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Enumerating Minimal Explanations by Minimal Hitting Set Computation

  • Ken Satoh
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4092)

Abstract

We consider the problem of enumerating minimal explanations in propositional theory. We propose a new way of characterizing the enumeration problem in terms of not only the number of explanations, but also the number of unexplanations. Maximal unexplanations are a maximal set of abducible formulas which cannot explain the observation given a background theory. In this paper, we interleavingly enumerate not only minimal explanations but also maximal unexplanations. To best of our knowledge, there has been no algorithm which is characterized in terms of such maximal unexplanations. We propose two algorithms to perform this task and also analyze them in terms of query complexity, space complexity and time complexity.

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References

  1. 1.
    Bailey, J., Stuckey, P.J.: Discovery of Minimal Unsatisfiable Subsets of Constraints Using Hitting Set Dualization. In: Proc. of PADL 2005, pp. 174–186 (2005)Google Scholar
  2. 2.
    Eiter, T., Gottlob, G.: The Complexity of Logic-Based Abduction. Journal of the ACM 42(1), 3–42 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Eiter, T., Makino, K.: On Computing all Abductive Explanations. In: Proc. of AAAI 2002, pp. 62–67 (2002)Google Scholar
  4. 4.
    Eiter, T., Makino, K.: Generating All Abductive Explanations for Queries on Propositional Horn Theories. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 197–211. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Eiter, T., Makino, K.: Abduction and the Dualization Problem. In: Grieser, G., Tanaka, Y., Yamamoto, A. (eds.) DS 2003. LNCS, vol. 2843, pp. 1–20. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Fredman, M.L., Khachiyan, L.: On the Complexity of Dualization of Monotone Disjunctive Normal Forms. Journal of Algorithms 21(3), 618–628 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gunopulos, D., Khardon, R., Mannila, H., Toivonen, H.: Data mining, Hypergraph Transversals, and Machine Learning. In: Proc. of PODS 1997, pp. 209–216 (1997)Google Scholar
  8. 8.
    Gunopulos, D., Khardon, R., Mannila, H., Saluja, S., Toivonen, H., Sharm, R.S.: Discovering all most specific sentences. ACM Trans. Database Syst. 28(2), 140–174 (2003)CrossRefGoogle Scholar
  9. 9.
    Kakas, A.C., Kowalski, R., Toni, F.: The Role of Abduction in Logic Programming. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming 5, pp. 235–324. Oxford University Press, Oxford (1998)Google Scholar
  10. 10.
    Kautz, H., Kearns, M., Selman, B.: Horn Approximations of Empirical Data. Artificial Intelligence 74, 129–245 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mannila, H., Toivonen, T.: On an Algorithm for Finding All Interesting Sentences. In: Cybernetics and Systems, The Thirteen European Meeting on Cybernetics and Systems Research, vol. II, pp. 973–978 (1996)Google Scholar
  12. 12.
    Reiter, R.: A Theory of Diagnosis from First Principles. Artificial Intelligence 32, 57–95 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Satoh, K., Uno, T.: Enumerating Maximal Frequent Sets Using Irredundant Dualization. In: Grieser, G., Tanaka, Y., Yamamoto, A. (eds.) DS 2003. LNCS, vol. 2843, pp. 256–268. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Satoh, K., Uno, T.: Enumerating Minimal Revised Specification using Dualization. In: Proc. of the third workshop on Learning with Logics and Logics for Learning, pp. 19–23 (2005)Google Scholar
  15. 15.
    Uno, T.: A Practical Fast Algorithm for Enumerating Minimal Set Coverings. In: SIGAL 1983, Information Processing Society of Japan (in Japanese), pp. 9–16 (2002)Google Scholar
  16. 16.
    Uno, T., Satoh, K.: Detailed Description of an Algorithm for Enumeration of Maximal Frequent Sets with Irredundant Dualization. In: Online CEUR Workshop Proceedings of the ICDM 2003 Workshop on Frequent Itemset Mining Implementations (FIMI 2003) (2003), http://sunsite.informatik.rwth-aachen.de/Publications/CEUR-WS//Vol-90/satoh.pdf

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ken Satoh
    • 1
  • Takeaki Uno
    • 1
  1. 1.National Institute of Informatics and Sokendai 

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