Probabilistic and Causal Satisfiability: the Impact of Marginalization

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, which extends previous results for the lower levels of the PCH. Finally, we consider constrained models that are restricted to a given Bayesian network, a Directed Acyclic Graph structure, or a small polynomial size. The complexity of languages on the interventional level is increased to the complexity of counterfactual languages without such a constraint, that is, linear languages become NEXP-complete. On the other hand, the complexity on the counterfactual level does not change. The constraint on the size reduces the complexity of the interventional and counterfactual languages to NEXP-complete.