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Categorical Methods in Qualitative Reasoning: The Case for Weak Representations

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Spatial Information Theory (COSIT 2005)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 3693))

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Abstract

This paper argues for considering qualitative spatial and temporal reasoning in algebraic and category-theoretic terms. A central notion in this context is that of weak representation (WR) of the algebra governing the calculus. WRs are ubiquitous in qualitative reasoning, appearing both as domains of interpretation and as constraints. Defining the category of WRs allows us to express the basic notion of satisfiability (or consistency) in a simple way, and brings clarity to the study of various variants of consistency. The WRs of many popular calculi are of interest in themselves. Moreover, the classification of WRs leads to non-trivial model-theoretic results. The paper provides a not-too-technical introduction to these topics and illustrates it with simple examples.

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References

  1. Allen, J.F.: Maintaining knowledge about temporal intervals. Comm. of the ACM 26(11), 832–843 (1983)

    Article  MATH  Google Scholar 

  2. Anger, F.D., Mitra, D., Rodriguez, R.V.: Temporal Constraint Networks in Nonlinear Time. In: Proc. of the ECAI 1998 Workshop on Spatial and Temporal Reasoning (W22), Brighton, UK, pp. 33–39 (1998)

    Google Scholar 

  3. Balbiani, P., Condotta, J.-F., Ligozat, G.: On the Consistency Problem for the INDU Calculus. In: Proceedings of TIME-ICTL-2003, Cairns, Australia (2003)

    Google Scholar 

  4. Broxvall, M., Jonsson, P.: Towards a complete classification of tractability in point algebras for nonlinear time. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 129–143. Springer, Heidelberg (1999)

    Google Scholar 

  5. Condotta, J.-F., Tripakis, S., Ligozat, G.: Ultimately periodic qualitative constraint networks (2005)

    Google Scholar 

  6. Cristani, M.: The complexity of reasoning about spatial congruence. J. Artif. Intell. Res (JAIR) 11, 361–390 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Cristani, M., Hirsch, R.: The complexity of constraint satisfaction problems for small relation algebras. Artif. Intell. 156(2), 177–196 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Düntsch, I.: Relation algebras and their application in qualitative spatial reasoning. Technical Report CS-03-07, Brock University, St. Catarines, Ontario (2003)

    Google Scholar 

  9. Egenhofer, M., Rodríguez, A.: Relation algebras over containers and surfaces: An ontological study of a room space. Spatial Cognition and Computation 1, 155–180 (1999)

    Article  Google Scholar 

  10. Gerevini, A., Nebel, B.: Qualitative spatio-temporal reasoning with RCC-8 and Allen’s interval calculus: Computational complexity. In: Proc. of ECAI 2002, pp. 312–316 (2002)

    Google Scholar 

  11. Hirsch, R., Hodkinson, I.: Relation Algebras by Games. In: Studies in Logic and the Foundations of Mathematics, vol. 147. North Holland, Amsterdam (2002)

    Google Scholar 

  12. Ladkin, P.B., Maddux, R.D.: On Binary Constraint Problems. Journal of the ACM 41(3), 435–469 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ligozat, G.: Weak Representations of Interval Algebras. In: Proc. of AAAI 1990, pp. 715–720 (1990)

    Google Scholar 

  14. Ligozat, G.: On generalized interval calculi. In: Proc. of AAAI 1991, pp. 234–240 (1991)

    Google Scholar 

  15. Ligozat, G., Mitra, D., Condotta, J.-F.: Spatial and temporal reasoning: beyond Allen’s calculus. AI Communications 17(4), 223–233 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Ligozat, G., Renz, J.: What is a qualitative calculus? a general framework. In: Zhang, C., Guesgen, H.W., Yeap, W.-K. (eds.) PRICAI 2004. LNCS (LNAI), vol. 3157, pp. 53–64. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  17. Ligozat, G.: Reasoning about cardinal directions. Journal of Visual Languages and Computing 1(9), 23–44 (1998)

    Article  Google Scholar 

  18. Ligozat, G.: Simple models for simple calculi. In: Freksa, C., Mark, D.M. (eds.) COSIT 1999. LNCS, vol. 1661, pp. 173–188. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  19. Ligozat, G.: When Tables Tell It All: Qualitative Spatial and Temporal reasoning Based on Linear Orderings. In: Montello, D.R. (ed.) COSIT 2001. LNCS, vol. 2205, pp. 60–75. Springer, Heidelberg (2001)

    Google Scholar 

  20. Ligozat, G., Renz, J.: Problems with local consistency for Qualitative Calculi. In: Proceedings of ECAI 2004, Valencia, Spain, pp. 1047–1048 (2004)

    Google Scholar 

  21. Maddux, R.: Some varieties containing relation algebras. Trans. Amer. Math. Soc. 272(2), 501–526 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pujari, A.K., Vijaya Kumari, G., Sattar, A.: INDU: An Interval and Duration Network. In: Australian Joint Conf. on Artificial Intelligence, pp. 291–303 (1999)

    Google Scholar 

  23. Randell, D., Cui, Z., Cohn, T.: A spatial logic based on regions and connection. In: Neumann, B. (ed.) Proc. of KR 1992, San Mateo, CA, pp. 165–176 (1992)

    Google Scholar 

  24. Renz, J.: A spatial odyssey of the interval algebra: 1. Directed intervals. In: IJCAI, pp. 51–56 (2001)

    Google Scholar 

  25. Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6, 73–89 (1941)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. LIMSI-CNRS, Université Paris-Sud, F-91403 Cedex, Orsay, France

    Gérard Ligozat

Authors
  1. Gérard Ligozat
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Editor information

Editors and Affiliations

  1. School of Computing, University of Leeds, LS2 9JT, Leeds, United Kingdom

    Anthony G. Cohn

  2. Department of Geography, State University of New York at Buffalo, 105 Wilkeson Quad, 14261, Buffalo, NY, USA

    David M. Mark

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© 2005 Springer-Verlag Berlin Heidelberg

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Ligozat, G. (2005). Categorical Methods in Qualitative Reasoning: The Case for Weak Representations. In: Cohn, A.G., Mark, D.M. (eds) Spatial Information Theory. COSIT 2005. Lecture Notes in Computer Science, vol 3693. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556114_17

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  • DOI: https://doi.org/10.1007/11556114_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28964-7

  • Online ISBN: 978-3-540-32020-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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