Related papers: Second Quantized Kolmogorov Complexity
Using the theory of Kolmogorov complexity the notion of facticity {\phi}(x) of a string is defined as the amount of self-descriptive information it contains. It is proved that (under reasonable assumptions: the existence of an empty machine…
The concept of effective complexity of an object as the minimal description length of its regularities has been initiated by Gell-Mann and Lloyd. The regularities are modeled by means of ensembles, that is probability distributions on…
In this chapter, a statistical measure of complexity is introduced and some of its properties are discussed. Also, some straightforward applications are shown.
In this paper we investigate a quantity called conditional entropy of ordinal patterns, akin to the permutation entropy. The conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
Permutation entropy measures the complexity of deterministic time series via a data symbolic quantization consisting of rank vectors called ordinal patterns or just permutations. The reasons for the increasing popularity of this entropy in…
An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…
We study the compressibility of enumerations in the context of Kolmogorov complexity, focusing on strong and weak forms of compression and their gain: the amount of auxiliary information embedded in the compressed enumeration. The existence…
We study structure of pure morphic and morphic sequences and prove the following result: the subword complexity of arbitrary morphic sequence is either $\Theta(n^{1+1/k})$ for some $k\in\mathbb N$, or is $O(n \log n)$.
Normalized information distance (NID) uses the theoretical notion of Kolmogorov complexity, which for practical purposes is approximated by the length of the compressed version of the file involved, using a real-world compression program.…
While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two…
We introduce quantum-K ($QK$), a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are…
We propose a new type of entropic descriptor that is able to quantify the statistical complexity (a measure of complex behaviour) by taking simultaneously into account the average departures of a system's entropy S from both its maximum…
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…
Complexity of patterns is a key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal…
The (Mahler-Popken) complexity $\| n \|$ of a natural number $n$ is the smallest number of ones that can be used via combinations of multiplication and addition to express $n$, with parentheses arranged in such a way so as to form legal…
A formal definition of epsilon-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov's idea of the complexity of an object. A definition of epsilon-complexity…
A theorem of A.A. Brudno says that the Kolmogorov-Sinai entropy of a subshift X over $\mathbb{N}$ with respect to an ergodic measure $\mu$ equals the asymptotic Kolmogorov complexity of almost every word $\omega$ in X. The purpose of this…
Special scattered subwords, in which the gaps are of length from a given set, are defined. The scattered subword complexity, which is the number of such scattered subwords, is computed for rainbow words.
We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on $2^\omega$, the so-called proper sequences. Our main results are as follows: (1) We…