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.

本文提出了一种基于应用学习 (LeAp) 框架的方法，通过明确的知识学习和应用来增强现有模型的推理能力，并形成了一种将知识 “学习” 和 “应用” 有机结合的知识图谱，提高了推理过程的解释性以及问题理解和符号推理能力。