Methods for discerning and measuring Kolmogorov Complexity are discussed and their relationships explored. A computationally efficient method of using Lempel Ziv 78 Universal compression algorithm to estimate complexity is introduced. 1 Abstract—Methods for discerning and measuring Kolmogorov Complexity are discussed and their relationships explored. A computationally efficient method of using ...

Kolmogorov complexity has intellectual roots in the areas of information theory, computability theory and probability theory. Despites its remarkably simple basis, it has some striking applications in Complexity Theory. The subject was developed by the Russian mathematician Andrei N. Kolmogorov (1903–1987) as an approach to the notion of random sequences and to provide an algorithmic approach t...

We continue the investigation of the path-connected geometry on the Cantor space and the related notions of dilution and compressibility described in [1]. These ideas are closely related to the notions of effective Hausdorff and packing dimensions of reals, and we argue that this geometry provides the natural context in which to study them. In particular we show that every regular real can be m...

What follows is a survey of the field of Kolmogorov complexity. Kolmogorov complexity, named after the mathematician Andrej Nikolaevič Kolmogorov, is a measure of the algorithmic complexity of a certain object (represented as a string of symbols) in terms of how hard it is to describe it. This measure dispenses with the need to know the probability distribution that rules a certain object (in g...

We introduce the study of Kolmogorov complexity with error. For a metric d, we define Ca(x) to be the length of a shortest program p which prints a string y such that d(x, y) ≤ a. We also study a conditional version of this measure Ca,b(x|y) where the task is, given a string y′ such that d(y, y′) ≤ b, print a string x′ such that d(x, x′) ≤ a. This definition admits both a uniform measure, where...

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...