Martin-Löf Randomness

Abstract

In this chapter, we will introduce three cornerstone approaches to the definition of algorithmic randomness for infinite sequences. (i) The computational paradigm: Random sequences are those whose initial segments are all hard to describe, or, equivalently, hard to compress. This approach is probably the easiest one to understand in terms of the previous sections. (ii) The measure-theoretic paradigm: Random sequences are those with no " effectively rare " properties. If the class of sequences satisfying a given property is an effectively null set, then a random sequence should not have this property. This approach is the same as the stochastic paradigm: a random sequence should pass all effective statistical tests. (iii) The unpredictability paradigm: This approach stems from what is probably the most intuitive conception of randomness, namely that one should not be able to predict the next bit of a random sequence, even if one knows all preceding bits, in the same way that a coin toss is unpredictable even given the results of previous coin tosses. We will see that all three approaches can be used to define the same notion of randomness, which is called Martin-Löf randomness or 1-randomness. This notion was the first successful attempt to capture the idea of a random infinite sequence, and is still the best known and most studied of the various definitions proposed to date, in great part because it enables us to develop a rich and appealing mathematical theory. For