Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz

TL;DR

This work analyzes satisfiability across Pearl's Causal Hierarchy (PCH), focusing on languages that include marginalization via a summation operator Σ. It proves a strict progression in hardness: ${NP^{PP}}$ at the probabilistic level, ${PSPACE}$ at the interventional level, and ${NEXP}$ at the counterfactual level for linear languages with Σ, establishing the first clear separation of PCH-level complexities. For full polynomial languages, counterfactual satisfiability matches the probabilistic and causal levels, solving an open problem; additionally, polynomial languages at the counterfactual level are shown to be ${succ\text{-}\exists\mathbb{R}}$-complete, strengthening connections to real-algebraic decision problems. Collectively, the results clarify the computational landscape of causal reasoning and inform algorithm design, indicating a need for heuristics or exponential-time approaches in the presence of Σ and polynomial terms.

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.

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

TL;DR

This work analyzes satisfiability across Pearl's Causal Hierarchy (PCH), focusing on languages that include marginalization via a summation operator Σ. It proves a strict progression in hardness: at the probabilistic level, at the interventional level, and at the counterfactual level for linear languages with Σ, establishing the first clear separation of PCH-level complexities. For full polynomial languages, counterfactual satisfiability matches the probabilistic and causal levels, solving an open problem; additionally, polynomial languages at the counterfactual level are shown to be -complete, strengthening connections to real-algebraic decision problems. Collectively, the results clarify the computational landscape of causal reasoning and inform algorithm design, indicating a need for heuristics or exponential-time approaches in the presence of Σ and polynomial terms.

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.
Paper Structure (17 sections, 12 theorems, 20 equations, 1 algorithm)

This paper contains 17 sections, 12 theorems, 20 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Languages of Causal Hierarchy
  4. Satisfiability for PCH Languages and Relevant Complexity Classes
  5. The Increasing Complexity of Satisfiability in PCH for Linear Languages with Marginalization
  6. The Probabilistic (Observational) Level
  7. The Interventional (Causal) Level
  8. The Counterfactual Level
  9. The Complexity of Satisfiability for Polynomial Languages with Marginalization
  10. Discussion
  11. Technical Details and Proofs
  12. Proofs of Section\ref{['sec:increasing:prob']}
  13. Proofs of Section\ref{['sec:increasing:causal']}
  14. Proofs of Section\ref{['sec:increasing:counter']}
  15. Proofs of Section \ref{['sec:succR:compl:satsumpoly']}
  16. ...and 2 more sections

Key Result

Lemma 3

$\hbox{\sc Sat}^{\text{\it base}{\langle{\Sigma}\rangle}}_{\text{${\cal L}_1$}}$ is $\mathsf{NP}$$^{\mathsf{PP}}$-hard.

Theorems & Definitions (24)

  • Lemma 3
  • proof
  • Theorem 4
  • Remark 5
  • Proposition 6
  • Lemma 7
  • proof
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 14 more