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European Workshop on Logics in Artificial Intelligence

JELIA 2008: Logics in Artificial Intelligence pp 206–218Cite as

Counting Complexity of Minimal Cardinality and Minimal Weight Abduction

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5293))

Abstract

Abduction is an important method of non-monotonic reasoning with many applications in artificial intelligence and related topics. In this paper, we concentrate on propositional abduction, where the background knowledge is given by a propositional formula. We have recently started to study the counting complexity of propositional abduction. However, several important cases have been left open, namely, the cases when we restrict ourselves to solutions with minimal cardinality or with minimal weight. These cases – possibly combined with priorities – are now settled in this paper. We thus arrive at a complete picture of the counting complexity of propositional abduction.

This work was partially supported by the Austrian Science Fund (FWF), project P20704-N18.

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References

  1. Durand, A., Hermann, M., Kolaitis, P.G.: Subtractive reductions and complete problems for counting complexity classes. Theoretical Computer Science 340(3), 496–513 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Eiter, T., Gottlob, G.: The complexity of logic-based abduction. Journal of the Association for Computing Machinery 42(1), 3–42 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Eiter, T., Makino, K.: On computing all abductive explanations. In: Proc. 18th AAAI, Edmonton, Alberta, Canada, pp. 62–67. AAAI Press, Menlo Park (2002)

    Google Scholar 

  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)

    Google Scholar 

  5. Fagin, R., Ullman, J.D., Vardi, M.Y.: On the semantics of updates in databases. In: Proc. 2nd PODS, Atlanta, Georgia, USA, pp. 352–365. ACM Press, New York (1983)

    Google Scholar 

  6. Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman and Co., New York (1979)

    MATH  Google Scholar 

  7. Gasarch, W.I., Krentel, M.W., Rappoport, K.J.: OptP as the normal behavior of NP-complete problems. Mathematical Systems Theory 28(6), 487–514 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hemaspaandra, L.A., Vollmer, H.: The satanic notations: Counting classes beyond #P and other definitional adventures. SIGACT News 26(1), 2–13 (1995)

    Article  Google Scholar 

  9. Hermann, M., Pichler, R.: Counting complexity of propositional abduction. In: Veloso, M.M. (ed.) Proc. 20th IJCAI 2007, Hyderabad, India, pp. 417–422. AAAI Press, Menlo Park (2007)

    Google Scholar 

  10. Hermann, M., Pichler, R.: Complexity of counting the optimal solutions. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 149–159. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  11. Herzig, A., Lang, J., Marquis, P., Polacsek, T.: Updates, actions, and planning. In: Nebel, B. (ed.) Proc. 17th IJCAI 2001, Seattle, Washington, USA, pp. 119–124. Morgan Kaufmann, San Francisco (2001)

    Google Scholar 

  12. Kakas, A.C., Mancarella, P.: Database updates through abduction. In: McLeod, D., Sacks-Davis, R., Schek, H.-J. (eds.) Proc. 16th VLDB, Brisbane, Queensland, Australia, pp. 650–661. Morgan Kaufmann, San Francisco (1990)

    Google Scholar 

  13. Kozen, D.C.: The design and analysis of algorithms. In: Counting problems and #P, ch. 26, pp. 138–143. Springer, Heidelberg (1992)

    Google Scholar 

  14. Krentel, M.W.: The complexity of optimization problems. Journal of Computer and System Sciences 36(3), 490–509 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Krentel, M.W.: Generalizations of OptP to the polynomial hierarchy. Theoretical Computer Science 97(2), 183–198 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Papatheodorou, I., Kakas, A.C., Sergot, M.J.: Inference of gene relations from microarray data by abduction. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 389–393. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  17. Peng, Y., Reggia, J.A.: Abductive inference models for diagnostic problem solving. Springer, Heidelberg (1990)

    MATH  Google Scholar 

  18. Poole, D.: Probabilistic Horn abduction and bayesian networks. Artificial Intelligence 64(1), 81–129 (1993)

    Article  MATH  Google Scholar 

  19. Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. LIX (CNRS, UMR 7161), École Polytechnique, 91128, Palaiseau, France

    Miki Hermann

  2. Institut für Informationssysteme, Technische Universität Wien, A-1040, Wien, Austria

    Reinhard Pichler

Authors
  1. Miki Hermann
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  2. Reinhard Pichler
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Editor information

Editors and Affiliations

  1. Department of Computer Science, TU Dresden, Germany

    Steffen Hölldobler

  2. Institut für Theoretische Informatik, TU Dresden, Germany

    Carsten Lutz

  3. Institute of Philosophy, Dresden University of Technology, Dresden, Germany

    Heinrich Wansing

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Hermann, M., Pichler, R. (2008). Counting Complexity of Minimal Cardinality and Minimal Weight Abduction. In: Hölldobler, S., Lutz, C., Wansing, H. (eds) Logics in Artificial Intelligence. JELIA 2008. Lecture Notes in Computer Science(), vol 5293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87803-2_18

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  • DOI: https://doi.org/10.1007/978-3-540-87803-2_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-87802-5

  • Online ISBN: 978-3-540-87803-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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