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AI（NeSy）允许神经网络以逻辑形式利用符号背景知识，已被证明有助于有限数据场景下的学习，并能促进分布外数据的推理。概率NeSy侧重于将神经网络与逻辑和概率论相结合，这进一步支持不确定性下的学习。当前概率NeSy系统（如DeepProbLog）的主要局限在于它们仅限于有限概率分布，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。这使其能够在逻辑约束下支持离散与连续概率分布的推理和学习。我们的主要贡献包括：1）DeepSeaProbLog的语义及其对应推理算法；2）一种经证明渐近无偏的学习算法；3）一系列展示我们方法通用性的实验。