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Towards Mathematical Morpho-Logics

  • Isabelle Bloch
  • Jérôme Lang
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 90)

Abstract

In this paper, we suggest a new way to process information represented in a logical framework based on mathematical morphology. We show how the basic morphological operations can be expressed in a logical setting. We give some properties, show some links with revision and fusion, and ideas illustrate possible use of morpho-logics to approximation, reasoning and decision.

Keywords

Belief Revision Integrity Constraint Mathematical Morphology Propositional Formula Maximal Ball 
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.

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References

  1. 1.
    C. E. Alchourrón, P. Gärdenfors, and D. Makinson. On the Logic of Theory Change: Partial Meet Contraction and Revision Functions. Journal of Symbolic Logic, 50: 510–530, 1985.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    O. Bailleux and P. Marquis. DISTANCE-SAT: Complexity and Algorithms. In National Conference on Artificial Intelligence AAAI’99, pages 642–647, 1999.Google Scholar
  3. 3.
    I. Bloch. On Links between Mathematical Morphology and Rough Sets. Pattern Recognition, 33 (9): 1487–1496, 2000.MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Bloch. Using Mathematical Morphology Operators as Modal Operators for Spatial Reasoning. In ECAI 2000, Workshop on Spatio-Temporal Reasoning, pages 73–79, Berlin, Germany, 2000.Google Scholar
  5. 5.
    I. Bloch and H. Maître. Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition, 28 (9): 1341–1387, 1995.MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Dalal. Investigations into a Theory of Knowledge Base Revision: Preliminary Report. In AAAI’88, pages 475–479, 1988.Google Scholar
  7. 7.
    D. Dubois, F. Esteva, P. Garcia, L. Godo, and H. Prade. A Logical Approach to Interpolation based on Similarity Relations. International Journal of Approximate Reasoning, 17 (1): 1–36, 1997.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    F. Esteva, P. Garcia, and L. Godo. A Modal Account of Similarity-Based Reasoning. International Journal of Approximate Reasoning, 16: 235–260, 1997.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    M. L. Ginsberg, A. J. Parkes, and A. Roy. Supermodels and Robustness. In Fifteenth National Conference on Artificial Intelligence AAAI’98, pages 334339, Madison, Wisconsin, July 1998.Google Scholar
  10. 10.
    G. E. Hughes and M. J. Cresswell. An Introduction to Modal Logic. Methuen, London, UK, 1968.MATHGoogle Scholar
  11. 11.
    H. Katsuno and A. O. Mendelzon. Propositional Kowledge Base Revision and Minimal Change. Artificial Intelligence, 52: 263–294, 1991.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    S. Konieczny and R. Pino-Pérez. On the Logic of Merging. In 6th International Conference on Principles of Knowledge Representation and Reasoning,pages 488–498, Trento, Italy, 1998.Google Scholar
  13. 13.
    S. Konieczny and R. Pino-Pérez. Merging with Integrity Constraints. In A. Hunter and S. Parsons, editors, ECSQARU’99, volume 1638 of LNCS, pages 232–244, London, July 1999. Springer.Google Scholar
  14. 14.
    C. Lafage and J. Lang. Représentation logique de préférences pour la décision de groupe. In RFIA 2000, volume III, pages 267–276, Paris, France, February 2000.Google Scholar
  15. 15.
    C. Lantuejoul and F. Maisonneuve. Geodesic Methods in Image Analysis. Pattern Recognition, 17 (2): 177–187, 1984.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    G. Matheron. Eléments pour une théorie des milieux poreux. Masson, Paris, 1967.Google Scholar
  17. 17.
    G. Matheron. Random Sets and Integral Geometry. Wiley, New-York, 1975.MATHGoogle Scholar
  18. 18.
    R. Pino-Pérez and C. Uzcâtegui. Jumping to Explanations versus jumping to Conclusions. Artificial Intelligence, 111: 131–169, 1999.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    M. Schmitt and J. Mattioli. Morphologie mathématique. Masson, Paris, 1994.Google Scholar
  20. 20.
    J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.MATHGoogle Scholar
  21. 21.
    J. Serra. Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press (J. Serra Ed. ), London, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Isabelle Bloch
    • 1
  • Jérôme Lang
    • 2
  1. 1.ENST-TSI, CNRS URA 820ParisFrance
  2. 2.IRIT-UPSToulouseFrance

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