arXiv subjects

Alexander Shen

Publications and source records attributed to Alexander Shen.

At least 37 records · Page 2Linked to original sources

arXiv2022

Inequalities for space-bounded Kolmogorov complexity

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997), as well as for sizes of subgroups and projections of sets (Chan, Yeung, Romashchenko, Shen, Vereshchagin, 1998--2002). This parallelism started with the Kolmogorov-Levin formula (1968) for the complexity of pairs of strings with logarithmic precision. Longpré (1986) proved a version of this formula for space-bounded complexities. In this paper we prove an improved version of Longpré's result with a tighter space bound, using Sipser's trick (1980). Then, using this space bound, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead.

cs.IT↗

arXiv2022

27 Open Problems in Kolmogorov Complexity

The paper proposes open problems in classical Kolmogorov complexity. Each problem is presented with background information and thus the article also surveys some recent studies in the area.

cs.IT↗

arXiv2021

Individual codewords

Algorithmic information theory translates statements about classes of objects into statements about individual objects; it defines individual random sequences, effective Hausdorff dimension of individual points, amount of information in individual strings, etc. We observe that a similar translation is possible for list-decodable codes.

cs.IT↗

arXiv2021

Mathematical works of Vladimir A. Uspensky: A commentary

Vladimir Andreevich Uspensky [1930-2018] was one of the Soviet pioneers of the theory of computation and mathematical logic in general (and my teacher and thesis advisor). This paper is the survey of his mathematical works and their influence. (His achievements in linguistics and his organizational role are outside the scope of this survey.) The Russian version follows the English one.

math.HO↗

arXiv2020

Complexity of majorants

The minimal Kolmogorov complexity of a total computable function that exceeds everywhere all total computable functions of complexity at most $n$, is $2^{n+O(1)}$. If we replace "everywhere" by "for all sufficiently large inputs", the answer is $n+O(1)$.

math.LO↗

arXiv2020

Elections and statistics: the case of "United Russia", 2009-2020

This survey contains statistics on elections in Russia published in different places and available online. This data is discussed from the viewpoint of statistical model selection. The current version is updated including the materials up to July, 2020 voting on constitutional changes, Belarus 2020 elections and papers that appeared in 2020; most of the data are not consistent with the assumption of fair elections.

stat.AP↗

arXiv2020

Automatic Kolmogorov complexity, normality and finite state dimension revisited

It is well known that normality can be described as incompressibility via finite automata. Still the statement and the proof of this result as given by Becher and Heiber (2013) in terms of "lossless finite-state compressors" do not follow the standard scheme of Kolmogorov complexity definition (an automaton is used for compression, not decompression). We modify this approach to make it more similar to the traditional Kolmogorov complexity theory (and simpler) by explicitly defining the notion of automatic Kolmogorov complexity and using its simple properties. Using this characterization and a sufficient condition for normality in terms of Kolmogorov complexity derived from it, we provide easy proofs for classical results about normal sequences (Champernown, Wall, Piatetski-Shapiro, Besicovitch, Copeland, Erdos et al.) Then we extend this approach to finite state dimension. We show that the block entropy definition of the finite state dimension remains the same if non-aligned blocks are used. Then we provide equivalent definitions in terms of automatic complexity, superadditive bounds for Kolmogorov complexity, calibrated superadditive functions and finite state a priori probability and use them to give simple proofs for known results about finite state dimension, and for Agafonov's result saying that normality is preserved by automatic selection rules as well as the results of Schnorr and Stimm that relate normality to finite state martingales. Some results of this paper were presented at the Fundamentals in Computing Theory conferences in 2017 and 2019. Preliminary version of this paper (that does not mention the finite state dimension) was published in arxiv in~2017 (see the previous version of this submission).

cs.IT↗

arXiv2019

Information Distance Revisited

We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [1] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. It was claimed by Mahmud [11] that the prefix version of information distance equals max(K(x|y), K(y|) + O(1) (this equality with logarithmic precision was one of the main results of the paper by Bennett, Gács, Li, Vitanyi, and Zurek). We show that this claim is false, but does hold if the information distance is at least super logarithmic.

cs.IT↗

arXiv2019

Rod in a train: a mechanical problem of H.Whitney, or Much Ado About Nothing

In 1941 a mechanical problem about a rod in a moving train (there is a initial position such that rod does not touch the floor while train is moving) was published by R.Courant and H.Robbins in their popular book "What is mathematics?" and attributed to H.Whitney. Many mathematicians, including G.E.Littlewood, A.Broman, T.Poston, I.Stewart, V.Arnold, commented on this problem and its solution based on a continuity argument, and created a lot of confusion. In this paper we follow these developments and discuss at what extent the objections were justified. (In Russian)

math.HO↗

arXiv2019

Hilbert's Error?

It is well known that a center of a given circle cannot be constructed using only a straightedge and that this was proven by David Hilbert. Still it is not so clear what kind of object is proven to be non-existing. We analyze different attempts to define a geometric construction appearing in the literature and observe that none of them is really satisfactory and that Hilbert's proof needs to be corrected (as noted by Akopyan and Fedorov)

math.HO↗

arXiv2019

Random noise increases Kolmogorov complexity and Hausdorff dimension

Consider a binary string $x$ of length $n$ whose Kolmogorov complexity is $αn$ for some $α<1$. We want to increase the complexity of $x$ by changing a small fraction of bits in $x$. This is always possible: Buhrman, Fortnow, Newman and Vereshchagin (2005) showed that the increase can be at least $δn$ for large $n$ (where $δ$ is some positive number that depends on $α$ and the allowed fraction of changed bits). We consider a related question: what happens with the complexity of $x$ when we randomly change a small fraction of the bits (changing each bit independently with some probability $τ$)? It turns out that a linear increase in complexity happens with high probability, but this increase is smaller than in the case of arbitrary change. We note that the amount of the increase depends on $x$ (strings of the same complexity could behave differently), and give an exact lower and upper bounds for this increase (with $o(n)$ precision). The proof uses the combinatorial and probabilistic technique that goes back to Ahlswede, Gács and Körner (1976). For the reader's convenience (and also because we need a slightly stronger statement) we provide a simplified exposition of this technique, so the paper is self-contained.

cs.IT↗

arXiv2018

Priority arguments and separation problems

Different constructions in the recursion theory use the so-called priority arguments. A general scheme was suggested by A.~Lachlan. Based on his work, we define the notion of a priority-closed class of requirements. Then, for a specific priority construction, we need to check only that all requirements we want to satisfy belong to some priority-closed class (defined in game terms). This game version of Lachlan's approach is used to present some results about recursively inseparable sets obtained by the author.

math.LO↗

arXiv2018

Axiomatic approach to the theory of algorithms and relativized computability

It is well known that many theorems in recursion theory can be "relativized". This means that they remain true if partial recursive functions are replaced by functions that are partial recursive relative to some fixed oracle set. Uspensky formulates three "axioms" called "axiom of computation records", "axiom of programs'" and "arithmeticity axiom". Then, using these axioms (more precisely, two first ones) he proves basic results of the recursion theory. These two axioms are true also for the class of functions that are partial recursive relative to some fixed oracle set. Also this class is closed under substitution, primitive recursion and minimization ($μ$-operator); these (intuitively obvious) closure properties are also used in the proofs. This observation made by Uspensky explains why many theorems of recursion theory can be relativized. It turns out that the reverse statement is also true: all relativizable results follow from the first two axioms and closure properties. Indeed, \emph{every class of partial functions that is closed under substitution, primitive recursion and minimization that satisfies the first two axioms is the class of functions that are partial recursive relative to some oracle set $A$}. This is the main result of the present article.

math.LO↗

arXiv2018

Algorithmic identification of probabilities is hard

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better approximations of $p$ as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that $p$ is a computable real, but one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter $p$ (in the form of a Turing code). Can one do such a thing uniformly on all sequences that are random for computable Bernoulli measures, or even on a `large enough' fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures.

math.LO↗

arXiv2018

Algorithms and Geometric Constructions

It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to an intuitive idea of a geometric construction as a special kind of an `algorithm' using restricted means (straightedge and/or compass). However, the formalization is not obvious, and different descriptions existing in the literature are far from being complete and clear. We discuss the history of this notion and a possible definition in terms of a simple game

math.HO↗

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

Plain stopping time and conditional complexities revisited

In this paper we analyze the notion of "stopping time complexity", informally defined as the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic (2016). It turns out that plain stopping time complexity of a binary string $x$ could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading $x$ on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing $x$; (c)~the conditional complexity $C(x|x*)$ where $x$ in the condition is understood as a prefix of an infinite binary sequence while the first $x$ is understood as a terminated binary string; (d) as a minimal upper semicomputable function $K$ such that each binary sequence has at most $2^n$ prefixes $z$ such that $K(z)<n$; (e) as $\max C^X(x)$ where $C^X(z)$ is plain Kolmogorov complexity of $z$ relative to oracle $X$ and the maximum is taken over all extensions $X$ of $x$. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition $y$ and the object $x$ may differ. We also answer an open question from Chernov, Hutter and~Schmidhuber.

cs.CC↗

arXiv2017

Dimension 1 sequences are close to randoms

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-Löf random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly $s$-random sequence. Further, for any $s<t$, every sequence of effective dimension $s$ can be changed on density at most $H^{-1}(t)-H^{-1}(s)$ of its bits to produce a sequence of effective dimension $t$, and this bound is optimal.

math.LO↗