On the Amount of Nonconstructivity in Learning Recursive Functions

  • Rūsiņš Freivalds
  • Thomas Zeugmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


Nonconstructive proofs are a powerful mechanism in mathematics. Furthermore, nonconstructive computations by various types of machines and automata have been considered by e.g., Karp and Lipton [] and Freivalds []. They allow to regard more complicated algorithms from the viewpoint of much more primitive computational devices. The amount of nonconstructivity is a quantitative characterization of the distance between types of computational devices with respect to solving a specific problem.

In the present paper, the amount of nonconstructivity in learning of recursive functions is studied. Different learning types are compared with respect to the amount of nonconstructivity needed to learn the whole class of general recursive functions. Upper and lower bounds for the amount of nonconstructivity needed are proved.


inductive inference recursive functions nonconstructivity 


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  1. 1.
    Adleman, L.M., Blum, M.: Inductive inference and unsolvability. The Journal of Symbolic Logic 56(3), 891–900 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bārzdiņš, J.M.: Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set. Soviet Mathematics Doklady 9, 1251–1254 (1968)zbMATHGoogle Scholar
  3. 3.
    Blum, M.: A machine independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Case, J., Smith, C.: Comparison of identification criteria for machine inductive inference. Theoretical Computer Science 25(2), 193–220 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cholak, P., Downey, R., Fortnow, L., Gasarch, W., Kinber, E., Kummer, M., Kurtz, S., Slaman, T.A.: Degrees of inferability. In: Proc. 5th Annual ACM Workshop on Computational Learning Theory, pp. 180–192. ACM Press, New York (1992)CrossRefGoogle Scholar
  6. 6.
    Damm, C., Holzer, M.: Automata that take advice. In: Hájek, P., Wiedermann, J. (eds.) MFCS 1995. LNCS, vol. 969, pp. 149–158. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  7. 7.
    Erdős, P.: Some remarks on the theory of graphs. Bulletin of the American Mathematical Society 53(4), 292–294 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Freivald, R.V., Wiehagen, R.: Inductive inference with additional information. Elektronische Informationsverarbeitung und Kybernetik 15(4), 179–185 (1979)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Freivald, R.: Minimal gödel numbers and their identification in the limit. In: Bečvář, J. (ed.) MFCS 1975. LNCS, vol. 32, pp. 219–225. Springer, Heidelberg (1975)Google Scholar
  10. 10.
    Freivalds, R.: Inductive inference of recursive functions: Qualitative theory. In: Bārzdiņš, J., Bjørner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 77–110. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  11. 11.
    Freivalds, R.: Amount of nonconstructivity in finite automata. In: Maneth, S. (ed.) CIAA 2009. LNCS, vol. 5642, pp. 227–236. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Freivalds, R.: Amount of nonconstructivity in deterministic finite automata. Theoretical Computer Science 411(38-39), 3436–3443 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gold, E.M.: Limiting recursion. The Journal of Symbolic Logic 30, 28–48 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gold, E.M.: Language identification in the limit. Inform. Control 10(5), 447–474 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jain, S., Osherson, D., Royer, J.S., Sharma, A.: Systems that Learn: An Introduction to Learning Theory, 2nd edn. MIT Press, Cambridge (1999)Google Scholar
  16. 16.
    Kalmár, L.: On the reduction of the decision problem. First Paper. Ackermann prefix, a single binary predicate. The Journal of Symbolic Logic 4(1), 1–9 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Karp, R.M., Lipton, R.J.: Turing machines that take advice. L’ Enseignement Mathématique 28, 191–209 (1982)MathSciNetzbMATHGoogle Scholar
  18. 18.
    КинϬер, Е.Б.: О лредельном синтезе почти минимальных геделевских номеров. In: Bārzdiņš, J. (ed.) Теория Алгоритмов и Программ, vol. I, pp. 212–223. Latvian State University (1974)Google Scholar
  19. 19.
    Kummer, M., Stephan, F.: On the structure of the degrees of inferability. Journal of Computer and System Sciences 52(2), 214–238 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Landweber, L.H., Robertson, E.L.: Recursive properties of abstract complexity classes. Journal of the ACM 19(2), 296–308 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pepis, J.: Ein Verfahren der mathematischen Logik. The Journal of Symbolic Logic 3(2), 61–76 (1938)CrossRefzbMATHGoogle Scholar
  22. 22.
    Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967); reprinted, MIT Press (1987)zbMATHGoogle Scholar
  23. 23.
    Trakhtenbrot, B.A., Barzdin, Y.M.: Finite Automata, Behavior and Synthesis. North Holland, Amsterdam (1973)zbMATHGoogle Scholar
  24. 24.
    Wiehagen, R.: Zur Theorie der Algorithmischen Erkennung. Dissertation B, Humboldt-Universität zu Berlin (1978)Google Scholar
  25. 25.
    Zeugmann, T., Zilles, S.: Learning recursive functions: A survey. Theoretical Computer Science 397(1-3), 4–56 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Rūsiņš Freivalds
    • 1
  • Thomas Zeugmann
    • 2
  1. 1.Institute of Mathematics and Computer Science, University of Latvia, Raiņa Bulvāris 29, Riga, LV-1459Latvia
  2. 2.Division of Computer ScienceHokkaido UniversitySapporoJapan

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