Related papers: On the Kolmogorov-Chaitin Complexity for short seq…
In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal…
A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse…
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…
Muchnik's theorem about simple conditional descriprion states that for all words $a$ and $b$ there exists a short program $p$ transforming $a$ to $b$ that has the least possible length and is simple conditional on $b$. This paper presents a…
For any quantum algorithm given by a path in the space of unitary operators we define the computational complexity as the typical computational time associated with the path. This time is defined using a quantum time estimator associated…
We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2log n upper bound on the Kolmogorov complexity of initial segments of r.e.\ sets is optimal and characterize…
This paper establishes an abstract Korovkin-type approximation theorem in general spaces, extending the framework of approximation theory to accommodate broader contexts. A critical result supporting this theorem is the proof that any…
We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing…
In quantum chemistry, the price paid by all known efficient model chemistries is either the truncation of the Hilbert space or uncontrolled approximations. Theoretical computer science suggests that these restrictions are not mere…
According to Kolmogorov complexity, every finite binary string is compressible to a shortest code -- its information content -- from which it is effectively recoverable. We investigate the extent to which this holds for infinite binary…
We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple…
We propose novel algorithms for sequence prediction based on ideas from stringology. These algorithms are time and space efficient and satisfy mistake bounds related to particular stringological complexity measures of the sequence. In this…
Finding an Approximate Longest Common Substring (ALCS) within a given set $S=\{s_1,s_2,\ldots,s_m\}$ of $m \ge 2$ strings is a key problem in computational biology, such as identifying related mutations across multiple genetic sequences. We…
In this research we present a quantum circuit for estimating algorithmic complexity using the coding theorem method. This accelerates inferring algorithmic structure in data for discovering causal generative models. The computation model is…
The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that…
We start by an introduction to the basic concepts of computability theory and the introduction of the concept of Turing machine and computation universality. Then se turn to the exploration of trade-offs between different measures of…
Let $ K(X_1, \ldots, X_n)$ and $H(X_n | X_{n-1}, \ldots, X_1)$ denote the Kolmogorov complexity and Shannon's entropy rate of a stationary and ergodic process $\{X_i\}_{i=-\infty}^\infty$. It has been proved that \[ \frac{K(X_1, \ldots,…
In an equilibrium system, the Kolmogorov-Sinai entropy, $h_{\mathrm{KS}}$, equals the sum of the positive Lyapunov exponents, the exponential rates of divergence of infinitesimal perturbations. Kinetic theory may be used to calculate the…
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A…
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…