On Fri, Jul 9, 2010 at 11:37 AM, Ben Goertzel <[email protected]> wrote:
> > I don't think Solomonoff induction is a particularly useful direction for > AI, I was just taking issue with the statement made that it is not capable > of correct prediction given adequate resources... Pi is not computable. It would take infinite resources to compute it. However, because Pi approaches a limit, the theory of limits can be used to show that it can be refined to any limit that is possible and since it consistently approaches a limit it can be used in general theorems that can be proven through induction. You can use *computed values* of pi in a general theorem as long as you can show that the usage is valid by using the theory of limits. I think I figured out a way, given infinite resources, to write a program that could "compute" Solomonoff Induction. However, since it cannot be shown (or at least I don't know anyone who has ever shown) that the probabilities approaches some value (or values) as a limit (or limits), this program (or a variation on this kind of program) could not be used to show that it can be: 1. computed to any specified degree of precision within some finite number of steps. 2. proven through the use of mathematical induction. The proof is based on the diagonal argument of Cantor, but it might be considered as variation of Cantor's diagonal argument. There can be no one to one *mapping of the computation to an usage* as the computation approaches infinity to make the values approach some limit of precision. For any computed values there is always a *possibility* (this is different than Cantor) that there are an infinite number of more precise values (of the probability of a string (primary sample space or compound sample space)) within any two iterations of the computational program (formula). So even though I cannot disprove what Solomonoff Induction might be given infinite resources, if this superficial analysis is right, without a way to compute the values so that they tend toward a limit for each of the probabilities needed, it is not a usable mathematical theorem. What uncomputable means is that any statement (most statements) drawn from it are matters of mathematical conjecture or opinion. It's like opinioning that the Godel sentence, given infinite resources, is decidable. I don't think the question of whether it is valid for infinite resources or not can be answered mathematically for the time being. And conclusions drawn from uncomputable results have to be considered dubious. However, it certainly leads to other questions which I think are more interesting and more useful. What is needed to promote greater insight about the problem of conditional probabilities in complicated situations where the probability emitters and the elementary sample space may be obscured by the use of complicated interactions and a preliminary focus on compound sample spaces? Are there theories, which like asking questions about the givens in a problem, that could lead toward a greater detection of the relation between the givens and the primary probability emitters and the primary sample space? Can a mathematical theory be based solely on abstract principles even though it is impossible to evaluate the use of those abstractions with examples from the particulars (of the abstractions)? How could those abstract principles be reliably defined so that they aren't too simplistic? 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
