$\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP}\cdot\oplus\mathrm{P}$ and a Random-Oracle Proof of Toda's Theorem

2026-09-28

AI Summary

This study addresses the equivalence between the parity class and the bounded-error probabilistic polynomial-time class relative to a random oracle, along with the challenges in proving Toda's theorem in this setting. By leveraging exponential correlation bounds, it establishes that $\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP} \cdot \oplus\mathrm{P}$. Furthermore, building upon the Regan-Royer framework, this work completes the random oracle proof for the first half of Toda's theorem while eliminating prior constraints. It proposes a novel approach that collapses the polynomial hierarchy into $\oplus\mathrm{P}$ level by level without shifting probability quantifiers, achieved by integrating pseudorandom generators, the Valiant-Vazirani lemma, and Papadimitriou-Zachos techniques. Ultimately, this research formulates an analog of the "almost-equal" theorem for parity classes and refines the logical foundation underlying the random oracle proof of $\mathrm{PH} \subseteq \mathrm{BP} \cdot \oplus\mathrm{P}$.

Abstract

Using the recent exponential correlation bounds of Chattopadhyay, Hatami, Lee, Lovett, Tal and Viola between $\mathbb{F}_2$-polynomials and the XOR of majorities, we show that $\mathrm{Almost}\text{-}\oplus\mathrm{P} = \mathrm{BP}\cdot\oplus\mathrm{P}$, where $\mathrm{Almost}\text{-}\oplus\mathrm{P}$ is the class of languages that lie in $\oplus\mathrm{P}^R$ with probability one for a random oracle $R$. This is the parity analogue of Bennett and Gill's $\mathrm{Almost}\text{-}\mathrm{P} = \mathrm{BPP}$ and Nisan and Wigderson's $\mathrm{Almost}\text{-}\mathrm{PH} = \mathrm{PH}$. The key ingredient is a pseudorandom generator with polynomial seed length that fools $\mathbb{F}_2$-polynomials of polynomial degree on exponentially many variables. As an application we complete a random-oracle proof of the first half of Toda's theorem, $\mathrm{PH} \subseteq \mathrm{BP}\cdot\oplus\mathrm{P}$, following an approach of Regan and Royer. Relative to a random oracle, the polynomial hierarchy collapses into $\oplus\mathrm{P}$ by applying Valiant-Vazirani and Papadimitriou-Zachos level by level, with no probabilistic quantifier ever moved through an oracle. Our result then removes the oracle. We compare this argument with the simple proof of Toda's theorem by Fortnow (2009).

Problem

• Computational Complexity
• Random Oracle
• Toda's Theorem
• Parity
• Polynomial Hierarchy

Innovation

• Pseudorandom Generator
• Random Oracle
• Toda's Theorem
• Parity Complexity
• Polynomial Hierarchy