Hierarchies within TFNP: building blocks and collapses

Abstract: We initiate the study of complexity classes $\mathsf{A^B}$ where $\mathsf{A}$ and $\mathsf{B}$ are both $\mathsf{TFNP}$ subclasses. For example, we consider complexity classes of the form $\mathsf{PPP^{PPP}}$, $\mathsf{PPAD^{PPA}}$, and $\mathsf{PPA^{PLS}}$. We define complete problems for such classes, and show that they belong in $\mathsf{TFNP}$. These definitions require some care, since unlike a class like $\mathsf{PPA^{NP}}$, where the $\mathsf{NP}$ oracle defines a function, in $\mathsf{PPA^{PPP}}$, the oracle is for a search problem with many possible solutions. Intuitively, the definitions we introduce quantify over all possible instantiations of the $\mathsf{PPP}$ oracle. With these notions in place, we then show that several $\mathsf{TFNP}$ subclasses are self-low. In particular, $\mathsf{PPA^{PPA}} = \mathsf{PPA}$, $\mathsf{PLS^{PLS}} = \mathsf{PLS}$, and $\mathsf{LOSSY^{LOSSY}} = \mathsf{LOSSY}$. These ideas introduce a novel approach for classifying computational problems within $\mathsf{TFNP}$, such as the problem of deterministically generating large primes.