In recent years there has been a push to integrate symbolic AI and deep learning, as it is argued that the strengths and weaknesses of these approaches are complementary. One such trend in the literature are weakly supervised learning techniques that use operators from fuzzy logics. They employ prior background knowledge described in logic to benefit 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 used in deep learning to integrate reasoning and learning. In this paper, we analyze how a large collection of logical operators from the fuzzy logic literature behave in a differentiable setting. We find large differences between the formal properties of these operators that are of crucial importance in a differentiable learning setting. We show that many of these operators, including some of the best known, are highly unsuitable for use in a differentiable learning setting. A further finding concerns the treatment of implication in these fuzzy logics, with a strong imbalance between gradients driven by the antecedent and the consequent of the implication. Finally, we empirically show that it is possible to use Differentiable Fuzzy Logics for semi-supervised learning. However, to achieve the most significant performance improvement over a supervised baseline, we have to resort to non-standard combinations of logical operators which perform well in learning, but which no longer satisfy the usual logical laws. We end with a discussion on extensions to large-scale problems.

中文翻译：

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分析可微的模糊逻辑算子

近年来，人们一直在推动将符号人工智能和深度学习相结合，因为有人认为这些方法的优缺点是互补的。文献中的一种趋势是使用模糊逻辑运算符的弱监督学习技术。他们利用逻辑中描述的先验背景知识来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深度学习中使用的常规损失函数中，以整合推理和学习。在本文中，我们分析了来自模糊逻辑文献的大量逻辑运算符在可微设置中的行为。我们发现这些运算符的形式属性之间存在巨大差异，这在可微学习环境中至关重要。我们表明，许多这些算子，包括一些最著名的算子，非常不适合在可微分学习环境中使用。进一步的发现涉及对这些模糊逻辑中蕴涵的处理，在由前件驱动的梯度和蕴涵的后果之间存在严重的不平衡。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。然而，为了在监督基线上实现最显着的性能改进，我们必须求助于在学习中表现良好但不再满足通常逻辑规律的逻辑运算符的非标准组合。