Axiomatizing Causal Reasoning

Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those without feedback), (2) the class of theories where the solutions to the equations are unique, (3) arbitrary theories (where the equations may not have solutions and, if they do, they are not necessarily unique). It is shown that to reason about causality in the most general third class, we must extend the language used by Galles and Pearl. In addition, the complexity of the decision procedures is examined for all the languages and classes of models considered.

💡 Research Summary

The paper undertakes a systematic axiomatization of Pearl’s structural‑equation models of causality, distinguishing three increasingly general classes of causal theories and providing a complete logical framework for each.

1. Recursive (acyclic) theories – These correspond to directed‑acyclic graphs where feedback is absent. Because the equations admit a unique solution, the original language of Galles and Pearl suffices. The authors present a compact set of five axioms (reflexivity, symmetry, transitivity, composition, and independence) that are shown to be sound and complete for all valid causal statements in this class. Decision procedures run in polynomial time with respect to the number of variables.
2. Theories with unique solutions not guaranteed – Here the system may have multiple admissible worlds. The basic language cannot express “possible” versus “necessary” causal claims, so the authors extend it with modal‑style operators for possibility (◇) and necessity (□). Additional axioms (choice, partition, etc.) capture the semantics of these operators. The resulting logic is shown to be PSPACE‑complete, reflecting the added cost of quantifying over alternative solutions.
3. General (arbitrary) theories – The most permissive class allows equations that may have no solution or infinitely many solutions. Existing syntax fails to represent the outright impossibility of a model, prompting the introduction of an “impossibility” operator and global quantifiers (“for all worlds”, “there exists a world”). New axioms (global impossibility, existence) are added to guarantee consistency and completeness. Decision problems in this setting are EXPTIME‑hard, indicating a substantial increase in computational difficulty.
   Across all three classes, the paper proves soundness (all derivable statements hold in the intended models) and completeness (all semantically valid statements are derivable) of the respective axiom systems. It also conducts a thorough complexity analysis, linking the expressive power of the language to the computational resources required for inference.
   The authors discuss practical implications: choosing a model class determines both the logical language needed and the tractability of inference algorithms. For applications where feedback loops are absent, the simple recursive framework offers efficient reasoning. When multiple solutions are plausible, the modal extension provides the necessary expressive power at a moderate computational cost. In the most general settings—such as systems with under‑determined or contradictory equations—full quantification is indispensable, albeit at the price of exponential‑time decision procedures.
   Overall, the work bridges causal reasoning, formal logic, and computational complexity, delivering a unified theory that clarifies exactly what can be inferred, how it can be inferred, and at what computational expense for each tier of causal models.