Bruno Marchal writes:> >> Let us define what is a computable infinite sequence. A sequence is> >> computable if there is a program (a machine) which generates> >> specifically the elements of that sequence in the right order, and> >> nothing else. The set of programs is enumerable, but by Cantor theorem> >> the set of *all* sequences is not enumerable. So the set of computable> >> sequences is almost negligible compared to the arbitrary one.> >>> >> Does it mean there is no program capable of generating a non > >> computable> >> sequence?> >>> >> Not at all. A universal dovetailer generates all the infinite> >> sequences. The computable one, (that is, those nameable by special> >> purpose, specific, program) and the non computable one (how? by> >> generating them all).> >>> >> I give another example of the same subtlety. One day a computer> >> scientist told me that it was impossible to write a program of n bits> >> capable of generating an incompressible finite sequence or string of> >> length m with m far greater than n. I challenge him.> >> Of course, what is true is that there is no program of n bit capable > >> of> >> generating that m bits incompressible string, AND ONLY, SPECIFICALLY,> >> THAT STRING.> >> But it is really easy to write a little program capable of generating> >> that incompressible string by letting him generate ALL strings: the> >> program COUNT is enough.> >>> >> I think this *is* the main line of the *everything* list, or a> >> miniature version of it if you want.> >> > Yes, and there are many related examples, like Borges' library; I > > would include> > the computations that might be hiding in noise as another such example.> > > We will have to go back on this. I would compare the Borges' library > (or its countable infinite generalization) as equivalent with the > counting algorithm. It generates all (finite) strings, and a lot of > computations can be considered as being embedded in those strings. > Still I consider that the counting algorithm is not equivalent with a > universal dovetailer. I will try to explain the difference with some > details, but roughly speaking, what the UD does, and what neither the > rock nor the counting algorithm really do, is that the UD generates > both the program codes and their finite and infinite running. Saying > that all computations are generated by the counting algorithm makes > sense only if we add a universal interpreter in the description.> I can anticipate that you will say this does not change anything from > inside. But remember that once we abandon the "physical" supervenience > thesis, we will replace it by the computational supervenience thesis > which ask us to be precise of what is a mathematical computations. We > can no more relate on notion of time space, energy, etc.> The easy way to do that consists in defining in some axiomatics what > will be necessary for both the existence of computations and the > existence of internal observer related to those computations. In that > case a counting algorithm is just equivalent to the first three axiom > of Peano Arithmetic. This is provably not enough to define or execute > all programs. It appears that by adding the definition of addition and > multiplication (and order) you get the turing universal level (and then > with the induction axioms you get the internal lobian observer. > Everything will emerge from (mathematical interrelation between those > beings).> I'm afraid I'm not clear, and I will think how to make this clearer > without going to much into the technics.It seems to me that abstract machines have been created for our benefit, rather like mathematical notation or human language. That is, they allow us to think about algorithms and to consider how we might build a physical machine to carry them out, even if this is not actually done in practice. If you do away with the possibility of physical implementation and if you consider only first person experience, what purpose is served by an abstract machine? The quotient of two numbers does not depend on a long division algorithm, or any algorithm running on any machine; it simply *is*.> > The> > significant thing in all these cases is that from the third person > > perspective, the> > information or computation is inaccessible. You need to have the book > > you want> > already before you can find it in the Library of Babel. However, if > > computations> > (or books) can be conscious, then they will still be conscious despite > > being unable> > to communicate with the world at the level of their implementation. > > The first person> > perspective makes these situations non-trivial.> > OK, but as far as they can communicate from their inside points of view > (btw: thanks for the spelling!) you are adding implicitly some addition > and multiplication laws. Once we stop to take for granted what exists > or not, such little nuance have some importance, especially for > deriving concretely physics from something else.Communicating with the outside world changes everything, but we can put a box around the whole system and declare it closed. This closed system (which may contain many interacting conscious entities) exists somewhere in the Library of Babel, in the output of the "count" program, in noise, in the decimal expansion of pi, etc., and even though we outside the the system cannot find it or interact with it, its inhabitants are going about their business regardless. > >> Now, when you run the UD, as far as you keep the discourse in the > >> third> >> person mode, everything remains enumerable, even in the limit.> >> But from the first person point of view, a priori the uncountable> >> stories, indeed generated by the UD, take precedence on the computable> >> one: thus the continua of white rabbits. This results from the lack of> >> any possibility from the first person point of view to locate herself> >> into UD*. Somehow the first person belongs to 2^aleph_zero histories > >> at> >> the start.> >> > Can you explain again why only the countable stories appear to the 3rd > > person> > but the 1st person sees the uncountable ones as well? Also, why should > > the> > white rabbits prefer the uncountable habitat?> > > The computable stories can be said to be generated by programs, and > thus are enumerable (countable).> But the UD generates not only all programs and their execution, it > generates also all data, including all streams, all oracle, etc.> Of course he never generates a complete streams in one strike, but by > dovetailing on the reals (which can be defined by a finite subroutine, > he can present such real streams through finite but enough big portion > to some program who "want to read them".> Now, for an external observer everything at any time (or number of > steps of the running of the UD) is countable. We can see finite portion > of the stream to be generated with some advance and then presented to > the program, then we have to wait a billion years (say) to see the > "same program" getting a larger portion of its stream datum.> Now, put yourself in front of a concrete UD, like in step 7 of the > version of UDA in 8 steps.> cf the pdf slide: >
http://iridia.ulb.ac.be/~marchal/publications/SANE2004Slide.pdf> To predict your "future history", and taking into account the first > person is unaware of delays of computation introduced by the UD (due to > its dovetaling), and taking into account that the UD generates all > programs infinitely often (necessarily), and taking into account that > the "probability measure" cannot bear on the computational states but > only on the (infinite) computational histories: all that entails that > you have to take into account ALL computational histories going through > your current state. Well, now take just into account the infinite > dumbness of the UD, which is that for all programs it will generate its > execution on all streams, and thus on all arbitrary sequences > (including first person descrition of white rabbits) which makes a non > countable set (!!!, by Cantor) then, well, then it *seems* we cannot > avoid white noise and flying pigs and all slight variants, etc... OK ? > (don't hesitate to say NO, I'm quick and it is not easy). Are you OK > with any "taking into account" described above?> Of course, such measure is a bit too much intuitive: a priori all > probabilities of histories add up, and we could a bit naively take this > as a refutation of comp. What refrains us to jump toward that > conclusion, is that such intuitive probabilities have not enough taken > into account the difference between the points of view, something any > self-referentially correct universal machine can be shown to be able to > do, thanks, not really to incompleteness, but thanks to the fact that > machine can reason about their own incompleteness (leading to the > arithmetical points of view/hypostases). This motivates then the AUDA > (Arithmetical version of UDA, ... or of Plotinus, actually).OK, I think I understand why you can see all the arbitrary sequences from inside the UD but not the outside (it relates to the irrelevance of delays from the inside), but I don't see why these extra sequences should be more likely to encode white rabbit universes than the 3rd person observable ones. Also, I still don't understand how you will avoid the white rabbits.> >> A similar "explosion of stories" appears with quantum mechanics, > >> except> >> that here the physicist as an easy answer: white rabbits and Potter> >> universe are eliminated through phase randomization (apparently).> >>> >> I am not satisfied by this answer if only because my motivation is to> >> understand where that quantum comes from.> >>> >> Is complex randomization of histories the only way to force normal> >> nature into the shorter path?> >>> >> Well, my point is that if we take comp seriously, we have to justify> >> the absence of rabbits from computer science. In case too much white> >> rabbits remains, comp would be false, and this would be an argument in> >> favor of materialism. But, when you interview a universal machine on> >> this question you can realize at least that this question is far from> >> being settled.> >> > If QM emerges from comp, does that solve the problem?> > > As far as QM is confirmed by nature, if QM can be justified by comp, it > would mean nature confirms comp, and thus its immateriality. In this > sense the problem would be solved: materialism would be false. If QM is > confirmed by nature (an infinite process, note) and if comp predicts > something different, then it will depend. Some circumstances could lead > to argument in favor of materialism. I mean I can speculate about this.Stathis Papaioannou
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Received on Fri Jan 26 2007 - 05:12:03 PST