Frequency Prediction of Functions

  • Kaspars Balodis
  • Ilja Kucevalovs
  • Rūsiņš Freivalds
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7119)

Abstract

Prediction of functions is one of processes considered in inductive inference. There is a “black box” with a given total function f in it. The result of the inductive inference machine F( < f(0), f(1), ⋯ ,f(n) > ) is expected to be f(n + 1). Deterministic and probabilistic prediction of functions has been widely studied. Frequency computation is a mechanism used to combine features of deterministic and probabilistic algorithms. Frequency computation has been used for several types of inductive inference, especially, for learning via queries. We study frequency prediction of functions and show that that there exists an interesting hierarchy of predictable classes of functions.

Keywords

Frequency Computation Recursive Function Inductive Inference Total Function Probabilistic Algorithm 
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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References

  1. 1.
    Farid, M.: Why Sometimes Probabilistic Algorithms Can Be More Effective. In: Wiedermann, J., Gruska, J., Rovan, B. (eds.) MFCS 1986. LNCS, vol. 233, pp. 1–14. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  2. 2.
    Apsītis, K., Freivalds, R., Kriķis, M., Simanovskis, R., Smotrovs, J.: Unions of Identifiable Classes of Total Recursive Functions. In: Jantke, K.P. (ed.) AII 1992. LNCS, vol. 642, pp. 99–107. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  3. 3.
    Austinat, H., Diekert, V., Hertrampf, U., Petersen, H.: Regular frequency computations. Theoretical Computer Science 330(1), 15–20 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bārzdiņš, J., Barzdin, Y.M.: Two theorems on limiting synthesis of functions. Theory of algorithms and programs 1, 82–88 (1974) (in Russian)Google Scholar
  5. 5.
    Bārzdiņš, J., Freivalds, R.: On the prediction of general recursive functions. Soviet Mathematics Doklady 13, 1224–1228 (1972)Google Scholar
  6. 6.
    Beigel, R., Gasarch, W.I., Kinber, E.B.: Frequency computation and bounded queries. Theoretical Computer Science 163(1/2), 177–192 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Case, J., Kaufmann, S., Kinber, E.B., Kummer, M.: Learning recursive functions from approximations. Journal of Computer and System Sciences 55(1), 183–196 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Degtev, A.N.: On (m,n)-computable sets. In: Moldavanskij, D.I., Gos, I. (eds.) Algebraic Systems, pp. 88–99. Universitet (1981)Google Scholar
  9. 9.
    Freivalds, R.: On the growth of the number of states in result of the determinization of probabilistic finite automata. Avtomatika i Vichislitel’naya Tekhnika (3), 39–42 (1982) (Russian)Google Scholar
  10. 10.
    Freivalds, R., Karpinski, M.: Lower Space Bounds for Randomized Computation. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 580–592. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  11. 11.
    Freivalds, R.: Complexity of Probabilistic Versus Deterministic Automata. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 565–613. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  12. 12.
    Freivalds, R.: Inductive Inference of Recursive Functions: Qualitative Theory. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 77–110. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  13. 13.
    Freivalds, R., Bārzdiņš, J., Podnieks, K.: Inductive Inference of Recursive Functions: Complexity Bounds. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 111–155. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Freivalds, R.: Models of Computation, Riemann Hypothesis, and Classical Mathematics. In: Rovan, B. (ed.) SOFSEM 1998. LNCS, vol. 1521, pp. 89–106. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  15. 15.
    Freivalds, R.: Non-constructive methods for finite probabilistic automata. International Journal of Foundations of Computer Science 19(3), 565–580 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Freivalds, R.: Amount of nonconstructivity in finite automata. Theoretical Computer Science 411(38-39), 3436–3443 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gold, E.M.: Language identification in the limit. Information and Control 10(5), 447–474 (1967)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Harizanova, V., Kummer, M., Owings, J.: Frequency computations and the cardinality theorem. The Journal of Symbolic Logic 57(2), 682–687 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hinrichs, M., Wechsung, G.: Time bounded frequency computations. Information and Computation 139, 234–257 (1997)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kinber, E.B.: Frequency calculations of general recursive predicates and frequency enumeration of sets. Soviet Mathematics Doklady 13, 873–876 (1972)MATHGoogle Scholar
  21. 21.
    Kinber, E.B.: On frequency real-time computations. In: Barzdin, Y.M. (ed.) Teoriya Algoritmov i Programm, vol. 2, pp. 174–182 (1973) (Russian)Google Scholar
  22. 22.
    Kinber, E.B.: Frequency computations in finite automata. Kibernetika 2, 7–15 (1976); Russian; English translation in Cybernetics 12, 179–187 (1976)Google Scholar
  23. 23.
    Kummer, M.: A proof of Beigel’s Cardinality Conjecture. The Journal of Symbolic Logic 57(2), 677–681 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Moore, E.F.: Gedanken-experiments on sequential machines. Automata Studies Ann. of Math. Studies (34), 129–153 (1956)MathSciNetGoogle Scholar
  25. 25.
    McNaughton, R.: The Theory of Automata, a Survey. Advances in Computers 2, 379–421 (1961)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3(2), 115–125 (1959)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. MIT Press (1987)Google Scholar
  28. 28.
    Rose, G.F.: An extended notion of computability. In: Abstracts of International Congress for Logic, Methodology and Philosophy of Science, p. 14 (1960)Google Scholar
  29. 29.
    Rose, G.F., Ullian, J.S.: Approximations of functions on the integers. Pacific Journal of Mathematics 13(2), 693–701 (1963)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Smullyan, R.M.: Theory of Formal Systems, Annals of Mathematics Studies, vol. (47), Princeton, NJ (1961)Google Scholar
  31. 31.
    Trakhtenbrot, B.A.: On the frequency computation of functions. Algebra i Logika 2, 25–32 (1964)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kaspars Balodis
    • 1
  • Ilja Kucevalovs
    • 1
  • Rūsiņš Freivalds
    • 1
  1. 1.Faculty of ComputingUniversity of LatviaRigaLatvia

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