The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: observational, interventional, and counterfactual, that reflect the progressive sophistication of human thought regarding causation. We investigate the computational complexity aspects of reasoning in this framework focusing mainly on satisfiability problems expressed in probabilistic and causal languages across the PCH. That is, given a system of formulas in the standard probabilistic and causal languages, does there exist a model satisfying the formulas? The resulting complexity changes depending on the level of the hierarchy as well as the operators allowed in the formulas (addition, multiplication, or marginalization). We focus on formulas involving marginalization that are widely used in probabilistic and causal inference, but whose complexity issues are still little explored. Our main contribution are the exact computational complexity results showing that linear languages (allowing addition and marginalization) yield NP^PP-, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. Moreover, we prove that the problem for the full language (allowing additionally multiplication) is complete for the class succ$\exists$R for languages on the highest, counterfactual level. Previous work has shown that the satisfiability problem is complete for succ$\exists$R on the lower levels leaving the counterfactual case open. Finally, we consider constrained models that are restricted to a small polynomial size. The constraint on the size reduces the complexity of the interventional and counterfactual languages to NEXP-complete.

翻译：Pearl因果层次（PCH）框架形式化了三种推理类型：观测性、介入性和反事实性，它们反映了人类关于因果思维的逐步深化。我们主要研究该框架内推理的计算复杂度方面，重点关注跨PCH的概率性和因果性语言中表述的可满足性问题。即：给定标准概率性和因果性语言中的公式系统，是否存在满足这些公式的模型？所得复杂度随层次级别及公式中允许的运算符（加法、乘法或边缘化）而变化。我们聚焦于广泛用于概率性和因果推理但复杂度问题仍鲜有探索的涉及边缘化的公式。我们的主要贡献是精确的计算复杂度结果：线性语言（允许加法和边缘化）根据PCH层次级别分别产生NP^PP-、PSPACE-和NEXP-完全的可满足性问题。此外，我们证明，对于完整语言（额外允许乘法），最高反事实层次上的问题对于succ∃R类是完全的。先前工作表明，较低层次上的可满足性问题对succ∃R是完全的，但反事实情形尚未解决。最后，我们考虑限制为小多项式规模的约束模型。规模约束将介入性和反事实性语言的复杂度降低至NEXP-完全。