> I previously posted here claiming that the human mind (and therefore an ideal AGI) entertains uncomputable models, counter to the > AIXI/Solomonoff model. There was little enthusiasm about this idea. :) Anyway, I hope I'm not being too annoying if I try to argue the point once again. This paper also argues the point: > > http://www.osl.iu.edu/~kyross/pub/new-godelian.pdf >
It looks like the paper hinges on: "None of this prior work takes account of G¨odel intuition, repeatedly communicated to Hao Wang, that human minds "converge to infinity" in their power, and for this reason surpass the reach of ordinary Turing machines." The thing to watch out for here is what Hegel described as the "spurious infinity" which is just the imagination thinking some imaginary quantity really big, but no matter how big, you always can envision "+1", but the result is always just another imaginary big number, to which you can add another "+1"... the point being that infinity is a idealistic quality, not a computable numeric quantity at all, ie., not numerical, we are talking about thought as such. I didn't read the whole paper, but the point I wanted to make was that Hegel takes up the issue of infinity in his Science of Logic, which I think is a good ontology in general because it mixes up a lot of issues AI struggles with, like the ideal nature of quality and quantity, and also infinity. Mike Archbold Seattle > The paper includes a study of the uncomputable "busy beaver" function up to x=6. The authors claim that their success at computing busy beaver strongly suggests that humans can hypercompute. > > I believe the authors take this to imply that AGI cannot succeed on current hardware; I am not suggesting this. Instead, I offer a fairly concrete way to make deductions using a restricted class of > uncomputable models, as an illustration of the idea (and as weak evidence that the general case can be embodied on computers). > > The method is essentially nonmonotonic logic. Computable predicates can be represented in any normal way (1st-order logic, lambda > calculus, a standard programming language...). Computably enumerable predicates (such as the halting problem) are represented by a default assumption of "false", plus the computable method of enumerating true cases. To reason about such a predicate, the system allocates however much time it can spare to trying to prove a case true; if at the end of that time it has not found a proof by the enumeration method, it considers it false. (Of course it could come back later and try > harder, too.) Co-enumerable predicates are similarly assumed true until a counterexample is found. > > Similar methodology can extend the class of uncomputables we can handle somewhat farther. Consider the predicate "all turing machines of class N halt", where N is a computably enumerable class. Neither the true cases nor the false cases of this predicate are computably enumerable. Nonetheless, we can characterize the predicate by assuming it is true until a counterexample is "found": a turing machine that doesn't seem to halt when run as long as we can afford to run it. If our best efforts (within time constraints) fail to find such a > machine, then we stick with the default assumption "true". (A > simplistic nonmonotonic logic can't quite handle this: at any stage of the search, we would have many turing machines still at their default status of "nonhalting", which would make the predicate seem > always-false; we need to only admit assumptions that have been > "hardened" by trying to disprove them for some amount of time.) > > This may sound "so easy that a Turing machine could do it". And it is, for set cutoffs. But the point is that an AGI that only considered computable models, such as AIXI, would never converge to the correct model in a world that contained anything uncomputable, whereas it seems a human could. (AIXI would find turing machines that were > ever-closer to the right model, but could never see that there was a simple pattern behind these ever-larger machines.) > > I hope that this makes my claim sound less extreme. Uncomputable models of the world are not really so hard to reason about, if we're comfortable with a logic that makes probably-true conclusions rather than definitely-true. > > > ------------------------------------------- > agi > Archives: http://www.listbox.com/member/archive/303/=now > RSS Feed: http://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > http://www.listbox.com/member/?&; Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: http://www.listbox.com/member/archive/303/=now RSS Feed: http://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: http://www.listbox.com/member/?member_id=8660244&id_secret=106510220-47b225 Powered by Listbox: http://www.listbox.com
