This invited Descriptional Complexity of Formal Systems lecture provides an opportunity to raise awareness of the tight connection that exists between Kolmogorov Complexity and Circuit Complexity, and to argue that this connection is useful and interesting. Kolmogorov complexity has been used to provide examples of complete sets for classes such as EXP and PSPACE that are fundamentally differen...

We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2 log n upper bound on the Kolmogorov complexity of initial segments of r.e. sets is optimal and characterize the T-degrees of r.e. sets which attain this bound. The main part of the paper is concerned with instance complexity of r.e. sets. We construct a nonrecursive ...

In the paper, we introduce a kind of method to solve the nonlinear filtering problem. Firstly, we review the basic filtering problem and the reduction from robust Duncan-Mortensen-Zakai equation to Kolmogorov equation. Then we use the difference discrete method to solve the Kolmogorov equation. The result is given to prove that the solution of the difference scheme convergences pointwise to the...

Randomness extraction is the process of constructing a source of randomness of high quality from one or several sources of randomness of lower quality. The problem can be modeled using probability distributions and min-entropy to measure their quality and also by using individual strings and Kolmogorov complexity to measure their quality. Complexity theorists are more familiar with the first ap...

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 in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last app...

The problem of Information Assurance is approached from the point of view of Kolmogorov Complexity and Minimum Message Length criteria. Several theoretical results are obtained, possible applications are discussed and a new metric for measuring complexity is introduced. Utilization of Kolmogorov Complexity like metrics as conserved parameters to detect abnormal system behavior is explored. Data...

In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the short...

How much do we have to change a string to increase its Kolmogorov complexity. We show that we can increase the complexity of any non-random string of length n by flipping O( √ n) bits and some strings require Ω( √ n) bit flips. For a given m, we also give bounds for increasing the complexity of a string by flipping m bits. By using constructible expanding graphs we give an efficient algorithm t...

One can prove X1 is not regular using the pumping lemma. One can prove X2 is not regular either by using the pumping lemma (a version with bounds on the prefix) or by contradiction: if X2 is regular than X2 ∩ a∗b∗ = X1 is regular. One cannot prove X3 non-regular with the pumping theorem directly; however one can prove its regular by contradiction: if X3 is regular than X3 = X2 is regular. We wi...

We consider (in the framework of algorithmic information theory) questions of the following type: construct a message that contains different amounts of information for recipients that have (or do not have) certain a priori information. Assume, for example, that the recipient knows some string a, and we want to send her some information that allows her to reconstruct some string b (using a). On...