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Are computers ready to solve this cumbersome math problem that everyone knows?


In a sense, the computer and the Collatz conjecture are a perfect match. First, as Jeremy Avigad, a logician and professor of philosophy at Carnegie Mellon, points out, the concept of an iterative algorithm is at the heart of computer science, and Collatz sequences are an example of an iterative algorithm and step-by-step accordingly. Similar to a deterministic rule, showing that a process has ended is a common problem in computer science. “Computer scientists often want to know that their algorithm has terminated, that is, they always return a response,” says Avigad. Heule and his collaborators are leveraging this technology to combat the Collatz assumption, which is really just a termination issue.

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

Jeffrey Lagarias

Heule’s specialty is a computer program that determines whether there is a solution to a formula or problem given a set of constraints, with a computational tool called the “SAT solver” or “satisfaction” solver. Although very important, in the case of a mathematical difficulty, a SAT solver first needs the problem translated or represented in terms that the computer understands. As Yolcu, a PhD student from Heule, said: “Representation is very important.”

It’s a long shot, but worth a try

“There’s no way this is going to work,” thought Aaronson when Heule first mentioned dealing with Collatz with an SAT solver. But Heule was easily convinced that it was worth a try, as he saw subtle ways to make this old problem flexible. He had noticed that a community of computer scientists used SAT solvers to successfully find termination proofs for an abstract computational representation called a “rewrite system.” It was a long shot, but suggested to Aaronson that turning the Collatz conjecture into a rewrite system might make it possible to obtain a termination proof for Collatz (Aaronson had previously helped transform the Riemann hypothesis into a computational system by encoding the Riemann hypothesis in a small Turing machine). That evening, Aaronson designed the system. “It was like homework, it was a fun exercise,” she says.

“I was literally fighting a Terminator – at least a termination theorem prover.”

Scott Aaronson

Aaronson’s system captured the Collatz problem with 11 rules. If researchers can obtain a proof of termination for this similar system by applying these 11 rules in any order, it will prove the correctness of the Collatz conjecture.

Heule tried with state-of-the-art tools to prove that rewrite systems were terminated, and it didn’t work – disappointing if not surprising. “While these tools are optimized for problems that can be resolved in a minute, any approach to solving Collatz requires days, if not years, of computation that would likely take years,” Heule says. This provided motivation to improve their approach and apply their own tools to turn the rewrite problem into an SAT problem.

collatz rewrite rules
A representation of the 11-rule rewrite system for the Collatz conjecture.

MARIJN HEULE

Aaronson thought it would be much easier to unravel the system by removing one of the 11 rules – leaving a “Collatz-like” system that was a litmus test for the larger target. He posted a human versus computer challenge: The first to solve all subsystems with 10 rules wins. Aaronson tried by hand. Heule was experimented with by the SAT solver: He coded the system into a satisfiability problem – again with another layer of intelligent representation, translating the system into the computer’s language of variables that can be 0s and 1s – and then let the SAT solver run on kernels. Looking for proof of termination.

collatz visualization
The system here follows the Collatz order for the initial value 27-27, with 1 in the upper left corner of the diagonal tier, 1 in the lower right. There are 71 steps instead of 111, as the researchers used a different but equivalent version of the Collatz algorithm: if the number is even then divide by 2; otherwise multiply by 3, add 1 and divide the result by 2.

MARIJN HEULE

Both managed to prove that the system ends with a variety of 10 rule sets. Sometimes it was a trivial undertaking for both man and program. Heule’s automated approach took a maximum of 24 hours. Aaronson’s approach required several hours or even a day of significant intellectual effort—a set of 10 rules he could never prove, although he firmly believed he could achieve with more effort. “I was literally fighting a Terminator,” Aaronson says – “at least a proof of termination theorem.”

Passenger has since fine-tuned the SAT solver, calibrating the vehicle to better suit the nature of the Collatz problem. These tricks made all the difference by speeding up proofs of termination for 10 rule subsystems and reducing runtimes to just seconds.

“The main question remaining,” Aaronson says, “is the entire set of 11? You try to run the system on the full set and it runs forever, so maybe that shouldn’t shock us, because that’s the Collatz problem.”

As Heule saw, most research in automatic reasoning is blind to problems that require a lot of computation. But he believes these problems can be solved based on his previous discoveries. There are others converted Collatz as rewrite system, but it is a strategy of using a fine-tuned SAT solver at scale with formidable computing power that can gain traction towards a proof.

So far, Heule Collatz has conducted its research using around 5,000 cores (the processing units that power computers; consumer computers have four or eight cores). As an Amazon Scholar, he received an open invitation from Amazon Web Services to access “virtually unlimited” resources – one million cores. But he is reluctant to use significantly more.

“I want a sign that this is a realistic attempt,” he says. Otherwise, Heule feels he will waste resources and trust. “I don’t need 100% confidence, but I would like to have some evidence that there is a reasonable chance that it will actually succeed.”

Supercharging a transformation

“The beauty of this automated method is that you can turn on the computer and wait,” says mathematician Jeffrey Lagarias of the University of Michigan. He has been playing with Collatz for nearly fifty years and is the keeper of knowledge, compiling annotated bibliographies and preparing a book on the subject, “Ultimate Challenge.“For Lagarias, the automated approach makes sense. 2013 paper By Princeton mathematician John Horton Conway, who thought the Collatz problem could be among an elusive class of problems that are true and “undecidable”—but not once demonstrably undecidable. As Conway puts it: “… it may even be that the claim that they are not provable is itself not provable, and so on.”

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



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