In his book Proving Darwin: Making Biology Mathematical (Knopf Doubleday 2012), Gregory Chaitin offers a "toy model" to demonstrate that progressive evolution works in principle. The idea is that DNA behavior is very similar to that of cyber software.
http://pantheon.knopfdoubleday.com/2012/05/08/proving-darwin-by-gregory-chaitin/
Computation of numbers then becomes the work of his evolution system. A mathematician, he finds, can "intelligently design" numbers that get very close to a Busy Beaver number with work growing at n. An exhaustive search of all possible computable numbers under some BB(n) requires exponential work. But, he found, his system of a climbing random walk arrived at numbers close to BB(n) on the order of n^2.
Chaitin posits a Turing-style oracle to sieve out the "less fit" (lower) numbers and the dud algorithms (those that get stuck without producing a number). The oracle represents the filtering of natural selection.
Caveat:
His system requires random mutations of relatively high information value. These "algorithmic mutations" alter multi-node sets. Point mutations, he found, were unproductive. Hence, he has not succeeded in demonstrating how the DNA system itself might have evolved.
Chaitin says he was concerned that there existed no mathematical justification for evolution. But, this assertion gives pause. The existence of the universal Turing machine would seem to demonstrate that progressive evolution is possible, though not necessarily highly probable. But granting that Chaitin was focused on probability, we can agree that if a system is of a high enough order, the probability of progressive evolution is strong. So in that respect, one may agree that Darwin has been proved right.
However, there's an old saying among IT people: "Garbage in, garbage out." The probability that random inputs or alterations will yield increased functionality is remote. One cannot say that Darwin has been proved right about the origin of life.
Remark
It must be acknowledged that in microbiological matters, probabilities need not always follow a routine independence multiplication rule. In cases where random matching is important, we have the number 0.63 turning up quite often.
For example, if one has n addressed envelopes and n identically addressed letters are randomly shuffled and then put in the envelopes, what is the probability that at least one letter arrives at the correct destination? The surprising answer is that it is the sum 1 - 1/2! + 1/3! ... up to n. For n greater than 10 the probability converges near 63%.
That is, we don't calculate, say 11^-11 (3.5x10^-15), or some routine binomial combinatorial multiple, but we have that our series approximates very closely 1 - e^-1 = 0.63.
Similarly, suppose one has eight distinct pairs of socks randomly strewn in a drawer and thoughtlessly pulls out six one by one. What is the probability of at least one matching pair?
The first sock has no match. The probability the second will fail to match the first is 14/15. The probability for the third failing to match is 12/14 and so on until the sixth sock. Multiplying all these probabilities to get the probability of no match at all yields 32/143. Hence the probability of at least one match is 1 - 32/143 or about 78%.
These are minor points, perhaps, but they should be acknowledged when considering probabilities in an evolutionary context.
