“How the Slowest Computer Programs Illuminate Math’s Fundamental Limits: The Goal of the ‘Busy Beaver’ Game Is to Find the Longest-Running Computer Program. Its Pursuit Has Surprising Connections to Some of the Most Profound Questions and Concepts in Mathematics”, 2020-12-10 (; similar):
“In math, there is a very permeable boundary between what’s an amusing recreation and what is actually important”, said Scott Aaronson, a theoretical computer scientist at the University of Texas, Austin who recently published a survey of progress in “BusyBeaverology.” The recent work suggests that the search for long-running computer programs can illuminate the state of mathematical knowledge, and even tell us what’s knowable. According to researchers, the busy beaver game provides a concrete benchmark for evaluating the difficulty of certain problems, such as the unsolved Goldbach conjecture and Riemann hypothesis. It even offers a glimpse of where the logical bedrock underlying math breaks down. The logician Kurt Gödel proved the existence of such mathematical terra incognita nearly a century ago. But the busy beaver game can show where it actually lies on a number line, like an ancient map depicting the edge of the world.
…For instance, if you’re only allowed one rule, and you want to ensure that the Turing machine halts, you’re forced to include the halt instruction right away. The busy beaver number of a one-rule machine, or BB(1), is therefore 1. But adding just a few more rules instantly blows up the number of machines to consider. Of 6,561 possible machines with two rules, the one that runs the longest—six steps—before halting is the busy beaver. But some others simply run forever. None of these are the busy beaver, but how do you definitively rule them out? Turing proved that there’s no way to automatically tell whether a machine that runs for a thousand or a million steps won’t eventually terminate.
That’s why finding busy beavers is so hard. There’s no general approach for identifying the longest-running Turing machines with an arbitrary number of instructions; you have to puzzle out the specifics of each case on its own. In other words, the busy beaver game is, in general, “uncomputable.” Proving that BB(2) = 6 and that BB(3) = 21 was difficult enough that Radó’s student Shen Lin earned a doctorate for the work in 1965. Radó considered BB(4) “entirely hopeless”, but the case was finally solved in 1983. Beyond that, the values virtually explode; researchers have identified a five-rule Turing machine, for instance, that runs for 47,176,870 steps before stopping, so BB(5) is at least that big. BB(6) is at least 7.4 × 1036,534. Proving the exact values “will need new ideas and new insights, if it can be done at all”, said Aaronson.
…The Goldbach conjecture, for instance, asks whether every even integer greater than 2 is the sum of two primes. Proving the conjecture true or false would be an epochal event in number theory, allowing mathematicians to better understand the distribution of prime numbers. In 2015, an anonymous GitHub user named Code Golf Addict published code for a 27-rule Turing machine that halts if—and only if—the Goldbach conjecture is false. It works by counting upward through all even integers greater than 4; for each one, it grinds through all the possible ways to get that integer by adding two others, checking whether the pair is prime. When it finds a suitable pair of primes, it moves up to the next even integer and repeats the process. If it finds an even integer that can’t be summed by a pair of prime numbers, it halts. Running this mindless machine isn’t a practical way to solve the conjecture, because we can’t know if it will ever halt until it does. But the busy beaver game sheds some light on the problem. If it were possible to compute BB(27), that would provide a ceiling on how long we’d have to wait for the Goldbach conjecture to be settled automatically. That’s because BB(27) corresponds to the maximum number of steps this 27-rule Turing machine would have to execute in order to halt (if it ever did). If we knew that number, we could run the Turing machine for exactly that many steps. If it halted by that point, we’d know the Goldbach conjecture was false. But if it went that many steps and didn’t halt, we’d know for certain that it never would—thus proving the conjecture true…In 2016, Aaronson established a similar result in collaboration with Yuri Matiyasevich and Stefan O’Rear. They identified a 744-rule Turing machine that halts if and only if the Riemann hypothesis is false
…In 2016, he and his graduate student Adam Yedidia specified a 7,910-rule Turing machine that would only halt if ZF set theory is inconsistent. This means BB(7,910) is a calculation that eludes the axioms of ZF set theory. Those axioms can’t be used to prove that BB(7,910) represents one number instead of another, which is like not being able to prove that 2 + 2 = 4 instead of 5…“So much of math can be encoded as a question of, ‘Does this Turing machine halt or not?’” Aaronson said. “If you knew all the busy beaver numbers, then you could settle all of those questions.”