From Probability to Counterfactuals: the Increasing Complexity of Satisfiability in Pearl's Causal Hierarchy
This work addresses foundational computational challenges in causal inference for researchers in AI and theoretical computer science, providing new complexity results that clarify the inherent difficulty of reasoning at different causal levels.
The paper tackles the computational complexity of satisfiability problems across Pearl's Causal Hierarchy, proving that languages with addition and marginalization yield NP^PP, PSPACE-, and NEXP-complete complexities from probabilistic to counterfactual reasoning, showing a strictly increasing trend, while full languages maintain the same complexity across all levels, resolving an open problem.
The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: probabilistic (i.e. purely 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? Our main contribution is to prove the exact computational complexities showing that languages allowing addition and marginalization (via the summation operator) yield NP^PP, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. These are the first results to demonstrate a strictly increasing complexity across the PCH: from probabilistic to causal and counterfactual reasoning. On the other hand, in the case of full languages, i.e. allowing addition, marginalization, and multiplication, we show that the satisfiability for the counterfactual level remains the same as for the probabilistic and causal levels, solving an open problem in the field.