Mathematical reasoning is one of the crucial abilities of general artificial intelligence, which requires machines to master mathematical logic and knowledge from solving problems. However, existing approaches are not transparent (thus not interpretable) in terms of what knowledge has been learned and applied in the reasoning process. In this paper, we propose a general Learning by Applying (LeAp) framework to enhance existing models (backbones) in a principled way by explicit knowledge learning. In LeAp, we perform knowledge learning in a novel problem-knowledge-expression paradigm, with a Knowledge Encoder to acquire knowledge from problem data and a Knowledge Decoder to apply knowledge for expression reasoning. The learned mathematical knowledge, including word-word relations and word-operator relations, forms an explicit knowledge graph, which bridges the knowledge "learning" and "applying" organically. Moreover, for problem solving, we design a semantics-enhanced module and a reasoning-enhanced module that apply knowledge to improve the problem comprehension and symbol reasoning abilities of any backbone, respectively. We theoretically prove the superiority of LeAp's autonomous learning mechanism. Experiments on three real-world datasets show that LeAp improves all backbones' performances, learns accurate knowledge, and achieves a more interpretable reasoning process.

翻译：数学推理是通用人工智能的关键能力之一，它要求机器从解题过程中掌握数学逻辑与知识。然而，现有方法在推理过程中未能清晰呈现所学知识与实际应用（因此缺乏可解释性）。本文提出一种通用的"学习即应用"（LeAp）框架，通过显式知识学习以原则性方式增强现有模型（骨干网络）。在LeAp中，我们采用新颖的"问题-知识-表达"范式开展知识学习，通过知识编码器从问题数据中获取知识，并由知识解码器将知识应用于表达推理。所习得的数学知识（包括词-词关系与词-算子关系）形成显式知识图谱，有机衔接知识的"学习"与"应用"。针对问题求解，我们分别设计了语义增强模块与推理增强模块，通过知识应用提升任意骨干网络的问题理解能力与符号推理能力。理论分析证明了LeAp自主学习机制的优势。在三个真实数据集上的实验表明，LeAp不仅提升了所有骨干网络的性能，还习得了精确知识，并实现了更具可解释性的推理过程。