On the Amount of Nonconstructivity in Learning Formal Languages from Positive Data

  • Sanjay Jain
  • Frank Stephan
  • Thomas Zeugmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)


Nonconstructive computations by various types of machines and automata have been considered by e.g., Karp and Lipton [18] and Freivalds [9, 10]. They allow to regard more complicated algorithms from the viewpoint of 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.

This paper studies the amount of nonconstructivity needed to learn classes of formal languages from positive data. Different learning types are compared with respect to the amount of nonconstructivity needed to learn indexable classes and recursively enumerable classes, respectively, of formal languages from positive data. Matching upper and lower bounds for the amount of nonconstructivity needed are shown.


Turing Machine Formal Language Initial Segment Recursive Function Inductive Inference 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angluin, D.: Finding patterns common to a set of strings. Journal of Computer and System Sciences 21(1), 46–62 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Angluin, D.: Inductive inference of formal languages from positive data. Information and Control 45(2), 117–135 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barzdin, J.: Inductive inference of automata, functions and programs. In: Proc. of the 20th International Congress of Mathematicians, Vancouver, Canada, pp. 455–460 (1974); republished in Amer. Math. Soc. Transl. 109 (2), 107– 112 (1977)Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Beyersdorff, O., Köbler, J., Müller, S.: Proof systems that take advice. Information and Computation 209(3), 320–332 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cook, S., Krajiček, J.: Consequences of the provability of NP ⊆ P/poly. The Journal of Symbolic Logic 72(4), 1353–1371 (2007)zbMATHCrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Erdős, P.: Some remarks on the theory of graphs. Bulletin of the American Mathematical Society 53(4), 292–294 (1947)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Freivalds, R.: Amount of nonconstructivity in deterministic finite automata. Theoretical Computer Science 411(38-39), 3436–3443 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Freivalds, R., Zeugmann, T.: On the Amount of Nonconstructivity in Learning Recursive Functions. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 332–343. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Freivalds, R., Zeugmann, T.: Co–Learning of Recursive Languages from Positive Data. In: Bjørner, D., Broy, M., Pottosin, I.V. (eds.) PSI 1996. LNCS, vol. 1181, pp. 122–133. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  13. 13.
    Gold, E.M.: Language identification in the limit. Inform. Control 10(5), 447–474 (1967)zbMATHCrossRefGoogle Scholar
  14. 14.
    Jain, S., Kinber, E., Lange, S., Wiehagen, R., Zeugmann, T.: Learning languages and functions by erasing. Theoretical Computer Science 241(1-2), 143–189 (2000)MathSciNetzbMATHCrossRefGoogle 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.
    Jain, S., Stephan, F., Zeugmann, T.: On the amount of nonconstructivity in learning formal languages from text. Tech. Rep. TCS-TR-A-12-55, Division of Computer Science, Hokkaido University (2012)Google Scholar
  17. 17.
    Jantke, K.P.: Monotonic and non-monotonic inductive inference. New Generation Computing 8(4), 349–360 (1991)zbMATHCrossRefGoogle Scholar
  18. 18.
    Karp, R.M., Lipton, R.J.: Turing machines that take advice. L’ Enseignement Mathématique 28, 191–209 (1982)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lange, S., Zeugmann, T.: Types of monotonic language learning and their characterization. In: Haussler, D. (ed.) Proc. 5th Annual ACM Workshop on Computational Learning Theory, pp. 377–390. ACM Press, New York (1992)CrossRefGoogle Scholar
  20. 20.
    Lange, S., Zeugmann, T.: Language learning in dependence on the space of hypotheses. In: Pitt, L. (ed.) Proceedings of the Sixth Annual ACM Conference on Computational Learning Theory, pp. 127–136. ACM Press, New York (1993)CrossRefGoogle Scholar
  21. 21.
    Lange, S., Zeugmann, T., Zilles, S.: Learning indexed families of recursive languages from positive data: A survey. Theoretical Computer Science 397(1-3), 194–232 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill (1967); reprinted, MIT Press 1987Google Scholar
  23. 23.
    Zeugmann, T., Lange, S.: A Guided Tour Across the Boundaries of Learning Recursive Languages. In: Lange, S., Jantke, K.P. (eds.) GOSLER 1994. LNCS (LNAI), vol. 961, pp. 190–258. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  24. 24.
    Zeugmann, T., Minato, S., Okubo, Y.: Theory of Computation. Corona Publishing Co, Ltd. (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sanjay Jain
    • 1
  • Frank Stephan
    • 2
  • Thomas Zeugmann
    • 3
  1. 1.Department of Computer ScienceNational University of SingaporeSingapore
  2. 2.Department of Computer Science and Department of MathematicsNational University of SingaporeSingapore
  3. 3.Division of Computer ScienceHokkaido UniversityJapan

Personalised recommendations