General inductive inference types based on linearly-ordered sets

  • Andris Ambainis
  • Rūsiņš Freivalds
  • Carl H. Smith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


In this paper, we reconsider the definitions of procrastinating learning machines. In the original definition of Freivalds and Smith [FS93], constructive ordinals are used to bound mindchanges. We investigate the possibility of using arbitrary linearly ordered sets to bound mindchanges in a similar way. It turns out that using certain ordered sets it is possible to define inductive inference types more general than the previously known ones. We investigate properties of the new inductive inference types and compare them to other types.


Maximal Element Recursive Function Inductive Inference Identification Type Duality Principle 
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  1. [Amb95]
    A. Ambainis. The power of procrastination in inductive inference: how it depends on used ordinal notations, volume 904. Springer-Verlag, 1995. P. Vitányi, editor.Google Scholar
  2. [AS83]
    D. Angluin and C. H. Smith. Inductive inference: Theory and methods. Computing Surveys, 15:237–269, 1983.CrossRefGoogle Scholar
  3. [Aps94]
    K. Apsītis. S. Atikawa and K. Jantke, editors. Derived sets and inductive inference, volume 872. Springer-Verlag, 1994.Google Scholar
  4. [Chu38]
    A. Church. The constructive second number class. Bulliten of the AMS, 44:224–232, 1938.Google Scholar
  5. [CJS95]
    J. Case, S. Jain, and M. Suraj, Not-so-nearly-minimal-size program inference. In K. Jantke, editor, Proceedings of the GOSLER Workshop on Algorithmic Learning for Knowledge Processing, to appear 1995. Preliminary version presented November 1993, Dagstuhl Castle, Germany.Google Scholar
  6. [CK37]
    A. Church and S. Kleene. Formal definitions in the theory of ordinal numbers. Fundamenta Mathematicae, 28:11–21, 1937.Google Scholar
  7. [CS83]
    J. Case and C. Smith. Comparison of identification criteria for machine inductive inference. Theoretical Computer Science, 25(2):193–220, 1983.CrossRefGoogle Scholar
  8. [FS93]
    R. Freivalds and C. Smith. On the power of procrastination for machine learning. Information and Computation, 107:237–271, 1993.Google Scholar
  9. [Gol67]
    E. M. Gold. Language identification in the limit. Information and Control, 10:447–474, 1967.CrossRefGoogle Scholar
  10. [Kle38]
    S. Kleene. On notation for ordinal numbers. Journal of Symbolic Logic, 3:150–155, 1938.Google Scholar
  11. [Erd43]
    P.Erdos. Some remarks on set theory, Annals of Mathematics(2), vol.44(1943), pp.643–646Google Scholar
  12. [FFG95]
    L. Fortnow, R.Freivalds, W.I.Gasarch, M. Kummer, S.A. Kurtz, C. Smith, F. Stephan. Measure, category and learnng theory. Lecture Notes in Computer Science, vol. 944(1995), pp. 558–569Google Scholar
  13. [Fre91]
    R. Freivalds. Inductive inference of recursive functions: qualitative theory. Lecture Notes in Computer Science, Springer, vol. 502(1991), pp. 77–100Google Scholar
  14. [Lis81]
    L.Lisagor. The Banach-Mazur game. Translated vaersion of Matematiceskij Sbornik, vol. 38(1981), pp. 201–206Google Scholar
  15. [Lut92]
    J. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, vol.44(1992), pp. 226–258Google Scholar
  16. [Meh73]
    K. Mehlhorn. On the size of sets of computable functions. proceedings of the 14th Annual Symposium on Switching and Automata Theory, 1973, IEEE Computer Society, pp. 190–196Google Scholar
  17. [Rog67]
    H. Rogers. Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967Google Scholar
  18. [Sie34]
    W. Sierpinski. Sur la dualite entre ka premiere categorie et la mesue nulle, Fundamenta Mathemaicae, vol.22(1934), pp.276–280Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Andris Ambainis
    • 1
    • 3
  • Rūsiņš Freivalds
    • 1
    • 3
  • Carl H. Smith
    • 2
    • 3
  1. 1.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia
  2. 2.Department of Computer ScienceUniversity of MarylandCollege ParkUSA
  3. 3.Department of Mathematics, Computer Science, Physics and AstronomyUniversity of AmsterdamTV Amsterdamthe Netherlands

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