In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept {\em para\-metrized probabilistic graphical model (PPGM)} to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain $D$ (a set of objects) can be represented by the set $\mathbf{W}_D$ of all first-order structures (for a suitable signature) with domain $D$. Using a formal logic we can describe events on $\mathbf{W}_D$. By combining a logic and a PPGM we can also define a probability distribution $\mathbb{P}_D$ on $\mathbf{W}_D$ and use it to compute the probability of an event. We consider a logic, denoted $PLA$, with truth values in the unit interval, which uses aggregation functions, such as arithmetic mean, geometric mean, maximum and minimum instead of quantifiers. However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence $\varphi$, converge as the size of $D$ tends to infinity. The convergence result is obtained by showing that every formula $\varphi(x_1, \ldots, x_k)$ which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula $\psi(x_1, \ldots, x_k)$ without aggregation functions.

翻译：在统计关系人工智能（统计关系AI）这一结合逻辑与统计学派的人工智能与机器学习分支中，研究者借助**参数化概率图模型（PPGM）**对随机变量间的（条件）依赖关系进行建模，并在“可能世界”空间中对事件进行概率推断。以$D$（对象集合）为基础域的可能世界集可表示为域$D$上所有一阶结构（针对适当签名）的集合$\mathbf{W}_D$。通过形式逻辑，我们能够描述$\mathbf{W}_D$上的事件；结合逻辑与PPGM，还可定义$\mathbf{W}_D$上的概率分布$\mathbb{P}_D$，并用于计算事件概率。本文研究一种称为$PLA$的逻辑，其真值取自单位区间，采用算术平均、几何平均、最大值和最小值等聚合函数替代量词。然而，计算效率问题成为制约统计关系AI方法在实践应用中广泛推广的障碍。我们通过证明以下结论解决该问题：在特定假设下（关于PPGM及语句$\varphi$），所述概率将随着域规模$D$趋于无穷而收敛。该收敛结果通过论证仅包含“可允许”聚合函数（如算术平均、几何平均、最大值和最小值）的公式$\varphi(x_1, \ldots, x_k)$渐近等价于不含聚合函数的公式$\psi(x_1, \ldots, x_k)$而得到。