SoylentNews

takyon writes:

Decades-Old Computer Science Conjecture Solved in Two Pages

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.

Original Submission