Characterization of Kurtz Randomness by a Differentiation Theorem
- 113 Downloads
Brattka, Miller and Nies (2012) showed that some major algorithmic randomness notions are characterized via differentiability. The main goal of this paper is to characterize Kurtz randomness by a differentiation theorem on a computable metric space. The proof shows that integral tests play an essential part and shows that how randomness and differentiation are connected.
KeywordsDifferentiability The differentiation theorem Kurtz randomness Integral test Computable metric space
The author thanks Jason Rute and André Nies for the useful comments. The author also appreciates the anonymous referees for their careful reading and many comments. This work was partly supported by GCOE, Kyoto University and JSPS KAKENHI 23740072.
- 5.Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability (2012, submitted) Google Scholar
- 8.Freer, C., Kjos-Hanssen, B., Nies, A.: Computable aspects of Lipshitz functions. In preparation Google Scholar
- 14.Kurtz, S.A.: Randomness and genericity in the degrees of unsolvability. PhD thesis, University of Illinois at Urbana-Champaign (1981) Google Scholar
- 15.Lebesgue, H.: Leçons sur l’Intégration et la Recherche des Fonctions Primitives. Gauthier-Villars, Paris (1904) Google Scholar
- 17.Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Graduate Texts in Computer Science. Springer, New York (2009) Google Scholar
- 19.Pathak, N., Rojas, C., Simpson, S.G.: Schnorr randomness and the Lebesgue Differentiation Theorem. Proc. Am. Math. Soc. (2012, to appear) Google Scholar