Turing’s diagonalization proof is a model of this sport the place the questions run by means of the infinite record of potential algorithms, repeatedly asking, “Can this algorithm solve the problem we’d like to prove uncomputable?”

“It’s sort of ‘infinity questions,’” Williams mentioned.

To win the sport, Turing wanted to craft an issue the place the reply is not any for each algorithm. That meant figuring out a selected enter that makes the primary algorithm output the improper reply, one other enter that makes the second fail, and so forth. He discovered these particular inputs utilizing a trick just like one Kurt Gödel had lately used to prove that self-referential assertions like “this statement is unprovable” spelled hassle for the foundations of arithmetic.

The key perception was that each algorithm (or program) will be represented as a string of 0s and 1s. That means, as within the instance of the error-checking program, that an algorithm can take the code of one other algorithm as an enter. In precept, an algorithm may even take its personal code as an enter.

With this perception, we are able to outline an uncomputable drawback just like the one in Turing’s proof: “Given an input string representing the code of an algorithm, output 1 if that algorithm outputs 0 when its own code is the input; otherwise, output 0.” Every algorithm that tries to resolve this drawback will produce the improper output on at the least one enter—particularly, the enter comparable to its personal code. That means this perverse drawback can’t be solved by any algorithm in any respect.

What Negation Can’t Do

Computer scientists weren’t but by means of with diagonalization. In 1965, Juris Hartmanis and Richard Stearns tailored Turing’s argument to prove that not all computable issues are created equal—some are intrinsically tougher than others. That outcome launched the sector of computational complexity concept, which research the problem of computational issues.

But complexity concept additionally revealed the boundaries of Turing’s opposite methodology. In 1975, Theodore Baker, John Gill, and Robert Solovay proved that many open questions in complexity concept can by no means be resolved by diagonalization alone. Chief amongst these is the well-known P versus NP drawback, which asks whether or not all issues with simply checkable options are additionally simple to resolve with the correct ingenious algorithm.

Diagonalization’s blind spots are a direct consequence of the excessive degree of abstraction that makes it so highly effective. Turing’s proof didn’t contain any uncomputable drawback that may come up in apply—as a substitute, it concocted such an issue on the fly. Other diagonalization proofs are equally aloof from the actual world, to allow them to’t resolve questions the place real-world particulars matter.

“They handle computation at a distance,” Williams mentioned. “I imagine a guy who is dealing with viruses and accesses them through some glove box.”

The failure of diagonalization was an early indication that fixing the P versus NP drawback could be a long journey. But regardless of its limitations, diagonalization stays one of many key instruments in complexity theorists’ arsenal. In 2011, Williams used it along with a raft of different methods to prove {that a} sure restricted mannequin of computation couldn’t resolve some terribly laborious issues—a outcome that had eluded researchers for 25 years. It was a far cry from resolving P versus NP, nevertheless it nonetheless represented main progress.

If you wish to show that one thing’s not potential, don’t underestimate the ability of simply saying no.

Original story reprinted with permission from Quanta Magazine, an editorially impartial publication of the Simons Foundation whose mission is to boost public understanding of science by masking analysis developments and traits in arithmetic and the bodily and life sciences.