Related papers: On the polynomial depth of various sets of random …
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal…
In this paper we introduce a new formulation of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of prefixes of infinite sequences from…
Consider a binary string $x$ of length $n$ whose Kolmogorov complexity is $\alpha n$ for some $\alpha<1$. We want to increase the complexity of $x$ by changing a small fraction of bits in $x$. This is always possible: Buhrman, Fortnow,…
Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements in the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical…
It is discussed and surveyed a numerical method proposed before, that alternative to the usual compression method, provides an approximation to the algorithmic (Kolmogorov) complexity, particularly useful for short strings for which…
Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and…
Given a reference computer, Kolmogorov complexity is a well defined function on all binary strings. In the standard approach, however, only the asymptotic properties of such functions are considered because they do not depend on the…
We characterize the monotone bounded depth formula complexity for graph homomorphism and colored isomorphism polynomials using a graph parameter called the cost of bounded product depth baggy elimination tree. Using this characterization,…
Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic…
The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the…
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 randomness rate of an infinite binary sequence is characterized by the sequence of ratios between the Kolmogorov complexity and the length of the initial segments of the sequence. It is known that there is no uniform effective procedure…
We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures,…
All strings with low mutual information with the halting sequence will have flat Kolmogorov Structure Functions, in the context of Algorithmic Statistics. Assuming the Independence Postulate, strings with non-negligible information with the…
We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be…
A selection of the relevant theorems of Probability Theory that comes directly from Kolmogorov's axioms, Set Theory basic results, definitions and rules of inference are listed and proven in a systematic approach, aiming the student who…
In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonngeative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve…
Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…
Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric condition that guarantees that it agrees…
Alice and Bob are given two correlated n-bit strings x_1 and, respectively, x_2, which they want to losslessly compress and send to Zack. They can either collaborate by sharing their strings, or work separately. We show that there is no…