It was mentioned by Kolmogorov (1968, IEEE Trans. Inform. Theory 14, 662 664) that the properties of algorithmic complexity and Shannon entropy are similar. We investigate one aspect of this similarity. Namely, we are interested in linear inequalities that are valid for Shannon entropy and for Kolmogorov complexity. It turns out that (1) all linear inequalities that are valid for Kolmogorov complexity are also valid for Shannon entropy and vice versa; (2) all linear inequalities that are valid for Shannon entropy are valid for ranks of finite subsets of linear spaces; (3) the opposite statement is not true; Ingleton's inequality (1971,``Combinatorial Mathematics and Its Applications,'' pp. 149 167. Academic Press, San Diego) is valid for ranks but not for Shannon entropy; (4) for some special cases all three classes of inequalities coincide and have simple description. We present an inequality for Kolmogorov complexity that implies Ingleton's inequality for ranks; another application of this inequality is a new simple proof of one of Ga cs Ko rner's results on common information (1973, Problems Control Inform. Theory 2, 149 162).

Abstract. In this paper we prove a countable set of non-Shannon-type linear information inequalities for entropies of discrete random variables, i.e., information inequalities which cannot be reduced to the "basic" inequality I(X : Y |Z) ≥ 0. Our results generalize the inequalities of Z. Zhang and R. Yeung (1998) who found the first examples of non-Shannon-type information inequalities.

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many fields, ranging from logic (the Entscheidungsproblem) to physics (quasicrystals).We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann's self-reproducing automata; similar ideas were also used by P. Gács in the context of error-correcting computations.This construction is rather flexible, so it can be used in many ways. We show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (in which any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, to characterize effectively closed one-dimensional subshifts in terms of two-dimensional subshifts of finite type (an improvement of a result by M. Hochman), to construct a tile set that has only complex tilings, and to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter, we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: There exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.Some of these results were included in the DLT extended abstract [10] and in the ICALP extended abstract [11]. * Supported in part by ANR EMC ANR-09-BLAN-0164-01, NAFIT ANR-08-EMER-008-01, and RFBR 09-01-00709-a grants.

In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions.