Limit on the computational power of $\mathrm{C}$-random strings

TL;DR

The paper constructs a universal decompressor demonstrating that the Halting Problem remains undecidable by polynomial-time oracle machines with access to certain Kolmogorov complexity-based random string sets, addressing a question in computational complexity.

Contribution

It provides a new construction of a universal decompressor that shows limitations of Kolmogorov complexity-based oracles in deciding the Halting Problem.

Findings

Halting Problem cannot be decided by polynomial-time oracle machines with access to the set of C-random strings.

Constructs a universal decompressor for Kolmogorov complexity.

Addresses a problem posed by Eric Allender about the power of Kolmogorov complexity oracles.

Abstract

We construct a universal decompressor $U$ for plain Kolmogorov complexity $\mathrm{C}_U$ such that the Halting Problem cannot be decided by any polynomial-time oracle machine with access to the set of random strings $R_{\mathrm{C}_U} = \{x : \mathrm{C}_U(x) \ge |x|\}$. This result resolves a problem posed by Eric Allender regarding the computational power of Kolmogorov complexity-based oracles.