Super Slow Computer Programs Reveal Math’s Fundamental Limits

December 13, 2020

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Programmers normally want to minimize the time their code takes to execute. But in 1962, the Hungarian mathematician Tibor Radó posed the opposite problem. He asked: How long can a simple computer program possibly run before it terminates? Radó nicknamed these maximally inefficient but still functional programs “busy beavers.”

Original story reprinted with permission from *Quanta Magazine*, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.

Finding these programs has been a fiendishly diverting puzzle for programmers and other mathematical hobbyists ever since it was popularized in *Scientific American*’s “Computer Recreations” column in 1984. But in the last several years, the busy beaver game, as it’s known, has become an object of study in its own right, because it has yielded connections to some of the loftiest concepts and open problems in mathematics.

“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.

An Uncomputable Computer Game

The busy beaver game is all about the behavior of Turing machines—the primitive, idealized computers conceived by Alan Turing in 1936. A Turing machine performs actions on an endless strip of tape divided into squares. It does so according to a list of rules. The first rule might say:

Each rule has this forking choose-your-own-adventure style. Some rules say to jump back to previous rules; eventually there’s a rule containing an instruction to “halt.” Turing proved that this simple kind of computer is capable of performing any possible calculation, given the right instructions and enough time.

As Turing noted in 1936, in order to compute something, a Turing machine must eventually halt—it can’t get trapped in an infinite loop. But he also proved that there’s no reliable, repeatable method for distinguishing machines that halt from machines that simply run forever—a fact known as the halting problem.

The busy beaver game asks: Given a certain number of rules, what’s the maximum number of steps that a Turing machine can take before halting?

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.”

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