Advertisement

Structured Frequency Algorithms

  • Kaspars BalodisEmail author
  • Jānis Iraids
  • Rūsiņš Freivalds
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9076)

Abstract

B.A. Trakhtenbrot proved that in frequency computability (introduced by G. Rose) it is crucially important whether the frequency exceeds \(\frac{1}{2}\). If it does then only recursive sets are frequency-computable. If the frequency does not exceed \(\frac{1}{2}\) then a continuum of sets is frequency-computable. Similar results for finite automata were proved by E.B. Kinber and H. Austinat et al. We generalize the notion of frequency computability demanding a specific structure for the correct answers. We show that if this structure is described in terms of finite projective planes then even a frequency \(O(\frac{\sqrt{n}}{n})\) ensures recursivity of the computable set. We also show that with overlapping structures this frequency cannot be significantly decreased. We also introduce the notion of graph frequency computation and prove sufficient conditions for a graph \(G\) such that a continuum of sets can be \(G\)-computed.

References

  1. 1.
    Ablaev, F., Freivalds, R.: 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.
    Austinat, H., Diekert, V., Hertrampf, U., Petersen, H.: Regular frequency computations. Theoret. Comput. Sci. 330(1), 15–21 (2005). Insightful TheoryCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Degtev, A.N.: On \((m, n)\)-computable sets. In: Moldavanskij, D.I. (ed.), Algebraic Systems, pp. 88–99. Ivanovo Gos. Universitet, (1981) (In Russian)Google Scholar
  4. 4.
    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
  5. 5.
    Hall Jr., M.: Combinatorial Theory, 2nd edn. Wiley, New York (1986)zbMATHGoogle Scholar
  6. 6.
    Harizanov, V., Kummer, M., Owings, J.: Frequency computations and the cardinality theorem. J. Symb. Log. 57, 682–687 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Hinrichs, M., Wechsung, G.: Time bounded frequency computations. In: Proceedings of Twelfth Annual IEEE Conference on Computational Complexity, 1997 (Formerly: Structure in Complexity Theory Conference), pp. 185–192. IEEE (1997)Google Scholar
  8. 8.
    Kinber, E.B.: Frequency calculations of general recursive predicates and frequency enumerations of sets. Sov. Math. 13, 873–876 (1972)zbMATHGoogle Scholar
  9. 9.
    Kinber, E.B.: Frequency computations in finite automata. Cybern. Sys. Anal. 12(2), 179–187 (1976)CrossRefGoogle Scholar
  10. 10.
    König, D.: Sur les correspondances multivoques des ensembles. Fundamenta Math. 8(1), 114–134 (1926)zbMATHGoogle Scholar
  11. 11.
    McNaughton, R.: The theory of automata, a survey. Adv. Comput. 2, 379–421 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Rabin, M.O.: Probabilistic automata. Inf. Control 6(3), 230–245 (1963)CrossRefzbMATHGoogle Scholar
  13. 13.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Rose, G.F.: An extended notion of computability. In: International Congress for Logic, Methodology and Philosophy of Science, Stanford, California (1960)Google Scholar
  15. 15.
    Tantau, T.: Towards a cardinality theorem for finite automata. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 625–636. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  16. 16.
    Trakhtenbrot, B.A.: On the frequency computation of functions. Algebra i Logika 2(1), 25–32 (1964). In RussianGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Kaspars Balodis
    • 1
    • 2
    Email author
  • Jānis Iraids
    • 1
    • 2
  • Rūsiņš Freivalds
    • 1
    • 2
  1. 1.Faculty of ComputingUniversity of LatviaRigaLatvia
  2. 2.Institute of Mathematics and Computer ScienceUniversity of LatviaRigaLatvia

Personalised recommendations