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

翻译：我们通过证明复杂度类PWPP（多项式弱鸽笼原理）在黑盒设定下对适应性图灵归约不封闭，确立了适应性碰撞查找查询严格强于非适应性查询。此前已知PWPP对非适应性图灵归约封闭（Jeřábek 2016）。我们通过引入NESTED-COLLISION问题——定义在一对收缩函数上的自然碰撞查找问题——展示了这一黑盒分离。我们证明，尽管该问题可通过两次对PWPP谕示机的适应性调用求解，却无法通过高效的黑盒非适应性归约规约至典型PWPP完全问题COLLISION。