Gateways to Tractability for Satisfiability in Pearl's Causal Hierarchy

TL;DR

This paper explores the computational complexity of satisfiability in Pearl's Causal Hierarchy, identifying conditions under which the problem becomes tractable using parameterized algorithms and structural insights.

Contribution

It introduces the first fixed-parameter and XP-algorithms for satisfiability in key causal fragments, along with hardness results delineating tractability limits.

Findings

Fixed-parameter algorithms for probabilistic and counterfactual satisfiability

Structural characterizations enable new algorithmic approaches

Hardness results define the boundaries of tractability

Abstract

Pearl's Causal Hierarchy (PCH) is a central framework for reasoning about probabilistic, interventional, and counterfactual statements, yet the satisfiability problem for PCH formulas is computationally intractable in almost all classical settings. We revisit this challenge through the lens of parameterized complexity and identify the first gateways to tractability. Our results include fixed-parameter and XP-algorithms for satisfiability in key probabilistic and counterfactual fragments, using parameters such as primal treewidth and the number of variables, together with matching hardness results that map the limits of tractability. Technically, we depart from the dynamic programming paradigm typically employed for treewidth-based algorithms and instead exploit structural characterizations of well-formed causal models, providing a new algorithmic toolkit for causal reasoning.