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^{\Sigma} 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^{\Sigma}.

翻译：为刻画 Pearl 因果层次中概率与因果推理可满足性问题的计算复杂性，arXiv:2305.09508 [cs.AI] 引入了一个名为 succ-$\exists$R 的新自然复杂性类。该类可视为基于实数存在理论（ETR）的经典复杂性类 $\exists$R 的简洁变体。与 $\exists$R 类似，succ-$\exists$R 是 NEXP 与 EXPSPACE（即 NP 与 PSPACE 的指数版本）之间的中间复杂性类。本研究的主要贡献包含三个方面。首先，我们通过非确定性实数随机存取机刻画 succ-$\exists$R 类，并建立了实数随机存取机的结构复杂性理论结果，包括转化定理与层次定理。特别地，我们证明了 $\exists$R 与 succ-$\exists$R 的分离性。其次，我们研究了存在性二阶逻辑片段与概率独立性逻辑的模型检测及可满足性问题。我们证明了其中多个问题具有 succ-$\exists$R 完全性，而此前已知的最佳复杂度下界与上界分别为 NEXP 困难性与 EXPSPACE。第三，虽然 succ-$\exists$R 可通过扩展了指数求和机制及指数规模变量索引功能的常规（非简洁）ETR 实例来刻画，但本文证明了当仅添加指数求和算子时，对应的复杂性类 $\exists$R^{\Sigma} 包含于 PSPACE 中。我们推测该包含关系是严格的，因为该类等价于在多项式时间非确定性实数随机存取机上增加 VNP 预言机。反之，在 ETR 中添加指数乘积算子将得到 PSPACE。此外，我们研究了具有小模型要求的概率推理可满足性问题，并证明该问题对 $\exists$R^{\Sigma} 具有完全性。