The Kolmogorov complexity of real numbers
We consider for a real number α the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other.
More precisely, we show that the complexity of the length l · logr b prefix of the base r expansion of α is the same (up to an additive constant) as the logr b-fold complexity of the length l prefix of the base b expansion of α.
Then we use this fact to derive complexity theoretic proofs for the base independence of the randomness of real numbers and for some properties of Liouville numbers.
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- [CC96]C. Calude and C. Câmpeanu, Are Binary Codings Universal?, Complexity 1 (1996) 15, 47–50.Google Scholar
- [CJ94]C. Calude and H. Jürgensen, Randomness as an invariant for number representations. In: Results and Trends in Theoretical Computer Science (H. Maurer, J. Karhumäki, G. Rozenberg, Eds.), Lecture Notes in Comput. Sci., Vol. 812, Springer-Verlag, Berlin, 1994, 44–66.Google Scholar
- [HW98]P. Hertling and K. Weihrauch, Randomness Spaces. in: Automata, Languages and Programming, Proc. 25th Int. Colloq. ICALP’98, (K. G. Larsen, S. Skyum and G. Winskel, Eds.), Lecture Notes in Comput. Sci., Vol. 1443, Springer-Verlag, Berlin, 1998, 796–807.Google Scholar
- [JT88]H. Jürgensen and G. Thierrin, Some structural properties of ω-languages. in: Sbornik 13th Nat. School “Applications of Mathematics in Technology”, Sofia, 1988, 56–63.Google Scholar
- [St93]L. Staiger, Kolmogorov complexity and Hausdorff dimension. Inform. and Comput. 103 (1993) 2, 159–194. Preliminary version in: “Fundamentals of Computation Theory” (J. Csirik, J. Demetrovics and F. Gècseg Eds.), Lecture Notes in Comput. Sci., No. 380, Springer-Verlag, Berlin 1989, 334–343.zbMATHCrossRefMathSciNetGoogle Scholar
- [We92]K. Weihrauch, The Degrees of Discontinuity of some Translators Between Representations of the Real Numbers. Informatik-Bericht 129, FernUniversität Hagen 1992.Google Scholar