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Abduction and the Dualization Problem

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Book cover Discovery Science (DS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2843))

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Abstract

Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines. While computing some abductive explanation for a literal χ with respect to a set of abducibles A from a Horn propositional theory Σ is intractable under the traditional representation of Σ by a set of Horn clauses, the problem is polynomial under model-based theory representation, where Σ is represented by its characteristic models. Furthermore, computing all the (possibly exponentially) many explanations is polynomial-time equivalent to the problem of dualizing a positive CNF, which is a well-known problem whose precise complexity in terms of the theory of NP-completeness is not known yet. In this paper, we first review the monotone dualization problem and its connection to computing all abductive explanations for a query literal and some related problems in knowledge discovery. We then investigate possible generalizations of this connection to abductive queries beyond literals. Among other results, we find that the equivalence for generating all explanations for a clause query (resp., term query) χ to the monotone dualization problem holds if χ contains at most k positive (resp., negative) literals for constant k, while the problem is not solvable in polynomial total-time, i.e., in time polynomial in the combined size of the input and the output, unless P=NP for general clause resp. term queries. Our results shed new light on the computational nature of abduction and Horn theories in particular, and might be interesting also for related problems, which remains to be explored.

This work was supported in part by the Austrian Science Fund (FWF) Project Z29-N04, by a TU Wien collaboration grant, and by the Scientific Grant in Aid of the Ministry of Education, Science, Sports, Culture and Technology of Japan.

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References

  1. Angluin, D.: Queries and Concept Learning. Machine Learning 2, 319–342 (1996)

    Google Scholar 

  2. Bioch, C., Ibaraki, T.: Complexity of identification and dualization of positive Boolean functions. Information and Computation 123, 50–63 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boros, E., Crama, Y., Hammer, P.L.: Polynomial-time inference of all valid implications for Horn and related formulae. Ann. Mathematics and Artificial Intelligence 1, 21–32 (1990)

    Article  MATH  Google Scholar 

  4. Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: Partial and multiple transversals of a hypergraph. SIAM J. Computing 30, 2036–2050 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: All minimal integer solutions for a monotone system of linear inequalities. SIAM Journal on Computing 31, 1624–1643 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 133–141. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: An intersection inequality for discrete distributions and related generation problems. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 543–555. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  8. Boros, E., Gurvich, V., Hammer, P.L.: Dual subimplicants of positive Boolean functions. Optimization Methods and Software 10, 147–156 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brewka, G., Dix, J., Konolige, K.: Nonmonotonic Reasoning – An Overview. CSLI Lecture Notes, vol. 73. CSLI Publications, Stanford University (1997)

    MATH  Google Scholar 

  10. Bshouty, N.H.: Exact Learning Boolean Functions via the Monotone Theory. Information and Computation 123, 146–153 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bylander, T.: The monotonic abduction problem:A functional characterization on the edge of tractability. In: Proc. 2nd International Conference on Principles of Knowledge Representation and Reasoning (KR 1991), pp. 70–77 (1991)

    Google Scholar 

  12. Crama, Y.: Dualization of regular boolean functions. Discrete App. Math. 16, 79–85 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. de Kleer, J.: An assumption-based truth maintenance system. Artif. Int. 28, 127–162 (1986)

    Article  Google Scholar 

  14. Dechter, R., Pearl, J.: Structure identification in relational data. Artificial Intelligence 58, 237–270 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. del Val, A.: On some tractable classes in deduction and abduction. Artificial Intelligence 116(1-2), 297–313 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. del Val, A.: The complexity of restricted consequence finding and abduction. In: Proc. 17th National Conference on Artificial Intelligence (AAAI 2000), pp. 337–342 (2000)

    Google Scholar 

  17. Domingo, C., Mishra, N., Pitt, L.: Efficient read-restricted monotone CNF/DNF dualization by learning with membership queries. Machine Learning 37, 89–110 (1999)

    Article  MATH  Google Scholar 

  18. Dowling, W., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn theories. Journal of Logic Programming 3, 267–284 (1984)

    Article  MathSciNet  Google Scholar 

  19. Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. Technical Report CD-TR 91/16, Christian Doppler Laboratory for Expert Systems, TU Vienna, Austria (January 1991)

    Google Scholar 

  20. Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing 24(6), 1278–1304 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Eiter, T., Gottlob, G.: The complexity of logic-based abduction. Journal of the ACM 42(1), 3–42 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eiter, T., Gottlob, G.: Hypergraph transversal computation and related problems in logic and AI. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 549–564. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  23. Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM Journal on Computing 32(2), 514–537 (2003) (Preliminary paper in Proc. ACM STOC 2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eiter, T., Makino, K.: On computing all abductive explanations. In: Proc. 18th National Conference on Artificial Intelligence (AAAI 2002), pp. 62–67 (2002); Preliminary Tech. Rep. INFSYS RR-1843-02-04, Institut für Informationssysteme, TU Wien, April 2002

    Google Scholar 

  25. 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)

    Chapter  Google Scholar 

  26. Eiter, T., Ibaraki, T., Makino, K.: Computing intersections of Horn theories for reasoning with models. Artificial Intelligence 110(1-2), 57–101 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Eshghi, K.: A tractable class of abduction problems. In: Proc. 13th International Joint Conference on Artificial Intelligence (IJCAI 1993), pp. 3–8 (1993)

    Google Scholar 

  28. Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms 21, 618–628 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Friedrich, G., Gottlob, G., Nejdl, W.: Hypothesis classification, abductive diagnosis, and therapy. In: Gottlob, G., Nejdl, W. (eds.) Expert Systems in Engineering. LNCS, vol. 462, pp. 69–78. Springer, Heidelberg (1990)

    Google Scholar 

  30. Gaur, D.R., Krishnamurti, R.: Self-duality of bounded monotone boolean functions and related problems. In: Arimura, H., Sharma, A.K., Jain, S. (eds.) ALT 2000. LNCS (LNAI), vol. 1968, pp. 209–223. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  31. Gunopulos, D., Khardon, R., Mannila, H., Toivonen, H.: Data mining, hypergraph transversals, and machine learning. In: Proc. 16th ACMSymposium on Principles of Database Systems (PODS 1996), pp. 209–216 (1993)

    Google Scholar 

  32. Inoue, K.: Linear resolution for consequence finding. Artif. Int. 56(2-3), 301–354 (1992)

    Article  MATH  Google Scholar 

  33. Johnson, D.S.: A Catalog of Complexity Classes. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, ch. 2 Elsevier, Amsterdam (1990)

    Google Scholar 

  34. Kautz, H., Kearns, M., Selman, B.: ReasoningWith Characteristic Models. In: Proc. 11th National Conference on Artificial Intelligence (AAAI 1993), pp. 34–39 (1993)

    Google Scholar 

  35. Kautz, H., Kearns, M., Selman, B.: Horn approximations of empirical data. Artificial Intelligence 74, 129–245 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kavvadias, D., Papadimitriou, C., Sideri, M.: On Horn envelopes and hypergraph transversals. In: Ng, K.W., et al. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 399–405. Springer, Heidelberg (1993)

    Google Scholar 

  37. Kavvadias, D.J., Stavropoulos, E.C.: Monotone Boolean dualization is in co-NP[log2n]. Information Processing Letters 85, 1–6 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kean, A., Tsiknis, G.: Assumption based reasoning and Clause Management Systems. Computational Intelligence 8(1), 1–24 (1992)

    Article  Google Scholar 

  39. Kean, A., Tsiknis, G.: Clause Management Systems (CMS). Computational Intelligence 9(1), 11–40 (1992)

    Article  Google Scholar 

  40. Khardon, R.: Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research 3, 349–372 (1995)

    MATH  Google Scholar 

  41. Khardon, R., Mannila, H., Roth, D.: Reasoning with examples: Propositional formulae and database dependencies. Acta Informatica 36(4), 267–286 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Khardon, R., Roth, D.: Reasoning with models. Artif. Int. 87(1/2), 187–213 (1996)

    Article  MathSciNet  Google Scholar 

  43. Khardon, R., Roth, D.: Defaults and relevance in model-based reasoning. Artificial Intelligence 97(1/2), 169–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  44. Levesque, H.: Making believers out of computers. Artificial Intelligence 30, 81–108 (1986)

    Article  MathSciNet  Google Scholar 

  45. Lovász, L.: Combinatorial optimization: Some problems and trends. DIMACS Technical Report 92-53, RUTCOR, Rutgers University (1992)

    Google Scholar 

  46. Lucchesi, C.L., Osborn, S.: Candidate Keys for Relations. Journal of Computer and System Sciences 17, 270–279 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  47. Makino, K., Ibaraki, T.: The maximum latency and identification of positive Boolean functions. SIAM Journal on Computing 26, 1363–1383 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  48. Makino, K., Ibaraki, T.: A fast and simple algorithm for identifying 2-monotonic positive Boolean functions. Journal of Algorithms 26, 291–305 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mannila, H., Räihä, K.-J.: Design by Example: An application of Armstrong relations. Journal of Computer and System Sciences 22(2), 126–141 (1986)

    Article  Google Scholar 

  50. Mannila, H., Räihä, K.-J.: Algorithms for inferring functional dependencies. Technical Report A-1988-3, University of Tampere, CS Dept., Series of Publ. A (April 1988)

    Google Scholar 

  51. Marquis, P.: Consequence Finding Algorithms. In: Gabbay, D., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems. Algorithms for Uncertainty and Defeasible Reasoning, vol. V, pp. 41–145. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  52. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  53. Papadimitriou, C.H.: NP-Completeness:Aretrospective. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 2–6. Springer, Heidelberg (1997)

    Google Scholar 

  54. Peirce, C.S.: Abduction and Induction. In: Buchler, J. (ed.) Philosophical Writings of Peirce, ch. 11. Dover, NewYork (1955)

    Google Scholar 

  55. Poole, D.: Explanation and prediction: An architecture for default and abductive reasoning. Computational Intelligence 5(1), 97–110 (1989)

    Article  MathSciNet  Google Scholar 

  56. Reiter, R., de Kleer, J.: Foundations of assumption-based truth maintenance systems: Preliminary report. In: Proc. 6th National Conference on Artificial Intelligence (AAAI 1987), pp. 183–188 (1982)

    Google Scholar 

  57. Selman, B., Levesque, H.J.: Abductive and default reasoning: A computational core. In: Proc. 8th National Conference on Artificial Intelligence (AAAI 1990), July 1990, pp. 343–348 (1990)

    Google Scholar 

  58. Selman, B., Levesque, H.J.: Support set selection for abductive and default reasoning. Artificial Intelligence 82, 259–272 (1996)

    Article  MathSciNet  Google Scholar 

  59. Zanuttini, B.: New polynomial classes for logic-based abduction. Journal of Artificial Intelligence Research (2003) (to appear)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Informationssysteme, Technische Universität Wien, Favoritenstraße 9-11, A-1040, Wien, Austria

    Thomas Eiter

  2. Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Kazuhisa Makino

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  1. Thomas Eiter
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  2. Kazuhisa Makino
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Editors and Affiliations

  1. FG Knowledge Engineering, FB Informatik, Technical University Darmstadt, Hochschulstr. 10, 64289, Darmstadt

    Gunter Grieser

  2. Meme Media Laboratory, Hokkaido University, N13 W8, 0608628, Sapporo, Japan

    Yuzuru Tanaka

  3. Graduate School of Informatics, Kyoto University Yoshida Honmachi, Sakyo-ku, 606-850, Kyoto, Japan

    Akihiro Yamamoto

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Eiter, T., Makino, K. (2003). Abduction and the Dualization Problem. In: Grieser, G., Tanaka, Y., Yamamoto, A. (eds) Discovery Science. DS 2003. Lecture Notes in Computer Science(), vol 2843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39644-4_1

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  • DOI: https://doi.org/10.1007/978-3-540-39644-4_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20293-6

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