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.

翻译：从低层次描述抽象到更具解释性的高层次描述，并理想地保持因果结构，是科学实践、因果推断问题以及鲁棒、高效且可解释的人工智能的基础。我们提出了一种将低层次与高层次模型之间的抽象作为自然变换的一般性描述，重点关注因果模型的情形。这为因果抽象提供了一种新的形式化框架，统一了文献中的多个概念，包括构造性因果抽象、Q-τ一致性、基于交换干预的抽象以及“分布式”因果抽象。我们的方法基于范畴论形式化，并使用具有给定查询集和语义的组合模型的一般概念，其语义可在幺半范畴、cd-范畴或马尔可夫范畴中定义；因果模型及其查询（如干预）是特例。我们识别了两种基本的抽象概念：向下抽象将查询从高层次映射到低层次；向上抽象将具体查询（如Do-干预）从低层次映射到高层次。尽管通常以后者形式呈现，我们展示了常见的因果抽象如何更根本地通过前者来理解。我们的方法还引导我们考虑一种新的、更强的“组件级”抽象概念，应用于模型的各个组件。特别地，这产生了一种新颖的、加强形式的机制级构造性因果抽象，我们为其证明了刻画定理。最后，我们展示了抽象可推广到其他组合模型，包括那些由量子电路实现量子语义的模型，并初步探索了量子组合电路模型与高层次经典因果模型之间的抽象，作为可解释量子人工智能的一种手段。