Causal and Compositional Abstraction

TL;DR

This work develops a unified, category-theoretic account of causal abstraction, modeling abstractions as natural transformations between high- and low-level query semantics within compositional models. It introduces downward and upward abstraction directions, type alignments, and two core abstraction notions that subsume many existing causal abstractions, including constructive, exact transformations, interchange, and counterfactual forms. The framework extends to component-level and mechanism-level abstractions, and it generalizes to quantum circuits, enabling abstractions between quantum and classical causal models and paving the way for explainable quantum AI. By grounding abstractions in string diagrams and monoidal categories, the paper provides precise, diagrammatic consistency conditions and characterizations, along with a structured pathway to distributed and quantum generalizations. The results offer a principled, scalable language for reasoning about abstraction across levels of description with potential implications for AI interpretability, causal representation learning, and quantum information processing.

Abstract

Abstracting from a low level to a more explanatory high level of description, and ideally while preserving causal structure, is fundamental to scientific practice, to causal inference problems, and to robust, efficient and interpretable AI. We present a general account of abstractions between low and high level models as natural transformations, focusing on the case of causal models. This provides a new formalisation of causal abstraction, unifying several notions in the literature, including constructive causal abstraction, Q-$τ$ consistency, abstractions based on interchange interventions, and `distributed' causal abstractions. Our approach is formalised in terms of category theory, and uses the general notion of a compositional model with a given set of queries and semantics in a monoidal, cd- or Markov category; causal models and their queries such as interventions being special cases. We identify two basic notions of abstraction: downward abstractions mapping queries from high to low level; and upward abstractions, mapping concrete queries such as Do-interventions from low to high. Although usually presented as the latter, we show how common causal abstractions may, more fundamentally, be understood in terms of the former. Our approach also leads us to consider a new stronger notion of `component-level' abstraction, applying to the individual components of a model. In particular, this yields a novel, strengthened form of constructive causal abstraction at the mechanism-level, for which we prove characterisation results. Finally, we show that abstraction can be generalised to further compositional models, including those with a quantum semantics implemented by quantum circuits, and we take first steps in exploring abstractions between quantum compositional circuit models and high-level classical causal models as a means to explainable quantum AI.