On Oscillation-Free Chaitin h-Random Sequences

The present paper generalises results by Tadaki [12] and Calude et al. [1] on oscillation-free partially random infinite strings. Moreover, it shows that oscillation-free partial Chaitin randomness can be separated from oscillation-free partial strong Martin-Löf randomness by Π 1 -definable sets of infinite strings. In the papers [11] and [2] several relaxations of randomness were defined. Subsequently, in [8] these were shown to be essentially different. The variants of partial randomness were characterised by different means such as Martin-Löf tests [11, 2], Solovay tests [11, 8] and prefix [11] or a priori complexity [2]. Using description complexity partial randomness of an infinite string ξ was defined by linear lower bounds on the complexity of the n-length prefix ξ n, that is, an infinite string was referred to as ε-random provided the complexity of ξ n was lower bounded by ε · n−O(1). In general, the mentioned papers did not require an upper bound on the complexity, except for [11] where an asymptotic upper bound was considered. For the case of a priori complexity, the papers [9, 7] gave a description of infinite oscillation free ε-random strings where the upper complexity bound matches the lower bound up to an additive constant. For the case of prefix complexity the construction of similar infinite strings was accomplished in [12, 1]. The construction in [1] uses ε-universal prefix machines. Here it was observed in Theorem 6 that there are different (inequivalent) types of ε-universal machines. In recent publications, based on Hausdorff’s original paper [5] the concept of partial randomness was refined to functions of the logarithmic scale [6] or to more general gauge functions [10]. Here we showed that for a priori complexity and computable gauge functions h : Q→ IR there are oscillation-free h-random infinite strings. The aim of the present paper is to show that, similarly to the results of [10], also in the case of prefix complexity one can refine ε-randomness to oscillationfree h-randomness. Moreover, our investigations reveal the reason of the paradox of [1, Theorem 6]. Cast into the language of gauge functions (cf. [4, 10]) the papers [12, 1] considered only the scale h(t) = t, ε ∈ (0, 1) computable, which results in complexity bounds of the form ε · n+O(1). The present paper refines this scale to a much larger class of gauge functions including also non-computable ones. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 132 (2011)