Alfageometry: AI system at the Olympics level for geometry

Our AI system exceeds the latest approach to geometry problems, developing AI reasoning in mathematics

Reflecting the Olympic spirit of ancient Greece, International Mathematical Olympics It is a contemporary arena for the world of high school mathematicians in the world. The competition not only shows young talents, but appeared as a basis for advanced AI systems in mathematics and reasoning.

In an article published today in NatureWe introduce alphageometry, the AI ​​system, which solves the problems complex geometry at a level approaching the gold of the Human Olympics – a breakthrough in AI performance. In the comparative test of 30 problems of the Olympics Olympics geometry, Alfageometry solved 25 in the standard time limit of the Olympics. For comparison, the previous most modern system solved 10 of these geometry problems, and the average human gold medalist solved problems 25.9.

AI systems often struggle with complex problems in geometry and mathematics due to lack of reasoning skills and training data. The alphageometry system combines the predictive power of a neural language model with a deductive engine associated with a rule that works in tandem to find solutions. By developing a method of generating a large pool of synthetic training data – 100 million unique examples – we can train alphageometry without any interpersonal demonstrations, supporting the bottleneck of the data.

Thanks to alphageometry, we show AI growing ability to logically reason and discover and verify new knowledge. Solving geometry problems at the Olympics level is an important milestone in developing deep mathematical reasoning on the road to more advanced and general AI systems. We are open Code and model of alphageometryAnd I hope that together with other tools and approaches in synthetic data generation and training, it helps to open new opportunities in mathematics, science and artificial intelligence.

Alfageometry adopts a neuro-symbolic approach

Alphageometry is a neuro-symbolic system consisting of a neural language model and a symbolic deductive engine that works to find evidence for complex geometry claims. Similar to the idea “Thinking, quick and slow“, One system provides quick,” intuitive “ideas, and the other, more intentional, rational decision making.

Since language models lead in identifying general patterns and relationships in data, they can quickly predict potentially useful constructs, but often they do not have the possibility of strict reasoning or explaining their decisions. On the other hand, symbolic deductive engines are based on formal logic and use clear rules to draw conclusions. They are rational and explanatory, but they can be “slow” and inelastic – especially in the case of large, complex problems on their own.

Alfageometry model leads its symbolic deductive engine towards probable solutions to geometry problems. The problems of the Olympics geometry are based on diagrams that require new geometric constructs, which should be added before they can be solved, such as points, lines or wheels. The alphageometry model predicts which new constructs would be most useful to add, with an infinite number of possibilities. These tips help to fill the gaps and allow the symbolic engine to make further deductions regarding the diagram and closure of the solution.

Generating 100 million examples of synthetic data

Geometry is based on understanding space, distance, shape and relative positions and is fundamental to art, architecture, engineering and many other fields. People can learn geometry with a pen and paper, studying diagrams and using existing knowledge to discover new, more sophisticated geometric properties and relationships. Our synthetic approach to generating data imitates this large -scale knowledge building, enabling us to train alphageometry from scratch, without any interpersonal demonstrations.

Using highly parallel calculations, the system began by generating a billion random diagrams of geometric objects and exhaustively obtained all the relationships between points and lines in each scheme. Alfageometry found all the evidence contained in each scheme, and then they worked back to find out what additional constructions, if at all, were needed to receive this evidence. We call this process “symbolic deduction and tracking”.

This huge pool of data has been filtered to exclude similar examples, which results in the final set of training data of 100 million unique examples of various difficulties, of which nine million presented the added constructs. With so many examples of the way these constructions led to evidence, the Alfageometers language model is able to present good suggestions for new constructions when they are presented with the problems of the Olympics geometry.

Pioneering mathematical reasoning with AI

The solution to every problem of the Olympics provided by Alfageometry has been checked and verified by the computer. We also compared its results with previous methods and and with human performance at the Olympics. In addition, Evan Chen, a mathematics coach and former medalist of Olympiad Gold, assessed the choice of alphageometry solutions for us.

Chen said: “The output of alphageometry is impressive, because it is both verifiable and clean. Earlier solutions and problems with competition based on evidence were sometimes hit (the output is sometimes only correct and need human controls). Alfageometry has no such weakness: its solutions have a structure that is possible to zerify the machine. Nevertheless, its output is still obvious. Computer program that solved by Brute-Furte: Pages of tedious algebra calculations.

Because each Olympics have six problems, of which only two focus on geometry, alphageometry can only be used up to one third of problems at a given Olympics. Nevertheless, the very ability of geometry makes it the first AI model in the world capable of transferring the bronze threshold of the IMO medal in 2000 and 2015.

In geometry, our system is approaching the IMO medalist standard, but we have an eye on an even larger reward: the reasoning of new generation AI systems. Given the wider potential of training AI from scratch with large synthetic data, this approach can shape the way and the future systems discover new knowledge in mathematics and more.

Alfageometry is based on Google Deepmind and Google Research to pioneering mathematical reasoning with AI – from exploration of the beauty of pure mathematics to Solving mathematical and scientific problems with language models. We recently introduced Foundsearch, which made the first discoveries in open problems in mathematical sciences using large language models.

Our long -term goal is to build AI systems that can generalize in various mathematical fields, developing sophisticated problem solving and reasoning, on which general AI systems will depend, while expanding the boundaries of human knowledge.