A Church-Turing Thesis for Randomness?

We discuss the difficulties in stating an analogue of the Church-Turing thesis for algorithmic randomness. We present one possibility and argue that it cannot occupy the same position in the study of algorithmic randomness that the Church-Turing thesis does in computability theory. We begin by observing that some evidence comparable to that for the Church-Turing thesis does exist for this statement: in particular, there are other reasonable formalizations of the intuitive concept of randomness that lead to the same class of random sequences (the Martin-Lof random sequences). However, we consider three properties that we would like a random sequence to satisfy and find that the Martin-Lof random sequences do not necessarily possess these properties to a greater degree than other types of random sequences, and we further argue that there is no more appropriate version of the ChurchTuring thesis for algorithmic randomness. This suggests that consensus around a version of the Church-Turing thesis in this context is unlikely.