Separations with Immunity Relative to a Random Oracle in Computational Complexity

2026-10-04

AI Summary

This study addresses the longstanding absence of a unified and rigorous characterization for strong separations within the Polynomial Hierarchy (PH) relative to random oracles. To this end, it introduces a “rare-or-wrong” technique to construct immune languages in the random oracle model, utilizing them as witnesses to achieve strong separations between complexity classes. The primary contribution is a proof that, with probability one, every level of PH contains a language immune to its adjacent level, thereby establishing strong separations across all levels. This work significantly strengthens prior separation results by unifying and generalizing classical findings such as those of Bennett and Gill, ultimately deepening the theoretical understanding of oracle separation mechanisms in computational complexity theory.

Abstract

Rossman, Servedio, and Tan proved that the polynomial hierarchy is infinite relative to a random oracle. This paper strengthens this celebrated result by considering \emph{strong separations}, witnessed by immune languages. The main result of the paper shows that with respect to a random oracle the levels of the polynomial hierarchy separate with immunity. Specifically, with probability one over a random oracle $A$, for every $k\geq 1$ there is a language in $\Sigma_{k}^{P,A}$ that is immune to $\Pi_{k}^{P,A}$, and, symmetrically, a language in $\Pi_{k}^{P,A}$ that is immune to $\Sigma_{k}^{P,A}$. We thus extend classical results due to Bennett and Gill and Vereshchagin. We accomplish this by developing a"rare-or-wrong"approach to strong separations. We highlight the flexibility of the approach by strengthening other separations with respect to a random oracle from the complexity-theoretic literature.

Problem

• Computational Complexity
• Random Oracle
• Polynomial Hierarchy
• Immune Languages
• Strong Separations

Innovation

• Random Oracle
• Polynomial Hierarchy
• Immune Languages
• Strong Separations
• Rare-or-Wrong Approach