Inequalities for space-bounded Kolmogorov complexity

TL;DR

This paper improves the understanding of space-bounded Kolmogorov complexity by establishing tighter bounds and demonstrating that all linear inequalities valid for complexities or entropies also hold in the space-bounded setting with polynomial overhead.

Contribution

It provides an improved version of Longpré's space-bounded complexity formula using Sipser's trick and shows the universality of linear inequalities for space-bounded complexities.

Findings

Tighter space bounds for Kolmogorov complexity formulas.

Linear inequalities for complexities extend to space-bounded complexities.

Polynomial space overhead suffices for inequality validity.

Abstract

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of strings (Hammer et al., 1997), as well as for sizes of subgroups and projections of sets (Chan, Yeung, Romashchenko, Shen, Vereshchagin, 1998--2002). This parallelism started with the Kolmogorov-Levin formula (1968) for the complexity of pairs of strings with logarithmic precision. Longpr\'e (1986) proved a version of this formula for space-bounded complexities. In this paper we prove an improved version of Longpr\'e's result with a tighter space bound, using Sipser's trick (1980). Then, using this space bound, we show that every linear inequality that is true for complexities or entropies, is also true for space-bounded Kolmogorov complexities with a polynomial space overhead.