In this paper, I discuss two logics for weighted finite structures: first-order logic with summation (FO(SUM)) and its recursive extension IFP(SUM). These logics originate from foundational work by Grädel, Gurevich, and Meer in the 1990s. In recent joint work with Standke, Steegmans, and Van den Bussche, we have investigated these logics as query languages for machine learning models, specifically neural networks, which are naturally represented as weighted graphs. I present illustrative examples of queries to neural networks that can be expressed in these logics and discuss fundamental results on their expressiveness and computational complexity.

翻译：本文探讨了两种加权有限结构的逻辑：带求和的一阶逻辑（FO(SUM）及其递归扩展IFP(SUM）。这些逻辑源于Grädel、Gurevich和Meer在20世纪90年代的基础性工作。在近期与Standke、Steegmans和Van den Bussche的合作研究中，我们将其作为机器学习模型（特别是神经网络）的查询语言进行了探究，因为神经网络天然地可表示为加权图。本文展示了用这些逻辑表达神经网络查询的示例，并讨论了其在表达能力和计算复杂性方面的基本结果。