On the computational power of $C$-random strings

TL;DR

This paper establishes that there exists a universal decompressor $U$ for which the halting problem $H$ lies in $\textbf{P}^{R_U}$, where $R_U$ is the set of strings with non-negligible plain Kolmogorov complexity under $U$. The authors construct $U$ by combining a $V$ with even conditional complexity and a matrix-based fingerprinting approach, relating halting to $R_U$ through carefully designed descriptions and length gaps. They then give a probabilistic polynomial-time algorithm with oracle access to $R_U$ that decides $H$ with high probability, using a hash-like scheme built from random matrices and a prime-based scaffolding to ensure correct discrimination between halting and non-halting instances. The result strengthens the understanding of the computational power of $R_U$ for plain Kolmogorov complexity and connects randomness, hashing, and halting in a novel way, with implications for upper bounds on complexity classes relative to Kolmogorov-based oracles.

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.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2: Kolmogorov-Solomonoff
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 8