Prefix-Like Complexities and Computability in the Limit

  • Alexey Chernov
  • Jürgen Schmidhuber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)

Abstract

Computability in the limit represents the non-plus-ultra of constructive describability. It is well known that the limit computable functions on naturals are exactly those computable with the oracle for the halting problem. However, prefix (Kolmogorov) complexities defined with respect to these two models may differ. We introduce and compare several natural variations of prefix complexity definitions based on generalized Turing machines embodying the idea of limit computability, as well as complexities based on oracle machines, for both finite and infinite sequences.

Keywords

Kolmogorov complexity limit computability generalized Turing machine non-halting computation 

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References

  1. 1.
    Asarin, E., Collins, P.: Noisy Turing Machines. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1031–1042. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Becher, V., Figueira, S.: Kolmogorov Complexity for Possibly Infinite Computations. J. Logic, Language and Information 14(2), 133–148 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Becher, V., Figueira, S., Nies, A., Picchi, S.: Program Size Complexity for Possibly Infinite Computations. Notre Dame J. Formal Logic 46(1), 51–64 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Burgin, M.S.: Inductive Turing Machines. Soviet Math. Doklady 27(3), 730–734 (1983)MATHGoogle Scholar
  5. 5.
    Calude, C.S., Pavlov, B.: Coins, Quantum Measurements, and Turing’s Barrier. Quantum Information Processing 1(1–2), 107–127 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Case, J., Jain, S., Sharma, A.: On Learning Limiting Programs. In: Proc. of COLT 1992, pp. 193–202. ACM Press, New York (1992)Google Scholar
  7. 7.
    Chaitin, G.J.: A Theory of Program Size Formally Identical to Information Theory. Journal of the ACM 22, 329–340 (1975)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chernov, A., Schmidhuber, J.: Prefix-like Complexities of Finite and Infinite Sequences on Generalized Turing Machines. Technical Report IDSIA-11-05, Manno (Lugano), Switzerland (2005)Google Scholar
  9. 9.
    Durand, B., Shen, A., Vereshchagin, N.: Descriptive Complexity of Computable Sequences. Theoretical Computer Science 271(1–2), 47–58 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Etesi, G., Nemeti, I.: Non-Turing Computations via Malament-Hogarth Space-Times. International Journal of Theoretical Physics 41, 341 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Freivald, R.V.: Functions Computable in the Limit by Probabilistic Machines. In: Proc. of the 3rd Symposium on Mathematical Foundations of Computer Science, pp. 77–87. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  12. 12.
    Gács, P.: On the Relation between Descriptional Complexity and Algorithmic Probability. Theoretical Computer Science 22(1–2), 71–93 (1983)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gold, E.M.: Limiting Recursion. J. Symbolic Logic 30(1), 28–46 (1965)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hayashi, S., Nakata, M.: Towards Limit Computable Mathematics. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 125–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Kolmogorov, A.N.: Three Approaches to the Quantitative Definition of Information. Problems of Information Transmission 1(1), 1–11 (1965)MathSciNetMATHGoogle Scholar
  16. 16.
    Levin, L.A.: Laws of Information (Nongrowth) and Aspects of the Foundation of Probability Theory. Problems of Information Transmission 10(3), 206–210 (1974)Google Scholar
  17. 17.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Heidelberg (1997)CrossRefMATHGoogle Scholar
  18. 18.
    Poland, J.: A Coding Theorem for Enumerating Output Machines. Information Processing Letters 91(4), 157–161 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schmidhuber, J.: Algorithmic Theories of Everything. Technical Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland (2000)Google Scholar
  20. 20.
    Schmidhuber, J.: Hierarchies of Generalized Kolmogorov Complexities and Nonenumerable Universal Measures Computable in the Limit. International J. Foundations of Computer Science 13(4), 587–612 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Shoenfield, J.R.: On Degrees of Unsolvability. Annals of Mathematics 69, 644–653 (1959)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Simpson, S.G.: Degrees of Unsolvability: A Survey of Results. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 631–652. North-Holland, Amsterdam (1977)CrossRefGoogle Scholar
  23. 23.
    Uspensky, V.A., Vereshchagin, N.K., Shen, A.: Lecture Notes on Kolmogorov Complexity (unpublished), http://lpcs.math.msu.su/~ver/kolm-book

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexey Chernov
    • 1
  • Jürgen Schmidhuber
    • 1
    • 2
  1. 1.IDSIAMannoSwitzerland
  2. 2.TU MunichGarching, MünchenGermany

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