Alan Turing and the Power of Negative Thinking

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Turing’s diagonalization impervious is simply a type of this crippled wherever nan questions tally done nan infinite database of imaginable algorithms, many times asking, “Can this algorithm lick nan problem we’d for illustration to beryllium uncomputable?”

“It’s benignant of ‘infinity questions,’” Williams said.

To triumph nan game, Turing needed to trade a problem wherever nan reply is nary for each algorithm. That meant identifying a peculiar input that makes nan first algorithm output nan incorrect answer, different input that makes nan 2nd 1 fail, and truthful on. He recovered those typical inputs utilizing a instrumentality akin to 1 Kurt Gödel had precocious utilized to prove that self-referential assertions for illustration “this connection is unprovable” spelled problem for nan foundations of mathematics.

The cardinal penetration was that each algorithm (or program) tin beryllium represented arsenic a drawstring of 0s and 1s. That means, arsenic successful nan illustration of nan error-checking program, that an algorithm tin return nan codification of different algorithm arsenic an input. In principle, an algorithm tin moreover return its ain codification arsenic an input.

With this insight, we tin specify an uncomputable problem for illustration nan 1 successful Turing’s proof: “Given an input drawstring representing nan codification of an algorithm, output 1 if that algorithm outputs 0 erstwhile its ain codification is nan input; otherwise, output 0.” Every algorithm that tries to lick this problem will nutrient nan incorrect output connected astatine slightest 1 input—namely, nan input corresponding to its ain code. That intends this perverse problem can’t beryllium solved by immoderate algorithm whatsoever.

What Negation Can’t Do

Computer scientists weren’t yet done pinch diagonalization. In 1965, Juris Hartmanis and Richard Stearns adapted Turing’s statement to prove that not each computable problems are created equal—some are intrinsically harder than others. That consequence launched nan section of computational complexity theory, which studies nan trouble of computational problems.

But complexity mentation besides revealed nan limits of Turing’s contrary method. In 1975, Theodore Baker, John Gill, and Robert Solovay proved that galore unfastened questions successful complexity mentation tin ne'er beryllium resolved by diagonalization alone. Chief among these is nan celebrated P versus NP problem, which asks whether each problems pinch easy checkable solutions are besides easy to lick pinch nan correct ingenious algorithm.

Diagonalization’s unsighted spots are a nonstop consequence of nan precocious level of abstraction that makes it truthful powerful. Turing’s impervious didn’t impact immoderate uncomputable problem that mightiness originate successful practice—instead, it concocted specified a problem connected nan fly. Other diagonalization proofs are likewise aloof from nan existent world, truthful they can’t resoluteness questions wherever real-world specifications matter.

“They grip computation astatine a distance,” Williams said. “I ideate a feline who is dealing pinch viruses and accesses them done immoderate mitt box.”

The nonaccomplishment of diagonalization was an early denotation that solving nan P versus NP problem would beryllium a agelong journey. But contempt its limitations, diagonalization remains 1 of nan cardinal devices successful complexity theorists’ arsenal. In 2011, Williams utilized it together pinch a raft of different techniques to prove that a definite restricted exemplary of computation couldn’t lick immoderate extraordinarily difficult problems—a consequence that had eluded researchers for 25 years. It was a acold outcry from resolving P versus NP, but it still represented awesome progress.

If you want to beryllium that something’s not possible, don’t underestimate nan powerfulness of conscionable saying no.

Original story reprinted pinch support from Quanta Magazine, an editorially independent publication of the Simons Foundation whose ngo is to heighten nationalist knowing of subject by covering investigation developments and trends successful mathematics and nan beingness and life sciences.