The Existential Theory of the Reals with Summation Operators
Markus Bläser, Julian Dörfler, Maciej Liskiewicz, Benito van der Zander
TL;DR
This work introduces and analyzes the existential theory of the reals with summation operators, focusing on the succinct variant $succ-\exists\mathbb{R}$ and the indexed variant $\Sigma_{vi}-\textsc{ETR}$. It provides a machine-theoretic characterization via nondeterministic real RAMs, establishes a separation between $\exists\mathbb{R}$ and $succ-\exists\mathbb{R}$, and situates these classes within the broader hierarchy ($\mathsf{NP}_{real}$, $\mathsf{NEXP}_{real}$, $\mathsf{PSPACE}$, $\mathsf{EXPSPACE}$). The paper also studies satisfiability and model-checking for fragments of existential second-order logic and probabilistic logics (including probabilistic independence logic), proving $succ-\exists\mathbb{R}$-completeness in several settings and showing containment results such as $\exists\mathbb{R}^{\Sigma} \subseteq PSPACE$. Additionally, it demonstrates that adding exponential products yields $PSPACE$, and establishes that probabilistic satisfiability with a small model remains complete for $\exists\mathbb{R}^{\Sigma}$ at the probabilistic layer. Overall, the results contribute a nuanced complexity landscape for real-number logics with summation, highlighting the boundary between succinct real-number computation and classical boolean complexity classes, and connecting logical formalisms to real RAM computation models.
Abstract
To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within the Pearl's Causal Hierarchy, arXiv:2305.09508 [cs.AI] introduce a new natural class, named succ-$\exists$R. This class can be viewed as a succinct variant of the well-studied class $\exists$R based on the Existential Theory of the Reals (ETR). Analogously to $\exists$R, succ-$\exists$R is an intermediate class between NEXP and EXPSPACE, the exponential versions of NP and PSPACE. The main contributions of this work are threefold. Firstly, we characterize the class succ-$\exists$R in terms of nondeterministic real RAM machines and develop structural complexity theoretic results for real RAMs, including translation and hierarchy theorems. Notably, we demonstrate the separation of $\exists$R and succ-$\exists$R. Secondly, we examine the complexity of model checking and satisfiability of fragments of existential second-order logic and probabilistic independence logic. We show succ-$\exists$R- completeness of several of these problems, for which the best-known complexity lower and upper bounds were previously NEXP-hardness and EXPSPACE, respectively. Thirdly, while succ-$\exists$R is characterized in terms of ordinary (non-succinct) ETR instances enriched by exponential sums and a mechanism to index exponentially many variables, in this paper, we prove that when only exponential sums are added, the corresponding class $\exists$R^Σ is contained in PSPACE. We conjecture that this inclusion is strict, as this class is equivalent to adding a VNP-oracle to a polynomial time nondeterministic real RAM. Conversely, the addition of exponential products to ETR, yields PSPACE. Additionally, we study the satisfiability problem for probabilistic reasoning, with the additional requirement of a small model and prove that this problem is complete for $\exists$R^Σ.