Happy Anniversary To Turing’s Paper



The anniversary of one of the greatest papers of all time



Alan Turing is Alan M. Turing is Alan Mathison Turing, OBE, FRS.

Today I wish to acknowledge that Turing’s famous paper is having its seventy-fifth anniversary.

The paper, “On Computable Numbers, with an Application to the Entscheidungsproblem,” was submitted to the Proceedings of the London Mathematical Society on May 28 1936, exactly seventy-five years ago. Like the sixtieth anniversary, the seventy-fifth is traditionally marked with diamonds.

Since 1917 congratulatory messages from the Monarch have been sent to those celebrating their 60th, 65th, and 70th wedding anniversaries, and every year after the 70th. Apparently the cards come with a special personal note from the Queen. They should also send cards to classic papers—perhaps this paper will get a card.

Turing had many other wonderful results, and I thought that on this great anniversary we might list some of the things named for Turing. See this for a discussion why so many things seem to be named “Gauss-X.” The following list proves that some were indeed named for others. Many of the things named for Turing are extremely important—the discovery of even one of these would have been enough to make one famous. Turing did them all.

Turing X

Here is a partial list of things named for Turing:

Turing Machine: This is the model of computation that is the best model of computation among all models of computation. That is pretty repetitive, but I wanted to make the point: Turing Machines are the model of computation. There are many alternative models: some based on string operations, some on registers, some based on various functional systems, and some on various types of logic. None seems as simple and natural as Turing’s model.

Turing Reduction: This is a relationship between computational problems. A problem is Turing reducible to provided there is an algorithm that can solve given as a subroutine. Emil Post first named the notion “Turing reducibility.”

Turing Degree: The notion of Turing reduction defines, in a natural way, an equivalence relation on sets of integers: two are equivalent if they are reducible to each other. This then defines what are called Turing Degrees. This also was named after Turing by Post.

The theory of degrees is one of the great areas of research in mathematics. The area has some amazing results, at least partially because the notion of what is computable relative to something else is very subtle. Turing proved, in terms of degrees, that there were two degrees: is used to denote those problems that can be computed and is those problems that are equivalent to the Halting Problem. Post, that’s Emil again, raised the beautiful question of was there anything between and ? The answer is yes, and the method of proof is called the priority method. This method has far-reaching consequences, and has been used in complexity theory by such experts as Harry Buhrman, Stephen Fenner, Lance Fortnow, and Leen Torenvliet in their paper.

The theory of degrees is still not completely understood, even though great progress has been made since Turing. In a sense our work on and is way below this theory. Almost always we are interested in problems that lie in the lowest degree . It reminds me of a quote by Stanislaw Ulam about combinatorics:

The infinite we shall do right away. The finite may take a little longer.

Turing Test: This is based on another famous idea of Turing. He defined his famous test, a test that could tell if a machine could fool a human. Today this idea is flipped around and used to detect machines or robots—this is the notion called CAPTCHA of Luis von Ahn, Manuel Blum, Nicholas Hopper, and John Langford.

Church-Turing Thesis: This states that all computable functions are computable by Turing machines. In modern complexity this has evolved in two directions. The major change is to try to define what is feasibly computable? Does polynomial time capture feasibility? or does one need to add randomness? or even add quantum operations? One of the major open problems in all of complexity theory is: Does quantum really add computational power?

Turing Switch: This is a type of boolean function that was invented by Jon Crowcroft, who named it after Turing. It acts as a kind of router—see the diagram below.

Good-Turing Estimation: This is a statistical method named after Turing and Irving Good. It concerns the estimation of the colors of balls in urns after sampling. Good was an assistant of Turing who went on to publish a paper that detailed the analysis of the method. Apparently it was used to help decode German ciphers during the war.

Among non-conceptual things, we have:

Alan Turing Way: This is a major road in Manchester, England, where he lived and worked in his last years. Information for a statue of Turing near the University of Manchester notes that a bridge on that road has also been named for him.

And finally:

Turing Award: This is, as we all know, the top award in Computer Science, and is often called the “Nobel Prize of Computer Science.” This years winner is Leslie Valiant, who is one of our own—a theorist. Of course Turing did not invent or discover this, but he no doubt would have won it had it been available when he was alive. Would have made a cool press headline: Alan Turing Wins Turing Award.

“Alan Turing Facts”?

There is an Internet phenomenon called Chuck Norris Facts. Norris is a martial arts specialist and tough-guy actor, and the “facts” imbue him with super-human powers for comic effect. For instance: “Chuck Norris can slam a revolving door.” “Chuck Norris eats nails as his breakfast cereal without any milk.” “There is no theory of evolution—just a list of critters Chuck Norris has allowed to live.”

There is now a similar site for Gauss Facts. These have a mathematical flavor: “Gauss can recite all the digits of backwards.” “The Monster Group is scared of Gauss.” `Gauss didn’t discover the normal distribution—instead Nature conformed to his will.” Relevant to our earlier column on Gauss is, “A mathematical discovery is just an unpublished contribution of Gauss.”

Can you come up with computer science-specific humorous hyperbole as “Alan Turing Facts”? Here are just a few to start; you can probably do better:

Alan Turing would have vetoed Moore’s Law.

Alan Turing could break any pseudo-random generator—with just a pencil, a single sheet of paper, and an hour or two.

Garry Kasparov didn’t lose to Deep Blue. It was Alan Turing inside the box running a Turing Machine.

Alan Turing had a secret way to solve the Halting Problem.

Open Problems

Did I miss some other mathematical things named for Turing? I know that there are countless other things named for him: buildings, labs, fellowships, seminars, and many other things.

[fixed typo “Chuch” before “Norris”—glad he didn’t discover that!:-]