#DeepMind AlphaGeometry

[jcmarchi]

AlphaGeometry2: The AI That Outperforms Human Olympiad Champions in Geometry

New Post has been published on https://thedigitalinsider.com/alphageometry2-the-ai-that-outperforms-human-olympiad-champions-in-geometry/

AlphaGeometry2: The AI That Outperforms Human Olympiad Champions in Geometry

Artificial intelligence has long been trying to mimic human-like logical reasoning. While it has made massive progress in pattern recognition, abstract reasoning and symbolic deduction have remained tough challenges for AI. This limitation becomes especially evident when AI is being used for mathematical problem-solving, a discipline that has long been a testament to human cognitive abilities such as logical thinking, creativity, and deep understanding. Unlike other branches of mathematics that rely on formulas and algebraic manipulations, geometry is different. It requires not only structured, step-by-step reasoning but also the ability to recognize hidden relationships and the skill to construct extra elements for solving problems.

For a long time, these abilities were thought to be unique to humans. However, Google DeepMind has been working on developing AI that can solve these complex reasoning tasks. Last year, they introduced AlphaGeometry, an AI system that combines the predictive power of neural networks with the structured logic of symbolic reasoning to tackle complex geometry problems. This system made a significant impact by solving 54% of International Mathematical Olympiad (IMO) geometry problems to achieve performance at par with silver medalists. Recently, they took it even further with AlphaGeometry2, which achieved an incredible 84% solve rate to outperform an average IMO gold medalist.

In this article, we will explore key innovations that helped AlphaGeometry2 achieve this level of performance and what this development means for the future of AI in solving complex reasoning problems. But before diving into what makes AlphaGeometry2 special, it’s essential first to understand what AlphaGeometry is and how it works.

AlphaGeometry: Pioneering AI in Geometry Problem-Solving

AlphaGeometry is an AI system designed to solve complex geometry problems at the level of the IMO. It is basically a neuro-symbolic system that combines a neural language model with a symbolic deduction engine. The neural language model helps the system predict new geometric constructs, while symbolic AI applies formal logic to generate proofs. This setup allows AlphaGeometry to think more like a human by combining the pattern recognition capabilities of neural networks, which replicate intuitive human thinking, with the structured reasoning of formal logic, which mimics human deductive reasoning abilities. One of the key innovations in AlphaGeometry was how it generated training data. Instead of relying on human demonstrations, it created one billion random geometric diagrams and systematically derived relationships between points and lines. This process created a massive dataset of 100 million unique examples, helping the neural model predict functional geometric constructs and guiding the symbolic engine toward accurate solutions. This hybrid approach enabled AlphaGeometry to solve 25 out of 30 Olympiad geometry problems within standard competition time, closely matching the performance of top human competitors.

How AlphaGeometry2 Achieves Improved Performance

While AlphaGeometry was a breakthrough in AI-driven mathematical reasoning, it had certain limitations. It struggled with solving complex problems, lacked efficiency in handling a wide range of geometry challenges, and had limitations in problem coverage. To overcome these hurdles, AlphaGeometry2 introduces a series of significant improvements:

Expanding AI’s Ability to Understand More Complex Geometry Problems

One of the most significant improvements in AlphaGeometry2 is its ability to work with a broader range of geometry problems. The former AlphaGeometry struggled with issues that involved linear equations of angles, ratios, and distances, as well as those that required reasoning about moving points, lines, and circles. AlphaGeometry2 overcomes these limitations by introducing a more advanced language model that allows it to describe and analyze these complex problems. As a result, it can now tackle 88% of all IMO geometry problems from the last two decades, a significant increase from the previous 66%.

A Faster and More Efficient Problem-Solving Engine

Another key reason AlphaGeometry2 performs so well is its improved symbolic engine. This engine, which serves as the logical core of this system, has been enhanced in several ways. First, it is improved to work with a more refined set of problem-solving rules which makes it more effective and faster. Second, it can now recognize when different geometric constructs represent the same point in a problem, allowing it to reason more flexibly. Finally, the engine has been rewritten in C++ rather than Python, making it over 300 times faster than before. This speed boost allows AlphaGeometry2 to generate solutions more quickly and efficiently.

Training the AI with More Complex and Varied Geometry Problems

The effectiveness of AlphaGeometry2’s neural model comes from its extensive training in synthetic geometry problems. AlphaGeometry initially generated one billion random geometric diagrams to create 100 million unique training examples. AlphaGeometry2 takes this a step further by generating more extensive and more complex diagrams that include intricate geometric relationships. Additionally, it now incorporates problems that require the introduction of auxiliary constructions—newly defined points or lines that help solve a problem, allowing it to predict and generate more sophisticated solutions

Finding the Best Path to a Solution with Smarter Search Strategies

A key innovation of AlphaGeometry2 is its new search approach, called the Shared Knowledge Ensemble of Search Trees (SKEST). Unlike its predecessor, which relied on a basic search method, AlphaGeometry2 runs multiple searches in parallel, with each search learning from the others. This technique allows it to explore a broader range of possible solutions and significantly improves the AI’s ability to solve complex problems in a shorter amount of time.

Learning from a More Advanced Language Model

Another key factor behind AlphaGeometry2’s success is its adoption of Google’s Gemini model, a state-of-the-art AI model that has been trained on an even more extensive and more diverse set of mathematical problems. This new language model improves AlphaGeometry2’s ability to generate step-by-step solutions due to its improved chain-of-thought reasoning. Now, AlphaGeometry2 can approach the problems in a more structured way. By fine-tuning its predictions and learning from different types of problems, the system can now solve a much more significant percentage of Olympiad-level geometry questions.

Achieving Results That Surpass Human Olympiad Champions

Thanks to the above advancements, AlphaGeometry2 solves 42 out of 50 IMO geometry problems from 2000-2024, achieving an 84% success rate. These results surpass the performance of an average IMO gold medalist and set a new standard for AI-driven mathematical reasoning. Beyond its impressive performance, AlphaGeometry2 is also making strides in automating theorem proving, bringing us closer to AI systems that can not only solve geometry problems but also explain their reasoning in a way that humans can understand

The Future of AI in Mathematical Reasoning

The progress from AlphaGeometry to AlphaGeometry2 shows how AI is getting better at handling complex mathematical problems that require deep thinking, logic, and strategy. It also signifies that AI is no longer just about recognizing patterns—it can reason, make connections, and solve problems in ways that feel more like human-like logical reasoning.

AlphaGeometry2 also shows us what AI might be capable of in the future. Instead of just following instructions, AI could start exploring new mathematical ideas on its own and even help with scientific research. By combining neural networks with logical reasoning, AI might not just be a tool that can automate simple tasks but a qualified partner that helps expand human knowledge in fields that rely on critical thinking.

Could we be entering an era where AI proves theorems and makes new discoveries in physics, engineering, and biology? As AI shifts from brute-force calculations to more thoughtful problem-solving, we might be on the verge of a future where humans and AI work together to uncover ideas we never thought possible.

[ibmathsresources]

AI Masters Olympiad Geometry

AI Masters Olympiad Geometry The team behind Google’s Deep Mind have just released details of a new AI system:  AlphaGeometry This has been specifically trained to solve classical geometry problems – and already is now at the level of a Gold Medalist at the International Olympiad (considering only geometry problems).  This is an incredible achievement – as in order to solve classical geometry…

[appslookup]

AlphaGeometry: DeepMind’s AI Masters Geometry Problems at Olympiad Levels

#AlphaGeometry is indeed an impressive feat in the advancement of AI! It represents a significant leap in the capability of AI to tackle complex geometry problems at the level of Olympiad-level competitions like the International Mathematical Olympiad (IMO). By AppsLookup

[er-10-media]

DeepMind представила «золотую» ИИ-модель AlphaGeometry2

New Post has been published on https://er10.kz/read/it-novosti/deepmind-predstavila-zolotuju-ii-model-alphageometry2/

DeepMind представила «золотую» ИИ-модель AlphaGeometry2

Компания DeepMind представила ИИ-модель AlphaGeometry2, которая, как утверждается, справляется с геометрическими задачами лучше, чем золотые медалисты Международной математической олимпиады (IMO).

AlphaGeometry2 является улучшенной версией системы AlphaGeometry, которую DeepMind выпустила в январе прошлого года. Разработчики утверждают, что их ИИ способен решить 84% всех задач по геометрии за последние 25 лет в рамках Международной математической олимпиады.

В лаборатории DeepMind считают, что ключ к созданию эффективного ИИ может лежать в открытии новых способов решения сложных задач по геометрии.

Доказательство математических теорем требует как рассуждений, так и умения выбирать из ряда возможных шагов. Если DeepMind не ошибется, эти навыки решения проблем могут стать полезным компонентом будущих моделей ИИ общего назначения.

Так, этим летом DeepMind продемонстрировала систему, которая в сочетании с AlphaGeometry2 и моделью ИИ для формальных математических рассуждений AlphaProof решила четыре из шести заданий из IMO 2024 года. Помимо задач по геометрии, подобные подходы могут быть распространены и на другие области математики и естественных наук – например в сложных инженерных расчетах. Конечно, есть и ограничения. Техническая особенность не позволяет AlphaGeometry2 решать задачи с переменным количеством точек, нелинейные уравнения и неравенства.

[moko1590m]

2024年12月25日 09時45分 OpenAIのo3モデルが数学の超難問データセット「FrontierMath」で25.2％のスコアを獲得した衝撃を数学者が語る インペリアル・カレッジ・ロンドンで純粋数学の教授を務める数学者のケビン・バザード氏が、OpenAIのo3モデルがFrontierMath問題データセットで25.2％のスコアを獲得したことについて解説するブログ記事を投稿しました。 Can AI do maths yet? Thoughts from a mathematician. | Xena https://xenaproject.wordpress.com/2024/12/22/can-ai-do-maths-yet-thoughts-from-a-mathematician/ 2024年12月20日に、OpenAIは新たな推論モデル「o3」シリーズを発表しました。OpenAIはo3モデルについて「これまで開発した中で最も高度な推論能力を持つ」と述べ、2025年の公開に向けて準備を進めています。 OpenAIが推論能力を大幅に強化した「o3」シリーズを発表、 推論の中でOpenAIの安全ポリシーについて「再考」する仕組みを導入 - GIGAZINE o3モデルはFrontierMathという問題データセットで25.2％のスコアを獲得したことが明らかになっています。FrontierMathは数百個の難しい数学の問題のデータセットで、問題そのものだけでなくデータセット全体の問題数なども秘密であり、AIが事前に問題をトレーニングしないよう注意深く設計されています。 FrontierMathの全ての問題は計算問題で、「証明せよ」などの形式の問題は含まれていないとのこと。公開されている5つのサンプル問題では答えが全て正の整数となっており、その他の問題についても「自動的に検証できる明確で計算可能な答えがある」とされています。問題の難易度はかなり高く、数学者のバザード氏でもサンプル問題のうち解けたのは2問だけで、別の1問については「取り組めば解けるかも」と思えたものの、残りの2問は「解けない」と思ったそうです。 FrontierMathの論文にはフィールズ賞受賞者などの著名な数学者による問題の難易度評価が記載されていますが、「極めて難しい」とコメントした上で、それぞれの問題の分野の専門家でなくては解答できないことを示唆しています。実際、バザード氏が解けた2問はバザード氏の専門分野の問題でした。 なお、実際の数学者は計算ではなく証明や証明のためのアイデアを考え出すことにほとんどの時間を使用するため、「計算で数値的な答えを出すことは独自の証明を思いつくことと完全に異なる」として数学力の計測に不適だとする数学者も存在します。しかし証明を採点するのはコストがかかるため、モデルが提出した答えが正答と一致するかどうかを確認するだけで採点できる計算問題が採用されているとのこと。 そうしたFrontierMathのテストに対し、OpenAIのo3モデルが25.2％ものスコアを獲得したことについてバザード氏は「衝撃を受けた」と述べました。 これまでAIは優秀な高校生が解くような「数学オリンピック形式」を得意としていることが明らかになっており、バザード氏は「多くの典型問題が出題される」という点で似ている大学の学部生レベルの数学の問題をAIが解けるようになることは疑っていませんでした。しかし、典型問題のレベルを超えて博士課程の初期レベルの問題に対し革新的なアイデアで対応するレベルの数学力をAIが獲得していることに対し、バザード氏は「かなり大きな飛躍が起きたように見える」とコメントしています。 ただし、FrontierMathを組み上げたEpoch AIのエリオット・グレイザー氏はデータセット内の問題の25％は数学オリンピック形式だと発言しています。公開されている5つのサンプル問題はいずれも数学オリンピックの形式とは全く異なるため、バザード氏はo3モデルがFrontierMathで25.2％のスコアを獲得したと聞いて非常に興奮したものの、25％が数学オリンピック形式と知って興奮は収まったとのこと。「今後、AIがFrontierMathで50％のスコアを獲得することを楽しみにしている」とバザード氏はコメントを残しました。 現在、AIの進歩は急速に進んでいますが、まだまだ道のりは遠く、やるべき事は山ほどあります。バザード氏はAIが「この定理を正しく証明し、その証明がなぜ機能するのかを人間が理解できる方法で説明せよ」というレベルの問題に対応できるほどの数学力を身につけることを期待しているとブログを締めくくりました。 この記事のタイトルとURLをコピーする ・関連記事 Microsoftが軽量なのにGPT-4oを圧倒的に上回る数学性��を発揮するAIモデル「Phi-4」をリリース - GIGAZINE AppleのAI研究者らが「今のAI言語モデルは算数の文章題への推論能力が小学生未満」と研究結果を発表 - GIGAZINE 数学オリンピックの問題で銀メダルレベルのスコアを残すAIを開発したとGoogle DeepMindが発表 - GIGAZINE OpenAIが「GPT-4o」を発表、人間と同等の速さでテキスト・音声・カメラ入力を処理可能で「周囲を見渡して状況判断」「数学の解き方を教える」「AI同士で会話して作曲」など多様な操作を実行可能 - GIGAZINE 数学を解ける言語モデル「Qwen2-Math」が登場、GPT-4o超えの数学性能 - GIGAZINE ・関連コンテンツ GPT-4oがAIベンチマークのARC-AGIで50％のスコアに到達、これまでの最高記録である34％を大幅に更新 Google DeepMindが数学オリンピックレベルの幾何学問題を解けるAI「AlphaGeometry」を発表、人間の金メダリストに近い性能を発揮 日本語対応マルチモーダルAI「Claude 3」はわずか2つのプロンプトで量子アルゴリズムを設計可能 DeepMindが開発したAIの「AlphaCode」がプログラミングコンテストで「平均」評価を獲得 Metaの大規模言語モデル「LLaMA」がChatGPTを再現できる可能性があるとさまざまなチャットAI用言語モデルのベンチマーク測定で判明 画像生成AI・Stable Diffusionのエンコーダーに見つかった致命的な欠陥とは？ OpenAIが4度目のブレイクスルーとなる数学ができるAI「Q*(キュースター)」で汎用人工知能開発の飛躍を目指す、アルトマンCEO解任騒動の一因か OpenAIが複雑な推論能力をもつAIモデル「OpenAI o1」と「OpenAI o1-mini」を発表、プログラミングや数学で高い能力を発揮

OpenAIのo3モデルが数学の超難問データセット「FrontierMath」で25.2％のスコアを獲得した衝撃を数学者が語る - GIGAZINE

[govindhtech]

AlphaProof: Google AI Systems To Think Like Mathematicians

AlphaProof and AlphaGeometry 2

Google AI systems advance towards thinking by making strides in maths. One question was answered in minutes, according to a blog post by Google, but other questions took up to three days to answer longer than the competition’s time limit. Nevertheless, the scores are among the highest achieved by an Al system in the competition thus far.

Google, a division of Alphabet, showcased two artificial intelligence systems that showed improvements in generative Al development the ability to solve challenging mathematical problems.

The current breed of AI models has had difficulty with abstract arithmetic since it demands more reasoning power akin to human intellect. These models operate by statistically anticipating the following word.

The company’s Al division, DeepMind, released data demonstrating that its recently developed Al models, namely AlphaProof and AlphaGeometry 2, answered four of every six questions in the 2024 International Math Olympiad, a well-known tournament for high school students.

One question was answered in minutes, according to a blog post by Google, but other questions took up to three days to answer longer than the competition’s time limit. Nevertheless, the scores are among the highest achieved by an Al system in the competition thus far.

AlphaZero

The business said that AlphaZero, another Al system that has previously defeated humans at board games like chess and go, and a version of Gemini, the language model underlying its chatbot of the same name, were combined to produce AlphaProof, a reasoning-focused system. Only five out of the more than 600 human competitors were able to answer the most challenging question, which was one of the three questions that AlphaProof answered correctly.

AlphaGeometry 2

AlphaGeometry 2 solved another math puzzle. It was previously reported in July that OpenAI, supported by Microsoft, was working on reasoning technology under the code name “Strawberry.” As Reuters first revealed, the project, originally known as Q, was regarded as such a breakthrough that several staff researchers warned OpenAI’s board of directors in a letter they wrote in November, stating that it could endanger humankind.

The top choice for document editing and proofreading is AlphaProof. The demand for accurate and efficient services is growing in the digital age. It stands out as a leading option, offering excellent services to guarantee your documents are flawless. In order to show why AlphaProof is unique in the industry, this article explores its features, advantages, and user experiences.

How does AlphaProof work?

AlphaProof a feature-rich online tool, handles all editing and proofreading needs. It offers specialized services to increase the quality and readability of your documents for professionals, students, and company owners. AlphaProof publishes technical documentation, corporate reports, creative writing, and academic essays.

Essential Elements of AlphaProof

Expert Proofreading

To fix typographical, punctuation, and grammar flaws in your documents, AlphaProof has a team of highly skilled proofreaders who carefully go over them. This guarantees that your text looks professional and is free of common mistakes.

Complex Editing

It provides sophisticated editing services in addition to basic proofreading. This entails streamlining the sentence structure, boosting readability overall, and strengthening coherence and flow. Better word selections and stylistic enhancements are also suggested by the editors.

Editors with specific expertise

AlphaProof recognizes that varying documents call for varying levels of competence. It boasts a diverse team of editors with skills in technical writing, business communication, academic writing, and creative writing. This guarantees that an individual possessing pertinent expertise and experience will evaluate your material.

Quick Resolution

Quick turnaround times are provided by AlphaProof to help you meet deadlines. You can choose 24-hour express service to ensure your document is available when you need it.

Easy-to-use interface

The AlphaProof platform boasts an intuitive interface that facilitates the uploading of documents, selection of services, and tracking of order status. From beginning to end, the procedure is simplified to offer a hassle-free experience.

Secrecy and Protection

The security and privacy of your papers are very important to it. The platform uses cutting-edge encryption technology to safeguard your data, and every file is handled with the highest care.

The Advantages of AlphaProof Use

Better Document Quality

The quality of your documents can be greatly improved by utilising it’s services. This can result in more professionalism in corporate communication, higher grades, and a more positive impression on your readers.

Reduce Effort and Time

Editing and proofreading can be laborious processes. With AlphaProof, you can focus on your primary responsibilities while professionals optimize your papers, saving you time and effort.

Customized Offerings

To address the unique requirements of various document formats, It offers customized services. AlphaProof may provide you with comprehensive editing for a research paper or expeditious proofreading for an email.

Knowledgeable Perspectives

The editor’s comments and recommendations on it can give you important information about your writing style and areas that need work. With time, this can assist you in improving as a writer.

A Boost in Self-Assurance

You may feel more confident in the calibre of your work if you know it has been expertly edited and proofread. For high-stakes papers like published articles, commercial proposals, and theses from academic institutions, this is especially crucial.

Customer Experiences

Scholars and Students

AlphaProof has proven to be a useful resource for numerous academics and students. A postgraduate student said, “AlphaProof enabled me to refine my thesis to the ideal level.” The final draft was error-free, and the editors’ suggestions were wise.”

Composers and Novelists

The specialized editing services provided by AlphaProof are valued by authors and creative writers. A budding writer said, “it’s editors understood my voice and style, providing feedback that improved my manuscript without altering my unique voice.”

In conclusion

With a variety of features and advantages to meet a wide range of demands, AlphaProof stands out as a top option for document editing and proofreading. It guarantees that your documents are flawless, saving you time and improving the calibre of your work. It does this through its skilled staff, quick return times, and intuitive interface.

Read more on govindhtech.com

[ai-news]

In a groundbreaking achievement, AI systems developed by Google DeepMind have attained a silver medal-level score in the 2024 International Mathematical Olympiad (IMO), a prestigious global competition for young mathematicians. The AI models, named #AI #ML #Automation

[iwan1979]

Google DeepMind: AI achieves silver-medal standard solving International Mathematical Olympiad problems

[jhavelikes]

s the name indicates, neuro-symbolic models combine neural networks, such as LLMs, with smaller, easier-to-interpret symbolic models to adapt LLMs to specific domains. One of the recent breakthroughs in neuro-symbolic models came from Google DeepMind with their work on AlphaGeometry, a model that was able to solve geometry problems at the level of math Olympiad gold medalists. AlphaGeometry combines a geometry symbolic model with an LLM used mostly for exploring possible solutions to a given problem. Can we expand the AlphaGeometry approach to mainstream use cases? Last week, a new AI startup called Symbolica emerged from stealth mode with $33 million in new funding from iconic VCs such as Khosla Ventures. Symbolica uses mathematical techniques such as category theory to build simpler models that are easier to manage and interpret. In their press release, Symbolica cited AlphaGeometry as one of the inspirations for their work.

(16) Neuro-Symbolic Models are Making a Comeback

[neuronetwork-ua]

🫢 АІ Deepmind розв'язує складні геометричні задачі AlphaGeometry - це нова нейромережа з відкритим вихідним кодом, яка розв'язуєзадачі з геометрії на рівні золотих медалістів. Вона розв'язала 25 олімпійськихзадачзгеометрії за стандартний час, випере��ивши 10 аналогічних систем у минулому.

[yes-deepbelievercollector]

AlphaGeometry ultrapassa alunos geniais de Geometria

A Google DeepMind, uma divisão de pesquisa em inteligência artificial (IA) da Google, lançou uma nova IA chamada AlphaGeometry que tem surpreendido especialistas em todo o mundo. Esta IA, especificamente treinada para resolver problemas de geometria, superou os melhores alunos mundiais nas Olimpíadas da Matemática Internacional, uma competição que reúne os melhores estudantes secundários de […]

[jhave]

AlphaGeometry: An Olympiad-level AI system for geometry - Google DeepMind

[mxmollusca]

By June Kim

January 17, 2024

Google DeepMind’s new AI system can solve complex geometry problems

Its performance matches the smartest high school mathematicians and is much stronger than the previous state-of-the-art system.

An abstract collage showcasing the concept of geometry.

Google DeepMind has created an AI system that can solve complex geometry problems. It’s a significant step towards machines with more human-like reasoning skills, experts say.

Geometry, and mathematics more broadly, have challenged AI researchers for some time. Compared with text-based AI models, there is significantly less training data for mathematics because it is symbol driven and domain specific, says Thang Wang, a coauthor of the research, which is published in Nature today.

Solving mathematics problems requires logical reasoning, something that most current AI models aren’t great at. This demand for reasoning is why mathematics serves as an important benchmark to gauge progress in AI intelligence, says Wang.

DeepMind’s program, named AlphaGeometry, combines a language model with a type of AI called a symbolic engine, which uses symbols and logical rules to make deductions. Language models excel at recognizing patterns and predicting subsequent steps in a process. However, their reasoning lacks the rigor required for mathematical problem-solving. The symbolic engine, on the other hand, is based purely on formal logic and strict rules, which allows it to guide the language model toward rational decisions.

These two approaches, responsible for creative thinking and logical reasoning respectively, work together to solve difficult mathematical problems. This closely mimics how humans work through geometry problems, combining their existing understanding with explorative experimentation.

DeepMind says it tested AlphaGeometry on 30 geometry problems at the same level of difficulty found at the International Mathematical Olympiad, a competition for top high school mathematics students. It completed 25 within the time limit. The previous state-of-the-art system, developed by the Chinese mathematician Wen-Tsün Wu in 1978, completed only 10.

“This is a really impressive result,” says Floris van Doorn, a mathematics professor at the University of Bonn, who was not involved in the research. “I expected this to still be multiple years away.”

Google DeepMind used a large language model to solve an unsolved math problem

They had to throw away most of what it produced but there was gold among the garbage.

DeepMind says this system demonstrates AI’s ability to reason and discover new mathematical knowledge.

“This is another example that reinforces how AI can help us advance science and better understand the underlying processes that determine how the world works,” said Quoc V. Le, a scientist at Google DeepMind and one of the authors of the research, at a press conference.

When presented with a geometry problem, AlphaGeometry first attempts to generate a proof using its symbolic engine, driven by logic. If it cannot do so using the symbolic engine alone, the language model adds a new point or line to the diagram. This opens up additional possibilities for the symbolic engine to continue searching for a proof. This cycle continues, with the language model adding helpful elements and the symbolic engine testing new proof strategies, until a verifiable solution is found.

To train AlphaGeometry's language model, the researchers had to create their own training data to compensate for the scarcity of existing geometric data. They generated nearly half a billion random geometric diagrams and fed them to the symbolic engine. This engine analyzed each diagram and produced statements about their properties. These statements were organized into 100 million synthetic proofs to train the language model.

Roman Yampolskiy, an associate professor of computer science and engineering at the University of Louisville who was not involved in the research, says that AlphaGeometry’s ability shows a significant advancement toward more “sophisticated, human-like problem-solving skills in machines.”

“Beyond mathematics, its implications span across fields that rely on geometric problem-solving, such as computer vision, architecture, and even theoretical physics,” said Yampoliskiy in an email.

However, there is room for improvement. While AlphaGeometry can solve problems found in “elementary” mathematics, it remains unable to grapple with the sorts of advanced, abstract problems taught at university.

“Mathematicians would be really interested if AI can solve problems that are posed in research mathematics, perhaps by having new mathematical insights,” said van Doorn.

Wang says the goal is to apply a similar approach to broader math fields. “Geometry is just an example for us to demonstrate that we are on the verge of AI being able to do deep reasoning,” he says.

[jcmarchi]

AI at the International Mathematical Olympiad: How AlphaProof and AlphaGeometry 2 Achieved Silver-Medal Standard

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AI at the International Mathematical Olympiad: How AlphaProof and AlphaGeometry 2 Achieved Silver-Medal Standard

Mathematical reasoning is a vital aspect of human cognitive abilities, driving progress in scientific discoveries and technological developments. As we strive to develop artificial general intelligence that matches human cognition, equipping AI with advanced mathematical reasoning capabilities is essential. While current AI systems can handle basic math problems, they struggle with the complex reasoning needed for advanced mathematical disciplines like algebra and geometry. However, this might be changing, as Google DeepMind has made significant strides in advancing an AI system’s mathematical reasoning capabilities. This breakthrough is made at the International Mathematical Olympiad (IMO) 2024. Established in 1959, the IMO is the oldest and most prestigious mathematics competition, challenging high school students worldwide with problems in algebra, combinatorics, geometry, and number theory. Each year, teams of young mathematicians compete to solve six very challenging problems. This year, Google DeepMind introduced two AI systems: AlphaProof, which focuses on formal mathematical reasoning, and AlphaGeometry 2, which specializes in solving geometric problems. These AI systems managed to solve four out of six problems, performing at the level of a silver medalist. In this article, we will explore how these systems work to solve mathematical problems.

AlphaProof: Combining AI and Formal Language for Mathematical Theorem Proving

AlphaProof is an AI system designed to prove mathematical statements using the formal language Lean. It integrates Gemini, a pre-trained language model, with AlphaZero, a reinforcement learning algorithm renowned for mastering chess, shogi, and Go.

The Gemini model translates natural language problem statements into formal ones, creating a library of problems with varying difficulty levels. This serves two purposes: converting imprecise natural language into precise formal language for verifying mathematical proofs and using predictive abilities of Gemini to generate a list of possible solutions with formal language precision.

When AlphaProof encounters a problem, it generates potential solutions and searches for proof steps in Lean to verify or disprove them. This is essentially a neuro-symbolic approach, where the neural network, Gemini, translates natural language instructions into the symbolic formal language Lean to prove or disprove the statement. Similar to AlphaZero’s self-play mechanism, where the system learns by playing games against itself, AlphaProof trains itself by attempting to prove mathematical statements. Each proof attempt refines AlphaProof’s language model, with successful proofs reinforcing the model’s capability to tackle more challenging problems.

For the International Mathematical Olympiad (IMO), AlphaProof was trained by proving or disproving millions of problems covering different difficulty levels and mathematical topics. This training continued during the competition, where AlphaProof refined its solutions until it found complete answers to the problems.

AlphaGeometry 2: Integrating LLMs and Symbolic AI for Solving Geometry Problems

AlphaGeometry 2 is the latest iteration of the AlphaGeometry series, designed to tackle geometric problems with enhanced precision and efficiency. Building on the foundation of its predecessor, AlphaGeometry 2 employs a neuro-symbolic approach that merges neural large language models (LLMs) with symbolic AI. This integration combines rule-based logic with the predictive ability of neural networks to identify auxiliary points, essential for solving geometry problems. The LLM in AlphaGeometry predicts new geometric constructs, while the symbolic AI applies formal logic to generate proofs.

When faced with a geometric problem, AlphaGeometry’s LLM evaluates numerous possibilities, predicting constructs crucial for problem-solving. These predictions serve as valuable clues, guiding the symbolic engine toward accurate deductions and advancing closer to a solution. This innovative approach enables AlphaGeometry to address complex geometric challenges that extend beyond conventional scenarios.

One key enhancement in AlphaGeometry 2 is the integration of the Gemini LLM. This model is trained from scratch on significantly more synthetic data than its predecessor. This extensive training equips it to handle more difficult geometry problems, including those involving object movements and equations of angles, ratios, or distances. Additionally, AlphaGeometry 2 features a symbolic engine that operates two orders of magnitude faster, enabling it to explore alternative solutions with unprecedented speed. These advancements make AlphaGeometry 2 a powerful tool for solving intricate geometric problems, setting a new standard in the field.

AlphaProof and AlphaGeometry 2 at IMO

This year at the International Mathematical Olympiad (IMO), participants were tested with six diverse problems: two in algebra, one in number theory, one in geometry, and two in combinatorics. Google researchers translated these problems into formal mathematical language for AlphaProof and AlphaGeometry 2. AlphaProof tackled two algebra problems and one number theory problem, including the most difficult problem of the competition, solved by only five human contestants this year. Meanwhile, AlphaGeometry 2 successfully solved the geometry problem, though it did not crack the two combinatorics challenges

Each problem at the IMO is worth seven points, adding up to a maximum of 42. AlphaProof and AlphaGeometry 2 earned 28 points, achieving perfect scores on the problems they solved. This placed them at the high end of the silver-medal category. The gold-medal threshold this year was 29 points, reached by 58 of the 609 contestants.

Next Leap: Natural Language for Math Challenges

AlphaProof and AlphaGeometry 2 have showcased impressive advancements in AI’s mathematical problem-solving abilities. However, these systems still rely on human experts to translate mathematical problems into formal language for processing. Additionally, it is unclear how these specialized mathematical skills might be incorporated into other AI systems, such as for exploring hypotheses, testing innovative solutions to longstanding problems, and efficiently managing time-consuming aspects of proofs.

To overcome these limitations, Google researchers are developing a natural language reasoning system based on Gemini and their latest research. This new system aims to advance problem-solving capabilities without requiring formal language translation and is designed to integrate smoothly with other AI systems.

The Bottom Line

The performance of AlphaProof and AlphaGeometry 2 at the International Mathematical Olympiad is a notable leap forward in AI’s capability to tackle complex mathematical reasoning. Both systems demonstrated silver-medal-level performance by solving four out of six challenging problems, demonstrating significant advancements in formal proof and geometric problem-solving. Despite their achievements, these AI systems still depend on human input for translating problems into formal language and face challenges of integration with other AI systems. Future research aims to enhance these systems further, potentially integrating natural language reasoning to extend their capabilities across a broader range of mathematical challenges.