Andrei E. Romashchenko

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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(More)
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(More)
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(More)
Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a program p transforming a to b that has the least possible length and is simple conditional on b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one,(More)
In 1997, Z. Zhang and R.W. Yeung found the first example of a conditional information inequality in four variables that is not “Shannon-type”. This linear inequality for entropies is called conditional (or constraint) since it holds only under condition that some linear equations are satisfied for the involved entropies. Later, the same(More)
We study the properties of the set of binary strings with the relation \the Kolmogorov complexity of x conditional to y is small". We prove that there are pairs of strings which have no greatest common lower bound with respect to this pre-order. We present several examples when the greatest common lower bound exists but its complexity is much less than(More)
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 topics 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(More)
We study conditional linear information inequalities, i.e., linear inequalities for Shannon entropy that hold for distributions whose joint entropies meet some linear constraints. We prove that some conditional information inequalities cannot be extended to any unconditional linear inequalities. Some of these conditional inequalities hold for almost(More)