Bruno Marchal wrote:
>
> On 02 Dec 2008, at 20:06, Brent Meeker wrote:
>
>> Bruno Marchal wrote:
>> ...
>>> ------------- technical footnote to be seen by technically inclined
>>> reader -------------------------------------
>>> (*) I think that not so much people here realize that the Universal
>>> Machine and the Universal Dovetailing are things very specific and
>>> non
>>> trivial. You can see an explicit Universal Dovetailer described in
>>> the
>>> language LISP by clicking on GEN et DU for a pdf here http://iridia.ulb.ac.be/~marchal/bxlthesis/consciencemecanisme.html
>>> Or better, thanks to the crazily formidable work of H. Putnam, M.
>>> Davis, J. Robinson, Y, Matiyasevitch, and with the help of J. Jones:
>>> here is a purely equational presentation of a universal machine in
>>> the
>>> integers:
>>>
>>> There are 31 unknowns ranging on the non negative integers (= 0
>>> included):
>>> A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, W, Z, U,
>>> Y, Al, Ga, Et, Th, La, Ta, Ph, and there are two parameters: Nu
>>> and X.
>>>
>>> The solution of the following system of diophantine equations define,
>>> taking together, one view, very precise here, of the mathematical
>>> object that I am talking about. I think the Mandelbrot set is
>>> another,
>>> one, and of course a dovetailer in Lisp, another one. Robinson
>>> Arithmetic gives yet another short one, expressible in first order
>>> logic with the symbol 0,S, +, *, and very few axioms, and it is the
>>> one needed to begin the interview of a lobian machine (which can
>>> "known" they are universal). Without allowing any other symbols than
>>> "=" and an implicit "E" quantifier, we can get a purely equational
>>> definition of such universal system: for those who remember the W_i,
>>> we have that X is in W_Nu (a universal relation) iff there exists
>>> numbers A, B, C, ... such that
>>>
>>>
>>> Nu = ((ZUY)^2 + U)^2 + Y
>>>
>>> ELG^2 + Al = (B - XY)Q^2
>>>
>>> Qu = B^(5^60)
>>>
>>> La + Qu^4 = 1 + LaB^5
>>>
>>> Th + 2Z = B^5
>>>
>>> L = U + TTh
>>>
>>> E = Y + MTh
>>>
>>> N = Q^16
>>>
>>> R = [G + EQ^3 + LQ^5 + (2(E - ZLa)(1 + XB^5 + G)^4 + LaB^5 + +
>>> LaB^5Q^4)Q^4](N^2 -N)
>>> + [Q^3 -BL + L + ThLaQ^3 + (B^5 - 2)Q^5] (N^2 - 1)
>>>
>>> P = 2W(S^2)(R^2)N^2
>>>
>>> (P^2)K^2 - K^2 + 1 = Ta^2
>>>
>>> 4(c - KSN^2)^2 + Et = K^2
>>>
>>> K = R + 1 + HP - H
>>>
>>> A = (WN^2 + 1)RSN^2
>>>
>>> C = 2R + 1 Ph
>>>
>>> D = BW + CA -2C + 4AGa -5Ga
>>>
>>> D^2 = (A^2 - 1)C^2 + 1
>>>
>>> F^2 = (A^2 - 1)(I^2)C^4 + 1
>>>
>>> (D + OF)^2 = ((A + F^2(D^2 - A^2))^2 - 1)(2R + 1 + JC)^2 + 1
>>>
>>> This is an explicit "theory of everything" acceptable for a
>>> computationalist. Assuming QM correct, Schroedinger equation (and the
>>> phenomenological quantum collapse) have to be derived from that, by
>>> those who believes in comp, or those who want to test comp.
>>> Such equations determine a "consciousness flux", and matter emerges
>>> in
>>> a precise way from observational invariance.
>>> No need, to understand this (at this stage). It can help to have
>>> images later to understand the difference between a computation,
>>> and a
>>> description of a computation, and how computations can emerge from
>>> number relation, and why this is non trivial. And things like that.
>> I don't remember the W_i, but without doing the math I can accept
>> that for a
>> given value of Nu=j the above equations pick out some values of X
>> which allow
>> them to be satisfied by integer values of A...Ph, and you express
>> this as X has
>> property W_j. But what does it mean to say W_Nu is a "universal
>> relation"? Has
>> any explicit solution to this set of equations been found?
>
>
>
> W_i is the ith recursively enumerable set (having fixed some universal
> system). It is the domain of the ith partial recursive function F_i.
> All computation can be reduce in a problem of the sort J belongs to W_i.
> For example, you can find a value of Nu so that positive X belongs to
> W_Nu if and only if X is a prime number. Indeed the set of prime
> numbers is recursively enumerable.
> The diophantine equation above is a universal (in the sense of Turing,
> or in the sense of Church thesis ...) diophantine equation.
Very interesting. So is the set of values which make W_Nu the set of primes known?
>
> This has solved negatively Hilbert 10th problem: is there an algorithm
> for solving diophantine equation?. Such an algorithm cannot exist
> because it would solve the halting problem, by the relation above.
>
> Note that if the variable range over the real numbers, such algorithm
> does exist (Sturm Liouville, Tarski). This shows that Diophantine
> equation on the reals are really much simpler than on the natural
> numbers or the integers. The real are an oversimplification of the
> natural!
>
> I guess this can help to understand why I do not think a rock can
> implement a complex computation. The intuition I guess comes here from
> the fact that even a point moving on a line is running over all
> descriptions of all computation (given that all infinite decimals
> strings can be associated to reals). This is true, but again a
> description of a computation is not (necessarily) a computation or a
> computable object. A computation is a more sophisticated object, and
> digitalness makes all the difference. In a rock, I don't see any
> working digitalness, nor even analogs of this digitalness.
Isn't this a matter of interpretation? Even the 1s and 0s of a digital computer
are not really digital - they are just approximately high or low voltage states
- and even in our most fundamental theory, i.e. quantum mechanics, they are
described by continuous functions on a Hilbert space. So we might consider the
motion of an atom in a rock to be zero is it less than some value and 1 if it is
greater.
Brent
>Even if
> Loop Gravity is correct and nature is digital, would a rock be to
> little to run any significant portion of a universal deployment, for
> example.
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>
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Received on Wed Dec 03 2008 - 13:31:22 PST