Stationary algorithmic probability

Kolmogorov complexity and algorithmic probability quantify the randomness and universal a priori probability of finite binary strings. Nevertheless, they share the disadvantage of depending on the choice of the universal computer which is used as a reference computer to count the program lengths. In this paper, we propose an approach to algorithmic probability that tries to eliminate this machine-dependence. Elaborating on the idea that computers with “atypical” algorithmic probability should be hard to emulate, we define the notion of emulation complexity. This naturally leads to a Markov process of universal computers that randomly emulate each other, yielding stationary probability distributions on the computers and finite binary strings. By proving symmetry relations with respect to input and output transformations, we show that properties of individual computers are successfully eliminated. Our approach is not limited to prefix-free computers, but can be applied to more general sets of computers. The question for what computer sets such stationary distributions exist remains open in general, but is answered in some special cases and is shown to be closely related to the aforementioned symmetry relations.