FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving

Xiaokai Zhang; Na Zhu; Yiming He; Jia Zou; Qike Huang; Xiaoxiao Jin; Yanjun Guo; Chenyang Mao; Yang Li; Zhe Zhu; Dengfeng Yue; Fangzhen Zhu; Yifan Wang; Yiwen Huang; Runan Wang; Cheng Qin; Zhenbing Zeng; Shaorong Xie; Xiangfeng Luo; Tuo Leng

arXiv · 2310.18021

FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving

View original source ↗Read source document ↗Export metadata ↓

Abstract

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.

Explore related subjects

cs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Xiaokai Zhang, Na Zhu, Yiming He, Jia Zou, Qike Huang, Xiaoxiao Jin, Yanjun Guo, Chenyang Mao, Yang Li, Zhe Zhu, Dengfeng Yue, Fangzhen Zhu, Yifan Wang, Yiwen Huang, Runan Wang, Cheng Qin, Zhenbing Zeng, Shaorong Xie, Xiangfeng Luo, Tuo Leng. 2024-02-15. FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving. https://arxiv.org/abs/2310.18021

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

InterPol: De-anonymizing LM Arena via Interpolated Preference Learning

Strict anonymity of model responses is a key for the reliability of voting-based leaderboards, such as LM Arena. While prior studies have attempted to compromise this assumption using simple statistical features like TF-IDF or bag-ofwords, these methods often lack the discriminative power to distinguish between stylistically similar or within-family models. To overcome these limitations and expose the severity of vulnerability, we introduce INTERPOL, a model-driven identification framework that learns to distinguish target models from others using interpolated preference data. Specifically, INTERPOL captures deep stylistic patterns that superficial statistical features miss by synthesizing hard negative samples through model interpolation and employing an adaptive curriculum learning strategy. Extensive experiments demonstrate that INTERPOL significantly outperforms existing baselines in identification accuracy. Furthermore, we quantify the real-world threat of our findings through ranking manipulation simulations on Arena battle data.

cs.AI↗

arXiv2026

Preregistered Belief Revision Contracts

Deliberative multi-agent systems allow agents to exchange messages and revise beliefs over time. While this interaction is meant to improve performance, it can also create dangerous conformity effects: agreement, confidence, prestige, or majority size may be treated as if they were evidence, producing high-confidence convergence to false conclusions. To address this, we introduce PBRC (Preregistered Belief Revision Contracts), a protocol-level mechanism that strictly separates open communication from admissible epistemic change. A PBRC contract publicly fixes first-order evidence triggers, admissible revision operators, a priority rule, and a fallback policy. A non-fallback step is accepted only when it cites a preregistered trigger and provides a nonempty witness set of externally validated evidence tokens. This ensures that every substantive belief change is both enforceable by a router and auditable after the fact. In this paper, (a) we prove that under evidential contracts with conservative fallback, social-only rounds cannot increase confidence and cannot generate purely conformity-driven wrong-but-sure cascades. (b) We show that auditable trigger protocols admit evidential PBRC normal forms that preserve belief trajectories and canonicalized audit traces. (c) We demonstrate that sound enforcement yields epistemic accountability: any change of top hypothesis is attributable to a concrete validated witness set. For token-invariant contracts, (d) we prove that enforced trajectories depend only on token-exposure traces; under flooding dissemination, these traces are characterized exactly by truncated reachability, giving tight diameter bounds for universal evidence closure. Finally, we introduce a companion contractual dynamic doxastic logic to specify trace invariants, and provide simulations illustrating cascade suppression, auditability, and robustness-liveness trade-offs.

cs.AI↗

arXiv2026

Anon: Extrapolating Adaptivity Beyond SGD and Adam

Adaptive optimizers such as Adam and non-adaptive methods like SGD exhibit distinct generalization capabilities across different architectures. Prior tunable optimizers attempt to bridge this gap by strictly interpolating between SGD and Adam, effectively confining adaptivity within the 0-to-1 bound. However, this restricted interpolation is fundamentally insufficient: we reveal that optimal adaptivity often requires extrapolation, such as negative adaptivity for classical CNNs and adaptivity of at least one ($γ\geq 1$) for Transformers. Extrapolating adaptivity theoretically violates the strict non-decreasing pre-conditioner assumption, often leading to divergence in existing methods. To break this barrier, we propose Anon, an optimizer that achieves fully continuous adaptivity extrapolation across the entire real-number spectrum. To guarantee provable stability in these out-of-bound regimes, we introduce Incremental Delay Update (IDU), a novel mechanism that bypasses hard max-tracking strategies. We theoretically establish Anon's convergence in both convex and non-convex settings. Empirically, by exploring previously unreachable adaptivity landscapes, Anon demonstrates highly competitive and scalable performance among state-of-the-art element-wise optimizers on representative image classification, diffusion, and large language modeling tasks.

cs.AI↗