The Quest to Find the Longest-Running Simple Computer Program

The original version of this story appeared in Quanta Magazine.

Imagine that personification gives you a database of 5 numbers: 1, 6, 21, 107 and—wait for it—47,176,870. Can you conjecture what comes next?

If you’re stumped, you’re not alone. These are nan first 5 engaged beaver numbers. They shape a series that’s intimately tied to 1 of nan astir notoriously difficult questions successful theoretical machine science. Determining nan values of engaged beaver numbers is simply a daunting situation that has attracted a cult pursuing among some master and amateur mathematicians for complete 60 years.

Researchers identified nan first 4 engaged beaver numbers successful nan 1960s and 1970s. The conspicuously larger 5th number, called BB(5), was only definitively pinned down past year, by a squad made up mostly of amateur mathematicians moving together successful an online organization called nan Busy Beaver Challenge.

No 1 knows really large BB(6) is. All we person are little limits—truly staggering ones. In 2022 engaged beaver hunters established that BB(6) must be, astatine a minimum, truthful ample that it’s virtually intolerable to constitute down successful mean decimal notation. Even if you someway carved a digit into each atom successful nan cosmos, you’d tally retired of atoms earlier making immoderate measurable progress.

“It’s measurement beyond thing that we could ever grasp aliases get our hands on,” said Scott Aaronson, a machine intelligence astatine nan University of Texas, Austin.

Busy beaver hunters person now discovered that this stupefyingly large number must beryllium moreover bigger. The uncovering comes from 1 of nan astir mysterious and prolific contributors to nan Busy Beaver Challenge, who proved a caller little limit connected BB(6) successful June and collapsed nan grounds again a specified 9 days later. The caller results make nan 2022 little bound look positively puny.

“I support connected being surprised,” said William Gasarch, a machine intelligence astatine nan University of Maryland. “Six is getting america into nan stratosphere of ample numbers.”

Beaver Trap

The notoriously difficult mobility down nan engaged beaver numbers is this: Given nan codification of a machine program, tin you show whether it will yet extremity aliases tally forever?

In 1936, nan pioneering logician Alan Turing proved that there’s nary cosmopolitan process for answering this question, which became known arsenic nan halting problem. Any method that useful for immoderate programs will neglect for others, and successful immoderate cases, nary method will work.

Turing proved this seminal consequence by inventing a general mathematical exemplary of computation successful which programs are represented by hypothetical devices now called Turing machines. Each Turing instrumentality performs computations successful discrete steps according to a unsocial database of elemental rules. The much rules a Turing instrumentality has, nan much analyzable its behaviour tin get, and nan harder it tin beryllium to find whether it will halt.

But conscionable really overmuch harder? In 1962, nan mathematician Tibor Radó invented a caller measurement to research this mobility done what he called the engaged beaver game. To play, commencement by choosing a circumstantial number of rules—call that number n. Your extremity is to find nan n-rule Turing instrumentality that runs nan longest earlier yet halting. This instrumentality is called nan engaged beaver, and nan corresponding engaged beaver number, BB(n), is nan number of steps that it takes.

In principle, if you want to find nan engaged beaver for immoderate fixed n, you conscionable request to do a fewer things. First, database retired each nan imaginable n-rule Turing machines. Next, usage a machine programme to simulate moving each machine. Look for telltale signs that machines will ne'er halt—for example, galore machines will autumn into infinite repeating loops. Discard each these non-halting machines. Finally, grounds really galore steps each different instrumentality took earlier halting. The 1 pinch nan longest runtime is your engaged beaver.

In practice, this gets tricky. For starters, nan number of imaginable machines grows quickly pinch each caller rule. Analyzing them each individually would beryllium hopeless, truthful you’ll request to constitute a civilization machine programme to categorize and discard machines. Some machines are easy to classify: They either halt quickly aliases autumn into easy identifiable infinite loops. But others tally for a agelong clip without displaying immoderate evident pattern. For these machines, nan halting problem deserves its fearsome reputation.

The much rules you add, nan much computing powerfulness you need. But brute unit isn’t enough. Some machines tally for truthful agelong earlier halting that simulating them measurement by measurement is impossible. You request clever mathematical tricks to measurement their runtimes.

“Technology improvements decidedly help,” said Shawn Ligocki, a package technologist and longtime engaged beaver hunter. “But they only thief truthful far.”

End of an Era

Busy beaver hunters started chipping distant astatine nan BB(6) problem successful earnest successful nan 1990s and 2000s, during an impasse successful nan BB(5) hunt. Among them were Shawn Ligocki and his father, Terry, an applied mathematician who ran their hunt programme successful nan disconnected hours connected powerful computers astatine Lawrence Berkeley National Laboratory. In 2007, they recovered a six-rule Turing instrumentality that collapsed nan grounds for nan longest runtime: The number of steps it took earlier halting had astir 3,000 digits. That’s a colossal number by immoderate mean measure. But it’s not excessively large to constitute down. In 12-point font, those 3,000 digits will conscionable astir screen a azygous expanse of paper.

Three years later, a Slovakian undergraduate machine subject student named Pavel Kropitz decided to tackle nan BB(6) hunt arsenic a elder thesis project. He wrote his ain hunt programme and group it up to tally successful nan inheritance connected a web of 30 computers successful a assemblage lab. After a period he recovered a instrumentality that ran acold longer than nan 1 discovered by nan Ligockis—a caller “champion,” successful nan lingo of engaged beaver hunters.

“I was lucky, because group successful nan laboratory were already complaining astir my CPU usage and I had to standard backmost a bit,” Kropitz wrote successful a nonstop connection speech connected nan Busy Beaver Challenge Discord server. After different period of searching, he collapsed his ain grounds pinch a instrumentality whose runtime had complete 30,000 digits—enough to capable astir 10 pages.

Kropitz’s instrumentality held nan BB(6) grounds for 12 years. Then, successful May 2022, Shawn Ligocki started a caller occupation wherever he had entree to a powerful machine cluster, and he decided to effort moving his aged codification connected newer hardware. Sure enough, he recovered a caller champion that hit Kropitz’s record. The find kicked disconnected a flurry of activity. Twice successful nan span of 2 weeks, Ligocki announced a new champion connected a engaged beaver mailing list. Each time, Kropitz hit his grounds wrong 3 days. Ligocki remembers his begetter marveling astatine really Kropitz pulled it off.

“He was joking that he imagines Pavel has already solved BB(6),” Ligocki said. “Whenever we find a champion, he conscionable goes and pulls retired of his container 1 that’s a small spot bigger.”

But nan past 2 machines that Ligocki and Kropitz discovered didn’t tally conscionable a spot longer than nan reigning champion—their runtimes were connected an wholly caller level.

To understand numbers this large, we request to spell backmost to nan acquainted mathematics of summation and multiplication. Start by adding up n copies of a number—that’s conscionable nan meaning of multiplication by n. If you alternatively multiply n copies of a number, that’s known arsenic exponentiation. So what happens if you many times exponentiate a number? That process defines a caller cognition called tetration, denoted by 2 arrows pointing up.

Tetration gets large fast. 10↑↑1 is conscionable 10. But 10↑↑2 is 1010, aliases 10 billion, and 10↑↑3 is 10 raised to nan 10-billionth power: a 1 followed by 10 cardinal zeros. To constitute retired each nan digits you’d request a stack of insubstantial a 1000 feet high. At 10↑↑4, you transverse a symbolic period wherever it’s nary longer a matter of uncovering capable paper—there are acold much digits than atoms successful nan universe.

When Ligocki hit Kropitz’s grounds for nan 2nd time, it was pinch a six-rule Turing instrumentality that ran for complete 10↑↑5 steps earlier halting. Kropitz countered pinch a instrumentality that ran for 10↑↑15 steps—that’s a building of tens 15 stories high. They’d near nan acquainted world of digits acold behind.

“That was nan extremity of an era,” Kropitz wrote complete nonstop message.

It was besides nan extremity of an era successful different respect. Until then, nan engaged beaver crippled had been a competition, and researchers mostly worked alone. Then nan Busy Beaver Challenge was formed, ushering successful a caller property of collaboration.

A New Class of Crazy

The Busy Beaver Challenge was founded successful 2022 by a machine subject postgraduate student named Tristan Stérin pinch nan definitive intent of rigorously proving nan existent worth of BB(5). In summertime 2024 nan group succeeded, pinch a cardinal publication from a mysterious newcomer known only by nan pseudonym mxdys.

News of nan consequence appeared successful Quanta, wherever Katelyn Doucette, an undergraduate machine subject student astatine Virginia Tech, happened to spot it. She soon joined nan Busy Beaver Challenge community, astatine first conscionable dropping successful to nan Discord server from clip to time. But successful May she made an breathtaking discovery, and since past she’s go 1 of nan astir progressive contributors to nan BB(6) hunt. “I’ve conscionable been benignant of hooked,” she said. “It’s specified a beautiful group of problems.”

In nan twelvemonth since putting nan finishing touches connected nan BB(5) proof, mxdys had steadily chipped distant astatine nan BB(6) problem, utilizing blase automated methods to categorize each but a fewer 1000 “holdout” machines. Doucette was rooting astir successful nan database of holdouts erstwhile she recovered one that looked particularly promising. Analyzing it further pinch immoderate thief from Shawn Ligocki, she discovered that its runtime was 2nd only to that of Kropitz’s reigning champion. What’s more, Doucette’s instrumentality belonged to a people of machines known arsenic shift overflow counters, which execute agelong runtimes utilizing a very different system than Kropitz’s champion.

“It’s breathtaking to spot that these engaged beavers person recovered a caller technology,” Ligocki said.

A fewer different engaged beaver hunters had previously discovered displacement overflow counters that halted aft a agelong time, but Doucette’s find made nan squad fishy that these machines were much plentiful than they’d realized. And if immoderate of nan first ones to beryllium studied had travel wrong striking region of Kropitz’s record, others would apt surpass it. Busy Beaver Challenge contributors rushed to analyse different displacement overflow counters, but mxdys sewage location first. On June 16, they announced nan find of a caller champion that halted aft 10↑↑107 steps—that is, its runtime is simply a building of tens that’s 10 cardinal stories high. Writing this number down arsenic a drawstring of digits is retired of nan question. But moreover penning it arsenic a building of powers gets dicey: In 12-point font, that statement of tens would agelong retired for astir 25 miles.

Kropitz, who saw nan news while distant connected vacation, accepted nan nonaccomplishment of his title graciously, penning successful nan Discord server, “unfortunately, one won’t beryllium performing my 3-day instrumentality this time.” It helped that he had a consolation prize. A period earlier, he’d nabbed nan grounds for nan longest-running known seven-rule Turing machine. At slightest for now, Kropitz is still connected nan leaderboard.

Beyond nan Biggest

The caller grounds didn’t past long. A week later, mxdys broke it again pinch a instrumentality whose runtime is—yet again—on a qualitatively caller level. To constitute it down successful its astir succinct form, we request to present an absurd mathematical cognition called pentation, represented by 3 arrows pointing up. Pentation is repeated tetration—in different words, it bears nan aforesaid narration to tetration arsenic tetration does to exponentiation.

The full number of steps that mxdys’ caller champion took earlier halting was greater than 2↑↑↑5, aliases 2↑↑(2↑↑(2↑↑(2↑↑2))). To unpack this expression, you activity outward from nan innermost parentheses: 2↑↑2 is 4, and 2↑↑4 is simply a spot complete 65,000. That leaves you pinch 2↑↑(2↑↑65,000), making nan tallness of nan last stack of 2s an incomprehensibly ample number. Forget astir penning a building of powers that stretches retired for miles aliases megaparsecs. Even this much compact notation nary longer fits successful nan universe.

This caller consequence is still conscionable a little limit connected BB(6)—the existent worth could beryllium moreover higher. Busy beaver hunters don’t expect to person a definitive reply anytime soon. The first motion of problem was a monstrous six-rule Turing instrumentality that nan squad has named Antihydra, discovered by mxdys past year.

Antihydra almost surely ne'er halts. But researchers haven’t been capable to beryllium it. And there’s a bully logic for that: A engaged beaver huntsman who goes by Racheline has shown that nan mobility of whether Antihydra halts is intimately related to a celebrated unsolved problem successful mathematics called nan Collatz conjecture. Since then, nan squad has discovered galore different six-rule machines pinch akin characteristics. Slaying nan Antihydra and its brethren will require conceptual breakthroughs successful axenic mathematics.

But to avid engaged beaver hunters, that’s nary logic to beryllium discouraged. There are still thousands of six-rule Turing machines to explore, each pinch its ain rich | behavior.

“For me, nan astir valid logic to do mathematics is because it’s fun. It is art,” Racheline wrote successful a Discord nonstop message. “There will ever beryllium thing caller to do.”

Original story reprinted pinch support from Quanta Magazine, an editorially independent publication of the Simons Foundation whose ngo is to heighten nationalist knowing of subject by covering investigation developments and trends successful mathematics and nan beingness and life sciences.