Theory of Computing Systems

, Volume 61, Issue 4, pp 1214–1236 | Cite as

On the Information Carried by Programs About the Objects they Compute

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Abstract

In computability theory and computable analysis, finite programs can compute infinite objects. Such objects can then be represented by finite programs. Can one characterize the additional useful information contained in a program computing an object, as compared to having the object itself? Having a program immediately gives an upper bound on the Kolmogorov complexity of the object, by simply measuring the length of the program, and such an information cannot usually be derived from an infinite representation of the object. We prove that bounding the Kolmogorov complexity of the object is the only additional useful information. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets. This article is an extended version of Hoyrup and Rojas (2015), including complete proofs and a new result (Theorem 9).

Keywords

Markov-computable Representation Kolmogorov complexity Ershov topology 

References

  1. 1.
    Čeitin, G.S.: Algorithmic operators in constructive metric spaces. Trudy Matematiki Instituta Steklov 67, 295–361 (1962). English translation: American Mathematical Society Translations, series 2, 64:1–80, 1967MathSciNetGoogle Scholar
  2. 2.
    de Brecht, M.: Quasi-polish spaces. Ann. Pure Appl. Logic 164(3), 356–381 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Freivalds, R, Wiehagen, R.: Inductive inference with additional information. J. Inf. Process. Cybern. 15, 179–185 (1979)MathSciNetMATHGoogle Scholar
  4. 4.
    Friedberg, R.M.: Un contre-exemple relatif aux fonctionnelles récursives. Comptes Rendus de l’Académie des Sciences 247, 852–854 (1958)MATHGoogle Scholar
  5. 5.
    Grassin, J.: Index sets in Ershov’s hierarchy. J. Symb. Logic 39, 97–104, 3 (1974)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grzegorczyk, A.: On the definitions of computable real continuous functions. Fundamenta Mathematicae 44, 61–71 (1957)MathSciNetMATHGoogle Scholar
  7. 7.
    Hertling, P.: Computable real functions: Type 1 computability versus Type 2 computability. In: CCA (1996)Google Scholar
  8. 8.
    Hoyrup, M., Rojas, C.: On the information carried by programs about the objects they compute. In: Mayr, E. W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pp 447–459. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  9. 9.
    Kreisel, G., Lacombe, D., Shøe nfield, J.R.: Fonctionnelles récursivement définissables et fonctionnelles récursives. Comptes Rendus de l’Académie des Sciences 245, 399–402 (1957)MATHGoogle Scholar
  10. 10.
    Kushner, Boris A.: The constructive mathematics of A. A. Markov. Amer. Math. Monthly 113(6), 559–566 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lacombe, D.: Extension de la notion de fonction récursive aux fonctions d?une ou plusieurs variables réelles I-III. Comptes Rendus Académie des Sciences Paris 240,241, 2478–2480,13–14,151–153 (1955)MathSciNetMATHGoogle Scholar
  12. 12.
    Markov, A.A.: On the continuity of constructive functions (russian). Uspekhi Mat. Nauk 9, 226–230 (1954)MathSciNetMATHGoogle Scholar
  13. 13.
    Myhill, J., Shepherdson, J.C.: Effective operations on partial recursive functions. Math. Logic Quart. 1(4), 310–317 (1955)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Marian, B.: Pour-El. A comparison of five “computable” operators. Math. Logic Quart. 6(15–22), 325–340 (1960)MATHGoogle Scholar
  15. 15.
    Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. Amer. Math. Soc. 74(2), 358–366 (1953)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rogers, H. Jr.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge (1987)MATHGoogle Scholar
  17. 17.
    Schnorr, C.P.: Optimal enumerations and optimal gödel numberings. Math. Syst. Theory 8(2), 182–191 (1974)CrossRefMATHGoogle Scholar
  18. 18.
    Schröder, M.: Extended admissibility. Theor. Comput. Sci. 284(2), 519–538 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Selivanov, V.L.: Index sets in the hyperarithmetical hierarchy. Siber. Math. J. 25, 474–488 (1984)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shapiro, N.: Degrees of computability. Trans. Amer. Math. Soc. 82, 281–299 (1956)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Spreen, D.: Representations versus numberings: On the relationship of two computability notions. Theor. Comput. Sci. 262(1), 473–499 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. 2(42), 230–265 (1936)MathSciNetMATHGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Inria Nancy Grand EstVillers-lès-NancyFrance
  2. 2.Departamento de MatemáticasUniversidad Andres BelloSantiagoChile

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