Solving Algebra Problems with Geometry Diagrams (APGDs) is still a challenging problem because diagram processing is not studied as intensively as language processing. To work against this challenge, this paper proposes a hologram reasoning scheme and develops a high-performance method for solving APGDs by using this scheme. To reach this goal, it first defines a hologram, being a kind of graph, and proposes a hologram generator to convert a given APGD into a hologram, which represents the entire information of APGD and the relations for solving the problem can be acquired from it by a uniform way. Then HGR, a hologram reasoning method employs a pool of prepared graph models to derive algebraic equations, which is consistent with the geometric theorems. This method is able to be updated by adding new graph models into the pool. Lastly, it employs deep reinforcement learning to enhance the efficiency of model selection from the pool. The entire HGR not only ensures high solution accuracy with fewer reasoning steps but also significantly enhances the interpretability of the solution process by providing descriptions of all reasoning steps. Experimental results demonstrate the effectiveness of HGR in improving both accuracy and interpretability in solving APGDs.

翻译：利用几何图解求解代数问题（APGDs）仍是一个具有挑战性的难题，因为图解处理的研究远不如语言处理那样深入。为应对这一挑战，本文提出一种全息图推理方案，并基于该方案开发了一种高性能的APGD求解方法。为实现此目标，首先定义了一种图结构形式的全息图，并提出全息图生成器将给定的APGD转换为全息图。该全息图完整表征APGD的全部信息，且可通过统一方式从中获取解题所需的关系。随后，全息图推理方法HGR利用预构建的图模型池推导出符合几何定理的代数方程。该方法可通过向模型池添加新图模型进行更新。最后，采用深度强化学习提升从模型池中选择模型的效率。完整的HGR方法不仅以较少的推理步骤保证了高求解精度，还通过提供所有推理步骤的描述显著增强了求解过程的可解释性。实验结果表明，HGR在提升APGD求解精度与可解释性方面均具有显著效果。