arXiv subjects

Nikolay Vereshchagin

Publications and source records attributed to Nikolay Vereshchagin.

18 recordsLinked to original sources

arXiv2026

Maximal Kolmogorov Complexity in a Hamming Ball

The minimal Kolmogorov complexity of a string within Hamming distance r of a given string x is the algorithmic rate-distortion function of x, and Vereshchagin and Vitanyi characterized completely which shapes it can have. This paper is about the opposite extreme. For a binary string x of length n let g_x(r) denote the maximal Kolmogorov complexity of a string within Hamming distance r of x; we study which values, and more generally which functions of r, this quantity can attain. First we characterize, up to an additive error O(log n), the possible values of the triple (C(x),r,g_x(r)): writing r_k for the radius of a Hamming ball of cardinality about 2^k, a triple (k,r,l) is realizable if and only if log V(r_k + r) < l < min{n, k+log V(r)}, where V(a) is the cardinality of a ball of radius a. In particular, for r_k+r > n/2 both bounds collapse to n and only l = n is realizable. The two ends of this interval correspond to the two extreme ways of placing a set of complexity k in the cube: a single Hamming ball, where the lower bound comes from Harper's isoperimetric inequality, and an error-correcting code, which for the intermediate parameters we relax to a family of centers with bounded covering multiplicity, in the spirit of list decoding. Then we turn to the function r -> g_x(r) as a whole: we establish four properties that it always has, and show that the minimal and the maximal functions consistent with these properties are both attained, for every complexity level k. Which intermediate profiles are attainable remains open.

cs.IT↗

arXiv2026

Matching Rules for Substitution and Hierarchical Tilings for any Substitution with Finite Local Complexity

The Goodman-Strauss theorem states that for ``almost every'' substitution $τ$, the family of substitution tilings is sofic, that is, it can be defined by local matching rules for some decoration of tiles. The conditions on the substitution that guarantee the soficity are quite complicated in the statement of the theorem. In this paper we propose a version of the Goodman-Strauss theorem with very simple conditions on the substitution: the family of substitution tilings must have finite local complexity (FLC), that is, the number of crowns that appear in $τ$-supertiles is finite. Like the original theorem, our theorem provides matching rules for all known substitution tilings. We also prove a similar theorem for the family of \emph{hierarchical} tilings associated with the given substitution. A tiling is called $τ$-hierarchical if it has a composition under $τ$, such that this composition also has a composition, and so on, infinitely many times. Every substitution tiling is hierarchical, but the converse is not always true.

math.DS↗

arXiv2026

A family of non-periodic tilings of the plane by right golden triangles

We study a family of substitution tilings with similar right triangles of two sizes which is obtained using the substitution rule introduced in [Danzer, L. and van Ophuysen, G. A species of planar triangular tilings with inflation factor $\sqrt{-τ}$. Res. Bull. Panjab Univ. Sci. 2000, 50, 1-4, pp. 137--175 (2001)]. In that paper, it is proved this family of tilings can be obtained from a local rule using decorated tiles. That is, that this family is \emph{sofic}. In the present paper, we provide an alternative proof of this fact. We use more decorated tiles than Danzer and van Ophuysen (22 in place of 10). However, our decoration of supertiles is more intuitive and our local rule is simpler.

math.CO↗

arXiv2026

A new simple family of non-periodic tilings with square tiles

We define a new family of non-periodic tilings with square tiles that is mutually locally derivable with some family of tilings with isosceles right triangles. Both families are defined by simple local rules, and the proof of their non-periodicity is as simple as that of the non-periodicity of Robinson's tilings. We also relate the construction to the classical chair tiling: forgetting the decoration, our tilings map almost one-to-one onto the chair, our substitution is essentially a square root of the chair substitution, and our local rules are perfect, defining exactly the family of substitution tilings.

math.CO↗

arXiv2025

Goodman-Strauss theorem revisited

The Goodman-Strauss theorem states that for ``almost every" substitution, the family of substitution tilings is sofic, that is, it can be defined by local rules for some decoration of tiles. The conditions on the substitution that guarantee the soficity are quite complicated. In this paper we propose a version of Goodman-Strauss theorem with simpler conditions on the substitution. Although the conditions are quite restrictive, we show that, in combination with two simple tricks (taking a sufficiently large power of the substitution and combining small tiles into larger ones), our version of Goodman-Strauss theorem can also prove the soficity of the family of substitution tilings for ``almost every'' substitution. We also prove a similar theorem for the family of hierarchical tilings associated with the given substitution. A tiling is called hierarchical if it has a composition under the substitution, such that this composition also has a composition, and so on, infinitely many times. Every substitution tiling is hierarchical, but the converse is not always true. Fernique and Ollinger formulated some conditions on the substitution that guarantee that the family of hierarchical tilings is sofic. However, their technique does not prove this statement under such general conditions as in their paper. In the present paper, we show that under the same assumptions, as for our version of the Goodman-Strauss theorem, their technique works.

math.CO↗

arXiv2024

Half-duplex communication complexity with adversary can be less than the classical communication complexity

Half-duplex communication complexity with adversary was defined in [Hoover, K., Impagliazzo, R., Mihajlin, I., Smal, A. V. Half-Duplex Communication Complexity, ISAAC 2018.] Half-duplex communication protocols generalize classical protocols defined by Andrew Yao in [Yao, A. C.-C. Some Complexity Questions Related to Distributive Computing (Preliminary Report), STOC 1979]. It has been unknown so far whether the communication complexities defined by these models are different or not. In the present paper we answer this question: we exhibit a function whose half-duplex communication complexity with adversary is strictly less than its classical communication complexity.

cs.CC↗

arXiv2023

Kolmogorov Last Discovery? (Kolmogorov and Algorithmic Statictics)

The last theme of Kolmogorov's mathematics research was algorithmic theory of information, now often called Kolmogorov complexity theory. There are only two main publications of Kolmogorov (1965 and 1968-1969) on this topic. So Kolmogorov's ideas that did not appear as proven (and published) theorems can be reconstructed only partially based on work of his students and collaborators, short abstracts of his talks and the recollections of people who were present at these talks. In this survey we try to reconstruct the development of Kolmogorov's ideas related to algorithmic statistics (resource-bounded complexity, structure function and stochastic objects).

math.LO↗

arXiv2023

On information content in certain objects

The fine approach to measure information dependence is based on the total conditional complexity CT(y|x), which is defined as the minimal length of a total program that outputs y on the input x. It is known that the total conditional complexity can be much larger than than the plain conditional complexity. Such strings x, y are defined by means of a diagonal argument and are not otherwise interesting. In this paper we investigate whether this happens also for some natural objects. More specifically, we consider the following objects: the number of strings of complexity less than n and the lex first string of length n and complexity at least n. It is known that they have negligible mutual conditional complexities. In this paper we prove that their mutual total conditional complexities may be large. This is the first example of natural objects whose plain conditional complexity is much less than the total one.

cs.IT↗

arXiv2021

Information disclosure in the framework of Kolmogorov complexity

We consider the network consisting of three nodes $1, 2, 3$ connected by two open channels $1\rightarrow 2$ and $1\rightarrow 3$. The information present in the node 1 consists of four strings $x,y,z,w$. The nodes $2, 3$ know $x,w$ and need to know $y,z$, respectively. We want to arrange transmission of information over the channels so that both nodes $2$ and $3$ learn what they need and the disclosure of information is as small as possible. By information disclosure we mean the amount of information in the strings transmitted through channels about $x,y,z,w$ (or about $x,w$). We are also interested in whether it is possible to minimize the disclosure of information and simultaneously minimize the length of words transferred through the channels.

cs.IT↗

arXiv2019

Proofs of conservation inequalities for Levin's notion of mutual information of 1974

In this paper we consider Levin's notion of mutual information in infinite 0-1-sequences, as defined in [Leonid Levin. Laws of Information Conservation (Nongrowth) and Aspects of the Foundation of Probability Theory. Problems of information transmission, vol. 10 (1974), pp. 206--211]. The respective information conservation inequalities were stated in that paper without proofs. Later some proofs appeared in the literature, however no proof of the probabilistic conservation inequality has been published yet. In this paper we prove that inequality and for the sake of completeness we present also short proofs of other properties of the said notion.

cs.IT↗

arXiv2019

Descriptive Complexity of Computable Sequences Revisited

The purpose of this paper is to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47--58]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence $α$: $C^{0'}(α)$, defined as the minimal length of a program with oracle $0'$ that prints $α$, and $M_{\infty}(α)$, defined as $\liminf C(α_{1:n}|n)$, where $α_{1:n}$ denotes the length-$n$ prefix of $α$ and $C(x|y)$ stands for conditional Kolmogorov complexity. We show that $C^{0'}(α)\le M_{\infty}(α)+O(1)$ and $M_{\infty}(α)$ is not bounded by any computable function of $C^{0'}(α)$, even on the domain of computable sequences.

math.LO↗

arXiv2018

On the structure of Ammann A2 tilings

We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete and Computational Geometry 20 (1998) 265-279]. By the same techniques we show that Ammann A2 tilings are not robust in the sense of [B. Durand, A. Romashchenko, A. Shen. Fixed-point tile sets and their applications, Journal of Computer and System Sciences, 78:3 (2012) 731--764].

math.LO↗

arXiv2017

Conditional Information Inequalities and Combinatorial Applications

We show that the inequality $H(A \mid B,X) + H(A \mid B,Y) \le H(A\mid B)$ for jointly distributed random variables $A,B,X,Y$, which does not hold in general case, holds under some natural condition on the support of the probability distribution of $A,B,X,Y$. This result generalizes a version of the conditional Ingleton inequality: if for some distribution $I(X: Y \mid A) = H(A\mid X,Y)=0$, then $I(A : B) \le I(A : B \mid X) + I(A: B \mid Y) + I(X : Y)$. We present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: assume that the edges of a bipartite graph are partitioned into $K$ matchings such that for each pair (left vertex $x$, right vertex $y$) there is at most one matching in the partition involving both $x$ and $y$; assume further that the degree of each left vertex is at least $L$ and the degree of each right vertex is at least $R$. Then $K\ge LR$. The second application is a new method to prove lower bounds for biclique coverings of bipartite graphs.

cs.IT↗

arXiv2017

Short lists with short programs in short time

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any standard Turing machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a O$(\log |x|)$-short program for $x$. We also show that there exists a computable function that maps every $x$ to a list of size $|x|^2$ containing a O$(1)$-short program for $x$. This is essentially optimal because we prove that for each such function there is a $c$ and infinitely many $x$ for which the list has size at least $c|x|^2$. Finally we show that for some standard machines, computable functions generating lists with $0$-short programs, must have infinitely often list sizes proportional to $2^{|x|}$.

cs.CC↗

arXiv2015

Algorithmic statistics revisited

The mission of statistics is to provide adequate statistical hypotheses (models) for observed data. But what is an "adequate" model? To answer this question, one needs to use the notions of algorithmic information theory. It turns out that for every data string $x$ one can naturally define "stochasticity profile", a curve that represents a trade-off between complexity of a model and its adequacy. This curve has four different equivalent definitions in terms of (1)~randomness deficiency, (2)~minimal description length, (3)~position in the lists of simple strings and (4)~Kolmogorov complexity with decompression time bounded by busy beaver function. We present a survey of the corresponding definitions and results relating them to each other.

cs.IT↗

arXiv2014

Short lists with short programs for functions

Let $\{ϕ_p\}$ be an optimal Gödel numbering of the family of computable functions (in Schnorr's sense), where $p$ ranges over binary strings. Assume that a list of strings $L(p)$ is computable from $p$ and for all $p$ contains a $ϕ$-program for $ϕ_p$ whose length is at most $\varepsilon$ bits larger that the length of the shortest $ϕ$-program for $ϕ_p$. We show that for infinitely many $p$ the list $L(p)$ must have $2^{|p|-\varepsilon-O(1)}$ strings. Here $\varepsilon$ is an arbitrary function of $p$.

math.LO↗

arXiv2010

Game interpretation of Kolmogorov complexity

The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation and showing that all "natural" properties are either true for all sufficiently powerful oracles or false for all sufficiently powerful oracles. This result is a simple consequence of Martin's determinacy theorem, but its proof is instructive: it shows how one can prove statements about Kolmogorov complexity by constructing a special game and a winning strategy in this game. This technique is illustrated by several examples (total conditional complexity, bijection complexity, randomness extraction, contrasting plain and prefix complexities).

math.LO↗

arXiv2008

Limit complexities revisited

The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x|n)$ (here $\KS(x|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $\KS^{\mathbf{0'}(x)$, the plain Kolmogorov complexity with $\mathbf{0'$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $\mathbf{0'}$ Martin-Löf randomness (called also 2-randomness) proved in (Miller, 2004): a sequence $ω$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $ω$ is a prefix of some string $y$ such that $\KS(y)\ge |y|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to 2-randomness was shown in (Miller, 2004) while proving another 2-randomness criterion (see also (Nies et al. 2005)): $ω$ is 2-random if and only if $\KS(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $ω$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.

cs.CC↗