Some Recent Progress in Algorithmic Randomness

Recently there has been exciting progress in our understanding of algorithmic randomness for reals, its calibration, and its connection with classical measures of complexity such as degrees of unsolvability. In this paper, I will give a biased review of (some of) this progress. In particular, I will concentrate upon randomness for reals. In this paper “real” will mean a member of Cantor space 2. This space is equipped with the topology where the basic clopen sets are [σ] = {σα : α ∈ 2}. Such clopen sets have measure 2. This space is measure-theoretically identical with the rational interval (0, 1), without being homeomorphic spaces. An important program which began in the early 20th Century was to give a proper mathematical foundation to notion of randomness. In terms of understanding this for probability theory, the work of Kolmogorov and others provides an adequate foundation. However, another key direction is to attempt to answer this question via notion of randomness in terms of algorithmic randomness. Here we try to capture the nature of randomness in terms of algorithmic considerations. (This is implicit in the work on Kollektivs in the fundamental paper of von Mises [88].) There are three basic approaches to algorithmic randomness. They are to characterize randomness in terms of algorithmic predictability (“a random real should have bits that are hard to predict”), algorithmic compressibility (“a random real should have segments that are hard to describe with short programs”), and measure theory (“a random real should pass all reasonable algorithmic statistical tests”). A classic example of the relationship between these three is given by the emergence of what is now called Martin-Löf randomness. For a real α = .a1a2 · · · ∈ 2, a consequence of the law of large numbers is that if α is to be random then lims a1+···+as s = 12 . Consider the null set of reals that fail such a test. Then Martin-Löf argued that a real α can only be random if it was not in such a null set. He argued that a random real should pass all such “effectively presented” statistical tests. Thus we define a Martin-Löf test as a computable collection