Conditional Separation as a Binary Relation. A Coq Assisted Proof

TL;DR

This paper extends the concept of d-separation in causality to infinite and cyclic graphs, characterizes it as a binary relation, and formalizes proofs using the Coq proof assistant, enabling more efficient reasoning.

Contribution

It introduces a novel binary relation perspective on d-separation, extending its applicability beyond acyclic graphs and formalizes the proofs with Coq.

Findings

Extended d-separation to infinite and cyclic graphs

Characterized d-separation as a binary relation

Formalized proofs using Coq proof assistant

Abstract

The concept of d-separation holds a pivotal role in causality theory, serving as a fundamental tool for deriving conditional independence properties from causal graphs. Pearl defined the d-separation of two subsets conditionally on a third one. In this study, we present a novel perspective by showing i) how the d-separation can be extended beyond acyclic graphs, possibly infinite, and ii) how it can be expressed and characterized as a binary relation between vertices. Compared to the typical perspectives in causality theory, our equivalence opens the door to more compact and computational proofing techniques, because the language of binary relations is well adapted to equational reasoning. Additionally, and of independent interest, the proofs of the results presented in this paper are checked with the Coq proof assistant.