A Tableau-Based System for Spatial Reasoning about Directional Relations Davide Bresolin 1,⋆ , Angelo Montanari 2 , Pietro Sala 2 , and Guido Sciavicco 3,⋆⋆ 1 Department of Computer Science, University of Verona, Verona, Italy 2 Department of Mathematics and Computer Science, University of Udine, Udine, Italy 3 Department of Information, Engineering and Communications, University of Murcia, Murcia, Spain Abstract. The management of qualitative spatial information is an im- portant research area in computer science and AI. Modal logic provides a natural framework for the formalization and implementation of qual- itative 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, respec- tively 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. 1 Introduction The main goal of qualitative spatial representation and reasoning techniques is to capture common-sense knowledge about space and to provide a calculus of spatial information without referring to a quantitative model. Even though quantita- tive models provide a more accurate description of spatial domains, qualitative models are often the best or the only choice. In many cases, indeed, there is a lack of quantitative models or existing ones turn out to be intractable. In addi- tion, qualitative models make it possible to cope with spatial data indeterminacy and to reason about incomplete spatial knowledge. The problem of representing and reasoning about qualitative spatial information can be viewed under three different points of view: (i) the algebraic perspective, that is, purely existential theories formulated as constraint satisfaction systems over jointly exhaustive and mutually disjoint sets of topological, directional, or combined relations; (ii) the first-order perspective, that is, first-order theories of topological, directional, or ⋆ Davide Bresolin has been partially supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant no. INFSO-ICT-223844. ⋆⋆ Guido Sciavicco has been partially supported by the Spanish projects PET2006-0406 and TIN2006-15460-C04-01. M. Giese and A. Waaler (Eds.): TABLEAUX 2009, LNAI 5607, pp. 123–137, 2009. c  Springer-Verlag Berlin Heidelberg 2009