Intuitionistic $j$-Do-Calculus in Topos Causal Models

2025-10-20

AI Summary

This paper addresses the limited applicability of classical do-calculus in non-classical logical settings. We propose a novel causal inference framework—$j$-stable causal models (TCMs)—grounded in intuitionistic logic and sheaf-theoretic topology. Methodologically, we define causal interventions as subobjects within a topos via a Lawvere–Tierney topology $j$, and formalize $j$-stable conditional independence and intervention propositions using Kripke–Joyal semantics, thereby constructing a complete $j$-do-calculus system. Our main contributions are threefold: (i) the first categorical generalization of do-calculus to an intuitionistic setting, yielding three sound inference rules; (ii) support for structure-preserving causal reasoning under locally defined truth values; and (iii) a theoretical foundation for $j$-cover-based, data-driven causal discovery and cohomological modeling on sheaves.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we generalize Pearl's do-calculus to an Intuitionistic setting called $j$-stable causal inference inside a topos of sheaves. Our framework is an elaboration of the recently proposed framework of Topos Causal Models (TCMs), where causal interventions are defined as subobjects. We generalize the original setting of TCM using the Lawvere-Tierney topology on a topos, defined by a modal operator $j$ on the subobject classifier $Ω$. We introduce $j$-do-calculus, where we replace global truth with local truth defined by Kripke-Joyal semantics, and formalize causal reasoning as structure-preserving morphisms that are stable along $j$-covers. $j$-do-calculus is a sound rule system whose premises and conclusions are formulas of the internal Intuitionistic logic of the causal topos. We define $j$-stability for conditional independences and interventional claims as local truth in the internal logic of the causal topos. We give three inference rules that mirror Pearl's insertion/deletion and action/observation exchange, and we prove soundness in the Kripke-Joyal semantics. A companion paper in preparation will describe how to estimate the required entities from data and instantiate $j$-do with standard discovery procedures (e.g., score-based and constraint-based methods), and will include experimental results on how to (i) form data-driven $j$-covers (via regime/section constructions), (ii) compute chartwise conditional independences after graph surgeries, and (iii) glue them to certify the premises of the $j$-do rules in practice

Problem

• Generalizing Pearl's do-calculus to intuitionistic logic in topos theory
• Defining causal interventions as j-stable morphisms using Lawvere-Tierney topology
• Establishing sound inference rules for causal reasoning in sheaf topoi

Innovation

• Generalizes do-calculus using Lawvere-Tierney topology in topos
• Introduces j-do-calculus with Kripke-Joyal semantics for local truth
• Defines j-stability for causal claims via internal intuitionistic logic