相关论文: Prefix and plain Kolmogorov complexity characteriz…

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive…

The linear complexity is a measure for the unpredictability of a sequence over a finite field and thus for its suitability in cryptography. In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion…

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every…

For every total recursive time bound $t$, a constant fraction of all compressible (low Kolmogorov complexity) strings is $t$-bounded incompressible (high time-bounded Kolmogorov complexity); there are uncountably many infinite sequences of…

We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) -…

Several classes of DNR functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of…

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts…

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program…

Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the…

Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information…

The even online Kolmogorov complexity of a string $x = x_1 x_2 \cdots x_{n}$ is the minimal length of a program that for all $i\le n/2$, on input $x_1x_3 \cdots x_{2i-1}$ outputs $x_{2i}$. The odd complexity is defined similarly. The sum of…

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…

Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known…

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$,…

For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO…

Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of size $n$. A sequence over $F$ of length $m$ is a \emph{$k$-radius sequence} if any two distinct elements of $F$ occur within distance $k$ of each other somewhere in the…

This paper contains an answer to the question of existence of regularities of the so called \textit{random in a broad sense} mass phenomena, asked by A. N. Kolmogorov in \cite{Kolmogorov}. It turns out that some family of finitely-additive…

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…

We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions…

A 1976 theorem of Chaitin can be used to show that arbitrarily dense sets of lengths n have a paucity of trivial strings (only a bounded number of strings of length n having trivially low plain Kolmogorov complexities). We use the…