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，研究者使用**参数化概率图模型**的概念来建模随机变量间的（条件）依赖性，并对“可能世界”空间上的事件进行概率推理。可能世界集由底层域$D$（对象集合）构成，可通过域$D$上所有一阶结构（针对适当签名）的集合$\mathbf{W}_D$表示。利用形式逻辑，我们可描述$\mathbf{W}_D$上的事件。通过结合逻辑与参数化概率图模型，我们还能在$\mathbf{W}_D$上定义概率分布$\mathbb{P}_D$，并以此计算事件概率。本文考虑一种真值位于单位区间的逻辑$PLA$，该逻辑使用聚合函数（如算术平均、几何平均、最大值、最小值）替代量词。然而，我们面临计算效率问题，这一障碍阻碍了统计关系AI方法在实践中的广泛运用。我们通过证明：在参数化概率图模型与语句$\varphi$满足特定假设的条件下，所述概率随域$D$规模趋于无穷大而收敛，从而解决该问题。收敛结果通过证明每个仅包含“可容许”聚合函数（如算术平均、几何平均、最大值、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$\psi(x_1, \ldots, x_k)$而获得。