Hologram Reasoning for Solving Algebra Problems with Geometry Diagrams

TL;DR

This paper tackles the challenge of solving algebra problems embedded in geometry diagrams (APGDs) by introducing hologram reasoning (HGR), which unifies problem text and diagrams into a global hologram and leverages a pool of graph models guided by deep reinforcement learning to select reasoning rules. The approach combines four components—parser, model pool, reasoner, and solution generator—and distinguishes between proving and property models to apply geometric theorems and derive equations, enhancing both accuracy and interpretability. Experimental results on Geometry3K show competitive accuracy, reduced reasoning steps, and superior solution transparency compared to baselines, with ablation studies confirming the importance of each component. The method holds promise for scalable, interpretable APGD solving and suggests future work in extending hologram reasoning to other problem domains and integrating with word algebra tasks.

Abstract

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.

Figures (6)

• Figure 1: An example of an algebra problem with a geometry diagram, illustrating the use of geometric theorems to derive algebraic equations and find the final solution.
• Figure 2: Overview of HGR which includes four components: 1) Parser, which processes the problem's text and diagram into a hologram; 2) Model pool, which contains a collection of predefined graph models representing different geometric theorems and reasoning patterns; 3) Reasoner, which selects appropriate graph models from the model pool, matches them to the global hologram, and applies operations to derive solutions; and 4) Solution generator, which outputs the final readable solution steps including Theorems (T), Relations (R), and Equations (E).
• Figure 3: An example of an APGD with corresponding global hologram.
• Figure 4: Examples of proving model and property model, showing the distinct roles and applications of each type.
• Figure 5: The iterative reasoning process using the proving model and property model.