Decades-Old Computer Science ‘Boolean Sensitivity’ Conjecture Solved in Two Pages

Long-time Slashdot reader Faizdog writes: The “sensitivity” conjecture stumped many top computer scientists, 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.

“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 Erds’ 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.

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