The broad concept of emergence is instrumental in various of the most challenging open scientific questions—yet, few quantitative theories of what constitutes emergent phenomena have been proposed. This article introduces a formal theory of causal emergence in multivariate systems, which studies the relationship between the dynamics of parts of a system and macroscopic features of interest. Our theory provides a quantitative definition of downward causation, and introduces a complementary modality of emergent behaviour—which we refer to as causal decoupling. Moreover, the theory allows practical criteria that can be efficiently calculated in large systems, making our framework applicable in a range of scenarios of practical interest. We illustrate our findings in a number of case studies, including Conway’s Game of Life, Reynolds’ flocking model, and neural activity as measured by electrocorticography.

涌现现象的广泛概念对于解决各种最具挑战性的开放科学问题很有帮助，然而，关于涌现现象的构成的定量理论却很少被提出。本文介绍了多元系统中因果涌现的形式理论，该理论研究系统各部分的动态与感兴趣的宏观特征之间的关系。我们的理论提供了向下因果关系的定量定义，并引入了紧急行为的补充模式——我们称之为因果脱钩。此外，该理论允许在大型系统中有效计算的实际标准，使我们的框架适用于一系列实际感兴趣的场景。我们在许多案例研究中阐述了我们的发现，包括康威的生命游戏、雷诺兹的集群模型以及通过皮层电图测量的神经活动。