We initiate the study of counting Markov Equivalence Classes (MEC) under logical constraints. MECs are equivalence classes of Directed Acyclic Graphs (DAGs) that encode the same conditional independence structure among the random variables of a DAG model. Observational data can only allow to infer a DAG model up to Markov Equivalence. However, Markov equivalent DAGs can represent different causal structures, potentially super-exponentially many. Hence, understanding MECs combinatorially is critical to understanding the complexity of causal inference. In this paper, we focus on analysing MECs of size one, with logical constraints on the graph topology. We provide a polynomial-time algorithm (w.r.t. the number of nodes) for enumerating essential DAGs (the only members of an MEC of size one) with arbitrary logical constraints expressed in first-order logic with two variables and counting quantifiers (C^2). Our work brings together recent developments in tractable first-order model counting and combinatorics of MECs.

翻译：我们首次研究了在逻辑约束下对马尔可夫等价类进行计数的问题。马尔可夫等价类是有向无环图的等价类，它们编码了DAG模型中随机变量之间相同的条件独立结构。观测数据只能推断出马尔可夫等价意义上的DAG模型。然而，马尔可夫等价的DAG可能表示不同的因果结构，其数量可能达到超指数级。因此，从组合角度理解MEC对于认识因果推断的复杂性至关重要。本文重点分析了规模为一的MEC，并对图拓扑施加了逻辑约束。我们提出了一种多项式时间算法（相对于节点数量），用于枚举具有任意逻辑约束的本质DAG（规模为一的MEC中的唯一成员），这些约束用带两个变量和计数量词的一阶逻辑表示。我们的工作结合了可处理一阶模型计数和MEC组合学的最新进展。