In a way, the pc and the Collatz conjecture are an ideal match. For one, as Jeremy Avigad, a logician and professor of philosophy at Carnegie Mellon notes, the notion of an iterative algorithm is on the basis of laptop science—and Collatz sequences are an instance of an iterative algorithm, continuing step-by-step in line with a deterministic rule. Similarly, exhibiting {that a} course of terminates is a standard downside in laptop science. “Computer scientists generally want to know that their algorithms terminate, which is to say, that they always return an answer,” Avigad says. Heule and his collaborators are leveraging that expertise in tackling the Collatz conjecture, which is basically only a termination downside.

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

Jeffrey Lagarias

Heule’s experience is with a computational software known as a “SAT solver”—or a “satisfiability” solver, a pc program that determines whether or not there’s a answer for a components or downside given a set of constraints. Though crucially, within the case of a mathematical problem, a SAT solver first wants the issue translated, or represented, in phrases that the pc understands. And as Yolcu, a PhD pupil with Heule, places it: “Representation matters, a lot.”

A longshot, however price a strive

When Heule first talked about tackling Collatz with a SAT solver, Aaronson thought, “There is no way in hell this is going to work.” But he was simply satisfied it was price a strive, since Heule noticed delicate methods to rework this previous downside which may make it pliable. He’d seen {that a} neighborhood of laptop scientists have been utilizing SAT solvers to efficiently discover termination proofs for an summary illustration of computation known as a “rewrite system.” It was a longshot, however he instructed to Aaronson that reworking the Collatz conjecture right into a rewrite system may make it attainable to get a termination proof for Collatz (Aaronson had beforehand helped rework the Riemann speculation right into a computational system, encoding it in a small Turing machine). That night, Aaronson designed the system. “It was like a homework assignment, a fun exercise,” he says.

“In a very literal sense I was battling a Terminator—at least a termination theorem prover.”

Scott Aaronson

Aaronson’s system captured the Collatz downside with 11 guidelines. If the researchers might get a termination proof for this analogous system, making use of these 11 guidelines in any order, that may show the Collatz conjecture true.

Heule tried with state-of-the-art instruments for proving the termination of rewrite methods, which didn’t work—it was disappointing if not so shocking. “These tools are optimized for problems that can be solved in a minute, while any approach to solve Collatz likely requires days if not years of computation,” says Heule. This offered motivation to hone their method and implement their very own instruments to rework the rewrite downside right into a SAT downside.

A illustration of the 11-rule rewrite system for the Collatz conjecture.


Aaronson figured it could be a lot simpler to unravel the system minus one of many 11 guidelines—leaving a “Collatz-like” system, a litmus take a look at for the bigger objective. He issued a human-versus-computer problem: The first to unravel all subsystems with 10 guidelines wins. Aaronson tried by hand. Heule tried by SAT solver: He encoded the system as a satisfiability downside—with one more intelligent layer of illustration, translating the system into the pc’s lingo of variables that may be both 0s and 1s—after which let his SAT solver run on the cores, looking for proof of termination.

collatz visualization
The system right here follows the Collatz sequence for the beginning worth 27—27 is on the prime left of the diagonal cascade, 1 is at backside proper. There are 71 steps, slightly than 111, for the reason that researchers used a distinct however equal model of the Collatz algorithm: if the quantity is even then divide by 2; in any other case multiply by 3, add 1, after which divide the end result by 2.


They each succeeded in proving that the system terminates with the assorted units of 10 guidelines. Sometimes it was a trivial enterprise, for each the human and this system. Heule’s automated method took at most 24 hours. Aaronson’s method required vital mental effort, taking just a few hours or perhaps a day—one set of 10 guidelines he by no means managed to show, although he firmly believes he might have, with extra effort. “In a very literal sense I was battling a Terminator,” Aaronson says—“at least a termination theorem prover.”

Yolcu has since fine-tuned the SAT solver, calibrating the software to raised match the character of the Collatz downside. These tips made all of the distinction—dashing up the termination proofs for the 10-rule subsystems and decreasing runtimes to mere seconds.

“The main question that remains,” says Aaronson, “is, What about the full set of 11? You try running the system on the full set and it just runs forever, which maybe shouldn’t shock us, because that is the Collatz problem.”

As Heule sees it, most analysis in automated reasoning has a blind eye for issues that require plenty of computation. But primarily based on his earlier breakthroughs he believes these issues will be solved. Others have remodeled Collatz as a rewrite system, but it surely’s the technique of wielding a fine-tuned SAT solver at scale with formidable compute energy which may acquire traction towards a proof.

So far, Heule has run the Collatz investigation utilizing about 5,000 cores (the processing items powering computer systems; shopper computer systems have 4 or eight cores). As an Amazon Scholar, he has an open invitation from Amazon Web Services to entry “practically unlimited” assets—as many as a million cores. But he’s reluctant to make use of considerably extra.

“I want some indication that this is a realistic attempt,” he says. Otherwise, Heule feels he’d be losing assets and belief. “I don’t need 100% confidence, but I really would like to have some evidence that there’s a reasonable chance that it’s going to succeed.”

Supercharging a change

“The beauty of this automated method is that you can turn on the computer, and wait,” says the mathematician Jeffrey Lagarias, of the University of Michigan. He’s toyed with Collatz for about fifty years and turn out to be keeper of the information, compiling annotated bibliographies and enhancing a guide on the topic, “The Ultimate Challenge.” For Lagarias, the automated method delivered to thoughts a 2013 paper by the Princeton mathematician John Horton Conway, who mused that the Collatz downside is likely to be amongst an elusive class of issues which might be true and “undecidable”—however without delay not provably undecidable. As Conway famous: “… it might even be that the assertion that they are not provable is not itself provable, and so on.”

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