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/ > > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
