# Are computers ready to solve this notoriously clunky mathematical problem?

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In a sense, computers and Kolatz’s conjecture are a perfect match. First of all, as Jeremy Avigad, a logician and professor of philosophy at Carnegie Mellon University, pointed out, the concept of iterative algorithms is the foundation of computer science-and the Collatz sequence is an example of iterative algorithms, which is determined according to Sex rules. Similarly, showing process termination is a common problem in computer science. “Computer scientists usually want to know that their algorithms will terminate, that is, they will always return an answer,” Avigad said. Heule and his collaborators are using this technology to solve the Collatz conjecture, which is actually just a termination problem.

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

Jeffrey Lagarrias

Heule’s specialty is the use of a calculation tool called the “SAT solver” or “satisfiability” solver, which is a computer program used to determine whether there is a solution for a given set of constraint formulas or problems. Although it is essential that when encountering a mathematical challenge, the SAT solver first needs to translate or represent the problem in terms that the computer can understand. As Yolcu, a PhD student at Heule, said: “Representation is important, a lot.”

**Long shot, but worth a try**

When Heule first mentioned the use of the SAT solver to solve Collatz, Aaronson thought, “This absolutely won’t work.” But he can easily believe that this is worth a try, because Heule sees the subtleties of changing this old problem. Method, which might make it flexible. He noticed that a community of computer scientists was using the SAT solver to successfully find a proof of termination for a computational abstract representation called a “rewrite system.” This is a long-term vision, but he suggested to Aaronson that transforming the Collatz conjecture into a rewriting system might get Collatz’s termination certificate (Aaronson had previously helped transform the Riemann hypothesis into a computing system, encoding it as A small Turing machine). That night, Aaronson designed this system. “It’s like a homework, a fun exercise,” he said.

Aaronson’s system captured the Collatz problem with 11 rules. If researchers can get the proof of termination of this similar system and apply these 11 rules in any order, it will prove that the Collatz conjecture is correct.

Heule tried to use the most advanced tools to prove the termination of the rewrite system, but it didn’t work—even if not so surprising, it was disappointing. “These tools are optimized for problems that can be solved within a minute, and any method to solve Collatz may require days or even years of calculation,” Heule said. This provides motivation to hone their methods and implement their own tools to turn rewriting problems into SAT problems.

Aaronson believes that it will be much easier to solve the system minus one of the 11 rules-leaving a “Collatz” system, which is a touchstone for larger goals. He initiated a human-computer battle: the first person to solve all subsystems with 10 rules wins. Aaronson tried it himself. Heule, who tried the SAT solver: He coded the system as a satisfiability problem—using another clever presentation layer to convert the system into computer variable terms, which can be 0 and 1—and then let his SAT solve it The processor runs on the kernel, looking for evidence of termination.

They all successfully proved that the system terminates with 10 different rules. Sometimes this is a trivial task for humans and programs. Heule’s automated method can take up to 24 hours. Aaronson’s method requires a lot of intellectual effort, several hours or even a day-a set of 10 rules that he never managed to prove, although he firmly believes that he can prove it with more effort. “Literally, I’m fighting the Terminator,” Aaronson said – “At least a prover of the termination theorem.”

Yolcu fine-tuned the SAT solver and calibrated the tool to better suit the nature of the Collatz problem. These techniques make everything different-speeding up the proof of termination of the 10 rule subsystem and reducing the running time to just a few seconds.

“The main remaining question,” Aaronson said, “Yes, how about the full set of 11? You try to run the system on the entire system, and it will run forever. This should probably not shock us, because that is Collat z problem.”

As Heule has seen, most automated reasoning research turns a blind eye to problems that require a lot of calculations. But based on his previous breakthroughs, he believes that these problems can be solved.Others have Deformed Kolatz Have a Rewrite systemBut this is a strategy for large-scale use of fine-tuned SAT solvers and powerful computing power, which may attract proof.

So far, Heule has used approximately 5,000 cores (processing units that power computers; consumer computers have four or eight cores) to run the Collatz survey. As an Amazon Scholar, he received a public invitation from Amazon Web Services to access “almost unlimited” resources-up to 1 million cores. But he is unwilling to use more.

“I want some signs that this is a realistic attempt,” he said. Otherwise, Heule felt that he would waste resources and trust. “I don’t need 100% confidence, but I really hope that there is some evidence that it has a reasonable chance of success.”

**Supercharged transformation**

“The beauty of this automated method is that you can turn on the computer and wait,” said Jeffrey Lagarrias, a mathematician at the University of Michigan. He played with Collatz for about 50 years, became the custodian of knowledge, edited annotated bibliography and edited a book on the subject,”The ultimate challenge.“For Lagarias, the automated method reminded me of a 2013 paper Proposed by the Princeton mathematician John Horton Conway, he believes that the Collatz problem may be a kind of elusive problem. These problems are real and “undecidable”-but at the same time they cannot The proof is undecidable. As Conway pointed out: “… it may even be that their unprovable assertions are themselves unprovable, and so on.”

“If Conway is right,” Lagarrias said, “whether it is automated or not, there is no evidence, and we will never know the answer.”

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