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

Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness

We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's Theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's Theorem where each half is computably random uniformly relative to the other

How powerful are integer-valued martingales?

In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model.Comment: Long version of the CiE 2010 paper

Comparing disorder and adaptability in stochasticity

In the literature, there are various notions of stochasticity which measure how well an algorithmically random set satisfies the law of large numbers. Such notions can be categorized by disorder and adaptability: adaptive strategies may use information observed about the set when deciding how to act, and disorderly strategies may act out of order. In the disorderly setting, adaptive strategies are more powerful than non-adaptive ones. In the adaptive setting, Merkle et al. showed that disorderly strategies are more powerful than orderly ones. This leaves open the question of how disorderly, non-adaptive strategies compare to orderly, adaptive strategies, as well as how both relate to orderly, non-adaptive strategies. In this paper, we show that orderly, adaptive strategies and disorderly, non-adaptive strategies are both strictly more powerful than orderly, non-adaptive strategies. Using the techniques developed to prove this, we also make progress towards the former open question by introducing a notion of orderly, ``weakly adaptable'' strategies which we prove is incomparable with disorderly, non-adaptive strategies

Some results on Kolmogorov-Loveland randomness

Whether Kolmogorov-Loveland randomness is equal to the Martin-Löf randomness is a well known open question in the field of algorithmic information theory. Randomness of infinite binary sequences can be defined in terms of betting strategies, a string is non-random if a computable betting strategy wins unbounded capital by successive betting on the sequence. For Martin-Löf randomness, a betting strategy makes a bet by splitting a set of sequences into any two clopen sets, and placing a portion of capital on one of them as a wager. Kolmogorov-Loveland betting strategies are more restricted, they bet on a value of the bit at some position they choose, which splits a set of sequences into two clopen sets, the sequences that have 0 at the chosen position and the sequences that have 1. In this thesis we consider betting strategies that when making a bet are restricted to split a set of sequences into two sets of equal uniform Lebesgue measure. We call this generalization of Kolmogorov-Loveland betting strategies the half-betting strategies. We show that there is a pair of such betting strategies such that for every non-Martin-Löf random sequence one of them wins unbounded capital (the pair is universal). Next, we define a finite betting game where the betting strategies bet on finite binary strings, and show that in this game Kolmogorov-Loveland betting strategies cannot increase capital by more than an arbitrary small amount on all strings on which the unrestricted betting strategy achieves arbitrary large capital. We also look at another relaxation of Kolmogorov-Loveland betting, where a betting strategy is allowed to access bits of the sequence within a set of positions a bounded number of times. We show that if this bound is less than ℓ - log ℓ for the first ℓ positions then a pair of such betting strategies cannot be universal. Furthermore, we show that, at least for some universal betting strategies, this bound is exponential

Kolmogorov-Loveland randomness and stochasticity

Lecture Notes in Computer Science3404422-43