A paper posted online this month has settled a nearly 30-year-old conjecture about the structure of the fundamental building blocks of computer circuits. This “sensitivity” conjecture has stumped many of the most prominent computer scientists over the years, yet the new proof is so simple that one researcher summed it up in a single tweet.

“This conjecture has stood as one of the most frustrating and embarrassing open problems in all of combinatorics and theoretical computer science,” wrote Scott Aaronson of the University of Texas, Austin, in a blog post. “The list of people who tried to solve it and failed is like a who’s who of discrete math and theoretical computer science,” he added in an email.

The conjecture concerns Boolean functions, rules for transforming a string of input bits (0s and 1s) into a single output bit. One such rule is to output a 1 provided any of the input bits is 1, and a 0 otherwise; another rule is to output a 0 if the string has an even number of 1s, and a 1 otherwise. Every computer circuit is some combination of Boolean functions, making them “the bricks and mortar of whatever you’re doing in computer science,” said Rocco Servedio of Columbia University.

Over the years, computer scientists have developed many ways to measure the complexity of a given Boolean function. Each measure captures a different aspect of how the information in the input string determines the output bit. For instance, the “sensitivity” of a Boolean function tracks, roughly speaking, the likelihood that flipping a single input bit will alter the output bit. And “query complexity” calculates how many input bits you have to ask about before you can be sure of the output.

Each measure provides a unique window into the structure of the Boolean function. Yet computer scientists have found that nearly all these measures fit into a unified framework, so that the value of any one of them is a rough gauge for the value of the others. Only one complexity measure didn’t seem to fit in: sensitivity.

In 1992, Noam Nisan of the Hebrew University of Jerusalem and Mario Szegedy, now of Rutgers University, conjectured that sensitivity does indeed fit into this framework. But no one could prove it. “This, I would say, probably was the outstanding open question in the study of Boolean functions,” Servedio said.

“People wrote long, complicated papers trying to make the tiniest progress,” said Ryan O’Donnell of Carnegie Mellon University.

Now Hao Huang, a mathematician at Emory University, has proved the sensitivity conjecture with an ingenious but elementary two-page argument about the combinatorics of points on cubes. “It is just beautiful, like a precious pearl,” wrote Claire Mathieu, of the French National Center for Scientific Research, during a Skype interview.

Aaronson and O’Donnell both called Huang’s paper the “book” proof of the sensitivity conjecture, referring to Paul Erdős’ notion of a celestial book in which God writes the perfect proof of every theorem. “I find it hard to imagine that even God knows how to prove the Sensitivity Conjecture in any simpler way than this,” Aaronson wrote.

A Sensitive Matter

Imagine, Mathieu said, that you are filling out a series of yes/no questions on a bank loan application. When you’re done, the banker will score your results and tell you whether you qualify for a loan. This process is a Boolean function: Your answers are the input bits, and the banker’s decision is the output bit.