Skip to main content
Log in

Learning concepts and their unions from positive data with refinement operators

Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Cite this article


This paper is concerned with a sufficient condition under which a concept class is learnable in Gold’s classical model of identification in the limit from positive data. The standard principle of learning algorithms working under this model is called the MINL strategy, which is to conjecture a hypothesis representing a minimal concept among the ones consistent with the given positive data. The minimality of a concept is defined with respect to the set-inclusion relation – the strategy is semantics-based. On the other hand, refinement operators have been developed in the field of learning logic programs, where a learner constructs logic programs as hypotheses consistent with given logical formulae. Refinement operators have syntax-based definitions – they are defined based on inference rules in first-order logic. This paper investigates the relation between the MINL strategy and refinement operators in inductive inference. We first show that if a hypothesis space admits a refinement operator with certain properties, the concept class will be learnable by an algorithm based on the MINL strategy. We then present an additional condition that ensures the learnability of the class of unbounded finite unions of concepts. Furthermore, we show that under certain assumptions a learning algorithm runs in polynomial time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. Angluin, D.: Finding patterns common to a set of strings. J. Comput. Syst. Sci. 21(1), 46–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angluin, D.: Inductive inference of formal languages from positive data. Inf. Control. 45(2), 117–135 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Angluin, D.: Inference of reversible languages. J. ACM 29(3), 741–765 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arimura, H., Shinohara, T., Otsuki, S.: A polynomial time algorithm for finding finite unions of tree pattern languages. In: Proceedings of the Second Internaitonal Workshop on Nonmonotonic and Inductive Logic, LNAI 659, pp. 118–131. Springer (1993)

  5. Arimura, H, Shinohara, T, Otsuki, S: Finding minimal generalizations for unions of pattern languages and its application to inductive inference from positive data. In: Enjalbert, P., Mayr, E., Wagner, K. (eds.) STACS 94, Lecture Notes in Computer Science, vol. 775, pp. 647–660. Springer, Berlin (1994)

    Google Scholar 

  6. Gold, E.M.: Language identification in the limit. Inf. Control. 10(5), 447–474 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jain, S., Ng, Y.K., Tay, T.S.: Learning languages in a union. J. Comput. Syst. Sci. 73, 89–108 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kobayashi, S.: Approximate identification, finite elasticity and lattice structure of hypothesis space. Tech. rep., Technical Report CSIM 96-04, Dept. of Compt. Sci. and Inform. Math., Univ. of Electro-Communications (1996)

  9. Laird, P.D.: Learning from good and bad data. Kluwer Academic Publishers, Norwell (1988)

    Book  MATH  Google Scholar 

  10. Lassez, J.-L., Maher, M.J., Marriott, K.: Unification Revisited. Foundations of Deductive Databases and Logic Programming, 587–625 (1988)

  11. Lehmann, J., Hitzler, P.: Concept learning in description logics using refinement operators. Mach. Learn. 78(1-2), 203–250 (2010)

    Article  MathSciNet  Google Scholar 

  12. Motoki, T., Shinohara, T., Wright, K.: The correct definition of finite elasticity: corrigendum to identification of unions. In: Proceedings of the fourth annual workshop on Computational learning theory, COLT ’91, p 375. Morgan Kaufmann Publishers Inc., San Francisco (1991)

  13. Ng, Y.K., Shinohara, T.: Inferring unions of the pattern languages by the most fitting covers. In: ALT, pp. 269–282 (2005)

  14. Okayama, T., Yoshinaka, R., Otaki, K., Yamamoto, A.: A sufficient condition for learning unbounded unions of languages with refinement operators. In: ISAIM (2014)

  15. Ouchi, S., Yamamoto, A.: Learning from positive data based on the MINL strategy with refinement operators. In: Proceedings of the 2009 international conference on New frontiers in artificial intelligence, JSAI-isAI’09, pp. 345–357. Springer, Berlin (2010)

  16. Plotkin, G.D.: A note on inductive generalization. Mach. Intell. 5, 153–163 (1970)

    MathSciNet  MATH  Google Scholar 

  17. Reidenbach, D.: A non-learnable class of E-pattern languages. Theor. Comput. Sci. 350(1), 91–102 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sakakibara, Y., Yokomori, T., Satoshi, K.: Keisanronteki Gakushuuriron (in Japanese. Computational Learning Theory). Baifukan (2001)

  19. Shapiro, E.Y.: Inductive inference of theories from facts. Research Report YALEU/DCS/RR-192,Yale University (1981)

  20. Shinohara, T., Arimura, H.: Inductive inference of unbounded unions of pattern languages from positive data. Theor. Comput. Sci. 241(1), 191–209 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Takami, R., Suzuki, Y., Uchida, T., Shoudai, T.: Polynomial time inductive inference of TTSP graph languages from positive data. IEICE Trans. 92-D(2), 181–190 (2009)

    Article  MATH  Google Scholar 

  22. Wright, K.: Identification of unions of languages drawn from an identifiable class. In: Proceedings of the second annual workshop on Computational learning theory, COLT ’89, pp. 328–333. Morgan Kaufmann Publishers Inc., San Francisco (1989)

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ryo Yoshinaka.

Additional information

Okayama and Ouchi have left Kyoto University

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ouchi, S., Okayama, T., Otaki, K. et al. Learning concepts and their unions from positive data with refinement operators. Ann Math Artif Intell 79, 181–203 (2017).

Download citation

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classifications (2010)