Neural-symbolic AI (NeSy) allows neural networks to exploit symbolic background knowledge in the form of logic. It has been shown to aid learning in the limited data regime and to facilitate inference on out-of-distribution data. Probabilistic NeSy focuses on integrating neural networks with both logic and probability theory, which additionally allows learning under uncertainty. A major limitation of current probabilistic NeSy systems, such as DeepProbLog, is their restriction to finite probability distributions, i.e., discrete random variables. In contrast, deep probabilistic programming (DPP) excels in modelling and optimising continuous probability distributions. Hence, we introduce DeepSeaProbLog, a neural probabilistic logic programming language that incorporates DPP techniques into NeSy. Doing so results in the support of inference and learning of both discrete and continuous probability distributions under logical constraints. Our main contributions are 1) the semantics of DeepSeaProbLog and its corresponding inference algorithm, 2) a proven asymptotically unbiased learning algorithm, and 3) a series of experiments that illustrate the versatility of our approach.

翻译：神经符号人工智能（NeSy）允许神经网络以逻辑形式利用符号背景知识。已有研究表明，该方法有助于在有限数据场景下进行学习，并能促进对分布外数据的推理。概率神经符号学致力于将神经网络与逻辑及概率论相融合，从而进一步支持不确定性下的学习。当前概率神经符号系统（如DeepProbLog）的主要局限性在于其仅能处理有限概率分布，即离散随机变量。相比之下，深度概率编程（DPP）在建模和优化连续概率分布方面表现出色。为此，我们提出DeepSeaProbLog——一种将DPP技术融入NeSy的神经概率逻辑编程语言。该方法能够在逻辑约束下同时支持离散与连续概率分布的推理与学习。我们的主要贡献包括：1) DeepSeaProbLog的语义及其对应的推理算法；2) 一种已验证渐近无偏的学习算法；3) 一系列验证该方法多才性的实验。