How This AI Breakthrough with Pure Mathematics and Reinforcement Learning Could Help Predict Future Crises
Imagine knowing that the stock market will probably block in three years, that extreme weather conditions will destroy your home in eight or that you will have a debilitating disease in 15 – but you can take measures now to protect yourself from these crises. Although the forecast of the future with certainty is always impossible, artificial intelligence could be closer to this, suggest certain experts. The predictions of such magnitude would require billions of connections in immense sets of data through huge distances or periods. Although such capacities are beyond current AI systems, a mathematical breakthrough described in a recent preparation document could provide indices to navigate such data and find more important models to reveal results that people could not predict otherwise.
To develop an AI system capable of doing such a difficult job, a team of researchers from the California Institute of Technology and other institutions used Andrews -Curtis conjecture – an intractable mathematical problem of group theory, an area that studies symmetry, structure and operations in mathematical groups. Proposed by mathematicians James Andrews and Morton Curtis in 1965, the conjecture suggests that such a complicated mathematical configuration could be reduced to its most basic form by a finite sequence of three movements. One way to visualize the conjecture is to imagine a vast labyrinth in which a player tries to connect all the points to a central point of “house”. The duration of any unique path could be unimaginably long and require millions, even billions of measures in the labyrinth, explains Sergei Gukov, the main author of the study and professor of mathematics in Caltech. “This is the reason why we have chosen this problem,” he says, “because it is a mathematical problem where, in order to progress, we are essentially forced to develop new AI systems that can adapt to this level of complexity.”
During the 60 years which followed the formulation of the Andrews-Curtis conjecture, the conjecture has never been proven or refuted. Prove that this would mean showing that each eligible description can be connected to the standard “home” description. Distribute that you should show a so-called counterexample in which there is no “path” to reach the conjecture. “A priori, we do not know if paths exist [for coordinates]And the objective is to try to prove or refute if a path exists or to find an example when a noting path, “explains the main author of the study, Ali Shehper, a principal researcher of the AI in Caltech. For decades, mathematicians tried to refute the conjecture by offering many counterexamples for which no way could be found. Potential counterexamples, thus showing that none of these proposals really refutes the conjecture.
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With the conjecture of Andrews-Curtis as a model, the team created a game: imagine a chess board but with a million or even a billion squares. As a player, you must reach a designated “home” square – using a toolbox of a few movements, similar to how each chess piece can be moved in a specific way. But it is a solitary game: you are the only player, and your work is to take any coordinated which is given to you and to determine if, using a combination of available movements as many times as necessary, you can reach the house. For coordinates closer to home, the task is not so difficult. But when the coordinates are distant, finding your way out of trials and errors could easily take a life, especially because you have no way to judge immediately if each step is crossed is on the right path until you reach the destination. The path is also much longer than the real distance between the two points. “To switch from A to B, you must travel thousands of kilometers in this complicated labyrinth, even if the real distance can be very small,” explains Gokov. “It is therefore like a devil designed the labyrinth.”
To train the AI to play the game, the Gukov team used the learning of strengthening, an automatic learning technique where an AI agent – a system that makes decisions and takes actions to achieve an objective – applies which actions work best by trials and errors and by receiving rewards or penalties. “If you just show the agent’s difficult problems at the beginning, that will not know what to do with them. But if you first show easier problems, it really helps,” says Shehper.
But to cross the immense spaces required by the conjecture of Andrews-Curtis, the small steps are not sufficient. The game addresses this problem using two AI agents with separate roles: a player and an observer. Looking at the player and evaluating his successes, the observer agent begins to combine basic movements in combinations, or “supermoves”, which the player can then use to make bigger leaps. As the player performs his available movements to excel in shorter paths, the observer learns to assess the difficulty of the contact details and to assess which supermoves will best serve the player; He then provides these supermoves strategically when the player is most likely to use them.
While the easiest coordinates may require up to 10 movements to reach “the house”, more difficult coordinates develop quickly in complexity. “Mathematically, we know that there are cases where it needs billions of movements, but we have not yet arrived with our AI system,” explains Shehper. “We are in the range of thousands of movements.”
Thousands of movements have nevertheless been sufficient to innovate on long-standing counterexamples at the Andrews-Curtis conjecture. Using the agentic AI system, the team was able to resolve large families of longtime potential counterexamples that have been open for 30 years. It has even progressed on a series of counter-examples that have existed for about four decades, reducing most of them to more simplified forms. A preparation study at the University of Liverpool has since independently confirmed the results of the Gukov team.
“What they have done was beyond the expectations I had” for what AI could do with the conjecture, explains Alexei Miasnikov, professor of mathematics at the Stevens Institute of Technology. Miasnikov, who conducted research on the conjecture of Andrews-Curtis and was not involved in the study of the Gukov team, says that their work has shown how useful machine strengthening could be for experimental mathematics. “This shows that you can get interesting results that you cannot get without a computer,” said Miasnikov. “I think much more interesting things will be developed soon. We are just at first. “
The Gukov team hopes to create tools for a wide range of mathematics problems and outside it, says Shehper. Current AI systems, such as Alphago (which plays Go) or Alphastar (which plays the video game Starcraft II), and even many large -language models, such as Openai GPT or XAI Grok, deal with problems that are known to be resolved, and they work to find more optimal solutions. “We know that failures and things are solved problems,” says Shehper. “A game ends, and you win or lose, and these systems actually find a better way to do it.” The objective of the team is to develop systems to solve the problems where mathematicians do not yet know if solutions exist – and where the path to assess if a response could be possible.
Gukov and Shehper hope that the new tools they develop can finally be applied to the predictions of the real world. Perhaps future AI models will be able to predict how complex machines could fail after years of use, how automated driving systems could produce rare but dangerous errors over long periods and how the disease could occur in an individual over the decades. They could potentially be applied to many areas, such as medicine, cryptography, finance and climate modeling. “You might say that we are developing AI systems for such applications,” says Gukov, “but first, we simply train them with mathematics. Mathematics are cheap, so we are not going to burn someone’s money or make bad predictions on hurricanes.”
As to prove or refute the conjecture of Andrews-Curtis itself, the AI system developed by the Gukov team is far from doing so-and it is not even the objective of the researchers. But by excluding counter-examples, their work provided new support for the conjecture. “The common belief in [mathematics] The community when we started this work was that the conjecture of Andrews -Curtis is probably false, so we must try to refute it, “explains Gukov.” But after spending several years on this conjecture, I started to believe that there may be a chance – a good luck – it is actually true. »»
