We study partitions of Fra¨ıssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin.

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last app...

Suppose τ is a cellular automaton over an amenable group and a finite alphabet. Celebrated Garden of Eden theorem states, that pre-injectivity of τ is equivalent to non-existence of Garden of Eden configuration. In this paper we will prove, that imposing some mild restrictions , we could add another equivalent assertion: non-existence of Garden of Eden configuration is equivalent to preservatio...

From 29.01.06 to 03.02.06, the Dagstuhl Seminar 06051 “Kolmogorov Complexity and Applications” was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar r...

Let SCRc = {σ ∈ 2n : K(σ) ≥ n + K(n) − c}, where K denotes prefix-free Kolmogorov complexity. These are the strings with essentially maximal prefix-free complexity. We prove that SCRc is not a Π1 set for sufficiently large c. This implies Solovay’s result that strings with maximal plain Kolmogorov complexity need not have maximal prefix-free Kolmogorov complexity, even up to a constant. We show...

From 29.01.06 to 03.02.06, the Dagstuhl Seminar 06051 “Kolmogorov Complexity and Applications” was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar r...

Randomness extraction is the process of constructing a source of randomness of high quality from one or several sources of randomness of lower quality. The problem can be modeled using probability distributions and min-entropy to measure their quality and also by using individual strings and Kolmogorov complexity to measure their quality. Complexity theorists are more familiar with the first ap...

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let RC, RKt, RKS, RKT be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measu...