On Fri, Jul 2, 2010 at 2:25 PM, Jim Bromer <jimbro...@gmail.com> wrote: > > There cannot be a one to one correspondence to the representation of >> the shortest program to produce a string and the strings that they produce. >> This means that if the consideration of the hypotheses were to be put into >> general mathematical form it must include the potential of many to one >> relations between candidate programs (or subprograms) and output strings. >> >
> But, there is also no way to determine what the "shortest" program is, > since there may be different programs that are the same length. That means > that there is a many to one relation between programs and program "length". > So the claim that you could just iterate through programs *by length* is > false. This is the goal of algorithmic information theory not a premise > of a methodology that can be used. So you have the diagonalization problem. > A counter argument is that there are only a finite number of Turing Machine programs of a given length. However, since you guys have specifically designated that this theorem applies to any construction of a Turing Machine it is not clear that this counter argument can be used. And there is still the specific problem that you might want to try a program that writes a longer program to output a string (or many strings). Or you might want to write a program that can be called to write longer programs on a dynamic basis. I think these cases, where you might consider a program that outputs a longer program, (or another instruction string for another Turing Machine) constitutes a serious problem, that at the least, deserves to be answered with sound analysis. Part of my original intuitive argument, that I formed some years ago, was that without a heavy constraint on the instructions for the program, it will be practically impossible to test or declare that some program is indeed the shortest program. However, I can't quite get to the point now that I can say that there is definitely a diagonalization problem. Jim Bromer ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
