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Black-Box PWPP Is Not Turing-Closed

Pavel Hubáček

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 (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.

Black-Box PWPP Is Not Turing-Closed

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 (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.
Paper Structure (11 sections, 5 theorems, 12 equations)

This paper contains 11 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. The curious case of $\mathsf{PWPP}$.
  3. Our result.
  4. Related Work
  5. Complete Problems for $\mathsf{PPP}$ and $\mathsf{PWPP}$.
  6. Black-Box Separations and Proof Complexity.
  7. Preliminaries
  8. Nested collision-finding
  9. Nested collision-finding is Strictly Harder Relative to a Random Oracle
  10. Roadmap of the lower bound.
  11. Learning collisions through query sets.

Key Result

Theorem 1

Relative to $\mathcal{O}$, $\textsc{NestedCollision}\xspace\not\le\textsc{Collision}\xspace$.

Theorems & Definitions (16)

  • Definition 1: $\mathsf{TFNP}$ and many-one reductions
  • Definition 2: $\textsc{Collision}$ and $\mathsf{PWPP}$
  • Definition 3: $\textsc{NestedCollision}$
  • proof
  • Definition 4: uniform nested functions
  • Theorem 1
  • Corollary 1
  • Definition 5: tainted inputs
  • Lemma 1: bounding tainted inputs
  • proof
  • ...and 6 more