论文标题
因果关系的组合
The Combinatorics of Causality
论文作者
论文摘要
我们介绍并探讨了“输入历史空间”的概念，这是一个广阔的组合对象家族，可用于建模输入依赖性，动态因果秩序。我们参考了大多数先前关于该主题的文献所采用的传统部分秩序和基于预订的因果秩序概念，以激励我们的定义，我们开始探索组合复杂性的新颖景观，这是我们对这些概念的概括提供的。 在此过程中，我们发现因果关系的细粒结构比我们先前认为的要复杂得多：在最简单的二进制输入情况下，可用的“因果关系完整”空间的数量从2个事件的7个事件增长到2644个事件，增加到3个事件，增加到4个事件的未知数（可能是十亿美元）。从角度来看，先前关于非本地性和上下文性的文献使用了3个事件的2644个可用空间中的一个，在确定的因果关系上使用了19个空间，从部分订单中得出，而无限期的因果关系仅使用6个，仅使用了6个，总计25个。 本文是三部曲中的第一部分：关于因果分布的隔离理论处理，在第2部分中详细介绍了“因果关系的拓扑” [ARXIV：2303.07148]，而相关经验模型形成的多面体则在第3部分中研究了。补充工作“在3个带有二进制输入的事件上的因果完整空间的分类”中提供了3个带有二进制输入的事件的2644个因果完整空间的详尽分类，以及用于分类的算法以及持续搜索的4个事件的算法和部分结果。
We introduce and explore the notion of "spaces of input histories", a broad family of combinatorial objects which can be used to model input-dependent, dynamical causal order. We motivate our definition with reference to traditional partial order- and preorder-based notions of causal order, adopted by the majority of previous literature on the subject, and we proceed to explore the novel landscape of combinatorial complexity made available by our generalisation of those notions. In the process, we discover that the fine-grained structure of causality is significantly more complex than we might have previously believed: in the simplest case of binary inputs, the number of available "causally complete" spaces grows from 7 on 2 events, to 2644 on 3 events, to an unknown number on 4 events (likely around a billion). For perspective, previous literature on non-locality and contextuality used a single one of the 2644 available spaces on 3 events, work on definite causality used 19 spaces, derived from partial orders, and work on indefinite causality used only 6 more, for a grand total of 25. This paper is the first instalment in a trilogy: the sheaf-theoretic treatment of causal distributions is detailed in Part 2, "The Topology of Causality" [arXiv:2303.07148], while the polytopes formed by the associated empirical models are studied in Part 3, "The Geometry of Causality" [arXiv:2303.09017]. An exhaustive classification of the 2644 causally complete spaces on 3 events with binary inputs is provided in the supplementary work "Classification of causally complete spaces on 3 events with binary inputs", together with the algorithm used for the classification and partial results from the ongoing search on 4 events.
