Computable reals

The stated purpose of Alan Turing's original 1936 paper on computation was to introduce the notion of computable real numbers, whose n^{th} digit for any n could be found by a Turing machine in a finite number of steps. Real numbers used in any explicit way in traditional mathematics are always computable in this sense. But as Turing pointed out, the overwhelming majority of all possible real numbers are not computable. For certainly there can be no more computable real numbers than there are possible Turing machines. But with his discovery of universality, Turing established that any Turing machine can be emulated by a single universal Turing machine with suitable initial conditions. And the point is that any such initial conditions can always be encoded as an integer.

As examples of non-computable reals that can readily be defined, Turing considered numbers whose successive digits are determined by the eventual behavior after an infinitely long time of a universal system with successive possible initial conditions (compare page 964). With two possible forms of behavior h[i] = 0 or 1 for initial condition i, an example of such a number is Sum[2^{-i} h[i], {i, ∞}]. Closely related is the total probability for each form of behavior, given for example by Sum[2^-(Ceiling[Log[2, i]]) h[i], {i, ∞}]. I suspect that many limiting properties of systems like cellular automata in general correspond to non-computable reals. An example is the average density of black cells after an arbitrarily long time. For many rules, this converges rapidly to a definite value; but for some rules it will wiggle forever as more and more initial conditions are included in the average.