X-Message-Number: 9265 Date: Wed, 11 Mar 98 12:31:54 From: Mike Perry <> Subject: Re: CryoNet #9261 Thomas writes, in regard to the "counterexample" to the Church-Turing thesis, > > In discussing the counterexample in depth I am a bit at a temporary handicap > here because I've packed away the issue of NATURE in which it was > presented, ... >And if Mike Perry puts the reference > on the net again, then we can all read it and have SOME background > for discussion. Here is the only reference I know about, other than what is listed in the paper itself, "Computation beyond the Turing Limit", by Hava T. Siegelmann, *Science* v.268 p. 545-8, 28 Apr '95 *Science* not *Nature*--is this the one you mean? If it is I have to say that it stops well short of producing a working, non-Turing computation. The author does suggest a way that such a computation *might* be done by bouncing particles off parabolic mirrors that have been given certain orientations in space. This is equivalent to his "analog-shift" device and also to a certain class of neural net. But it is not the same as a working system. The mirror setup that the author describes would encode non-Turing information in the x,y coordinates of the various mirrors the particles bounce to. It is easy to guess that this might be done, given that these coordinates could be real numbers whose digits might be non-Turing computable. The author further makes the point that it would *not* be necessary to have infinite precision in the positioning of the mirrors to carry forward the computation. However, the tradeoff is still severe: n bits might be needed for n steps of the computation! The exact wording is: "if one is interested in computing up to some time *q*, both the mirror system and the location of the particle bouncing there are not required to be described (or measured) with more than *q* bits." It is easy to see how that could quickly pass practical bounds. For 1000 steps the needed accuracy could be one part in 2^1000 or over 300 decimal places! > I did give the explicit reference to the paper in which this example was > presented on a previous Cryonet. As I recall, the only thing that was > done with the neural net was to make some of the links between nodes > incommensurable. This appears to be a reference to eq. 1 on p. 546, which gives the activation strength of a neuron on time step t+1 from the known activations of the neurons on step t. As far as I can see, the connection strengths would have to be more than just incommensurable, they would have to be non-computable real numbers! Otherwise, you would have no trouble in principle computing the activations at all times to as much (finite) precision as you needed for each step of the computation, and consequently, emulating it with a Turing machine. (The author notes that, if the connection strengths are limited to rational, ie commensurable numbers, Turing-computability is not violated, but doesn't comment on exactly when it *is* violated, other than to give a reference.) Anyway, to me this paper seems like an interesting exercise, and it may even be possible to build a kind of non-Turing machine based around it, or even something simpler. Why not build a machine, for instance, that computes the proton-electron mass ratio to successively higher precision? This number is about 1,836.1527... If it is constant but not Turing-computable (and nobody knows whether it is or not) then it might suffice. But then it might be computable after all (via a "theory of everything" say). Even if it isn't, I don't see that as a violation of the "digital" paradigm I've been advocating, just a kind of "infinitely inscribed tape" that we, with sufficient labor, might be able to read. More generally I conclude that, so far, neither the digital paradigm nor the Church-Turing thesis have been disproved. ... > > HOWEVER, there is a larger issue which I WILL discuss. Computers now, and > for some time in the future --- possibly to a time in which the entire > notion of what a brain or a computer is vanishes into the past, with both > ideas considered inadequate concepts --- will provide at best clearly > inaccurate models of brains. > My interest in comparisons between computers and brains is not for any detailed, comprehensive modeling--admittedly our computers today are very crude for this, and I don't seriously propose it. But instead it is for "in principle" type arguments, such as arguing that a brain, like any other phenomenon in our world, is just a (in this case very large) digital process. Another "in principle" argument is that artificial devices should one day be able to emulate people, i.e. uploading should become a reality. Not anytime soon though. > I very much do NOT mean to say that we cannot build devices which ARE > good models of brains, merely that they would not qualify as computers. I evidently take a broader view of "computer"--essentially, any information processing, digital device, however large or complex, including devices that haven't been built yet such as quantum computers, if they can be made. > I'll also add that I see no reason why the plan for my brain or your > brain could not be stored, statically, in a computer. (It's getting it > to work right where the problems come). > If the "plan" at one instant in time could be stored, then so could the plan at successive instants in time. A running computation that produced successive "plans" (which I maintain must form a denumerable set and thus should be computable successively, given other properties) would furnish the basis of a person-emulation such as I've referred to, i.e. uploading. All well beyond our powers today, but see you in 100 years. > Nor can we infer anything about the behavior of brains from that of > computers. Among other points major for cryonics, we aren't going to > get any idea of whether or not we can be brought back from present > suspensions by thinking along the lines of computer science. A big slab > of neuroscience will be needed here. > I think our "sciences" will have to advance considerably in any case. I can see a need for "computer science," "neuroscience" and maybe a lot besides. > I have been a professional programmer myself, in parallel computing, and > I would be the last to say that computers are not valuable. But I get > this feeling, hearing some people on Cryonet and elsewhere, that some > people now have decided they have a hammer (computers and all the theory > of computing behind them) and consequently everything else in the universe > is a nail. Me, I'm glad I have a hammer, but I've got lots of other tools > too. > "Computer science," suitably generalized, might furnish the basis of a theory of everything, i.e. a universal "hammer," and it might even be the best such basis. Time will tell. But if such a theory is possible, there will no doubt be equivalent approaches that at first look very different. Endless best, Mike Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=9265