On the Kolmogorov Complexity of Continuous Real Functions

  • Amin Farjudian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6735)


Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the behaviour of the sequence of Kolmogorov complexities of finitely-representable objects—such as rational numbers—used to approximate them. The idea will be taken further here by extending the definition to functions over real numbers. Any real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients. The asymptotic behaviour of the sequence of Kolmogorov complexities of the enclosures in such a sequence can be considered as a measure of practical suitability of the sequence as the candidate for representation of that real function. Based on that definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ real function has such a high-growth Kolmogorov complexity. Moreover, we will present an asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions.


Real Function Domain Theory Kolmogorov Complexity Continuous Real Function Invariant Ideal 
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.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford, Oxford (1994)Google Scholar
  2. 2.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society (new series) 21(1), 1–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Böhm, H.J., Cartwright, R., Riggle, M., O’Donnell, M.J.: Exact real arithmetic: A case study in higher order programming. In: Proceedings of the 1986 ACM Conference on LISP and Functional Programming, pp. 162–173. ACM, New York (1986); held at MIT, Cambridge, MACrossRefGoogle Scholar
  4. 4.
    Cai, J., Hartmanis, J.: On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. Journal of Computer and System Sciences 49(3), 605–619 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duracz, J.A., Končný, M.: Polynomial function enclosures and floating point software verification. In: CFV 2008, Sydney, Australia (August 2008)Google Scholar
  6. 6.
    Edalat, A., Lieutier, A.: Domain theory and differential calculus (functions of one variable). In: Proceedings of 17th Annual IEEE Symposium on Logic in Computer Science (LICS 2002), Copenhagen, Denmark, pp. 277–286 (2002)Google Scholar
  7. 7.
    Edalat, A., Sünderhauf, P.: A domain theoretic approach to computability on the real line. Theoretical Computer Science 210, 73–98 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Escardó, M.H.: PCF extended with real numbers: a domain theoretic approach to higher order exact real number computation. Ph.D. thesis, Imperial College (1997)Google Scholar
  9. 9.
    Farjudian, A.: On the Kolmogorov complexity of continuous real functions (2011), an extended abstract available at,
  10. 10.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains. In: Encycloedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge (2003)Google Scholar
  11. 11.
    Hunt, B.R., Sauer, T., Yorke, J.A.: Prevalence: A translation-invariant “almost every” on infinite-dimensional spaces. Bulletin of the American Mathematical Society 27(2), 217–238 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Konečný, M., Farjudian, A.: Compositional semantics of dataflow networks with query-driven communication of exact values. Journal of Universal Computer Science 16(18), 2629–2656 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Konečný, M., Farjudian, A.: Semantics of query-driven communication of exact values. Journal of Universal Computer Science 16(18), 2597–2628 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lambov, B.: Reallib: An efficient implementation of exact real arithmetic. Mathematical Structures in Computer Science 17(1), 81–98 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Springer, Heidelberg (1997)CrossRefzbMATHGoogle Scholar
  16. 16.
    Montaña, J.L., Pardo, L.M.: On Kolmogorov complexity in the real Turing machine setting. Information Processing Letters 67, 81–86 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)zbMATHGoogle Scholar
  18. 18.
    Müller, N.T.: The iRRAM: Exact arithmetic in C++. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Neumaier, A.: Taylor forms - use and limits. Reliable Computing 9(1), 43–79 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Solomonoff, R.: A formal theory of inductive inference. Information and Control 7(1), 1–22 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Solomonoff, R.: A formal theory of inductive inference. Information and Control 7(2), 224–254 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Staiger, L.: The Kolmogorov complexity of real numbers. Theoretical Computer Science 284(2), 455–466 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis, An Introduction. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    Ziegler, M., Koolen, W.M.: Kolmogorov complexity theory over the reals. Electronic Notes in Theoretical Computer Science 221, 153–169 (2008); Proceedings of the Fifth International Conference on Computability and Complexity in Analysis (CCA 2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Amin Farjudian
    • 1
  1. 1.Division of Computer ScienceUniversity of Nottingham NingboNingboChina

Personalised recommendations