Some structural complexity results for $\exists\mathbb R$

TL;DR

This paper explores the structural properties of the complexity class , providing results on its relation to NP, descriptive complexity, and the existence of problems within  that are not NP-complete, extending classical complexity theory results.

Contribution

It establishes analogues of classical theorems for , including oracle separations, Ladner's theorem for , and a descriptive complexity characterization.

Findings

Existence of oracles separating NP and .

Existence of  problems not NP-complete.

Characterization of  via descriptive complexity.

Abstract

The complexity class $\exists\mathbb R$, standing for the complexity of deciding the existential first order theory of the reals as real closed field in the Turing model, has raised considerable interest in recent years. It is well known that NP $ \subseteq \exists\mathbb R\subseteq$ PSPACE. In their compendium, Schaefer, Cardinal, and Miltzow give a comprehensive presentation of results together with a rich collection of open problems. Here, we answer some of them dealing with structural issues of $\exists\mathbb R$ as a complexity class. We show analogues of the classical results of Baker, Gill, and Solovay finding oracles which do and do not separate NP form $\exists\mathbb R$, of Ladner's theorem showing the existence of problems in $\exists\mathbb R \setminus$ NP not being complete for $\exists\mathbb R$ (in case the two classes are different), as well as a characterization of $\exists\mathbb R$ by means of descriptive complexity.