Slaman proposes to investigate the effective aspects of randomness. The central question to be answered is, "How can we evaluate the random content of an infinite binary sequence?" One case is well understood, that in which an infinite sequence X of 0's and 1's is random if and only if each digit is chosen independently and with equal probability for the values of 0 and 1. In this case, X's being effectively random has been equivalently characterized by Martin-Lof in terms of X's having the properties of almost all infinite sequences and by Kolmogorov and others in terms of X's being unpredictable and indescribable. Slaman proposes to study the non-uniform case, its associated criteria for effective randomness, and its possibilities for the sets of random sequences. Even the most basic questions are open. For example, for a given infi- nite binary sequence X, under what conditions does there exist a measure relative to which X is random? This is a classic mathematical problem, given an individual data set determine a distribution which would generate it. Qualitatively, a measure relative to which X is random concentrates on the nonrandom aspects of X and thereby separates those from the random ones. Other ways to quantify X's random content include the complexity of X's initial segments and X's ability to compute uniformly random sequences. The proposal is to investigate all of these and the relationships between them.
