Black-Box PWPP Is Not Turing-Closed

TL;DR

This paper proves that the complexity class PWPP is not closed under adaptive Turing reductions in the black-box setting by introducing a new collision-finding problem that separates adaptive and non-adaptive query powers.

Contribution

The paper establishes a black-box separation showing PWPP's non-closure under adaptive Turing reductions, introducing the NESTED-COLLISION problem as a key example.

Findings

Adaptive collision-finding queries are more powerful than non-adaptive ones.

PWPP is not closed under adaptive Turing reductions in the black-box setting.

The NESTED-COLLISION problem demonstrates this separation.

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 in the black-box setting. 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, it cannot be solved via an efficient black-box non-adaptive reduction to the canonical PWPP-complete problem COLLISION.