The Randomness Deficiency Function and the Shift Operator

Almost surely, the difference between the randomness deficiencies of two infinite sequences will be unbounded with respect to repeated iterations of the shift operator

Algorithmic information and incompressibility of families of multidimensional networks

This article presents a theoretical investigation of string-based generalized representations of families of finite networks in a multidimensional space. First, we study the recursive labeling of networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. We show that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. This universal algorithmic approach sets limitations and conditions for irreducible information content analysis in comparing networks with a large number of dimensions, such as multilayer networks. Nevertheless, we show that there are particular cases of infinite nesting families of finite multidimensional networks with a unified recursive labeling such that each member of these families is incompressible. From these results, we study network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph.Comment: Extended preprint version of the pape

On Algorithmic Statistics for space-bounded algorithms

Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple i.e. should have small Kolmogorov complexity and capture all the algorithmically discoverable regularities in the data. However this idea can not be used in practice because Kolmogorov complexity is not computable. In this paper we develop algorithmic statistics using space-bounded Kolmogorov complexity. We prove an analogue of one of the main result of `classic' algorithmic statistics (about the connection between optimality and randomness deficiences). The main tool of our proof is the Nisan-Wigderson generator.Comment: accepted to CSR 2017 conferenc

Universality, optimality, and randomness deficiency

A Martin-Löf test UU is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test VV there is a c∈ωc∈ω such that ∀n(Vn+c⊆Un)∀n(Vn+c⊆Un). We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAYLAY, the function that produces a bound for a given Martin-Löf random sequence's randomness deficiency. We prove several robustness and idempotence results concerning the Weihrauch degree of LAYLAY, and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAYLAY. Along similar lines we also study the principle RDRD, a variant of LAYLAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAYLAY

A Quantum Outlier Theorem

In recent results, it has been proven that all sampling methods produce outliers. In this paper, we extend these results to quantum information theory. Projectors of large rank must contain pure quantum states in their images that are outlying states. Otherwise, the projectors are exotic, in that they have high mutual information with the halting sequence. Thus quantum coding schemes that use projections, such as Schumacher compression, must communicate using outlier quantum states

Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $\Theta (n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $\Omega(n^2 \log n)$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $\Theta (n^3)$ bits on average. For worst-case static networks we prove a $\Omega(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea

K-trivial, K-low and MLR-low sequences: a tutorial

A remarkable achievement in algorithmic randomness and algorithmic information theory was the discovery of the notions of K-trivial, K-low and Martin-Lof-random-low sets: three different definitions turns out to be equivalent for very non-trivial reasons. This paper, based on the course taught by one of the authors (L.B.) in Poncelet laboratory (CNRS, Moscow) in 2014, provides an exposition of the proof of this equivalence and some related results. We assume that the reader is familiar with basic notions of algorithmic information theory.Comment: 25 page

Algorithmic statistics: forty years later

Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad ("non-stochastic") data appear "in real life"? Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde

Computable Bayesian Compression for Uniformly Discretizable Statistical Models

Supplementing Vovk and V'yugin's `if' statement, we show that Bayesian compression provides the best enumerable compression for parameter-typical data if and only if the parameter is Martin-L\"of random with respect to the prior. The result is derived for uniformly discretizable statistical models, introduced here. They feature the crucial property that given a~discretized parameter, we can compute how much data is needed to learn its value with little uncertainty. Exponential families and certain nonparametric models are shown to be uniformly discretizable

Complexity of complexity and strings with maximal plain and prefix Kolmogorov complexity

Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x of length n whose plain complexity C(x) has almost maximal conditional complexity relative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i = log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simple game-based proof of this result; modifying her argument, we get a better (and tight) bound log n - O(1). We also show the same bound for prefix-free complexity. Robert Solovay showed (Solovay 1975) that infinitely many strings x have maximal plain complexity but not maximal prefix complexity (among the strings of the same length): for some c there exist infinitely many x such that |x| - C(x) log^(2) |x| - c log^(3) |x|. In fact, the results of Solovay and Gacs are closely related. Using the result above, we provide a short proof for Solovay's result. We also generalize it by showing that for some c and for all n there are strings x of length n with n - C (x) < c and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a close upper bound K(K(n)|n) + O(1). Finally, we provide a direct game proof for Joseph Miller's generalization (Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set with c.e. complement) contains for every length a string of this length, then it contains infinitely many strings x such that |x| + K(|x|) - K(x) > log^(2) |x| + O(log^(3) |x|).Comment: 13 pages, 1 figur