Studia Logica

, Volume 69, Issue 3, pp 381–409

A Necessary Relation Algebra for Mereotopology

  • Ivo DÜntsch
  • Gunther Schmidt
  • Michael Winter
Article

Abstract

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T0 topological space with an additional "contact relation" C defined by xCy ⇔ x ∩ ≠ Ø

A (possibly) more general class of models is provided by the Region Connection Calculus (RCC) of Randell et al. We show that the basic operations of the relational calculus on a "contact relation" generate at least 25 relations in any model of the RCC, and hence, in any standard model of mereotopology. It follows that the expressiveness of the RCC in relational logic is much greater than the original 8 RCC base relations might suggest. We also interpret these 25 relations in the the standard model of the collection of regular open sets in the two-dimensional Euclidean plane.

mereology mereotopology relation algebra qualitative spatial reasoning 

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References

  1. [1]
    Asher, N. and Vieu, L. (1995), ‘Toward a geometry of common sense: A semantics and a complete axiomatization of mereotopology’, in C. Mellish, editor, IJCAI 95, Proceedings of the 14th International Joint Conference on Artificial Intelligence.Google Scholar
  2. [2]
    Bennett, B. (1997), ‘Logical representations for automated reasoning about spatial relationships’, Doctoral dissertation, School of Computer Studies, University of Leeds.Google Scholar
  3. [3]
    Bennett, B. (1998), ‘Determining consistency of topological relations’, Constraints 3:213-225.Google Scholar
  4. [4]
    Bennett, B., Isli, A., and Cohn, A. (1997), ‘When does a composition table provide a complete and tractable proof procedure for a relational constraint language?’, in IJCAI 97, Proceedings of the 15th International Joint Conference on Artificial Intelligence.Google Scholar
  5. [5]
    Berghammer, R., Gritzner, T., and Schmidt, G. (1993), ‘Prototyping relational specifications using higher-order objects’, in J. Heering, K. Meinke, B. Möller, and T. Nipkow, editors, Proc. International Workshop on Higher Order Algebra, Logic and Term Rewriting (HOA '93), LNCS 816, pages 56-75, Springer.Google Scholar
  6. [6]
    Berghammer, R. and Karger, B. (1997), ‘Algorithms from relational specifications’, in KAHL, W., and Schmidt, G., editors, Relational Methods in Computer Science, Advances in Computing Science, Springer [9] pages 132-150.Google Scholar
  7. [7]
    Berghammer, R. and Zierer, H. (1986), ‘Relational algebraic semantics of deterministic and nondeterministic programs’, Theoret. Comp. Sci. 43:123-147.Google Scholar
  8. [8]
    Biacino, L. and Gerla, G. (1991), ‘Connection structures’, Notre Dame Journal of Formal Logic 32:242-247.Google Scholar
  9. [9]
    Brink, C., Kahl, W., and Schmidt, G., editors (1997), Relational Methods in Computer Science, Advances in Computing Science, Springer.Google Scholar
  10. [10]
    Chin, L. and Tarski, A. (1951), ‘Distributive and modular laws in the arithmetic of relation algebras’, University of California Publications, 1:341-384.Google Scholar
  11. [11]
    Clarke, B. L. (1981), ‘A calculus of individuals based on “connection”’, Notre Dame Journal of Formal Logic 22:204-218.Google Scholar
  12. [12]
    Cohn, A. G. (1997), ‘Qualitative spatial representation and reasoning techniques’, Research report, School of Computer Studies, University of Leeds.Google Scholar
  13. [13]
    DÜntsch, I. (2000), ‘Contact relation algebras’, in E. Orłowska and A. Szałas (eds.), Relational Methods in Computer Science Applications, pages 113-134, Berlin, Springer-Verlag.Google Scholar
  14. [14]
    DÜntsch, I., Wang, H., and McCloskey, S. (1999), ‘Relation algebras in qualitative spatial reasoning’, Fundamenta Informaticœ 39:229-248.Google Scholar
  15. [15]
    DÜntsch, I., Wang, H., and McCloskey, S. (2001), ‘A relation algebraic approach to the Region Connection Calculus’, Theoretical Computer Science 255:63-83.Google Scholar
  16. [16]
    Egenhofer, M., and Franzosa, R. (1991), ‘Point-set topological spatial relations’, International Journal of Geographic Information Systems 5(2):161-174.Google Scholar
  17. [17]
    Egenhofer, M., and Herring, J. (1991), ‘Categorizing binary topological relationships between regions, lines and points in geographic databases’, Tech. report, Department of Surveying Engineering, University of Maine.Google Scholar
  18. [18]
    Egenhofer, M., and RodrÍguez, A. (1999), ‘Relation algebras over containers and surfaces: An ontological study of a room space’, Spatial Cognition and Computation, to appear.Google Scholar
  19. [19]
    Egenhofer, M., and Sharma, J. (1992), ‘Topological consistency’, in Fifth International Symposium on Spatial Data Handling, Charleston, SC.Google Scholar
  20. [20]
    Engelking, R. (1977), General Topology, Monografie Matematyczne, Polish Scientific Publ., Warszawa.Google Scholar
  21. [21]
    Gotts, N. M. (1996a), ‘An axiomatic approach to topology for spatial information systems’, Research Report 96.25, School of Computer Studies, University of Leeds.Google Scholar
  22. [22]
    Gotts, N. M. (1996b), ‘Topology from a single primitive relation: Defining topological properties and relations in terms of connection’, Research Report 96.23, School of Computer Studies, University of Leeds.Google Scholar
  23. [23]
    JÓnsson, B. (1984), ‘Maximal algebras of binary relations’, Contemporary Mathematics 33:299-307.Google Scholar
  24. [24]
    JÓnsson, B. (1991), ‘The theory of binary relations’, in H. Andréka, J. D. Monk, and I. Németi, editors, Algebraic Logic, volume 54 of Colloquia Mathematica Societatis János Bolyai, pages 245-292, North Holland, Amsterdam.Google Scholar
  25. [25]
    JÓnsson, B., and Tarski, A. (1952), ‘Boolean algebras with operators II’, Amer. J. Math. 74:127-162.Google Scholar
  26. [26]
    Koppelberg, S. (1989), General Theory of Boolean Algebras, volume 1 of Handbook on Boolean Algebras, North Holland.Google Scholar
  27. [27]
    LeŚniewski, S. (1927–1931), ‘O podstawach matematyki’, Przegląd Filozoficzny 30-34.Google Scholar
  28. [28]
    LeŚniewski, S. (1983), ‘On the foundation of mathematics’, Topoi 2:7-52.Google Scholar
  29. [29]
    Lyndon, R. C. (1950), ‘The representation of relational algebras’, Annals of Mathematics (2) 51:707-729.Google Scholar
  30. [30]
    Maddux, R. (1982), ‘Some varieties containing relation algebras’, Transactions of the American Mathematical Society 272:501-526.Google Scholar
  31. [31]
    Pratt, I., and Schoop, D. (1998), ‘A complete axiom system for polygonal mereotopology of the real plane’, Journal of Philosophical Logic 27(6):621-658.Google Scholar
  32. [32]
    Pratt, I., and Schoop, D. (1999), ‘Expressivity in polygonal, plane mereotopology’, Journal of Symbolic Logic, (to appear).Google Scholar
  33. [33]
    Pratt, V. (1990), ‘Dynamic algebras as a well behaved fragment of relation algebras’, Dept. of Computer Science, Stanford.Google Scholar
  34. [34]
    Randell, D. A., Cohn, A., and Cui, Z. (1992), ‘Computing transitivity tables: A challenge for automated theorem provers’, in Proc CADE 11, pages 786-790, Springer Verlag.Google Scholar
  35. [35]
    Schmidt, G. (1981a), ‘Programs as partial graphs I: Flow equivalence and correctness’, Theoret. Comp. Sci. 15:1-25.Google Scholar
  36. [36]
    Schmidt, G. (1981b), ‘Programs as partial graphs II: Recursion’, Theoret. Comp. Sci. 15:159-179.Google Scholar
  37. [37]
    Schmidt, G., and StrÖhlein, T. (1989), Relationen und Graphen, Diskrete Mathematik für Informatiker, Springer. English version: Relations and Graphs. Discrete Mathematics for Computer Scientists, EATCS Monographs on Theoret. Comp. Sci., Springer (1993).Google Scholar
  38. [38]
    SchrÖder, E. (1895), Algebra der Logik, volume 3, Teubner, Leipzig.Google Scholar
  39. [39]
    Smith, T. P., and Park, K. K. (1992), ‘An algebraic approach to spatial reasoning,' International Journal of Geographical Information Systems 6:177-192.Google Scholar
  40. [40]
    Stell, J. (1997), ‘Personal communication’, October 30, 1997.Google Scholar
  41. [41]
    Tarski, A. (1935), ‘Zur Grundlegung der Boole'schen Algebra, I’, Fundamenta Mathematicœ 24:177-198.Google Scholar
  42. [42]
    Tarski, A. (1941), ‘On the calculus of relations’, Journal of Symbolic Logic 6:73-89.Google Scholar
  43. [43]
    Tarski, A., and Givant, S. (1987), A Formalization of Set Theory Without Variables, volume 41 of Colloquium Publications, Amer. Math. Soc., Providence.Google Scholar
  44. [44]
    Winter, M. (1998), Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdeterminismus in Programmiersprachen, PhD thesis, Fakultät für Informatik, Universität der Bundeswehr. ISBN 3-933214-11-4.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Ivo DÜntsch
    • 1
  • Gunther Schmidt
    • 2
  • Michael Winter
    • 2
  1. 1.School of Information and Software EngineeringUniversity of UlsterNewtonabbeyIreland
  2. 2.Department of Computer ScienceUniversity of the Federal Armed Forces MunichNeubibergGermany

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