# A Symbolic Character-Aware Model for Solving Geometry Problems

arXiv: https://arxiv.org/abs/2308.02823 · HTML: https://mlsanity.com/paper/2308.02823/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 35 references that are in the map.
- First posted on arXiv 5 Aug 2023.

## Builds on (most cited first, 18 of 18)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Visual Genome: Connecting Language and Vision Using Crowdsourced Dense Image Annotations](https://mlsanity.com/paper/1602.07332/) (2016)
- [Making the V in VQA Matter: Elevating the Role of Image Understanding in Visual Question Answering](https://mlsanity.com/paper/1612.00837/) (2016)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [A simple neural network module for relational reasoning](https://mlsanity.com/paper/1706.01427/) (2017)
- [Deep Modular Co-Attention Networks for Visual Question Answering](https://mlsanity.com/paper/1906.10770/) (2019)
- [Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding](https://mlsanity.com/paper/1810.02338/) (2018)
- [GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning](https://mlsanity.com/paper/2105.14517/) (2021)
- [TallyQA: Answering Complex Counting Questions](https://mlsanity.com/paper/1810.12440/) (2018)
- [Generate & Rank: A Multi-task Framework for Math Word Problems](https://mlsanity.com/paper/2109.03034/) (2021)
- [Learning to Reason Deductively: Math Word Problem Solving as Complex Relation Extraction](https://mlsanity.com/paper/2203.10316/) (2022)
- [Seeking Patterns, Not just Memorizing Procedures: Contrastive Learning for Solving Math Word Problems](https://mlsanity.com/paper/2110.08464/) (2021)
- [Plane Geometry Diagram Parsing](https://mlsanity.com/paper/2205.09363/) (2022)
- [Learning by Analogy: Diverse Questions Generation in Math Word Problem](https://mlsanity.com/paper/2306.09064/) (2023)

## Built on by (most cited first, 20 of 20)

- [G-LLaVA: Solving Geometric Problem with Multi-Modal Large Language Model](https://mlsanity.com/paper/2312.11370/) (2023)
- [A Comprehensive Survey of Scientific Large Language Models and Their Applications in Scientific Discovery](https://mlsanity.com/paper/2406.10833/) (2024)
- [LANS: A Layout-Aware Neural Solver for Plane Geometry Problem](https://mlsanity.com/paper/2311.16476/) (2023)
- [GeoX: Geometric Problem Solving Through Unified Formalized Vision-Language Pre-training](https://mlsanity.com/paper/2412.11863/) (2024)
- [Fuse, Reason and Verify: Geometry Problem Solving with Parsed Clauses from Diagram](https://mlsanity.com/paper/2407.07327/) (2024)
- [Learning to Solve Geometry Problems via Simulating Human Dual-Reasoning Process](https://mlsanity.com/paper/2405.06232/) (2024)
- [Pi-GPS: Enhancing Geometry Problem Solving by Unleashing the Power of Diagrammatic Information](https://mlsanity.com/paper/2503.05543/) (2025)
- [FGeo-HyperGNet: Geometric Problem Solving Integrating FormalGeo Symbolic System and Hypergraph Neural Network](https://mlsanity.com/paper/2402.11461/) (2024)
- [GOLD: Geometry Problem Solver with Natural Language Description](https://mlsanity.com/paper/2405.00494/) (2024)
- [GeoDANO: Geometric VLM with Domain Agnostic Vision Encoder](https://mlsanity.com/paper/2502.11360/) (2025)
- [GeoSDF: Plane Geometry Diagram Synthesis via Signed Distance Field](https://mlsanity.com/paper/2506.13492/) (2025)
- [A Survey of Deep Learning for Geometry Problem Solving](https://mlsanity.com/paper/2507.11936/) (2025)
- [Hologram Reasoning for Solving Algebra Problems with Geometry Diagrams](https://mlsanity.com/paper/2408.10592/) (2024)
- [Hilbert-Geo: Solving Solid Geometric Problems by Neural-Symbolic Reasoning](https://mlsanity.com/paper/2605.16385/) (2026)
- [GAPS: Geometry-Aware Problem Solver](https://mlsanity.com/paper/2401.16287/) (2024)
- [Foundation of Intelligence: Review of Math Word Problems from Human Cognition Perspective](https://mlsanity.com/paper/2510.21999/) (2025)
- [Enhancing Geometric Perception in VLMs via Translator-Guided Reinforcement Learning](https://mlsanity.com/paper/2602.22703/) (2026)
- [Non-Parametric Structural Priors for Geometry Theorem Prediction](https://mlsanity.com/paper/2603.04852/) (2026)
- [Verifiable Geometry Problem Solving: Solver-Driven Autoformalization and Theorem Proposing](https://mlsanity.com/paper/2606.27926/) (2026)
- [Euclid-Omni : A Unified Neuro-Symbolic Framework for Plane Geometry](https://mlsanity.com/paper/2608.14585/) (2026)

## Abstract (arXiv)

AI has made significant progress in solving math problems, but geometry problems remain challenging due to their reliance on both text and diagrams. In the text description, symbolic characters such as "triangleABC" often serve as a bridge to connect the corresponding diagram. However, by simply tokenizing symbolic characters into individual letters (e.g., 'A', 'B' and 'C'), existing works fail to study them explicitly and thus lose the semantic relationship with the diagram. In this paper, we develop a symbolic character-aware model to fully explore the role of these characters in both text and diagram understanding and optimize the model under a multi-modal reasoning framework. In the text encoder, we propose merging individual symbolic characters to form one semantic unit along with geometric information from the corresponding diagram. For the diagram encoder, we pre-train it under a multi-label classification framework with the symbolic characters as labels. In addition, we enhance the geometry diagram understanding ability via a self-supervised learning method under the masked image modeling auxiliary task. By integrating the proposed model into a general encoder-decoder pipeline for solving geometry problems, we demonstrate its superiority on two benchmark datasets, including GeoQA and Geometry3K, with extensive experiments. Specifically, on GeoQA, the question-solving accuracy is increased from 60.0% to 64.1%, achieving a new state-of-the-art accuracy; on Geometry3K, we reduce the question average solving steps from 6.9 down to 6.0 with marginally higher solving accuracy.