Book Review: Kolmogorov complexity and algorithmic randomness

Kolmogorov Complexity and Algorithmic Randomness

This paper aims to provide a minimal introduction to algorithmic randomness. In particular, we cover the equivalent 1-randomness and MartinLöf randomness. After a brief review of relevant concepts in computability, we develop the basic theory of Kolmogorov complexity, including the KC theorem and more general notion of information content measures. We then build two natural definitions of rando...

Algorithmic Randomness and Complexity

Algorithmic “Kolmogorov” Complexity

The information content or complexity of an object can be measured by the length of its shortest description. For instance the string ‘01010101010101010101010101010101’ has the short description “16 repetitions of 01”, while ‘11001000011000011101111011101100’ presumably has no simpler description other than writing down the string itself. More formally, the Algorithmic “Kolmogorov” Complexity (...

Computability, Algorithmic Randomness and Complexity

I think mathematical ability manifests itself in many different ways. In particular, mathematicians can be drawn to space and geometry, can have strong analytic intuition, can be drawn to formalism, they can be drawn to counting arguments, etc. There is definitely no unique type of mathematician. Maybe those mathematicians who are drawn to algorithmic thinking have found a home in computer scie...

Expanding Algorithmic Randomness to the Algebraic Approach to Quantum Physics: Kolmogorov Complexity and Quantum Logics

Physicists usually assume that events with a very small probability cannot occur. Kolmogorov complexity formalizes this idea for non-quantum events. We show how this formalization can be extended to quantum events as well. c ⃝2011 World Academic Press, UK. All rights reserved.