The Mysterious Trail to P versus NP: Inside Meta-Complexity

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Since 2007, researchers have been studying the open question of the P vs NP problem. Despite decades of effort, a resolution to the P versus NP question has remained elusive. By studying inward-looking questions, researchers have developed the field of meta-complexity which helps them understand the fundamental difficulty of computational problems. This trail has been long and winding, with many false turns and roadblocks, but researchers have seen the rewards in the form of newfound insight into computability and the potential to approach a seemingly unapproachable problem.

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In the first week of the fall semester in 2007, Marco Carmosino dragged himself to a math class required for all computer science majors at the University of Massachusetts, Amherst. Carmosino, a sophomore, was considering dropping out of college to design video games. Then the professor posed a simple question that would change the course of his life: How do you know math actually works? "That made me sit up and pay attention," recalled Carmosino, now a theoretical computer scientist at IBM. He signed up for an optional seminar on the work of Kurt Gödel, whose dizzying self-referential arguments first exposed the limits of mathematical reasoning and created the foundation for all future work on the fundamental limits of computation. It was a lot to take in. "I 100% did not understand," Carmosino said. "But I knew that I wanted to." .

P versus NP has been a conjecture since 1971

Today, even seasoned researchers find understanding in short supply when they confront the central open question in theoretical computer science, known as the P versus NP problem. In essence, that question asks whether many computational problems long considered extremely difficult can actually be solved easily (via a secret shortcut we haven’t discovered yet), or whether, as most researchers suspect, they truly are hard. At stake is nothing less than the nature of what’s knowable.

Meta-complexity shows a relationship between the difficulty of proving something and the difficulty of the question

Despite decades of effort by researchers in the field of computational complexity theory — the study of such questions about the intrinsic difficulty of different problems — a resolution to the P versus NP question has remained elusive. And it’s not even clear where a would-be proof should start.

"There’s no road map," said Michael Sipser, a veteran complexity theorist at the Massachusetts Institute of Technology who spent years grappling with the problem in the 1980s. "It’s like you’re going into the wilderness." .

Meta-complexity is a field of theoretical computer science

It seems that proving that computational problems are hard to solve is itself a hard task. But why is it so hard? And just how hard is it? Carmosino and other researchers in the subfield of meta-complexity reformulate questions like this as computational problems, propelling the field forward by turning the lens of complexity theory back on itself.

"You might think, ‘OK, that’s kind of cool. Maybe the complexity theorists have gone crazy,’" said Rahul Ilango, a graduate student at MIT who has produced some of the most exciting recent results in the field.

The P vs NP problem is related to the mathematics of logic and computability theory

By studying these inward-looking questions, researchers have learned that the hardness of proving computational hardness is intimately tied to fundamental questions that may at first seem unrelated. How hard is it to spot hidden patterns in apparently random data? And if truly hard problems do exist, how often are they hard? .

"It’s become clear that meta-complexity is close to the heart of things," said Scott Aaronson, a complexity theorist at the University of Texas, Austin.

The language of meta-complexity questions is complexity theory

This is the story of the long and winding trail that led researchers from the P versus NP problem to meta-complexity. It hasn’t been an easy journey — the path is littered with false turns and roadblocks, and it loops back on itself again and again. Yet for meta-complexity researchers, the rewards have been OBVIOUS — sparkling insight into the nature of computability, and a renewed faith in our ability to approach a seemingly unapproachable problem.

The goal of meta-complexity studies is to map the difficulty of a problem

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