I have Matiyasevich's paper on the MRDP theorem. I have not as yet read it. I have had this idea that a general scheme for quantum eigenvalues could by Diophantine. This would then be a sort of universal dovetailer of all possible physical states. Unfortunately this is an area I have thought about some, but as yet have never endeavored to pursue.
LC On Saturday, June 16, 2018 at 7:18:41 PM UTC-5, Jason 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. > > 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? > > 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? > > 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
