On Oscillation-free ε-random Sequences

Abstract

In this paper we discuss three notions of partial randomness or ε-randomness. ε-randomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [J. Reimann, and F. Stephan, On hierarchies of randomness tests, in: Mathematical Logic in Asia, Proceedings of the 9th Asian Logic Conference, Novosibirsk, World Scientific, Singapore 2006] proved, Tadaki [K. Tadaki, A generalization of Chaitin's halting probability Ωand halting self-similar sets, Hokkaido Math. J. 31 (2002), 219–253] and Calude et al. [C.S. Calude, L. Staiger, and S.A. Terwijn. On partial randomness, Annals of Applied and Pure Logic, 138 (2006), 20–30] proposed at least three different concepts of partial randomness.We show that all of them satisfy the natural requirement that any ε-non-null set contains an ε-random infinite word. This allows us to focus our investigations on the strongest one which is based on a priori complexity.We investigate this concept of partial randomness and show that it allows—similar to the random infinite words—oscillation-free (w.r.t. to a priori complexity) ε-random infinite words if only ε is a computable number. The proof uses the dilution principle.Alternatively, for certain sets of infinite words (ω-languages) we show that their most complex infinite words are oscillation-free ε-random. Here the parameter ε is also computable and depends on the set chosen.