# Fuse, Reason and Verify: Geometry Problem Solving with Parsed Clauses from Diagram

arXiv: https://arxiv.org/abs/2407.07327 · HTML: https://mlsanity.com/paper/2407.07327/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 28 of its 46 references that are in the map.
- First posted on arXiv 10 Jul 2024.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering](https://mlsanity.com/paper/2209.09513/) (2022)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Bottom-Up and Top-Down Attention for Image Captioning and Visual Question Answering](https://mlsanity.com/paper/1707.07998/) (2017)
- [ViLT: Vision-and-Language Transformer Without Convolution or Region Supervision](https://mlsanity.com/paper/2102.03334/) (2021)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [MathQA: Towards Interpretable Math Word Problem Solving with Operation-Based Formalisms](https://mlsanity.com/paper/1905.13319/) (2019)
- [LayoutLM: Pre-training of Text and Layout for Document Image Understanding](https://mlsanity.com/paper/1912.13318/) (2019)
- [Unifying Vision-and-Language Tasks via Text Generation](https://mlsanity.com/paper/2102.02779/) (2021)
- [TAT-QA: A Question Answering Benchmark on a Hybrid of Tabular and Textual Content in Finance](https://mlsanity.com/paper/2105.07624/) (2021)
- [Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning](https://mlsanity.com/paper/2105.04165/) (2021)
- [Dynamic Prompt Learning via Policy Gradient for Semi-structured Mathematical Reasoning](https://mlsanity.com/paper/2209.14610/) (2022)
- [Deep Modular Co-Attention Networks for Visual Question Answering](https://mlsanity.com/paper/1906.10770/) (2019)
- [IconQA: A New Benchmark for Abstract Diagram Understanding and Visual Language Reasoning](https://mlsanity.com/paper/2110.13214/) (2021)
- [GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning](https://mlsanity.com/paper/2105.14517/) (2021)
- [UniGeo: Unifying Geometry Logical Reasoning via Reformulating Mathematical Expression](https://mlsanity.com/paper/2212.02746/) (2022)
- [A Survey of Deep Learning for Mathematical Reasoning](https://mlsanity.com/paper/2212.10535/) (2022)
- [A Multi-Modal Neural Geometric Solver with Textual Clauses Parsed from Diagram](https://mlsanity.com/paper/2302.11097/) (2023)
- [Complex Knowledge Base Question Answering: A Survey](https://mlsanity.com/paper/2108.06688/) (2021)
- [Document Collection Visual Question Answering](https://mlsanity.com/paper/2104.14336/) (2021)
- [Explicit Bias Discovery in Visual Question Answering Models](https://mlsanity.com/paper/1811.07789/) (2018)
- [Plane Geometry Diagram Parsing](https://mlsanity.com/paper/2205.09363/) (2022)
- [A Symbolic Character-Aware Model for Solving Geometry Problems](https://mlsanity.com/paper/2308.02823/) (2023)
- [Sequence to General Tree: Knowledge-Guided Geometry Word Problem Solving](https://mlsanity.com/paper/2106.00990/) (2021)
- [Generalizing Math Word Problem Solvers via Solution Diversification](https://mlsanity.com/paper/2212.00833/) (2022)

## Built on by (most cited first, 13 of 13)

- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [Enhancing the Geometric Problem-Solving Ability of Multimodal LLMs via Symbolic-Neural Integration](https://mlsanity.com/paper/2504.12773/) (2025)
- [HiddenGuard: Fine-Grained Safe Generation with Specialized Representation Router](https://mlsanity.com/paper/2410.02684/) (2024)
- [a1: Steep Test-time Scaling Law via Environment Augmented Generation](https://mlsanity.com/paper/2504.14597/) (2025)
- [Pi-GPS: Enhancing Geometry Problem Solving by Unleashing the Power of Diagrammatic Information](https://mlsanity.com/paper/2503.05543/) (2025)
- [Large Language Models as Computable Approximations to Solomonoff Induction](https://mlsanity.com/paper/2505.15784/) (2025)
- [A Survey of Deep Learning for Geometry Problem Solving](https://mlsanity.com/paper/2507.11936/) (2025)
- [Beyond Symbolic Solving: Multi Chain-of-Thought Voting for Geometric Reasoning in Large Language Models](https://mlsanity.com/paper/2604.00890/) (2026)
- [Mathematical Reasoning in Large Language Models: Benchmarks, Architectures, Evaluation, and Open Challenges](https://mlsanity.com/paper/2605.19723/) (2026)
- [Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interaction](https://mlsanity.com/paper/2605.20743/) (2026)
- [BiNSGPS: Geometry Problem Solving via Bidirectional Neuro-Symbolic Interaction](https://mlsanity.com/paper/2606.04648/) (2026)
- [Verifiable Geometry Problem Solving: Solver-Driven Autoformalization and Theorem Proposing](https://mlsanity.com/paper/2606.27926/) (2026)
- [Reactivating Test-Time Scaling for Plane Geometry Problem Solving](https://mlsanity.com/paper/2608.30156/) (2026)

## Abstract (arXiv)

Geometry problem solving (GPS) requires capacities of multi-modal understanding, multi-hop reasoning and theorem knowledge application. In this paper, we propose a neural-symbolic model for plane geometry problem solving (PGPS), named PGPSNet-v2, with three key steps: modal fusion, reasoning process and knowledge verification. In modal fusion, we leverage textual clauses to express fine-grained structural and semantic content of geometry diagram, and fuse diagram with textual problem efficiently through structural-semantic pre-training. For reasoning, we design an explicable solution program to describe the geometric reasoning process, and employ a self-limited decoder to generate solution program autoregressively. To reduce solution errors, a multi-level theorem verifier is proposed to eliminate solutions that do not match geometric principles, alleviating the hallucination of the neural model. We also construct a large-scale geometry problem dataset called PGPS9K, containing fine-grained annotations of textual clauses, solution program and involved knowledge tuples. Extensive experiments on datasets Geometry3K and PGPS9K show that our PGPSNet solver outperforms existing symbolic and neural solvers in GPS performance, while maintaining good explainability and reliability, and the solver components (fusion, reasoning, verification) are all justified effective.