Kolmogorov complexity and generalized length functions

Abstract

Kolmogorov complexity measures the algorithmic complexity of a finite binary string $\sigma$ in terms of the length of the shortest description $\sigma^*$ of $\sigma$. Traditionally, the length of a string is taken to measure the amount of information contained in the string. However, we may also view the length of $\sigma$ as a measure of the cost of producing $\sigma$, which permits one to generalize the notion of length, wherein the cost of producing a 0 or a 1 can vary in some prescribed manner. In this article, we initiate the study of this generalization of length based on the above information cost interpretation. We also modify the definition of Kolmogorov complexity to use such generalized length functions instead of standard length. We further investigate conditions under which the notion of complexity defined in terms of a given generalized length function preserves some essential properties of Kolmogorov complexity. We focus on a specific class of generalized length functions that are intimately related to a specific subcollection of Bernoulli $p$-measures, namely those corresponding to the unique computable real $p\in(0,1)$ such that $p^k=1-p$, for integers $k\geq 1$. We then study randomness with respect to such measures, by proving a generalization version of the classic Levin-Schnorr theorem that involves $k$-length functions and then proving subsequent results that involve effective dimension and entropy.