Are computers ready to solve this unlikely math problem?

In a way, the computer and Collatz’s conjecture are perfect. On the one hand, Jeremy Avigad, a logician and professor of philosophy at Carnegie Mellon, warns that the notion of an iterative algorithm is at the core of computing – and the Collatz sequences are examples of an iterative algorithm, followed step by step. to a deterministic rule. Also, showing that a process is over is a common problem in computing. “Computer scientists generally want to know that their algorithms are terminated, which means that they always provide the answer,” says Avigad. Heule and his collaborators are using this technology to counter Collatz’s conjecture, which is really just the end problem.

“The beauty of this automated method is you can turn on the computer and wait.”

Jeffrey Lagarias

Heule’s specialization lies in a computer tool called a “SAT solver,” or a “satisfaction” solver, a computer program that determines whether there is a formula or solution to a problem given a set of limitations. Although crucial, in the case of a mathematical challenge, the SAT solver must return or replace the problem as understood by the computer. As Yuleu, a PhD student at Heule, puts it: “Representation matters.”

Long, but worth a try

When Heule referred to Collatz confronting an SAT decision-maker, Aaronson thought, “There’s nothing in hell that’s going to work.” But he easily believed it was worth a try, as he saw subtle ways to transform this old problem that could make Heul flexible. He realized that a community of computer scientists used SAT solvers to successfully find evidence for the end of an abstract representation of computing called a “rewriting system”. It took a long time, but he suggested to Aaron that turning Collatz’s conjecture into a rewriting system would allow him to get proof of Collatz’s termination (Aaronson had earlier helped Riemann’s hypothesis become a computational system, coded in a small Turing machine). That afternoon, Aaron designed the system. “Doing homework was like a fun exercise,” he says.

Aaron’s system caught the Collatz problem with 11 rules. If researchers could obtain proof of the end of this analog system by applying these 11 rules in any order, this would prove that Collatz’s conjecture is true.

Heule tried to prove that the end of rewriting systems was over, with cutting-edge tools that didn’t work – it was disappointing, though not so surprising. “These tools are optimized for problems that can be solved in a minute, and any approach to fixing Collatz is needed if it takes years of computing a day,” says Heul. This gave them the motivation to improve their vision and implement their tools to turn the rewriting problem into a SAT problem.

Aaronson thought it would be much easier to fix the system by removing one of the 11 rules – leaving a “Collatz-like” system at a test pace for a larger purpose. He challenged man and computer: the first to fix all subsystems with 10 rules wins. Aaronson tried his hand. Heule tested the SAT solver: he coded the system as a satisfaction problem — with a clearer layer of representation, turning the system into a language of possible variables 0 and 1 — and then let his SAT solver run in the cores. , looking for evidence of termination.

HEULE OF THE SEA

They both managed to prove that the system ends with 10 different sets of rules. At times, it was a trivial task, both for the human being and for the program. Heule’s automatic viewing required a maximum of 24 hours. Aaronson’s approach required a great deal of intellectual effort, even taking a few hours or a day – a set of 10 rules he never managed to prove, even though he thought he could do it with greater effort. “In a very literal sense I was struggling with a Terminator,” says Aaronson, “at least the proof of the end theorem.”

Yolcu has since adjusted the SAT solution, calibrating the tool to better suit the nature of the Collatz problem. These tricks affected all sides: accelerating the end tests for the 10-rule subsystems and reducing execution times by just one second.

“The main question he asks,” says Aaronson, “is this, what about the whole set of 11s? You try to run the whole set of systems and it works forever, which shouldn’t surprise us, because that’s the Collatz problem.”

As Heule sees it, most research on automated reasoning has a blind eye to problems that require a lot of computation. But he believes that based on previous advances, these problems can be solved. Others have transformed Collatz be rewrite the system, but SAT is a fine-grained strategy for handling at scale, with tremendous computing power that can achieve traction toward a proof.

To date, Heul has conducted Collatz research using about 5,000 cores (processing units that power computers; computer consumers have four or eight cores). As an Amazon Scholar, Amazon Web Services has an open invitation to access “almost unlimited” resources — even millions of cores. But he is reluctant to use it much more.

“I want a sign that the essay is realistic,” he says. Otherwise, Heul believes it would waste resources and trust. “I don’t need 100% confidence, but I would like to have evidence that there are reasonable opportunities to really be successful.”

Transformation overload

“The beauty of this automated method is you can turn on the computer and wait,” says mathematician Jeffrey Lagarias, of the University of Michigan. He has been playing with Collatz for about fifty years and has become a guardian of knowledge, collecting noted bibliographies and editing a book on the subject. “The ultimate challenge.For Lagarias, he recalled an automated approach 2013 paper At the hands of Princeton mathematician John Horton Conway, he thought that the Collatz problem might be among the elusive classes of problems that are real and “decisive,” but at the same time not entirely decisive. As Conway put it: “… it could happen that the assertion that is not provable is not even provable, and so on.”

“If Conway is right,” says Lagarias, “there will be no evidence, automated or not, and we will never know the answer.”