Kolmogorov-Loveland Sets and Advice Complexity Classes

TL;DR

This paper explores the relationship between Loveland complexity, a variant of Kolmogorov complexity, and advice complexity classes, revealing structural connections that impact hierarchy and natural proof properties.

Contribution

It establishes a structural link between Loveland sets and advice complexity classes, extending known properties of Kolmogorov sets to advice complexity theory.

Findings

Non-inclusion of Loveland sets implies hierarchy in advice classes

Immunity of Loveland sets indicates absence of natural proofs

Structural connections enable transfer of properties between concepts

Abstract

Loveland complexity is a variant of Kolmogorov complexity, where it is asked to output separately the bits of the desired string, instead of the string itself. Similarly to the resource-bounded Kolmogorov sets we define Loveland sets. We highlight a structural connection between resource-bounded Loveland sets and some advice complexity classes. This structural connection enables us to map to advice complexity classes some properties of Kolmogorov sets first noticed by Hartmanis and thoroughly investigated in Longpr\'e's thesis: 1. Non-inclusion properties of Loveland sets result in hierarchy properties on the corresponding advice complexity classes; 2. Immunity properties of Loveland sets result in the non-existence of natural proofs between the corresponding advice complexity classes, in the sense of Razborov & Rudich.