Extremal Relations between Additive Loss Functions and the Kolmogorov Complexity

AbstractConditions are presented under which the maximum of the Kolmogorov complexity (algorithmic entropy) K(ω1...ωN) is attained, given the cost $$\sum\limits_{i = 1}^N {} $$ f(ωi) of a message ω1...ωN. Various extremal relations between the message cost and the Kolmogorov complexity are also considered; in particular, the minimization problem for the function $$\sum\limits_{i = 1}^N {} $$ f(ωi) − θK(ω1...ωN) is studied. Here, θ is a parameter, called the temperature by analogy with thermodynamics. We also study domains of small variation of this function.