From Probability to Counterfactuals: the Increasing Complexity of Satisfiability in Pearl's Causal Hierarchy

TL;DR

This paper analyzes the computational complexity of reasoning within Pearl's Causal Hierarchy, revealing that satisfiability problems become strictly more complex at higher levels, with some cases remaining equally complex across levels.

Contribution

It provides the first precise complexity classifications for satisfiability in the probabilistic, causal, and counterfactual levels of Pearl's Causal Hierarchy, including resolving an open problem.

Findings

Satisfiability in probabilistic languages is NP^PP-complete.

Satisfiability in causal languages is PSPACE-complete.

Satisfiability in counterfactual languages with full operators remains NEXP-complete.

Abstract

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.