神经符号语义是支离破碎的:经典,模糊,概率和神经系统每个都通过自己的归纳规则定义真理。 NeSyCat, 扩展了 ULLER,，将它们纳入强单子中的 true, 参数的单一归纳定义和真值聚合结构中。 NeSyCat 迄今为止还缺乏对神经网络学习的谓词和函数的描述。我们提供 NeSyCat Torch 作为缺失的环节，并通过神经网络, 解释计算符号，在概率编程和基于张量的后端中实现框架。我们使用分布单子进行参考语义和度量评估,，并用单子对其进行补充，以实现数值稳定, 可微分训练: 对数半环上的惰性对数张量单子。为了批量, 的有效训练，我们还使用了批量 monad。公理是源代码: 以基于 monad 的 do-notation 编写一次, Monadic 绑定执行边缘化, 懒惰地修剪不需要的分支。在 MNIST 附加, 上，我们的 HaskTorch, JAX, 和 PyTorch 实现在速度和准确性, 方面优于 LTN 和 DeepProbLog，同时几乎达到 DeepStochLog 的准确性。然而，, 与 DeepStochLog, 不同，我们停留在一个适用于许多一阶 NeSy 方法的统一框架中。即, 构造在单子; 中是参数化的，用, 实例化它，例如, Giry 单子将方法扩展到连续概率(，这里计算出的神经表示留待将来工作)。
Neurosymbolic semantics is fragmented: classical, fuzzy, probabilistic and neural systems each define truth by their own inductive rules. NeSyCat, extending ULLER, subsumes them under a single inductive definition of truth, parametric in a strong monad and an aggregation structure on truth-values. NeSyCat has so far lacked an account of predicates and functions learned by neural networks. We provide NeSyCat Torch as the missing link and interpret computational symbols via neural networks, implementing the framework in probabilistic programming and tensor-based backends. We use the distribution monad for reference semantics and metric evaluation, and complement it by a monad for numerically stable, differentiable training: the lazy log-tensor monad over the log-semiring. For efficient training in batches, we furthermore employ a batch monad. The axioms are the source code: written once in monad-based do-notation, monadic bind performs marginalisation, lazily pruning unneeded branches. On MNIST addition, our HaskTorch, JAX, and PyTorch implementations outperform LTN and DeepProbLog in speed and accuracy, while achieving nearly the accuracy of DeepStochLog. However, unlike DeepStochLog, we stay in a uniform framework that applies to many first-order NeSy approaches. Namely, the construction is parametric in the monad; instantiating it with, e.g., the Giry monad extends the approach to continuous probability (working out a neural representation here is left for future work).
科目: 人工智能 (cs.AI); 机器学习 (cs.LG); 计算机科学逻辑 (cs.LO); 类别理论 (math.CT); 逻辑 (math.LO); 概率 (math.PR)
Subjects: Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Logic in Computer Science (cs.LO); Category Theory (math.CT); Logic (math.LO); Probability (math.PR)