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

2024-05-12

AI Summary

This paper investigates the computational complexity evolution of satisfiability across Pearl’s Causal Hierarchy (PCH)—namely, probabilistic, interventional, and counterfactual reasoning—over systems expressed in standard probabilistic and causal logic. Using causal logic modeling, semantics of probabilistic programs, and oracle Turing machine reductions, we establish precise complexity classifications. We prove that satisfiability is NP^PP-complete at the probabilistic level, PSPACE-complete at the interventional level, and NEXP-complete at the counterfactual level—demonstrating strict complexity growth with causal hierarchy. Moreover, we resolve the open problem concerning apparent non-monotonicity in the full counterfactual language: by incorporating multiplication into the logical syntax, we show that counterfactual satisfiability drops to PP-complete, thereby refuting the monotonic increase hypothesis. This work provides the first exact complexity characterization of satisfiability across all three layers of PCH, unifying foundational insights from probabilistic reasoning, causal inference, and computational complexity theory.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

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.

Problem

• Investigates computational complexity in Pearl's Causal Hierarchy
• Focuses on satisfiability problems in probabilistic and causal languages
• Proves increasing complexity from probabilistic to counterfactual reasoning

Innovation

• NP^PP complexity for probabilistic reasoning
• PSPACE-complete for interventional level
• NEXP-complete for counterfactual reasoning