Graph Surgery and the Do-Operator: A Precise Correspondence for Acyclic Structural Causal Models
| Source: arXiv AI
Tags: causal inference, do-operator, structural causal models, graph surgery, Pearl causality
A formal proof establishes that the do-operator in Pearl's causal calculus is exactly equivalent to graph surgery on acyclic structural causal models, providing a clean mathematical foundation linking the two most-used formalisms in causal AI research.
Details
Pearl's do-operator and graph surgery are the two dominant ways to formalize intervention in causal models, but their precise relationship has remained informal in most treatments. This paper gives a rigorous proof that for deterministic acyclic structural causal models (SCMs), applying do(X=x) produces exactly the same causal structure as performing graph surgery: Graph(F^ι) = Surg(Graph(F), T_ι). The main theorem characterizes this as a strict categorical equivalence, not merely a notational correspondence. The proof proceeds by showing that both operations produce the same modified DAG (interventional graph) and the same altered functional assignments — they are two different descriptions of the same mathematical object. For practitioners, this matters mostly as theoretical hygiene: code and reasoning that works in one formalism can be directly translated to the other without subtle correctness concerns at the boundary. For researchers, it closes a gap in the foundations of causal inference and makes it easier to reason across frameworks that have historically been developed in parallel. The result is limited to deterministic acyclic settings; extensions to stochastic or cyclic SCMs are noted as open problems.