Neural-symbolic integration aims to combine the connectionist subsymbolic
with the logical symbolic approach to artificial intelligence. In this paper,
we first define the answer set semantics of (boolean) neural nets and then
introduce from first principles a class of neural logic programs and show that
nets and programs are equivalent.

神经符号一体化试图将连接主义子符号方法与逻辑符号方法相结合以实现人工智能。本文首先定义了（布尔）神经网络的答案集语义，然后从第一原则介绍了一类神经逻辑程序，并证明了网络和程序的等价性。

神经逻辑程序与神经网络

Neural logic programs and neural nets

Neural-symbolic computing (NeSy), which pursues the integration of the
symbolic and statistical paradigms of cognition, has been an active research
area of Artificial Intelligence (AI) for many years. As NeSy shows promise of
reconciling the advantages of reasoning and interpretability of symbolic
representation and robust learning in neural networks, it may serve as a
catalyst for the next generation of AI. In the present paper, we provide a
systematic overview of the important and recent developments of research on
NeSy AI. Firstly, we introduce study history of this area, covering early work
and foundations. We further discuss background concepts and identify key
driving factors behind the development of NeSy. Afterward, we categorize recent
landmark approaches along several main characteristics that underline this
research paradigm, including neural-symbolic integration, knowledge
representation, knowledge embedding, and functionality. Then, we briefly
discuss the successful application of modern NeSy approaches in several
domains. Finally, we identify the open problems together with potential future
research directions. This survey is expected to help new researchers enter this
rapidly-developing field and accelerate progress towards data-and
knowledge-driven AI.

本文介绍了神经符号计算（NeSy）领域的重要和最新研究发展，包括研究历史，背景知识，关键驱动因素以及主要特征。文章讨论了该领域的一些重要应用并针对目前存在的问题提出了新的研究方向，以期进一步推动以数据和知识为驱动的 AI 的研究进展。

面向数据和知识驱动的人工智能：神经符号计算综述

Towards Data-and Knowledge-Driven Artificial Intelligence: A Survey on Neuro-Symbolic Computing

In the recent past, there has been a growing interest in Neural-Symbolic
Integration frameworks, i.e., hybrid systems that integrate connectionist and
symbolic approaches to obtain the best of both worlds. In this work we focus on
a specific method, KENN (Knowledge Enhanced Neural Networks), a Neural-Symbolic
architecture that injects prior logical knowledge into a neural network by
adding on its top a residual layer that modifies the initial predictions
accordingly to the knowledge. Among the advantages of this strategy, there is
the inclusion of clause weights, learnable parameters that represent the
strength of the clauses, meaning that the model can learn the impact of each
rule on the final predictions. As a special case, if the training data
contradicts a constraint, KENN learns to ignore it, making the system robust to
the presence of wrong knowledge. In this paper, we propose an extension of KENN
for relational data. One of the main advantages of KENN resides in its
scalability, thanks to a flexible treatment of dependencies between the rules
obtained by stacking multiple logical layers. We show experimentally the
efficacy of this strategy. The results show that KENN is capable of increasing
the performances of the underlying neural network, obtaining better or
comparable accuracies in respect to other two related methods that combine
learning with logic, requiring significantly less time for learning.

本研究介绍了一种名为 KENN 的神经符号一体化框架，其将先前的逻辑知识注入神经网络。通过增加一个残差层来修改初始预测，这种方法的优点之一是包含条款权重，这些可学习参数表示条款的强度和影响。本研究的扩展版本更适用于关系数据，并在实验中验证了其有效性和可扩展性。