Computational Properties of Qualitative Spatial Reasoning: First Results

While the computational properties of qualitative temporal reasoning have been analyzed quite thoroughly, the computational properties of qualitative spatial reasoning are not very well investigated. In fact, almost no completeness results are known for qualitative spatial calculi and no computational complexity analysis has been carried out yet. In this paper, we will focus on the so-called RCC approach and use Bennett's encoding of spatial reasoning in intuitionistic logic in order to show that consistency checking for the topological base relations can be done efficiently. Further, we show that path-consistency is sufficient for deciding global consistency. As a side-effect we prove a particular fragment of propositional intuitionistic logic to be tractable.

[1]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[2]  Alan K. Mackworth.  Consistency in Networks of Relations , 1977, Artif. Intell..

[3]  Bernhard Nebel,et al.  Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra , 1994, JACM.

[4]  Daniel Hernández,et al.  Qualitative Representation of Spatial Knowledge , 1994, Lecture Notes in Computer Science.

[5]  Peter van Beek,et al.  Exact and approximate reasoning about temporal relations 1 , 1990, Comput. Intell..

[6]  Anil Nerode,et al.  Some lectures on intuitionistic logic , 1990 .

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

[8]  Ugo Montanari,et al.  Networks of constraints: Fundamental properties and applications to picture processing , 1974, Inf. Sci..

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

[10]  David S. Johnson,et al.  A Catalog of Complexity Classes , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[11]  J. N. Crossley,et al.  Formal Systems and Recursive Functions , 1963 .

[12]  J. M. BocheŃski,et al.  The Axiomatic Method , 1965 .

[13]  Jörg Hudelmaier,et al.  An O(n log n)-Space Decision Procedure for Intuitionistic Propositional Logic , 1993, J. Log. Comput..

[14]  Saul A. Kripke,et al.  Semantical Analysis of Intuitionistic Logic I , 1965 .

[15]  J. Davenport.  Editor , 1960 .

[16]  A. Tarski.  What is Elementary Geometry , 1959 .

[17]  Anthony G. Cohn,et al.  Modelling Topological and Metrical Properties in Physical Processes , 1989, KR.

[18]  Max J. Egenhofer,et al.  Reasoning about Binary Topological Relations , 1991, SSD.

[19]  Henry A. Kautz,et al.  Constraint Propagation Algorithms for Temporal Reasoning , 1986, AAAI.

[20]  Piergiorgio Odifreddi,et al.  Logic and computer science , 1990 .

[21]  Anthony G. Cohn,et al.  A Spatial Logic based on Regions and Connection , 1992, KR.

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

[23]  Anthony G. Cohn,et al.  Computing Transivity Tables: A Challenge For Automated Theorem Provers , 1992, CADE.