Some structural complexity results for $\exists\mathbb R$
Klaus Meer, Adrian Wurm
TL;DR
This work investigates the structural properties of the existential theory of the reals $\exists\mathbb{R}$ beyond its known placement between $\text{NP}$ and $\text{PSPACE}$. It develops a descriptive and relativized framework, showing that $\exists\mathbb{R}$ can be characterized as $BP(NP_{\mathbb{R}}^0)$ and analyzed via discrete $\mathbb{R}$-structures, enabling a translation between real algebraic computations and logical definability. The authors prove oracle separations and a Ladner-like theorem, demonstrating the existence of $\exists\mathbb{R}$-complete phenomena not reducible to $NP$ under plausible assumptions, and provide detailed appendix proofs that connect BSS computation, existential second-order logic, and fixed-point formalisms. Overall, the paper extends classical NP-style results to the $\exists\mathbb{R}}$ setting, offering structural insights and a robust descriptive framework for real-algebraic decision problems with broad implications for complexity and logic.
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.