We'll find all its secrets and make them our own;
A yarn to weave and be woven.
We'll choose among the least likely chosen
And build a castle stone by stone.
|Alan Turing - Superhero
Of course, there are no superheroes in real life, but there was a man whose biography pretty much mirrors the superhero story — the British mathematician Alan Turing. Turing discovered his superpower, a genius for mathematical logic and computing, at the age of twenty-four when, as a graduate student, he published a paper entitled, "On Computable Numbers, with an Application to the Entscheidungsproblem.” I'll have more to say about the ideas in this paper later, but suffice to say that Turing's result is one of the most profound triumphs of rationality. Turing took his superpower into World War II and used it to battle the Nazis. He designed a computer called the "Bombe" that he and his team used to crack the German Enigma Code, thus freeing Allied shipping and supply lines from the dreaded U-Boats and saving England from the arch-villain Adolph Hitler.
|Replica of Turing's 'Bombe' Code Breaker
Turing's remarkable 1936 paper defined both the beginning and the end of a new branch of mathematics — computation theory. In the first third of the paper, he formally defined a simple but robust computer now called a Turing Machine (TM). In the second third of the paper he proved that his little machine is universal, that it is capable of performing any computation any other computer can perform. In fact, any machine that can reasonably be called a computer, past or future, laptop or supercomputer, is no more powerful than a simple TM. Modern computers may be and generally are faster in performing some computations, but however elaborate they may be, a TM can accomplish the same thing, eventually. Turing's goal in this was to find an intuitive but formal definition of the phrase "by mechanical means," a phrase that is used quite a lot in mathematics and science.
If Turing had stopped and published the first two thirds of his paper, it would still have been an incredible achievement. It provides us with a universally applicable, easy to understand definition of what we mean by the word "algorithm", an orderly, step-by-step, procedure for doing something like verifying the proof of a mathematical theorem. Turing conjectured that we mean a computer program, specifically a Turing Machine, and today most mathematicians agree with his conjecture. Any effective method for doing something, any method that does not require ingenuity or creativity, exists if and only if there is a TM whose steps encode that method. That's an amazingly general concept which is widely known as the Church-Turing Thesis (Church was Turing's teacher, among many other accomplishments).
But Turing didn't stop after just defining what computation means in a formal sense. The final section of his paper went on to prove there are problems, infinitely many of them, that are not soluble by any TM, and therefore no deterministic procedure exists for finding all the answers. In particular, he proved there are infinitely many real numbers whose digits cannot be computed by any TM, or by any mechanical means whatsoever. I wish I could give you a concrete example of such an uncomputable number, but I cannot, and no one can. Turing proved they are out there, but to provide a finite recipe for producing one of them is fundamentally impossible! It turns out to be quite difficult to identify individual objects floating in the sea of randomness, and when we do find them they're often fuzzy and difficult to crisply discern.
Though we are unable to provide a recipe to produce all the digits of any given uncomputable number, Turing did define a specific problem that he proved could not be solved by any algorithm — the Halting Problem. Is it possible to write a compiler that, given the source code and input data of any other program, will tell us whether the program will ever halt on that input data? The answer is, no such program can possibly exist because it would lead to a logical paradox akin to the "everything I say is a lie" conundrum. Much has already been written about the halting problem, so I won't dwell on it here. It will, however, come up again when we talk about randomness in mathematics.
Our working, informal definition of a random number is any number that is (i.e. whose digits are) universally unpredictable. We've seen that nature provides copious examples of quantities whose measures at the nano-scale are random in this sense. Turing's result shows there are numbers that cannot be computed by any finite algorithm and the Church-Turing Thesis allows us to extend that to mean such numbers cannot be produced by any effective procedure whatsoever. In other words, the shortest, most compact way to describe such a number is to simply list all its digits!
Now you may think these unnameable, unspeakable numbers are rare and difficult to find, and you'd be half right about that. But they are most assuredly not rare. In fact, if you threw a very, very sharp dart and hit the real number line, the probability of hitting such an unnameable number is 1. There is a countably infinite number of nameable numbers, but an uncountably infinite number of unnameable ones. Gregory Chaitin, an American mathematician and computing theorist whom we will talk more about in the chapter on randomness in mathematics, has a name for these real numbers whose shortest possible exact descriptions are the numbers themselves. He calls them algorithmically random, but admits the concept may actually be the best definition of randomness in all its generality.
In physics, we began with the ambition to discover a small set of general principles with which all the universe could be explained, only to find that most of the universe is in fact beyond human grasp. Now, in the more orderly and man-made world of computation, we find that almost everything is uncomputable. There are infinitely many numbers, infinitely many problems, for which even the most powerful and sophisticated computer will be all but useless. Random numbers, in the sense discussed here, cannot be computed and can only come from some source outside the computer.
Turing completed his PhD dissertation at Princeton after the war. While accepting his own conclusion that some problems, like the Halting Problem, were just not tractable, he still wondered what a computer would look like if it could solve such problems, if it could somehow perform magic. In his dissertation, he augmented his TMs to include an "oracle," a magic device that could answer certain unanswerable questions. For example, a Halting Oracle was assumed to be able to discern, somehow, whether any given TM will eventually halt or whether it will run forever. Surprisingly, he found that such super-Turing machines are still limited in their abilities. A super-TM augmented with a halting oracle, for example, still suffers an inability to solve the Halting Problem as it applies to super-TMs. Modern computer scientists still use the idea of oracles to study the limits of computing. A random oracle, a function that always generates the same truly random number when given the same input, is widely used as an ideal cryptographic hash function. In Teaching Robots to Sneeze, we examined how to build what we might call a truly random oracle, which has no input parameters.
Last week, nearly sixty years after his suicide, Queen Elizabeth II granted Alan Turing a posthumous pardon for his supposed crimes. I've noticed that much of the reporting and discussion of this important event has focused on Turing's accomplishments and discoveries, which I find slightly ironic given that he spent so much of his short life contemplating the undiscoverable, the unknowable. Legendary superheroes often have a quiet place, a lair, in which to meditate and seek inspiration. Superman had his Fortress of Solitude, Batman his Batcave, and Alan Turing had the Divine Random.