Learning acyclic first-order horn sentences from entailment

  • Hiroki Arimura
Session 12
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1316)

Abstract

This paper considers the problem of learning an unknown first-order Horn sentence H* from examples of Horn clauses that H* either implies or does not imply. Particularly, we deal with a subclass of first-order Horn sentences ACH(k), called acyclic constrained Horn programs of constant arity k. ACH(k) allows recursions, disjunctive definitions, and the use of function symbols. We present an algorithm that exactly identifies every target Horn program H* in ACH(k) in polynomial time in p, m and n using O(pmnk+1) entailment equivalence queries and O(pm2n2k+1) request for hint queries, where p is the number of predicates, m is the number of clauses contained in H* and n is the size of the longest counterexample. This algorithm combines saturation and least general generalization operators to invert resolution steps. Next, using the technique of replacing request for hint queries with entailment membership queries, we have a polynomial time learning algorithm using entailment equivalence and entailment membership queries for a subclass of ACH(k). Finally, we show that any algorithm which learns ACH(k) using entailment equivalence and entailment membership queries makes μ(mnk) queries, and that the use of entailment cannot be eliminated to learn ACH(k) even with both equivalence and membership queries for ground atoms are allowed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Angluin. Finding patterns common to a set of strings. In Proc. the 11th Annual Symposium on Theory of Computing, pp. 130–141, 1979.Google Scholar
  2. 2.
    D. Angluin, Learning with hints, In Proc. COLT'88 (1988) 223–237.Google Scholar
  3. 3.
    D.Z Angluin, Queries and concept learning. Machine Learning, 2:319–342, 1988.Google Scholar
  4. 4.
    D. Angluin, M. Frazier, L. Pitt, Learning conjunctions of Horn clauses, Machine Learning 9 (1992) 147–164.Google Scholar
  5. 5.
    D. Angluin and M. Kharitonov, When won't membership queries help?, JOSS 50 (1995) 336–355.Google Scholar
  6. 6.
    H. Arimura, Completeness of depth-bounded resolution for weakly reducing programs, Software Science and Engineering, World Scientific Series in Computer Science Vol. 31, 1991, 227–245.Google Scholar
  7. 7.
    H. Arimura, T. Shinohara, S. Otsuki. Finding minimal generalizations for unions of pattern languages and its application to inductive inference from positive data. In Proc. the 11th STACS, LNCS 775, (1994) 649–660.Google Scholar
  8. 8.
    H. Arimura, H. Ishizaka, T. Shinohara, Learning unions of tree patterns using queries, In Proc. the 6th ALT, LNAI 997, (1995). (To appear in TCS)Google Scholar
  9. 9.
    W. Cohen, Cryptographic limitations on learning one-clause logic programs, In Proc. AAAI'93 (1993) 80–85.Google Scholar
  10. 10.
    W. Cohen and H. Hirsh, Learnability of description logics, In Proc. COLT'92 (1992) 116–127.Google Scholar
  11. 11.
    M. Frazier, L. Pitt, Learning from entailment: an application to propositional Horn sentences In Proc. 10th Int. Conf. Machine Learning (1993) 120–127.Google Scholar
  12. 12.
    M. Frazier, L. Pitt, CLASSIC learning In Proc. COLT'94 (1994) 23–34.Google Scholar
  13. 13.
    H. Ishizaka, H. Arimura, and T. Shinohara, Finding tree patterns consistent with positive and negative examples using queries, In Proc. ALT'94, LNAI 872 (1994) 317–332.Google Scholar
  14. 14.
    N. Littlestone. Learning quickly when irrelevant attributes abound: A new linearthreshold algorithm. Machine Learning, 2, 285–318, 1988.Google Scholar
  15. 15.
    J. W. Lloyd, Foundation of Logic Programming, Springer-Verlag, 2nd. ed., (1987)Google Scholar
  16. 16.
    W. Maass and G. Turin, Lower bound methods and separation results for on-line learning models, Machine Learning 9 (1992) 107–145.Google Scholar
  17. 17.
    S. Miyano, A. Shinohara, T. Shinohara, Which classes of elementary formal systems are polynomial-time learnable?, In Proc. ALT'91 (1991) 139–150.Google Scholar
  18. 18.
    Nienhuys-Cheng and de Wolf, The subsumption Theorem for Several Forms of Resolution, In Proc. CSN95 (1995) 143–154Google Scholar
  19. 19.
    C. D. Page and A. M. Frisch, Generalization and learnability: a study of constrained atoms, In Inductive Logic Programming, Academic Press (1992) 29–61.Google Scholar
  20. 20.
    L. Pitt and M. K. Warmuth, Prediction preserving reduction, J. Comput. System Sci. 41 (1990) 430–467.Google Scholar
  21. 21.
    G. D. Plotkin, A note on inductive generalization, In Machine Intell., 5 (Edinburgh Univ. Press, 1970) 153–163.Google Scholar
  22. 22.
    C. Rouveirol and J. F. Puget, Beyond inversion of resolution, In Proc. 7th Int. Conf. Machine Learning (1990) 122–130.Google Scholar
  23. 23.
    J. D. Ullman and A. V. Gelder, Parallel complexity of logical query programs, Algorithmica 3 (1988) 5–42.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Hiroki Arimura
    • 1
  1. 1.Department of InformaticsKyushu UniversityKasugaJapan

Personalised recommendations