Combining Models of Approximation with Partial Learning

  • Ziyuan Gao
  • Frank Stephan
  • Sandra Zilles
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9355)


In Gold’s framework of inductive inference, the model of partial learning requires the learner to output exactly one correct index for the target object and only the target object infinitely often. Since infinitely many of the learner’s hypotheses may be incorrect, it is not obvious whether a partial learner can be modified to “approximate” the target object.

Fulk and Jain (Approximate inference and scientific method. Information and Computation 114(2):179–191, 1994) introduced a model of approximate learning of recursive functions. The present work extends their research and solves an open problem of Fulk and Jain by showing that there is a learner which approximates and partially identifies every recursive function by outputting a sequence of hypotheses which, in addition, are also almost all finite variants of the target function.

The subsequent study is dedicated to the question how these findings generalise to the learning of r.e. languages from positive data. Here three variants of approximate learning will be introduced and investigated with respect to the question whether they can be combined with partial learning. Following the line of Fulk and Jain’s research, further investigations provide conditions under which partial language learners can eventually output only finite variants of the target language.


Target Object Target Language Recursive Function Inductive Inference Learning Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45(2), 117–135 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bārzdiņs̆, J.: Two theorems on the limiting synthesis of functions. In: Theory of Algorithms and Programs, vol. 1, pp. 82–88. Latvian State University (1974) (in Russian)Google Scholar
  3. 3.
    Case, J., Lynes, C.: Machine inductive inference and language identification. In: Nielsen, M., Schmidt, E.M. (eds.) Automata, Languages and Programming. LNCS, vol. 140, pp. 107–115. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  4. 4.
    Case, J., Smith, C.: Comparison of identification criteria for machine inductive inference. Theoretical Computer Science 25, 193–220 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fulk, M., Jain, S.: Approximate inference and scientific method. Information and Computation 114, 179–191 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, Z., Jain, S., Stephan, F.: On conservative learning of recursively enumerable languages. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 181–190. Springer, Heidelberg (2013) Google Scholar
  7. 7.
    Gao, Z., Stephan, F., Zilles, S.: Partial learning of recursively enumerable languages. In: Jain, S., Munos, R., Stephan, F., Zeugmann, T. (eds.) ALT 2013. LNCS, vol. 8139, pp. 113–127. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  8. 8.
    Gold, E.M.: Language identification in the limit. Information and Control 10, 447–474 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jain, S., Osherson, D., Royer, J.S., Sharma, A.: Systems that learn: an introduction to learning theory. MIT Press (1999)Google Scholar
  10. 10.
    Jain, S., Martin, E., Stephan, F.: Learning and classifying. Theoretical Computer Science 482, 73–85 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martin, E., Osherson, D.N.: Elements of scientific inquiry. MIT Press (1998)Google Scholar
  12. 12.
    Osherson, D.N., Stob, M., Weinstein, S.: Learning strategies. Information and Control 53, 32–51 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Osherson, D.N., Stob, M., Weinstein, S.: Systems that learn: an introduction to learning theory for cognitive and computer scientists. MIT Press (1986)Google Scholar
  14. 14.
    Rogers, Jr., H.: Theory of recursive functions and effective computability. MIT Press (1987)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of ReginaReginaCanada
  2. 2.Department of MathematicsNational University of SingaporeSingaporeSingapore
  3. 3.Department of Computer ScienceNational University of SingaporeSingaporeSingapore

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