The AI community is increasingly putting its attention towards combining symbolic and neural approaches, as it is often argued that the strengths and weaknesses of these approaches are complementary. One recent trend in the literature is weakly supervised learning techniques that employ operators from fuzzy logics. In particular, these use prior background knowledge described in such logics to help the training of a neural network from unlabeled and noisy data. By interpreting logical symbols using neural networks, this background knowledge can be added to regular loss functions, hence making reasoning a part of learning.

We study, both formally and empirically, how a large collection of logical operators from the fuzzy logic literature behave in a differentiable learning setting. We find that many of these operators, including some of the most well-known, are highly unsuitable in this setting. A further finding concerns the treatment of implication in these fuzzy logics, and shows a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Furthermore, we introduce a new family of fuzzy implications (called sigmoidal implications) to tackle this phenomenon. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning, and compare how different operators behave in practice. We find that, to achieve the largest performance improvement over a supervised baseline, we have to resort to non-standard combinations of logical operators which perform well in learning, but no longer satisfy the usual logical laws.

AI 社区越来越关注符号和神经方法的结合，因为人们经常认为这些方法的优点和缺点是互补的。文献中最近的一个趋势是采用模糊逻辑运算符的弱监督学习技术。特别是，这些使用在此类逻辑中描述的先验背景知识来帮助从未标记和嘈杂的数据中训练神经网络。通过使用神经网络解释逻辑符号，可以将此背景知识添加到常规损失函数中，从而使推理成为学习的一部分。

我们从形式上和经验上研究了来自模糊逻辑文献的大量逻辑运算符在可微学习环境中的行为。我们发现其中许多运算符，包括一些最著名的运算符，都非常不适合这种设置。进一步的发现涉及这些模糊逻辑中蕴涵的处理，并表明由先行词驱动的梯度与蕴涵的后果之间存在严重的不平衡。此外，我们引入了一系列新的模糊含义（称为 sigmoidal 含义）来解决这种现象。最后，我们凭经验表明可以将可微模糊逻辑用于半监督学习，并比较不同算子在实践中的表现。我们发现，为了在监督基线上实现最大的性能改进，