Weighted First Order Model Counting (WFOMC) computes the weighted sum of the models of a first order theory on a domain of a given finite size. WFOMC has emerged as a fundamental tool for probabilistic inference. Algorithms for WFOMC that run in polynomial time w.r.t. the domain size are called lifted inference algorithms. Such algorithms have been developed for multiple extensions of FO$^2$(the fragment of First Order Logic with two variables) for the special case of symmetric weight functions. In this paper, instead of developing a specific algorithm, we derive a closed form formula for WFOMC in FO$^2$. The three key advantages of our proposal are: (i) it deals with existential quantifiers without introducing negative weights; (ii) it easily extends to FO$^2$ with cardinality constraints and counting quantifiers (aka C$^2$); finally, (iii) it supports WFOMC for a class of weight functions strictly larger than symmetric weight functions, which can model count distributions, without introducing complex or negative weights.

中文翻译：

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具有基数约束和计数量词的 FO2 中的加权模型计数：封闭形式公式

加权一阶模型计数 (WFOMC) 在给定有限大小的域上计算一阶理论模型的加权和。WFOMC 已成为概率推理的基本工具。在多项式时间内运行的 WFOMC 算法称为提升推理算法。此类算法是为 FO$^2$（具有两个变量的一阶逻辑的片段）的多个扩展而开发的，用于对称权重函数的特殊情况。在本文中，我们没有开发特定的算法，而是推导出了 FO$^2$ 中 WFOMC 的闭式公式。我们提议的三个主要优点是：（i）它处理存在量词而不引入负权重；(ii) 它很容易扩展到具有基数约束和计数量词（又名 C$^2$）的 FO$^2$；最后，