General linear relations between different types of predictive complexity

In this paper we introduce a general method of establishing tight linear inequalities between different types of predictive complexity. Predictive complexity is a generalization of Kolmogorov complexity and it bounds the ability of an algorithm to predict elements of a sequence. Our method relies upon probabilistic considerations and allows us to describe explicitly the sets of coefficients which correspond to true inequalities. We apply this method to two particular types of predictive complexity, namely, logarithmic complexity, which coincides with a variant of Kolmogorov complexity, and square-loss complexity, which is interesting for applications.