Imagine knowing that the stock market will likely crash in three years, that extreme weather will destroy your home in eight or that you will have a debilitating disease in 15—but that you can take steps now to protect yourself from these crises. Although predicting the future with certainty will always be impossible, artificial intelligence could come close to doing so, some experts suggest. Predictions of such magnitude would require making billions of connections in immense datasets across enormous distances or time periods. Though such capabilities are beyond current AI systems, a mathematical breakthrough described in a recent preprint paper might provide clues for navigating such vast data and finding the larger patterns within it to reveal outcomes that people wouldn’t otherwise be able to predict.
To develop an AI system capable of doing such difficult work, a team of researchers at the California Institute of Technology and other institutions used the Andrews-Curtis conjecture—an intractable math problem from group theory, a field that studies symmetry, structure and operations in mathematical groups. Proposed by mathematicians James Andrews and Morton Curtis in 1965, the conjecture suggests that any such complicated mathematical configuration might be reduced to its most basic form by a finite sequence of three moves. One way to visualize the conjecture is to picture a vast maze in which a player is trying to connect all points to a central “home” point. The length of any single path could be unimaginably long and require taking millions or even billions of steps in the maze, says Sergei Gukov, the recent study’s senior author and a professor of mathematics at Caltech. “That was the reason we picked this problem,” he says, “because it’s a mathematical problem where, in order to make any progress, we basically are forced to develop new AI systems which can adapt to this level of complexity.”
In the 60 years since the Andrews-Curtis conjecture was formulated, the conjecture has never been proved or disproved. Proving it would mean showing that every eligible description can be connected to the single standard “home” description. Disproving it would require showing a so-called counterexample in which there is no “path” to achieving the conjecture. “A priori, it’s not known whether paths exist [for coordinates], and the goal is to try to prove or disprove whether a path exists or to find one example where a path does notexist,” says the study’s lead author Ali Shehper, a senior AI researcher at Caltech. For decades, mathematicians have attempted to disprove the conjecture by proposing many counterexamples for which no paths could be found—at least until now. The team made its breakthrough by finding complete or partial paths for a number of such unresolved potential counterexamples, thus showing that none of these proposals actually refutes the conjecture.
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With the Andrews-Curtis conjecture as its model, the team created a game: Picture a chesslike board but with a million or even a billion squares. As player, you must reach a designated “home” square—using a toolbox of just a few moves, similar to how each chess piece can be moved in specific ways. But this is a solitary game: you are the only player, and your job is to take any coordinate you are given and determine whether, using some combination of the available moves as many times as necessary, you can reach home. For coordinates closer to home, the task isn’t so hard. But when the coordinates are far-flung, finding your way by trial and error could easily take a lifetime, especially because you have no way of immediately judging whether each step taken is on the right path until you reach the destination. The path is also much longer than the actual distance between the two points. “In order to go from A to B you have to go thousands of miles in this complicated maze, even though the actual distance can be very small,” Gukov says. “So it’s like a devil designed the maze.”
To train AI to play the game, Gukov’s team used reinforcement learning, a machine-learning technique where an AI agent—a system that makes decisions and takes actions to achieve a goal—learns which actions work best through trial and error and by receiving rewards or penalties. “If you just show the agent hard problems in the beginning, it won’t know what to do with them. But if you show it easier problems first, then that really helps,” Shehper says.
But to cross the immense spaces required by the Andrews-Curtis conjecture, small steps aren’t enough. The game addresses this problem by using two AI agents with distinct roles: a player and an observer. By watching the player and evaluating its successes, the observer agent begins to combine basic moves into combinations, or “supermoves,” which the player can then use to make bigger leaps. As the player executes its available moves to excel at the shorter paths, the observer learns to evaluate the difficulty of the coordinates and to gauge which supermoves will best serve the player; it then provides those supermoves strategically when the player is most likely to be able to use them.
Whereas the easier coordinates can require as few as 10 moves to reach “home,” more difficult coordinates rapidly grow in complexity. “Mathematically it’s known that there exist cases where it needs billions of moves, but we have not gotten there yet with our AI system,” Shehper says. “We are in the range of thousands of moves.”
Thousands of moves have nonetheless been enough to break ground on some long-standing counterexamples to the Andrews-Curtis conjecture. Using the agentic AI system, the team was able to solve large families of longstanding potential counterexamples that had been open for 30 years. It even made progress on a series of counterexamples that have existed for about four decades, reducing most of them to more simplified forms. A preprint study at the University of Liverpool has since independently confirmed the Gukov’s team’s results.
“What they did, it’s beyond the expectations that I had” for what AI could do with the conjecture, says Alexei Miasnikov, a professor of mathematics at the Stevens Institute of Technology. Miasnikov, who has conducted research on the Andrews-Curtis conjecture and was not involved in the study by Gukov’s team, says their work has shown how useful machine reinforcement might be for experimental math. “It shows that you can get interesting results that you can’t get without a computer,” Miasnikov says. “I think much more interesting things will be developed soon. We are just at the beginning.”
Gukov’s team hopes to create tools for a broad range of problems in math and outside of it, Shehper says. 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’s GPT or xAI’s Grok, deal with problems that are known to be solvable, and they work to find more optimal solutions. “We know that chess and Go are solvable problems,” Shehper says. “A game ends, and you win or lose, and these systems are actually just finding a better way of doing that.” The team’s goal is to develop systems to tackle problems where mathematicians don’t yet know if solutions even exist—and where the path to evaluating whether an answer might be possible is incalculably long.
Gukov and Shehper hope the new tools they develop can ultimately be applied to real-world predictions. Perhaps future AI models will be able to foresee how complex machines might fail after years of use, how automated driving systems might produce rare but dangerous errors over long periods and how sickness might arise in an individual over decades. They could potentially be applied to many fields, such as medicine, cryptography, finance and climate modeling. “You could say that we’re developing AI systems for such applications,” Gukov says, “but first we’re just training them with math. Math is cheap, so we’re not going to burn somebody’s money or make wrong predictions about hurricanes.”
As for proving or disproving the Andrews-Curtis conjecture itself, the AI system developed by Gukov’s team is far from being able to do so—and this isn’t even the researchers’ goal. But by ruling out counterexamples, their work has provided some new support for the conjecture. “The common belief in the [mathematics] community when we started this work was that the Andrews-Curtis conjecture is probably false, so therefore one should try to disprove it,” Gukov says. “But after spending several years on this conjecture, I have started believing that maybe there is a chance—a good chance—it is actually true.”
