Hierarchies within TFNP: building blocks and collapses

TL;DR

This paper explores the internal structure of TFNP subclasses, introduces hierarchies with oracle access, and proves collapses of these hierarchies, providing new tools for classifying problems like prime generation.

Contribution

It defines TFNP classes with oracle gates, proves certain hierarchies collapse, and offers a new framework for classifying and separating problems within TFNP.

Findings

Several TFNP subclasses are self-low, leading to hierarchy collapses.

Hierarchies such as PPA^PPA and PLS^PLS collapse to their base classes.

Prime number generation is in PPP^PPP under GRH.

Abstract

In all well-studied $\mathsf{TFNP}$ subclasses (e.g. $\mathsf{PPA}, \mathsf{PPP}$ etc.), the canonical complete problem takes as input a polynomial-size circuit $C: \{ 0, 1\}^n \rightarrow \{ 0, 1\}^m$ whose input-output behavior implicitly encodes an exponentially large object $G$, i.e. $C$ is the succinct (polynomial-size) representation of the exponential size object $G$. The goal is to find some particular substructure in $G$ which can be confirmed in polynomial time using queries to $C$. We initiate the study of classes of the form $\mathsf{A}^{\mathsf{B}}$ where both $\mathsf{A}$ and $\mathsf{B}$ are $\mathsf{TFNP}$ subclasses. In particular, we define complete problems for these classes that take as input a circuit $C$ which is allowed oracle gates to another $\mathsf{TFNP}$ class. Beyond introducing definitions for $\mathsf{TFNP}$ oracle problems, our specific technical contributions include showing that several $\mathsf{TFNP}$ subclasses are self-low and hence their corresponding hierarchies collapse. In particular, $\mathsf{PPA^{PPA}} = \mathsf{PPA}$, $\mathsf{PLS^{PLS}} = \mathsf{PLS}$, and $\mathsf{LOSSY^{LOSSY}} = \mathsf{LOSSY}$. As an immediate consequence, we derive that when reducing to $\mathsf{PPA}$, one can always assume access to $\mathsf{PPA}$ -- and therefore factoring -- oracle gates. In addition to introducing a variety of hierarchies within $\mathsf{TFNP}$ that merit study in their own right, these ideas introduce a novel approach for classifying computational problems within $\mathsf{TFNP}$ and proving black-box separations. For example, we observe that the problem of deterministically generating large prime numbers, which has long resisted classification in a $\mathsf{TFNP}$ subclass, is in $\mathsf{PPP^{\mathsf{PPP}}}$ under the Generalized Riemann Hypothesis.