Mike A.: Well, if you're convinced that infinity and the uncomputable are imaginary things, then you've got a self-consistent view that I can't directly argue against. But are you really willing to say that seemingly understandable notions such as the problem of deciding whether a given Turing machine will eventually halt are nonsense, simply because we would need infinite time to verify that one doesn't halt?
Ben J.: "Step 3 requires human society to invent new concepts and techniques, and to thereby perform hypercomputation. I don't think that a computable nonmonotonic logic really solves this problem." I agree that nonmonotonic logic is not enough, not nearly. The point is just that since there are computable approximations of hypercomputers, it is not unreasonable to allow an AGI to reason about uncomputable objects. "My own interpretation of the work is that an individual person is no more powerful than a Turing machine (though, this point isn't discussed in the paper), but that society as a whole is capable of hypercomputation because we can keep drawing upon more resources to solve a problem: we build machines, we reproduce, we interact with and record our thoughts in the environment. Effectively, society as a whole becomes somewhat like a Zeus machine - faster and more complex with each moment." Something like this is mentioned in the paper as objection #4. But personally, I'd respond as follows: if a society of AGIs can hypercompute, then why not a single AGI with a society-of-mind style architecture? It is difficult to distinguish between a closely-linked society and a loosely-knit individual, where AI is concerned. So I argue that if a society can (and should) hypercompute, there is no reason to suspect that an individual can't (or shouldn't). On Mon, Jun 16, 2008 at 11:37 PM, Mike Archbold <[EMAIL PROTECTED]> wrote: >> I'm not sure that I'm responding to your intended meaning, but: all >> computers are in reality finite-state machines, including the brain >> (granted we don't think the real-number calculations on the cellular >> level are fundamental to intelligence). However, the finite state >> machines we call PCs are so large that it is convenient to pretend >> they have infinite memory; and when we do this, we get a machine that >> is equivalent in power to a Turing machine. But a turing machine has >> an infinite tape, so it cannot really exist (the real computer >> eventually runs out of memory). Similarly, I'm arguing that the human >> brain is so large in particular ways that it is convenient to treat it >> as an even more powerful machine (perhaps an infinite-time turing >> machine), despite the fact that such a machine cannot exist (we only >> have a finite amount of time to think). Thus a "spurious infinity" is >> not so spurious. > > Abrahm, > > Thanks for responding. You know, i might be in a bit over my head with > some of the terminology in your paper, so to apologize in advance, but > just to clarify: "spurious infinity" according to Hegel is the sleight of > hand the happens when quantity transitions surreptiously into a quality. > At some point counting up, we are simply not talking about any number at > all, but about a quality of being REALLY SUPER BIG as we make kind of a > leap. > > According to him when we talk about infinity we are talking about some > idea of a huge number (in this case of calculations) and to use a phrase > he liked: "imaginary being." So since I am kind of a Hegelian of sorts > when I scanned the paper it looked like it argued that it is not possible > to compute something that I had become convinced was imaginary anyway. > That would be true if you bought into Hegel's definition of infinity and I > realize there aren't a log of hegelians around. But, tomorrow I will read > further. > > Mike > > >> >> On Mon, Jun 16, 2008 at 9:19 PM, Mike Archbold <[EMAIL PROTECTED]> wrote: >>>> 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/?&; >>> 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/?&; >> 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/?&; > 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
