Black-Box PWPP Is Not Turing-Closed

Authors: Pavel Hubáček

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 (Jeřábek 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.