Not Every Domain of a Plain Decompressor Contains the Domain of a Prefix-Free One

TL;DR

This paper demonstrates that not every domain of a plain decompressor necessarily contains the domain of an optimal prefix-free decompressor, answering a question about the relationship between plain and prefix Kolmogorov complexities.

Contribution

It provides a negative answer to a longstanding open question about the relationship between plain and prefix-free Kolmogorov complexities.

Findings

Not all plain decompressor domains contain prefix-free decompressor domains.

The result clarifies the structural differences between plain and prefix-free Kolmogorov complexities.

The paper resolves a question posed by Calude et al. in 2008.

Abstract

C.Calude, A.Nies, L.Staiger, and F.Stephan posed the following question about the relation between plain and prefix Kolmogorov complexities (see their paper in DLT 2008 conference proceedings): does the domain of every optimal decompressor contain the domain of some optimal prefix-free decompressor? In this paper we provide a negative answer to this question.