On Tue, Jun 19, 2018 at 11:36 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:
> > On 17 Jun 2018, at 02:18, Jason Resch <jasonre...@gmail.com> wrote: > > In solving Hilbert's 10th problem > <https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem> in the > negative, the work of Martin Davis, Yuri Matiyasevich, Hilary Putnam and > Julia Robinson culminated in 1970 with the MRDP theorem > <https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem> > which concludes: > > *Every computably enumerable set has a representation as a Diophantine > equation <https://en.wikipedia.org/wiki/Diophantine_equation> (an equation > involving only integer coefficients and variables).* > > This shocked number theorists, because it meant simple equations involving > nothing more than a few integer variables have the full power of Turing > machines. In fact, it was shown by Yuri Matiyasevich that a universal > Diophantine equation can be made with as few as 9 unknowns. > > Some examples: > > - k is even if there exists a solution to: k - 2x = 0 > - k is a perfect square if there exists a solution to: k - x^2 = 0 > - k is a Fibonacci number if there exists a solution to: k^4 - k^2*x^2 > - x^4 - 1 = 0 > - (k+2) is a prime number if there exists a solution to the sum of: (these > 14 equations > <http://mathworld.wolfram.com/PrimeDiophantineEquations.html>) > - k is a LISP program having output n, if the equation having > variables: k, n, x1, x2, x3 ... x20000 (a polynomial having ~20,000 > variables <https://arxiv.org/pdf/math/0404335.pdf>) has a solution. > > The universality of Diophantine equations means there are polynomial > equations that compute things quite surprising, such as polynomials that > have solutions of 0, IFF: > > - One of the variables "k" is a valid MP3 file. > - One of the variables "k" is a JPEG image containing the image of a > cat (where the equation implements the same computation as a neural network > trained to recognize images of cats) > - For two of the variables "y" and "x", "y" equals a state of a chess > board after deep blue makes a move given a chess board with a state of "x". > - For two of the variables "y" and "x", "y" equals the state of the > Universal Dovetailer after performing "n" steps of execution. > > > The last example seems to suggest to me, that pure arithmetical truth, > concerning the solutions to equations, is identical to computation. That > is to say, certain mathematical statements carry with them (effectively) > Turing machines, and their executions. > > > Matiyazevic results is indeed quite impressive. It finishes an inquiry > begun by Davis and Putnam with important progress by Julia Robinson, and > eventually Matiyazevic got the proof, and its solved the 10th problem of > Hilbert: there is no mechanical procedure to tell if a diophantine > polynomial equation has a solution or not. (Assuming Church’s thesis, as > Matiyzevic explains well in a ten page section in his book). > > > > > > Just as all solutions to the deep-blue implementing equation is equivalent > to the computations that Deep blue makes when evaluating the board, and all > solutions to the cat recognizing equation are equivalent to the processing > done by the trained neural network, all solutions to the LISP equation are > equivalent to the execution of every possible LISP program (including the > UD). > > Does this our conscious experience might be a direct consequence of > Diophantine equations? > > > Yes. Although you could *equivalently* say that our conscious experience > is a direct consequence of the combinators laws Kxy = x and Sxyz = xz(yz). > Do you have some references that you would recommend for someone wanting to learn more about combinator laws and how they lead to universality? Is the above the same thing as a Y-combinator, or some more specific equation in lamda calculus or combinatorial logic? I wish to lean more. > Which is certainly shorter than providing a degree 4 universal Diophantine > equation, like below (I can’t resist): > > (unknowns range on the non negative integers (= 0 included) > 31 unknowns: 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 two parameters: Nu and X. > > X is in W_Nu iff phi_Nu(X) stop if and only if > > > I don't quite follow what W_Nu is here. I am guessing from the context that this means for a given machine/program Nu, and input X, this equation has a solution IFF Nu halts given X as input, but I am not sure what it means to say X is in W_Nu. > > > 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 > > END > > > Very nice. Where is this from? (Is there a paper that gives these equations?) > > > Can Diophantine equations for a single set of parameters model non-halting > programs like the UD, or one must consider the set of of all possible > parameters? > > > > In this case we model the recursively enumerable set, and they are > determined by by the existence of the solutions when you fix Nu and X, the > programs and the argument. Phi_Nu(X). But you can take a Nu which is itself > universal, and then look only at all solution for each X, or, make X a > dummy variable, and use a Nu such that phi_Nu(X) is a universal dovetailer. > Its “leaves” will enumerate the true sigma_1 sentences, which emulates all > dreams. > > > I guess the problem I am having is that Phi_Nu(X) doesn't halt, then it has no solution. So what can be said about programs like the UD which don't halt, other than that there exists no solution to the above equations when "Nu = the UD". Would it be better to establish the trace of the UD by having "Nu" be a program that outputs the state of the UD at time "X", thus the equation still captures the UD and all its computations, but now with a halting program whose solutions match the solutions to the Diophantine equation? But perhaps it's better to just leave Nu a free variable, and get all the halting computations as solutions. Is this what you mean by the "leaves of the UD"-- those programs that halt? How do you define a "sigma1 true sentence"; is a sigma1 sentence just statements/asserts of arithmetical relations that can be true or false? Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. 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