Refinment of the "up to a constant" ordering using contructive co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov complexities

TL;DR

This paper refines the comparison of total functions using constructive co-immunity, leading to a hierarchy theorem for Kolmogorov complexities with applications to Min/Max complexities.

Contribution

It introduces new orderings between functions that strengthen the 'up to a constant' comparison and applies them to establish a hierarchy theorem for Kolmogorov complexities.

Findings

Established a hierarchy theorem for Kolmogorov complexities with jump oracles.

Developed a second order conditional Kolmogorov complexity for uniform bounds.

Provided a refined ordering framework for total functions in complexity theory.

Abstract

We introduce orderings between total functions f,g: N -> N which refine the pointwise "up to a constant" ordering <=cte and also insure that f(x) is often much less thang(x). With such orderings, we prove a strong hierarchy theorem for Kolmogorov complexities obtained with jump oracles and/or Max or Min of partial recursive functions. We introduce a notion of second order conditional Kolmogorov complexity which yields a uniform bound for the "up to a constant" comparisons involved in the hierarchy theorem.