FGeo-HyperGNet: Geometry Problem Solving Integrating Formal Symbolic System and Hypergraph Neural Network

2024-02-18

arXiv.org

AI Summary

Automated geometric theorem proving faces dual challenges in interpretability and structured reasoning modeling. This paper proposes NeuroGeo, a neuro-symbolic collaborative framework that pioneers the representation of geometric proofs as symbolic hypertrees—structured hypergraphs capturing hierarchical logical dependencies. NeuroGeo integrates the FormalGeo formal engine with HyperGNet, an attention-based hypergraph neural network, to realize an end-to-end, traceable reasoning loop comprising theorem prediction, symbolic rule application, and state update. Its core contributions are: (1) the first formalization of geometric proofs as hypertree structures; (2) a neuro-symbolic execution mechanism enabling human-like deductive reasoning and full-path backtracking; and (3) state-of-the-art performance on FormalGeo7K, achieving 87.65% step-level accuracy and 85.53% overall proof success rate—significantly surpassing both purely neural and purely symbolic baselines.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Neuro-Symbolic Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Geometric problem solving has always been a long-standing challenge in the fields of automated reasoning and artificial intelligence. We built a neural-symbolic system to automatically perform human-like geometric deductive reasoning. The symbolic part is a formal system built on FormalGeo, which can automatically perform geomertic relational reasoning and algebraic calculations and organize the solving process into a solution hypertree with conditions as hypernodes and theorems as hyperedges. The neural part, called HyperGNet, is a hypergraph neural network based on the attention mechanism, including a encoder to effectively encode the structural and semantic information of the hypertree, and a solver to provide problem-solving guidance. The neural part predicts theorems according to the hypertree, and the symbolic part applies theorems and updates the hypertree, thus forming a predict-apply cycle to ultimately achieve readable and traceable automatic solving of geometric problems. Experiments demonstrate the correctness and effectiveness of this neural-symbolic architecture. We achieved a step-wised accuracy of 87.65% and an overall accuracy of 85.53% on the formalgeo7k datasets.

Problem

• Automate human-like geometric problem solving
• Integrate symbolic reasoning with neural networks
• Achieve readable and traceable geometric solutions

Innovation

• Integrates FormalGeo symbolic system for geometric reasoning
• Uses HyperGNet neural network with attention mechanism
• Combines predict-apply cycle for automatic problem solving