On the Kolmogorov Complexity of Continuous Real Functions
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.
KeywordsReal Function Domain Theory Kolmogorov Complexity Continuous Real Function Invariant Ideal
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- 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
- 5.Duracz, J.A., Končný, M.: Polynomial function enclosures and floating point software verification. In: CFV 2008, Sydney, Australia (August 2008)Google Scholar
- 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
- 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.Farjudian, A.: On the Kolmogorov complexity of continuous real functions (2011), an extended abstract available at, http://www.cs.nott.ac.uk/~avf/AuxFiles/2011-Farjudian-Kolmogorov-Real-Fun.pdf
- 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