As witnessed by recent advances in deep learning technologies, neural network models of very high complexity have been successfully applied in many data-rich domains. Challenges remain, however, if the amount of training data is severely limited, which is often the case due to the cost of acquiring such data or due to interest in systems that are constantly evolving thereby imposing natural limits on how much data can be collected. The core hypothesis explored in this paper is that data (to some degree) can be substituted by domain knowledge, not only addressing the limited data problem but also offering potential improvements in data-rich settings. For the representation of suitable domain theories, we propose Probabilistic Approximate Logic (PALO) to deal with the natural uncertainty associated with such representations and also to serve as a foundation for a new class of neuro-symbolic architectures, in which both neural and symbolic computations can be peacefully and synergistically integrated. Utilizing TensorFlow and Maude as neural and symbolic frameworks, respectively, we discuss our prototypical implementation of PALO in what we call the Logical Imagination Engine (LIME). By means of a small toy example, we convey a glimpse of its capabilities, but we also briefly discuss some real-world applications and how it may serve as a prototypical framework to explore a broader range of neuro-symbolic strategies in the future.

正如深度学习技术的最新进展所见证的那样，非常复杂的神经网络模型已成功应用于许多数据丰富的领域。然而，如果训练数据量受到严重限制，挑战仍然存在，这通常是由于获取此类数据的成本或对不断发展的系统的兴趣，从而对可以收集的数据量施加自然限制。本文探讨的核心假设是数据（在某种程度上）可以被领域知识取代，不仅可以解决数据有限的问题，而且可以在数据丰富的环境中提供潜在的改进。对于合适的领域理论的表示，我们提出概率近似逻辑 (PALO) 来处理与此类表示相关的自然不确定性，并作为一类新的神经符号架构的基础，其中神经和符号计算可以和平地协同集成。我们分别利用 TensorFlow 和 Maude 作为神经和符号框架，讨论了我们在所谓的逻辑想象引擎 (LIME) 中 PALO 的原型实现。通过一个小玩具示例，我们可以一瞥它的功能，但我们也简要讨论了一些现实世界的应用，以及它如何作为一个原型框架来探索未来更广泛的神经符号策略。其中神经和符号计算可以和平地和协同地集成。我们分别利用 TensorFlow 和 Maude 作为神经和符号框架，讨论了我们在所谓的逻辑想象引擎 (LIME) 中 PALO 的原型实现。通过一个小玩具示例，我们可以一瞥它的功能，但我们也简要讨论了一些现实世界的应用，以及它如何作为一个原型框架来探索未来更广泛的神经符号策略。其中神经和符号计算可以和平地和协同地集成。我们分别利用 TensorFlow 和 Maude 作为神经和符号框架，讨论了我们在所谓的逻辑想象引擎 (LIME) 中 PALO 的原型实现。通过一个小玩具示例，我们可以一瞥它的功能，但我们也简要讨论了一些现实世界的应用，以及它如何作为一个原型框架来探索未来更广泛的神经符号策略。