual German Conference on Artificial Intelligence, pp. 113–124. Springer (1999)
Randell, D.A., Cui, Z., Cohn, A.: A spatial logic based on regions and connection. In: Nebel, B., Rich, C., Swartout, W. (eds.) Proceedings of the Third International Conference on Principles of Knowledge Representation and Reasoning, pp. 165–176. Morgan Kaufmann, San Mateo, CA (1992)
Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame J. Form. Log. 31(4), 529–547 (1990)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morales, A., Navarrete, I. & Sciavicco, G. A new modal logic for reasoning about space: spatial propositional neighborhood logic. Ann Math Artif Intell 51, 1–25 (2007). https://doi.org/10.1007/s10472-007-9083-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-007-9083-0