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因果层级中概率与因果推理的可满足性问题的计算复杂性，文献arXiv:2305.09508 [cs.AI]引入了一个新的自然类，命名为succ-$\exists$R。该类可被视为基于实数存在性理论（ETR）的经典类$\exists$R的简洁变体。与$\exists$R类似，succ-$\exists$R是介于NEXP与EXPSPACE（即NP与PSPACE的指数版本）之间的中间类。本文的主要贡献有三方面。首先，我们利用非确定性实RAM机刻画了succ-$\exists$R类，并为实RAM建立了结构复杂性理论结果，包括转换定理与层次定理。值得注意的是，我们证明了$\exists$R与succ-$\exists$R的可分离性。其次，我们考察了存在二阶逻辑片段与概率独立逻辑的模型检测与可满足性问题复杂性。我们证明了其中若干问题属于succ-$\exists$R完全问题，而此前这些问题的已知最佳复杂度下界与上界分别为NEXP困难性与EXPSPACE。第三，尽管succ-$\exists$R可通过添加指数和与索引指数多个变量的机制来刻画普通（非简洁）ETR实例，本文证明了仅添加指数和时，对应类$\exists$R^{\Sigma}包含于PSPACE。我们猜想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RAM上添加VNP预言机。反之，在ETR中添加指数乘积将得到PSPACE。此外，我们研究了概率推理中附加小模型约束的可满足性问题，并证明该问题对$\exists$R^{\Sigma}是完全的。