Which hypotheses can be found with inverse entailment?

  • Akihiro Yamamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1297)


In this paper we give a completeness theorem of an inductive inference rule inverse entailment proposed by Muggleton. Our main result is that a hypothesis clause H can be derived from an example E under a background theory B with inverse entailment iff H subsumes E relative to B in Plotkin's sense. The theory B can be any clausal theory, and the example E can be any clause which is neither a tautology nor implied by B. The derived hypothesis H is a clause which is not always definite. In order to prove the result we give a declarative semantics for arbitrary consistent clausal theories, and show that SB-resolution, which was originally introduced by Plotkin, is a complete procedural semantics. The completeness is shown as an extension of the completeness theorem of SLD-resolution. We also show that every hypothesis H derived with saturant generalization, proposed by Rouveirol, must subsume E w.r.t. B in Buntine's sense. Moreover we show that saturant generalization can be obtained from inverse entailment by giving some restriction to it.


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  1. 1.
    H. Arimura. Learning Acyclic First-order Horn Sentences From Implication. To appear in the Proceedings of the 8th International Workshop on Algorithmic Learning Theory, 1997.Google Scholar
  2. 2.
    W. Buntine. Generalized Subsumption and Its Applications to Induction and Redundancy. Artificial Intelligence, 36:149–176, 1988.Google Scholar
  3. 3.
    W. W. Cohen. Pac-learning Recursive Logic Programs: Efficient Algorithms. J. of Artificial Intelligence Research, 2:501–539, 1995.Google Scholar
  4. 4.
    W. W. Cohen. Pac-learning Recursive Logic Programs: Negative Results. J. of Artificial Intelligence Research, 2:541–573, 1995.Google Scholar
  5. 5.
    K. Furukawa, T. Murakami, K. Ueno, T. Ozaki, and K. Shimazu. On a Sufficient Condition for the Exisitence of Most Specific Hypothesis in Progol. SIG-FAI-9603, 56–61, Resarch Reprot of JSAI, 1997.Google Scholar
  6. 6.
    K. Inoue. Linear Resolution for Consequence Finding. Artificial Intelligence, 56:301–353, 1992.Google Scholar
  7. 7.
    R. A. Kowalski. The Case for Using Equality Axioms in Automatic Demonstration. In Proceedings of the Symposium on Automatic Demonstaration (Lecture Notes in Mathematics 125), pages 112–127. Springer-Verlag, 1970. 8. J. W. Lloyd. Foundations of Logic Programming: Second, Extended Edition. Springer-Verlag, 1987.Google Scholar
  8. 9.
    S. Muggleton. Inverse Entailment and Progol. New Generation Computing, 13:245–286, 1995.Google Scholar
  9. 10.
    S.-H. Nienhuys-Cheng and Ronald de Wolf. The Subsumption Theorem in Inductive Logic Programming: Facts and Fallacies. In L. de Raedt, editor, Proceedings of the 5th International Workshop on Inductive Logic Programming, pages 147–160, 1994.Google Scholar
  10. 11.
    G. D. Plotkin. Automatic Methods of Inductive Inference. PhD thesis, Edinburgh University, 1971.Google Scholar
  11. 12.
    C. Rouveirol. Completeness for Inductive Procedures. In Proceedings of the 8th International Workshop on Machine Learning, pages 452–456. Morgan Kaufmann, 1991.Google Scholar
  12. 13.
    C. Rouveirol. Extentions of Inversion of Resolution Applied to Theory Completion. In S. Muggleton, editor, Inductive Logic Programming, pages 63–92. Academic Press, 1992.Google Scholar
  13. 14.
    A. Yamamoto. Improving Theories for Inductive Logic Programming Systems with Ground Reduced Programs. Submitted to New Generation Computing, 1996.Google Scholar
  14. 15.
    A. Yamamoto. Representing Inductive Inference with SOLD-Resolution. To appear in the Proceedings of the IJCAI'97 Workshop on Abduction and Induction in AI, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Akihiro Yamamoto
    • 1
  1. 1.Meme Media LaboratoryHokkaido UniversityJapan

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