Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers ($\mathrm{C^2}$) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in $\mathrm{C^2}$, or first order logic in general. In this work, we expand the domain liftability of $\mathrm{C^2}$ with multiple such properties. We show that any $\mathrm{C^2}$ sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.

翻译：加权一阶模型计数（WFOMC）是统计关系学习模型中概率推理的基础。由于WFOMC通常被认为是难处理的（$\#$P完全），因此允许多项式时间WFOMC的逻辑片段具有重要研究意义。这类片段被称为领域可提升的。近期研究表明，扩展了计数量词的一阶逻辑二变量片段（$\mathrm{C^2}$）具有领域可提升性。然而，现实世界数据的许多特性（如引文网络中的无环性和社交网络中的连通性）无法在$\mathrm{C^2}$或一般的一阶逻辑中建模。在本工作中，我们针对多种此类特性扩展了$\mathrm{C^2}$的领域可提升性。我们证明：当$\mathrm{C^2}$语句的某个关系被限制为表示有向无环图、连通图、树（或有向树）或森林（或有向森林）时，该语句仍保持领域可提升性。所有结果均基于一种新颖且通用的"分割计数"方法。除应用于概率推理外，我们的结果为组合结构的计数提供了通用框架，并拓展了离散数学文献中关于有向无环图、系统发育网络等的大量既有成果。