Black-Box PWPP Is Not Turing-Closed
For complexity theorists, this resolves a question about the closure properties of PWPP under adaptive reductions, showing a black-box separation.
The paper proves that adaptive collision-finding queries are more powerful than non-adaptive ones by showing that PWPP is not closed under adaptive Turing reductions in the black-box setting, via the NESTED-COLLISION problem which requires two adaptive calls but cannot be solved non-adaptively.
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 (JeÅábek 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.