Euclid-Omni : A Unified Neuro-Symbolic Framework for Plane Geometry
This work addresses the challenge of solving complex Euclidean geometry problems for AI reasoning systems, particularly those requiring a combination of diagram understanding, deduction, and algebraic computation, by providing a unified neuro-symbolic approach.
This paper introduces Euclid-Omni, a neuro-symbolic framework that combines a formal geometry system (Euclidea) with LLMs and VLMs to solve plane geometry problems. It achieves superior performance on calculation tasks using VLMs trained on synthetic data and is competitive with state-of-the-art systems on Olympiad-level proving problems with significantly less compute and training data.
Euclidean geometry is a compelling testbed for AI reasoning, as it demands the combination of intuitive diagram understanding, axiomatic deduction, and algebraic computation. Yet, existing approaches typically address only a subset of these abilities or struggle with competition-level problems. We introduce \textit{Euclid-Omni}, a unified neuro-symbolic framework that couples a formal geometry system with Large Language Models (LLMs) and Vision-Language Models (VLMs) to tackle both calculation- and proving-style problems, in formal and natural languages, up to Olympiad-level difficulty. At its core, we develop \textit{Euclidea}, a versatile symbolic geometry solver that automatically generates reasoning steps through deductive inference and algebraic computation. Building on this, we develop a data-generation pipeline that synthesizes symbolic problems and solutions, renders diagrams, and translates them into natural language, producing large-scale, diverse datasets for training LLMs and VLMs across a wide range of reasoning settings. Experiments show that VLMs trained on our synthetic data achieve superior performance on calculation tasks, and that LLMs combined with \textit{Euclidea} are competitive with state-of-the-art systems on Olympiad-level proving problems, despite using orders of magnitude less compute and training data. Code and scripts are publicly available at https://github.com/20171130/Euclid-Omni