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Gateways to Tractability for Satisfiability in Pearl's Causal Hierarchy

Robert Ganian, Marlene Gründel, Simon Wietheger

TL;DR

This work studies satisfiability in Pearl's Causal Hierarchy through parameterized complexity, identifying the first tractable cases by exploiting structural properties of causal models. Departing from standard dynamic programming, the authors develop a novel LP-based approach guided by primal treewidth and variable count to achieve fixed-parameter tractability for key fragments, while establishing tight hardness results that delineate the limits of tractability. Key contributions include FPT and XP algorithms for $ extsc{SAT}_{ ext{prob}}^{ ext{lin}}$ and $ extsc{SAT}_{ ext{counterfact}}^{ ext{lin}}$, plus hardness results for deeper fragments such as $ extsc{SAT}_{ ext{causal}}^{ ext{lin}}$, and a structural toolkit that maps SCMs onto tree-like topologies to enable efficient reasoning. The findings provide practical algorithmic strategies and theoretical limits for reasoning about probabilistic, interventional, and counterfactual statements in causal models, with potential extensions to marginalization and broader AI applications.

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.

Gateways to Tractability for Satisfiability in Pearl's Causal Hierarchy

TL;DR

This work studies satisfiability in Pearl's Causal Hierarchy through parameterized complexity, identifying the first tractable cases by exploiting structural properties of causal models. Departing from standard dynamic programming, the authors develop a novel LP-based approach guided by primal treewidth and variable count to achieve fixed-parameter tractability for key fragments, while establishing tight hardness results that delineate the limits of tractability. Key contributions include FPT and XP algorithms for and , plus hardness results for deeper fragments such as , and a structural toolkit that maps SCMs onto tree-like topologies to enable efficient reasoning. The findings provide practical algorithmic strategies and theoretical limits for reasoning about probabilistic, interventional, and counterfactual statements in causal models, with potential extensions to marginalization and broader AI applications.

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.

Paper Structure

This paper contains 6 sections, 8 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Satisfiability for Languages of PCH and Structural Insights
  4. Linear Inequalities over Probabilistic Expressions
  5. Linear Inequalities over Causal or Counterfactual Expressions
  6. Concluding Remarks

Key Result

Theorem 1

$\mathrm{arb}\textsc{SAT}_{\mathsf{prob}}^{\mathsf{base}}\xspace$ is -complete even if $G_\phi$ is edgeless and $d =~2$.

Figures (2)

  • Figure 1: Parameterized complexity of Satisfiability in the PCH based on the position (breadth/depth) in the expressivity matrix $M$. All results hold under well-established complexity assumptions and refer to an instance with $n$ observed variables over a domain of size $d$ such that the treewidth of the primal graph $G_\phi$ is $\mathop{\mathrm{tw}}\nolimits$. We note that the $\exists \mathbb{R}$-hardness of the $\poly$ fragment was already established by MosseII24, but not under the stated restrictions which rule out tractability in the parameterized setting.
  • Figure :

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 5
  • Example 6: Construction in \ref{['thm:tw_lin']}
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • Theorem 11