Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates expressions in the syntax into loss functions. These loss functions can then be used during training with standard gradient descent algorithms. The variety of existing DLs and the differing levels of formality with which they are treated makes a systematic comparative study of their properties and implementations difficult. This paper remedies this problem by suggesting a meta-language for defining DLs that we call the Logic of Differentiable Logics, or LDL. Syntactically, it generalises the syntax of existing DLs to FOL, and for the first time introduces the formalism for reasoning about vectors and learners. Semantically, it introduces a general interpretation function that can be instantiated to define loss functions arising from different existing DLs. We use LDL to establish several theoretical properties of existing DLs, and to conduct their empirical study in neural network verification.

翻译：可微分逻辑（DL）近期被提出作为一种训练神经网络满足逻辑规范的方法。DL由陈述规范的语法和将语法表达式转化为损失函数的解释函数组成。这些损失函数可在训练过程中与标准梯度下降算法配合使用。不同DL的多样性及其形式化程度的差异，使得对其性质与实现进行系统性比较研究变得困难。本文通过提出一种用于定义DL的元语言——称为"可微分逻辑的逻辑"（LDL）——来解决此问题。在语法层面，LDL将现有DL的语法泛化到一阶逻辑（FOL），并首次引入对向量和学习者进行推理的形式体系。在语义层面，LDL提出了一种通用解释函数，可通过实例化来定义源自不同现有DL的损失函数。我们利用LDL建立了若干现有DL的理论性质，并在神经网络验证中对其进行了实证研究。