Black-Box PWPP Is Not Turing-Closed

2026-02-27

AI Summary

This work investigates the closure of the complexity class PWPP under adaptive Turing reductions, with a focus on the computational gap between adaptive and non-adaptive collision-finding queries. By introducing a novel problem, NESTED-COLLISION, in the random oracle model and employing complexity-theoretic reduction techniques, the paper establishes—for the first time—that PWPP is not closed under adaptive Turing reductions. Specifically, NESTED-COLLISION can be solved using two adaptive queries to a PWPP oracle, yet it cannot be resolved by any black-box non-adaptive reduction to PWPP. This result provides a strict separation between the power of adaptive and non-adaptive query models in the context of PWPP, highlighting a fundamental limitation of non-adaptive access to collision-finding primitives.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We establish that adaptive collision-finding queries are strictly more powerful than non-adaptive ones by proving that the complexity class PWPP (Polynomial Weak Pigeonhole Principle) is not closed under adaptive Turing reductions relative to a random oracle. Previously, PWPP was known to be closed under non-adaptive Turing reductions (Je\v{r}\'abek 2016). We demonstrate this black-box separation by introducing the NESTED-COLLISION problem, a natural collision-finding problem defined on a pair of shrinking functions. We show that while this problem is solvable via two adaptive calls to a PWPP oracle, its random instances cannot be solved via a black-box non-adaptive reduction to the canonical PWPP-complete problem COLLISION.

Problem

• PWPP
• adaptive reductions
• collision-finding
• Turing-closed
• random oracle

Innovation

• PWPP
• adaptive reductions
• collision-finding
• black-box separation
• random oracle