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.

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. 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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