Ben, The most extreme case is if we happen to live in a universe with uncomputable physics, which of course would violate the AIXI assumption. This could be the case merely because we have physical constants that have no algorithmic description (but perhaps still have mathematical descriptions). As a concrete example, let's say some physical constant turns out to be a (whole-number) multiple of Chaitin's Omega. Omega cannot be computed, but it can be approximated (slowly), so we could after a long time suspect that we had determined the first 20 digits (although we would never know for sure!). If a physical constant turned out to match (some multiple of) these, we would strongly suspect that the rest of the digits matched as well.
(Of course, the actual value of Omega depends on the model of computation employed, so it would be very surprising indeed if the physical constant matched Omega for one of our standard computational models...) AIXI would never except this inductive evidence. This is similar to Wei Dai's argument about aliens offering humans a box that seems to be a halting oracle. I think there is a less extreme case to be considered (meaning, I think there is a broader way in which we might say AIXI cannot "understand" uncomputable entities the way we can), but the argument is probably clearer for the extreme case, so I will leave it at that for now. Clearly, this argument is very "type 2" at the moment. What I *really* would like to discuss is, as you put it, the set of sufficient mathematical axioms for (patially-)logic-based AGI such as OpenCogPrime. --Abram On Mon, Oct 20, 2008 at 9:45 AM, Ben Goertzel <[EMAIL PROTECTED]> wrote: > > I do not understand what kind of understanding of noncomputable numbers you > think a human has, that AIXI could not have. Could you give a specific > example of this kind of understanding? What is some fact about > noncomputable numbers that a human can understand but AIXI cannot? And how > are you defining "understand" in this context? > > I think uncomputable numbers can be indirectly useful in modeling the world > even if the world is fundamentally computable. This is proved by > differential and integral calculus, which are based on the continuum (most > of the numbers on which are uncomputable), and which are extremely handy for > analyzing real, finite-precision data ... more so, it seems, than > "computable analysis" variants. > > But, I think AIXI or other AI systems can understand how to apply > differential calculus in the same sense that humans can... > > And, neither AIXI nor a human can display a specific example of an > uncomputable number. But, both can understand the diagonalization > constructs that lead us to believe uncomputable numbers "exist" in some > sense of the word "exist" > > -- Ben G > > On Sun, Oct 19, 2008 at 9:33 PM, Abram Demski <[EMAIL PROTECTED]> wrote: >> >> Ben, >> >> How so? Also, do you think it is nonsensical to put some probability >> on noncomputable models of the world? >> >> --Abram >> >> On Sun, Oct 19, 2008 at 6:33 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote: >> > >> > But: it seems to me that, in the same sense that AIXI is incapable of >> > "understanding" proofs about uncomputable numbers, **so are we humans** >> > ... >> > >> > On Sun, Oct 19, 2008 at 6:30 PM, Abram Demski <[EMAIL PROTECTED]> >> > wrote: >> >> >> >> Matt, >> >> >> >> Yes, that is completely true. I should have worded myself more clearly. >> >> >> >> Ben, >> >> >> >> Matt has sorted out the mistake you are referring to. What I meant was >> >> that AIXI is incapable of understanding the proof, not that it is >> >> incapable of producing it. Another way of describing it: AIXI could >> >> learn to accurately mimic the way humans talk about uncomputable >> >> entities, but it would never invent these things on its own. >> >> >> >> --Abram >> >> >> >> On Sun, Oct 19, 2008 at 4:32 PM, Matt Mahoney <[EMAIL PROTECTED]> >> >> wrote: >> >> > --- On Sat, 10/18/08, Abram Demski <[EMAIL PROTECTED]> wrote: >> >> > >> >> >> No, I do not claim that computer theorem-provers cannot >> >> >> prove Goedel's Theorem. It has been done. The objection applies >> >> >> specifically to AIXI-- AIXI cannot prove goedel's theorem. >> >> > >> >> > Yes it can. It just can't understand its own proof in the sense of >> >> > Tarski's undefinability theorem. >> >> > >> >> > Construct a "predictive" AIXI environment as follows: the environment >> >> > output symbol does not depend on anything the agent does. However, >> >> > the agent >> >> > receives a reward when its output symbol matches the next symbol >> >> > input from >> >> > the environment. Thus, the environment can be modeled as a string >> >> > that the >> >> > agent has the goal of compressing. >> >> > >> >> > Now encode in the environment a series of theorems followed by their >> >> > proofs. Since proofs can be mechanically checked, and therefore found >> >> > given >> >> > enough time (if the proof exists), then the optimal strategy for the >> >> > agent, >> >> > according to AIXI is to guess that the environment receives as input >> >> > a >> >> > series of theorems and that the environment then proves them and >> >> > outputs the >> >> > proof. AIXI then replicates its guess, thus correctly predicting the >> >> > proofs >> >> > and maximizing its reward. To prove Goedel's theorem, we simply >> >> > encode it >> >> > into the environment after a series of other theorems and their >> >> > proofs. >> >> > >> >> > -- Matt Mahoney, [EMAIL PROTECTED] >> >> > >> >> > >> >> > >> >> > ------------------------------------------- >> >> > 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/?&; >> >> > Powered by Listbox: http://www.listbox.com >> >> > >> >> >> >> >> >> ------------------------------------------- >> >> 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/?&; >> >> Powered by Listbox: http://www.listbox.com >> > >> > >> > >> > -- >> > Ben Goertzel, PhD >> > CEO, Novamente LLC and Biomind LLC >> > Director of Research, SIAI >> > [EMAIL PROTECTED] >> > >> > "Nothing will ever be attempted if all possible objections must be first >> > overcome " - Dr Samuel Johnson >> > >> > >> > ________________________________ >> > agi | Archives | Modify Your Subscription >> >> >> ------------------------------------------- >> 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/?&; >> Powered by Listbox: http://www.listbox.com > > > > -- > Ben Goertzel, PhD > CEO, Novamente LLC and Biomind LLC > Director of Research, SIAI > [EMAIL PROTECTED] > > "Nothing will ever be attempted if all possible objections must be first > overcome " - Dr Samuel Johnson > > > ________________________________ > agi | Archives | Modify Your Subscription ------------------------------------------- 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=117534816-b15a34 Powered by Listbox: http://www.listbox.com
