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Abstraction and Complexity Measures

  • Conference paper
Abstraction, Reformulation, and Approximation (SARA 2007)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4612))

Abstract

Abstraction is fundamental for both human and artificial reasoning. The word denotes different activities and process, but all are intuitively related to the notion of complexity/simplicity, which is as elusive a notion as abstraction. From an analysis of the literature on abstraction and complexity it clearly appears that it is unrealistic to find definitions valid in all disciplines and for all tasks. Hence, we consider a particular model of abstraction, and try to investigate how complexity measures could be mapped to it. Preliminary results show that abstraction and complexity are not monotonically coupled notions, and that complexity may either increase or decrease with abstraction according to the definition of both and to the specificities of the considered domain.

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References

  1. Anderson, S., Revesz, P.Z.: Verifying the incorrectness of programs and automata. In: Zucker, J.-D., Saitta, L. (eds.) SARA 2005. LNCS (LNAI), vol. 3607, pp. 1–13. Springer, Heidelberg (2005)

    Google Scholar 

  2. Bennett, C.: Logical depth and physical complexity. In: Bennett, C. (ed.) The Universal Turing Machine: A Half-Century Survey, pp. 227–257 (1988)

    Google Scholar 

  3. Bredeche, N., Shi, Z., Zucker, J.-D.: Perceptual learning and abstraction in machine learning: an application to autonomous robotics. IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews 36(2), 172–181 (2006)

    Article  Google Scholar 

  4. Choueiry, B.Y., Iwasaki, Y., McIlraith, S.: Towards a practical theory of reformulation for reasoning about physical systems. Artificial Intelligence 162(1-2), 145–204 (2006)

    Article  Google Scholar 

  5. Feldman, D.P., Crutchfield, J.P.: Measures of statistical complexity: Why? Physics Letters A 238, 244–252 (1998)

    Article  MATH  Google Scholar 

  6. Ellman, T.: Synthesis of abstraction hierarchies for constraint satisfaction by clustering approximatively equivalent objects. In: International Conference on Machine Learning, Amherst, MA, Morgan Kaufmann, Seattle, Washington, USA (1993)

    Google Scholar 

  7. Gell-Mann, M., Lloyd, S.: Information measures, effective complexity, and total information. Complexity 2(1), 44–52 (1996)

    Article  Google Scholar 

  8. Giordana, A., Saitta, L.: Abstraction: a general framework for learning. In: Working notes of the AAAI Workshop on Automated Generation of Approximations and Abstraction, pp. 245–256, Boston, MA (1990)

    Google Scholar 

  9. Giunchiglia, F., Walsh, T.: A theory of abstraction. Artificial Intelligence 56(2-3), 323–390 (1992)

    Article  Google Scholar 

  10. Goldstone, R., Barsalou, L.: Reuniting perception and conception. Cognition 65, 231–262 (1998)

    Article  Google Scholar 

  11. Hobbs, J.: Granularity. In: Int. Joint Conf. on Artificial Intelligence, pp. 432–435 (1985)

    Google Scholar 

  12. Holte, R.C., Grajkowski, J., Tanner, B.: Hierarchical heuristic search revisited. In: SARA, pp. 121–133 (2005)

    Google Scholar 

  13. Holte, R.C., Mkadmi, T., Zimmer, R.M., MacDonald, A.J.: Speeding up problem-solving by abstraction: A graph-oriented approach. Artificial Intelligence 85, 321–361 (1996)

    Article  Google Scholar 

  14. Imielinski, T.: Domain abstraction and limited reasoning. In: Proceedings of the Intern. Joint Conf. on Artificial Intelligence, pp. 997–1003 (1987)

    Google Scholar 

  15. Shiner, J.S., Davison, M., Landsberg, P.T.: Simple measure for complexity. Physical review E 59, 1459–1464 (1999)

    Article  Google Scholar 

  16. Knoblock, C.: Learning hierarchies of abstraction spaces. In: 6th International Workshop on Machine Learning, pp. 241–245, Ithaca, NY (1989)

    Google Scholar 

  17. Koppel, M.: Complexity, depth and sophistication. Complex Systems 1, 1087–1091 (1987)

    MATH  Google Scholar 

  18. Lowry, M.: The abstraction/implementation model of problem reformulation. In: Int. Joint Conf. on Artificial Intelligence, pp. 1004–1010, Milano, Italy (1987)

    Google Scholar 

  19. Marczyk, J., Deshpande, B.: Measuring and tracking complexity in science. In: Proceedings of the 6th International Conference on Complex Systems (2006)

    Google Scholar 

  20. Mustiere, S., Zucker, J.-D., Saitta, L.: An abstraction-based machine learning approach to cartographic generalization. In: Spatial Data Handling 2000 (SDH), pp. 150–163, Beijing, China (2000)

    Google Scholar 

  21. Nayak, P., Levy, A.: A semantic theory of abstraction. In: IJCAI 1995. International Joint Conference on Artificial Intelligence, pp. 192–196 (1995)

    Google Scholar 

  22. Plaisted, D.: Theorem proving with abstraction. Artificial Intelligence 16, 47–108 (1981)

    Article  MATH  Google Scholar 

  23. Lopez-Ruiz, R., Mancini, H., Calbet, X.: A statistical measure of complexity. Physics Letters A 209, 209–321 (1995)

    Article  Google Scholar 

  24. Ravishankar, K.C., Prasad, B.G., Gupta, S.K., Biswas, K.K.: Dominant color region based indexing for cbir. In: iciap, 00:887 (1999)

    Google Scholar 

  25. Sacerdoti, E.: Planning in a hierarchy of abstraction spaces. Artificial Intelligence 5, 115–135 (1974)

    Article  MATH  Google Scholar 

  26. Saitta, L., Torasso, P., Torta, G.: Formalizing the abstraction process in model-based diagnosis. Tr cs-2006/34, Univ. of Torino, Italy (2006)

    Google Scholar 

  27. Saitta, L., Zucker, J-D.: Semantic abstraction for concept representation and learning. In: Proc. of the Intern. Symposium on Approximation, Reformulation and Abstraction (Asilomar, CA) (1998)

    Google Scholar 

  28. Saitta, L., Zucker, J.-D.: A model of abstraction in visual perception. Applied Artificial Intelligence 15(8), 761–776 (2001)

    Article  Google Scholar 

  29. Shalizi, C.: Methods and techniques of complex systems science: An overview. In: Complex Systems Science in Biomedicine, pp. 33–114. Springer, New York (2006)

    Chapter  Google Scholar 

  30. Sheeren, D., Mustiere, S., Zucker, J.-D.: Consistency assessment between multiple representations of geographical databases: a specification-based approach. In: Fisher, P. (ed.) Developments in Spatial Data Handling, pp. 617–628. Springer, Heidelberg (2004)

    Google Scholar 

  31. Subramanian, D.: A theory of justified reformulations. In: Paul, D. (ed.) Change of Representation and Inductive Bias, pp. 147–167. Kluwer Academic Publishers, Boston (1990)

    Google Scholar 

  32. Tenenberg, J.: Preserving consistency across abstraction mappings. In: Proceedings of IJCAI-87, pp. 1011–1014, Milan, Italy (1987)

    Google Scholar 

  33. Vitanyi, P.: Meaningful information. IEEE Transactions on Information Theory 52, 4617–4630 (2006)

    Article  Google Scholar 

  34. Wolpert, D., Macready, W.: Self-dissimilarity: An empirically observable complexity measure. In: Proc. of the Intern. Conf. on Complex Systems (Nashua, NH) (1997)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Università del Piemonte Orientale, Dipartimento di Informatica, Via Bellini 25/G, Alessandria, Italy

    Lorenza Saitta

  2. LIM&BIO, EA3502, Univ. Paris 13, 74 rue Marcel Cachin, 93017 Bobigny, France

    Jean-Daniel Zucker

  3. UR 079 GEODES, IRD, 32 avenue Henri Varagnat, 93143 Bondy, France

    Jean-Daniel Zucker

Authors
  1. Lorenza Saitta
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  2. Jean-Daniel Zucker
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Editor information

Ian Miguel Wheeler Ruml

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© 2007 Springer-Verlag Berlin Heidelberg

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Saitta, L., Zucker, JD. (2007). Abstraction and Complexity Measures. In: Miguel, I., Ruml, W. (eds) Abstraction, Reformulation, and Approximation. SARA 2007. Lecture Notes in Computer Science(), vol 4612. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73580-9_29

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  • DOI: https://doi.org/10.1007/978-3-540-73580-9_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73579-3

  • Online ISBN: 978-3-540-73580-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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