Advertisement

Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics

  • Dimiter Vakarelov
Part of the International Mathematical Series book series (IMAT, volume 5)

Abstract

In this paper, we present recent results in the region-based theory of space that concern algebras of regions, the corresponding topological and discrete models, and representation theory. We also discuss applications to Qualitative Spatial Reasoning (QSR), an actively developing branch of AI and Knowledge Representation (KR). In particular, we show how new results in some practically motivated areas of QSR and KR can be obtained by combining methods from such established classical disciplines as Boolean algebras, topology and logic.

Keywords

Topological Space Modal Logic Boolean Algebra Boolean Variable Completeness Theorem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ph. Balbiani (Ed.), Special Issue on Spatial Reasoning, J. Appl. Non-Classical Logics 12 (2002), No. 3–4.Google Scholar
  2. 2.
    Ph. Balbiani, T. Tinchev, and D. Vakarelov, Dynamic logic of region-based theory of discrete spaces, J. Appl. Non-Classical Logics (2006). [To appear]Google Scholar
  3. 3.
    Ph. Balbiani, G. Dimov, T. Tinchev and D. Vakarelov, Modal logics for region-based theory of space, 2006. [Submitted]Google Scholar
  4. 4.
    B. Bennett, A categorical axiomatization of region-based geometry, Fundam. Inform. 46, (2001), 145–158.MATHGoogle Scholar
  5. 5.
    L. Biacino and G. Gerla, Connection structures, Notre Dame J. Formal Logic 32 (1991), 242–247.MATHCrossRefGoogle Scholar
  6. 6.
    L. Biacino and G. Gerla, Connection structures: Grzegorczyk’s and Whitehead’s definition of point, Notre Dame J. Formal Logic 37 (1996), 431–439.MATHCrossRefGoogle Scholar
  7. 7.
    P. Blackburn, M. de Rijke, and Y. Venema, Modal Logic, Cambridge Univ. Press, 2001.Google Scholar
  8. 8.
    A. Chagrov and M. Zakharyaschev. Modal Logic, Oxford Univ. Press, 1997.Google Scholar
  9. 9.
    E. Čech, Topological Spaces, Interscience, 1966.Google Scholar
  10. 10.
    B. L. Clarke, A calculus of individuals based on’ connection, Notre Dame J. Formal Logic 22 (1981), 204–218.MATHCrossRefGoogle Scholar
  11. 11.
    B. L. Clarke, Individuals and points, Notre Dame J. Formal Logic 26 (1985), 61–75.MATHCrossRefGoogle Scholar
  12. 12.
    A. Cohn and S. Hazarika, Qualitative spatial representation and reasoning: An overview, Fundam. Inform. 46 (2001), 1–29.MATHGoogle Scholar
  13. 13.
    T. de Laguna, Point, line and surface as sets of solids, J. Philos. 19 (1922), 449–461.CrossRefGoogle Scholar
  14. 14.
    G. Dimov and D. Vakarelov, Topological representation of precontact algebras, In: ReLMiCS’2005, St. Catharines, Canada, February 22–26, 2005, Proceedings, W. MacCaul, M. Winter, and I. Düntsch (Eds.), Lect. Notes Commp. Sci. 3929 Springer, 2006, pp. 1–16.Google Scholar
  15. 15.
    G. Dimov and D. Vakarelov, Contact algebras and region-based theory of space. A proximity approach. I, Fundam. Inform. (2006). [To appear]Google Scholar
  16. 16.
    G. Dimov and D. Vakarelov, Contact algebras and region-based theory of space. A proximity approach. II, Fundam. Inform. (2006). [To appear]Google Scholar
  17. 17.
    I. Düntsch (Ed.), Special issue on Qualitative Spatial Reasoning, Fundam. Inform. 46 (2001).Google Scholar
  18. 18.
    I. Düntsch, W. MacCaul, D. Vakarelov, and M. Winter, Topological Representation of Contact Lattices, Lect. Notes Comput. Sci., Springer, 2006. [To appear]Google Scholar
  19. 19.
    I. Düntsch and D. Vakarelov, Region-based theory of discrete spaces: A proximity approach, In: Proceedings of Fourth International Conference Journées de l’informatique Messine, Metz, France, 2003, Nadif, M., Napoli, A., SanJuan, E., and Sigayret, A. (Eds.), pp. 123–129Google Scholar
  20. 20.
    I. Düntsch, H. Wang, and S. McCloskey, A relational algebraic approach to Region Connection Calculus, Theor. Comput. Sci. 255, (2001), 63–83.MATHCrossRefGoogle Scholar
  21. 21.
    I. Düntsch and M. Winter, A Representation theorem for Boolean Contact Algebras, Theor. Comput. Sci. (B) 347 (2005), 498–512.MATHCrossRefGoogle Scholar
  22. 22.
    I. Düntsch and M. Winter, Weak Contact Structures, Lect. Notes Comput. Sci. 3929, Springer, 2006. [To appear].Google Scholar
  23. 23.
    M. Egenhofer, R. Franzosa, Point-set topological spatial relations, Int. J. Geogr. Inform. Systems 5 (1991), 161–174.CrossRefGoogle Scholar
  24. 24.
    R. Engelking, General Topology, PWN, Warszawa, 1977.MATHGoogle Scholar
  25. 25.
    V. Efremovič, Infinitesimal spaces, Dokl. Akad. Nauk SSSR 76 (1951), 341–343.Google Scholar
  26. 26.
    V. V. Fedorčuk, Boolean δ-algebras and quasi-open mappings, Siberian Math. J. 14 (1973) 759–767.CrossRefGoogle Scholar
  27. 27.
    D. Gabbay, An irreflexivity lemma with applications to axiomatizations of conditions in tense frames, In: Aspects of Philosophical Logic, U. Moenich (Ed.), Reidel-Dordrecht, 1981, pp. 67–89.Google Scholar
  28. 28.
    D. Gabelaia, R. Konchakov, A. Kurucz, F. Wolter and M. Zakharyaschev, Combining spatial and temporal logics: expressiveness versus complexity, J. Artif. Intell. Res. 23 (2005), 167–243.MATHGoogle Scholar
  29. 29.
    A. Galton, The mereotopology of discete spaces, In: Spatial Information Theory, Proceedings of the International Conference COSIT’99, C. Freksa-D. M. Mark (Eds.), Lect. Notes Comput. Sci. 1661, Springer-Verlag, 1999, pp. 251–266.Google Scholar
  30. 30.
    A. Galton, Qualitative Spatial Change Oxford Univ. Press, 2000.Google Scholar
  31. 31.
    G. Gerla, Pointless geometries, In: Handbook of Incidence Geometry, F. Buekenhout (Ed.), Elsevier, 1995, pp. 1015–1031.Google Scholar
  32. 32.
    A. Grzegorczyk, Undecidability of some topological theories, Fundam. Math. 38 (1951), 137–152.Google Scholar
  33. 33.
    A. Grzegorczyk, The system of Leśnewski in relation to contemporary logical research, Stud. Log. 3 (1955), 77–95.CrossRefGoogle Scholar
  34. 34.
    A. Grzegorczyk, Axiomatization of geometry without points, Synthese textbf12 (1960), 228–235.Google Scholar
  35. 35.
    S. T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 28, (1949), 287–320.MATHGoogle Scholar
  36. 36.
    R. Konchakov, A. Kurucz, F. Wolter and M. Zakharyaschev, Spatial Logic + Temporal Logic = ? In: Logic of Space, M. Aiello, I. Pratt, and J. van Benthem (Eds.), 2006. [To appear]Google Scholar
  37. 37.
    S. Leader, Local proximity spaces, Math. Ann. 169 (1967), 275–281.MATHCrossRefGoogle Scholar
  38. 38.
    S. Leśnewski, 1927–1931. Collected works, S. J. Surma, J. T. J. Srzednicki, and D. I. Barmett (Eds.), PWN-Kluwer, Vols 1 and 2, 1992.Google Scholar
  39. 39.
    S. Li and M. Ying, Generalized Region Connection Calculus, Artif. Intell. 160 (2004), no. 1–2, 1–34.MATHCrossRefGoogle Scholar
  40. 40.
    C. Lutz and F. Wolter, Modal logics for topological relations, Logical Meth. Computer Sci. (2006). [To appear]Google Scholar
  41. 41.
    T. Mormann, Continuous lattices and Whiteheadian theory of space, Log. Log. Philos. 6 (1998), 35–54.MATHGoogle Scholar
  42. 42.
    S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge Univ. Press, 1970.Google Scholar
  43. 43.
    I. Pratt-Hartmann, Empiricism and racionalism in region-based theories of space, Funadam. Informa. 46, (2001), 159–186.MATHGoogle Scholar
  44. 44.
    I. Pratt-Hartmann, A Topological constraint language with component counting, J. Appl. Non-Classical Logics 12 (2002), no 3–4, 441–467.CrossRefGoogle Scholar
  45. 45.
    I. Pratt-Hartmann, First-order region-based theories of space, In: Logic of Space, M. Aiello, I. Pratt and J. van Benthem (Eds.), 2006. [To appear]Google Scholar
  46. 46.
    I. Pratt and D. Schoop, A complete axiom system for polygonal mereotopology of the real plane, J. Philos. Logic 27 (1998), no. 6, 621–661.MATHCrossRefGoogle Scholar
  47. 47.
    I. Pratt and D. Schoop, Expressivity in polygonal plane mereotopology, J. Symb. Log. 65 (2000), 822–838.MATHCrossRefGoogle Scholar
  48. 48.
    D. A. Randell, Z. Cui and A. G. Cohn, A spatial logic based on regions and connection, In: Proceedings of the 3rd International Conference Knowledge Representation and Reasoning, B. Nebel, W. Swartout, and C. Rich (Eds.), Morgan Kaufmann, 1992, pp. 165–176.Google Scholar
  49. 49.
    P. Roeper, Region-based topology, J. Philos. Logic 26 (1997), 251–309.MATHCrossRefGoogle Scholar
  50. 50.
    D. J. Schoop, Points in point-free mereotopology, Fundam. Inform. 46, (2001), 129–143.MATHGoogle Scholar
  51. 51.
    E. Shchepin, Real-valued functions and spaces close to normal, Siberian Math. J. 13 (1972), 820–830.CrossRefGoogle Scholar
  52. 52.
    R. Sikorski, Boolean Algebras, Springer, 1964.Google Scholar
  53. 53.
    P. Simons, PARTS. A Study in Ontology, Oxford, Clarendon Press, 1987.Google Scholar
  54. 54.
    J. Stell, Boolean connection algebras: A new approach to the Region Connection Calculus, Artif. Intell. 122 (2000), 111–136.MATHCrossRefGoogle Scholar
  55. 55.
    M. H. Stone, The theory of representations for Boolean algebras, Trans. Am. Math. Soc. 40, (1937), 37–111CrossRefGoogle Scholar
  56. 56.
    W. J. Thron, Proximity structures and grills, Math. Ann. 206 (1973), 35–62.MATHCrossRefGoogle Scholar
  57. 57.
    A. Tarski, Les fondements de la gsèomètrie des corps, In: First Polish Mathematical Congress, Lwø’w, 1927; English transl.: Foundations of the Geometry of Solids, In: Logic, Semantics, Metamathematics, J. H. Woodger (Ed.), Clarendon Press, 1956, pp. 24–29.Google Scholar
  58. 58.
    A. Tarski, On the foundations of Boolean Algebra, [in German] Fundam. Inform. 24, (1935), 177–198; English transl. in: Logic, Semantics, Metamathematics, J. H.Woodger (Ed.), Clarendon Press, 1956, pp. 320–341.MATHGoogle Scholar
  59. 59.
    A. Urquhart, A topological representation theory for lattices, Algebra Univers. 8, (1978), 45–58.MATHCrossRefGoogle Scholar
  60. 60.
    D. Vakarelov, Proximity modal logic, In: Proceedings of the 11th Amsterdam Colloquium, December 1997, pp. 301–308.Google Scholar
  61. 61.
    D. Vakarelov, G. Dimov, I. Düntsch, and B. Bennett, A proximity approach to some region-based theories of space, J. Appl. Non-Classical Logics 12 (2002), no. 3–4, 527–559.CrossRefGoogle Scholar
  62. 62.
    D. Vakarelov, I. Düntsch, and B. Bennett, A note on proximity spaces and connection based mereology, In: Proceedings of the 2nd International Conference on Formal Ontology in Information Systems (FOIS’01), 2001, C. Welty-B. Smith (Eds.), pp. 139–150.Google Scholar
  63. 63.
    H. de Vries, Compact Spaces and Compactifications, Van Gorcum, 1962Google Scholar
  64. 64.
    A. N. Whitehead, Process and Reality, New York, MacMillan, 1929.MATHGoogle Scholar
  65. 65.
    F. Wolter. and M. Zakharyaschev, Spatial representation and reasoning in RCC-8 with Boolean region terms, In: Proceedings of the 14th European Conference on Artificial Intelligence (ECAI 2000), Horn W. (Ed.), IOS Press, pp. 244–248.Google Scholar
  66. 66.
    G. H. von Wright, A new system of modal logic, In: Actes du XI Congrès International de Philosophie, Bruxelles 1953; Amsterdam 1953, Vol. 5, pp. 59–63.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Dimiter Vakarelov
    • 1
  1. 1.Sofia UniversitySofiaBulgaria

Personalised recommendations