Related papers: Kolmogorov complexity as a language
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…
This chapter does not deal with specific tools and techniques for managing complex systems, but proposes some basic concepts that help us to think and speak about complexity. We review classical thinking and its intrinsic drawbacks when…
We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings $x$ and $y$ is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having $x$ and the…
For the minimization of state-based systems (i.e. the reduction of the number of states while retaining the system's semantics), there are two obvious aspects: removing unnecessary states of the system and merging redundant states in the…
In this paper, we assess the complexity results of formalisms that describe the feature theories used in computational linguistics. We show that from these complexity results no immediate conclusions can be drawn about the complexity of the…
A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new ``normalized information distance'', based on the noncomputable notion of…
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…
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…
If a concept is not well defined, there are grounds for its abuse. This is particularly true of complexity, an inherently interdisciplinary concept that has penetrated very different fields of intellectual activity from physics to…
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 extend the concept of Krylov complexity to include general unitary evolutions involving multiple generators. This generalization enables us to formulate a framework for generalized Krylov complexity, which serves as a measure of the…
The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most…
The present paper presents and proves a proposition concerning the time complexity of finite languages. It is shown herein, that for any finite language (a language for which the set of words composing it is finite) there is a Turing…
These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following…
We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies…
Loveland complexity is a variant of Kolmogorov complexity, where it is asked to output separately the bits of the desired string, instead of the string itself. Similarly to the resource-bounded Kolmogorov sets we define Loveland sets. We…
If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…
In program semantics and verification, reasoning about loops is complicated by the need to produce two separate mathematical arguments: an invariant, for functional properties (ignoring termination); and a variant, for termination (ignoring…
One of the most fundamental problems in science is to define {\it quantitatively} the complexity of organized matters, i.e., {\it organized complexity}. Although many measures have been proposed toward this aim in previous decades, there is…
Symmetry of information establishes a relation between the information that x has about y (denoted I(x : y)) and the information that y has about x (denoted I(y : x)). In classical information theory, the two are exactly equal, but in…