Kolmogorov Complexity and Solovay Functions

Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality

Solovay functions and K-triviality

As part of his groundbreaking work on algorithmic randomness, Solovay demonstrated in the 1970s the remarkable fact that there are computable upper bounds of prefix-free Kolmogorov complexity $K$ that are tight on infinitely many values (up to an additive constant). Such computable upper bounds are called Solovay functions. Recent work of Bienvenu and Downey~[STACS 2009, LIPIcs 3, pp 147-158] indicates that Solovay functions are deeply connected with central concepts of algorithmic randomness such as $Omega$ numbers, K-triviality, and Martin-Loef randomness. In what follows, among other results we answer two open problems posed by Bienvenu and Downey about the definition of $K$-triviality and about the Gacs-Miller-Yu characterization of Martin-Loef randomness. The former defines a sequence A to be K-trivial if K(A|n) =^+ n-K(n). So both involve the noncomputable function K. As our main results we show that in both cases K(n) can be equivalently replaced by any Solovay function, and, what is more, that among all computable functions such a replacement is possible exactly for the Solovay functions. Moreover, similar statements hold for the larger class of all right-c.e. in place of the computable functions. These full characterizations, besides having significant theoretical interest on their own, will be useful as tools when working with K-trivial and Martin-Loef random sequences

Random semicomputable reals revisited

The aim of this expository paper is to present a nice series of results, obtained in the papers of Chaitin (1976), Solovay (1975), Calude et al. (1998), Kucera and Slaman (2001). This joint effort led to a full characterization of lower semicomputable random reals, both as those that can be expressed as a "Chaitin Omega" and those that are maximal for the Solovay reducibility. The original proofs were somewhat involved; in this paper, we present these results in an elementary way, in particular requiring only basic knowledge of algorithmic randomness. We add also several simple observations relating lower semicomputable random reals and busy beaver functions.Comment: 15 page