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Alan Turing and the Power of Negative Thinking

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An example of moving diagonally.

Turing’s strategy was based on a mathematical technique called diagonalization that has a distinguished history.

Credit: Kristina Armitage/Quanta Magazine

Algorithms have become ubiquitous. They optimize our commutes, process payments and coordinate the flow of Internet traffic. It seems that for every problem that can be articulated in precise mathematical terms, there's an algorithm that can solve it, at least in principle.

But that's not the case—some seemingly simple problems can never be solved algorithmically. The pioneering computer scientist Alan Turing proved the existence of such "uncomputable" problems nearly a century ago, in the same paper where he formulated the mathematical model of computation that launched modern computer science.

Turing proved this groundbreaking result using a counterintuitive strategy: He defined a problem that simply rejects every attempt to solve it.

"I ask you what you're doing, and then I say, 'No, I'm going to do something different,'" said Rahul Ilango, a graduate student at the Massachusetts Institute of Technology studying theoretical computer science.

From Quanta Magazine
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