The Hardness of Reasoning about Probabilities and Causality

We study formal languages which are capable of fully expressing quantitative probabilistic reasoning and do-calculus reasoning for causal effects, from a computational complexity perspective. We focus on satisfiability problems whose instance formulas allow expressing many tasks in probabilistic and causal inference. The main contribution of this work is establishing the exact computational complexity of these satisfiability problems. We introduce a new natural complexity class, named succ∃R, which can be viewed as a succinct variant of the well-studied class ∃R, and show that these problems are complete for succ∃R. Our results imply even stronger limitations on the use of algorithmic methods for reasoning about probabilities and causality than previous state-of-the-art results that rely only on the NP- or ∃R-completeness of the satisfiability problems for some restricted languages.