We consider (in the framework of algorithmic information theory) questions of the following type: construct a message that contains different amounts of information for recipients that have (or do not have) certain a priori information. Assume, for example, that the recipient knows some string a, and we want to send her some information that allows her to reconstruct some string b (using a). On...

In this survey we consider some results on Kolmogorov complexity whose proofs are based on interesting games. The close relation between Recursion theory, whose part is Kolmogorov complexity, and Game theory was revealed by Andrey Muchnik. In [10], he associated with every statement φ of Recursion theory a game Gφ that has the following properties. First, Gφ is a game with complete information ...

In analogy of classical Kolmogorov complexity we develop a theory of the algorithmic information in bits contained in any one of continuously many pure quantum states: quantum Kolmogorov complexity. Classical Kolmogorov complexity coincides with the new quantum Kolmogorov complexity restricted to the classical domain. Quantum Kolmogorov complexity is upper bounded and can be effectively approxi...

We call A weakly low for K if there is a c such that KA(σ) ≥ K(σ)−c for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K iff Chaitin’s Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity o...

Turbulence is often expressed in terms of either irregular or random fluid flows, without quantification. In this paper, a methodology to evaluate the randomness of the turbulence using measures based on the Kolmogorov complexity (KC) is proposed. This methodology is applied to experimental data from a turbulent flow developing in a laboratory channel with canopy of three different densities. T...

Peter Gacs showed [2] that for every n there exists a bit string x of length n whose plain complexity C (x) has almost maximal conditional complexity relative to x, i.e., C (C (x)|x)≥ logn− log logn−O(1). Following Elena Kalinina [3], we provide a gametheoretic proof of this result; modifying her argument, we get a better (and tight) bound logn−O(1). We also show the same bound for prefix-free ...

We study the properties of the intersection of independence systems. An independence system (V, I) is a k-system if for all S ⊆ V the ratio of the cardinality of the largest to the smallest maximal independent subset of S is at most k. We show that the intersection of a k1-system and a k2-system is a (k1 + k2)-system. As an application of our results, we show an improved approximation algorithm...