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Symmetry
Volume 18
Issue 4
10.3390/sym18040592
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

A Hybrid Framework for Automated Geometric Problem-Solving by Integrating Formal Symbolic Systems and Deep Learning

by
Zhengyu Hu
1,
Xiaokai Zhang
1,
Cheng Qin
1,2,
Yang Li
1 and
Tuo Leng
1,2,*
1
School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China
2
School of Future Technology, Shanghai University, Shanghai 200444, China
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(4), 592; https://doi.org/10.3390/sym18040592
Submission received: 25 February 2026 / Revised: 24 March 2026 / Accepted: 27 March 2026 / Published: 30 March 2026
(This article belongs to the Section Computer)
Versions Notes

Abstract

Geometric problem-solving (GPS) has been a long-standing challenge in the fields of formal mathematics and artificial intelligence. To address the limitations of unidirectional approaches, we developed a neuro-symbolic system that integrates forward and backward reasoning. The neural component employs a gating-enhanced attention network to select candidate theorems, guiding the heuristic search and pruning irrelevant branches. The symbolic component is a bidirectional solver built on FormalGeo, which performs rigorous geometric relational reasoning and algebraic computation. The neural component predicts the theorems based on the current problem state, while the symbolic component applies these theorems and updates the problem state. These two parts interact iteratively until the problem is solved. The solving process is organized as a graph structure where facts and goals serve as nodes and theorems as edges, thereby generating a human-readable solution. The proposed neuro-symbolic system achieved an 89.63% problem-solving success rate (PSSR) on the FormalGeo7K dataset, surpassing the previous best result.
Keywords: symmetry in geometry; formal mathematics; geometric problem-solving; bidirectional reasoning symmetry in geometry; formal mathematics; geometric problem-solving; bidirectional reasoning

Share and Cite

MDPI and ACS Style

Hu, Z.; Zhang, X.; Qin, C.; Li, Y.; Leng, T. A Hybrid Framework for Automated Geometric Problem-Solving by Integrating Formal Symbolic Systems and Deep Learning. Symmetry 2026, 18, 592. https://doi.org/10.3390/sym18040592

AMA Style

Hu Z, Zhang X, Qin C, Li Y, Leng T. A Hybrid Framework for Automated Geometric Problem-Solving by Integrating Formal Symbolic Systems and Deep Learning. Symmetry. 2026; 18(4):592. https://doi.org/10.3390/sym18040592

Chicago/Turabian Style

Hu, Zhengyu, Xiaokai Zhang, Cheng Qin, Yang Li, and Tuo Leng. 2026. "A Hybrid Framework for Automated Geometric Problem-Solving by Integrating Formal Symbolic Systems and Deep Learning" Symmetry 18, no. 4: 592. https://doi.org/10.3390/sym18040592

APA Style

Hu, Z., Zhang, X., Qin, C., Li, Y., & Leng, T. (2026). A Hybrid Framework for Automated Geometric Problem-Solving by Integrating Formal Symbolic Systems and Deep Learning. Symmetry, 18(4), 592. https://doi.org/10.3390/sym18040592

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