On the computational power of $C$-random strings

TL;DR

This paper demonstrates that there exists a universal decompressor for which the Halting problem is solvable within polynomial time with access to the set of strings with high Kolmogorov complexity, addressing a question posed by Eric Allender.

Contribution

It establishes the existence of a universal decompressor making the Halting problem solvable in polynomial time relative to a specific high-complexity set, linking Kolmogorov complexity to computational power.

Findings

Halting problem is in P^{R_U} for some universal U.

Defines R_U as strings with high Kolmogorov complexity.

Connects Kolmogorov complexity with computational complexity classes.

Abstract

Denote by $H$ the Halting problem. Let $R_U: = \{ x | C_U(x) \ge |x|\}$, where $C_U(x)$ is the plain Kolmogorov complexity of $x$ under a universal decompressor $U$. We prove that there exists a universal $U$ such that $H \in P^{R_U}$, solving the problem posted by Eric Allender.