Overview
This study presents a machine-checked, axiom-free formalization of the 1-topos core of topos causal models. This formalization utilizes Cubical Agda, building upon a previously verified probability monad and do-calculus. The work addresses aspects of topos causal models, which typically involve recasting causal inference within a topos, where a causal world is represented as a presheaf, and an intervention is modeled as a sub-model characterized by a map into the subobject classifier $\Omega$. Reasoning in this framework employs Kripke-Joyal forcing within an intuitionistic internal language.
Research Context
Topos causal models have been developed primarily on paper, with central claims often stated rather than formally proved. This research aims to provide a rigorous, machine-verified foundation for these models. The existing framework conceives of interventions as named by a characteristic map into the subobject classifier $\Omega$. The method of reasoning relies on Kripke-Joyal forcing, executed within an intuitionistic internal language. The prior development of this area involved a probability monad and a do-calculus, which serve as foundational components for the current Cubical Agda formalization.
Approach
The research employed Cubical Agda to machine-check the 1-topos core of topos causal models. This approach ensures an axiom-free development. The formalization incorporated a previously verified probability monad and do-calculus. The entire development typechecks under Agda's --safe flag, with the ordered field discharged at $\mathbb{Q}$. Specific components machine-checked include the classifier of sieves along with its classification theorem, the pullback mechanism for collating local mechanisms, and the Kripke-Joyal forcing clauses.
Findings
- The research identified a contextuality obstruction within the topos causal model program. This obstruction manifests as pairwise-consistent local causal data that lacks a global model. It is detected by a degree-one holonomy class. This obstruction represents an aspect not treated by the existing program.
- The study delimited the claim that interventions are exclusively modeled by the subobject classifier $\Omega$. It clarified that while an intervention and an observation name the same subobject, $\Omega$ fixes the target of a do-operation rather than the operation itself. The do-operation is characterized as a surgery on the kernels, where interventional and observational laws diverge on a confounder.
- The research settled the question of the modal unit. It established that inflationarity is derivable from $j\top = \top$ and naturality, negating the necessity of a fourth axiom for its derivation.
- The machine-checked formalization successfully verified several core components. This includes the classifier of sieves with its associated classification theorem, the mechanism of pullback for collating local mechanisms, and the specific clauses pertaining to Kripke-Joyal forcing.
Why This Matters
This work provides an axiom-free, machine-checked formalization for a core aspect of topos causal models. By formalizing and verifying these models in Cubical Agda, it offers a rigorous foundation for theoretical advancements in causal inference. The identification of a contextuality obstruction highlights a specific limitation in the current scope of topos causal models, which may guide future theoretical developments.
Potential Applications
While the source does not explicitly detail applications beyond theoretical formalization, the rigorous, machine-checked framework could support future developments in formal methods for causal inference. The clarification of how interventions are modeled could influence the design of formal systems where intervention mechanisms are central.
Key Limitations Mentioned by Researchers
The researchers indicate that type-level sheafification and a directed do-calculus are areas for future work. The current development assumes no axioms and functions under Agda's --safe flag, with the ordered field discharged at $\mathbb{Q}$.