On 12 Jan 2014, at 17:50, Richard Ruquist wrote:
On Sun, Jan 12, 2014 at 11:32 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 12 Jan 2014, at 15:30, Richard Ruquist wrote:
Bruno: Those machines are enumerable. There is an enumeration of
all of them: m_0, m_1, m_2, m_3, m_4, ...
Richard: We are in close agreement if the digital machines are each
a Calabi-Yau CY Compact Manifold that can be enumerated.
Bruno: So, you can fix one universal language, like a base, and
identify each machine with a number.
Richard: Agreed presuming that the base is an m_i and the unique
universal language to that machine involves all other machines.
Bruno: Each programming language, or computers boolean net,
correspond to some m_i, and are universal m_i, as they can imitate
all others machines (accepting Church thesis).
Richard: You seem to be identifying each machine with a programming
language that has the property of imitating all other enumerated
machine.
Is it sheer coincidence that for more than one string theory
consideration, each CY machine relects or perceives (or perhaps it
can be said is conscious of) all other machines. So I conjecture
that the CY machines satisfy the Church Thesis. Can that be proven
or falsified?
Wow! Pretty difficult question. To prove this you need not just to
enumerate the objects, but to define how they compute: what they do
when presenting data. What would be a data for a CY machines? Could
a CY machines never stop? What would that mean? can you give me a CY
which generates the Fibonacci numbers?
Thanks to a work by Rogers, an enumeration of machine m_i is Turing
universal, if each partial computable phi_i is computed by some m_i,
and if the list of the corresponding phi_i obeys the two rules:
1) Universal machine existence: there is a u such that phi_u(x, y) =
phi_x(y) (U emulates x, for all x, on any y).
1) seems almost obvious if each machine perceives all others yet has
a unique perception..
Most m_i are not universal. Only the m_u are, which are those
computing the phi_u, capable of emulating all phi_i (as phi_u(i, x) =
phi_i(x)).
Having a unique perception will define your 1-person, but not your
universality. Universality is cheap, and the CY might be universal,
but I doubt that this is obvious.
In case you insist that it is obvious, just gives me the CY computing
the factorial function. Better: give me a program written in LISP
emulating the CY computing factorial(5).
2) Automated Parametrization: all computable functions with n
arguments (x, y, z, t, ...) can be transformed into a function of
n-1 arguments by some function SMN fixing his argument to some value:
phi_i(x, y, z, t, ...) = phi_SMN(x) (y, z, t, ...). Note that SMN
is a metaprogram: it acts on the indices of the phi_i.
2) I do not understand. No wait. I am getting a glimmer. Lets
suppose phi_i(x,y,z,t...) were the laws of physics.
No phi_i at all computes the physical laws, as the physical laws
emerges from all computations (or from our relative ignorance on which
computations supports us below our substitution level).
SMN just says that there is a program capable of doing the
parametrization. For example you give it a program computing the
addition x+y, and you give it x = 4, the parametrization program
(S21, here) will output the code of a program computing (4 + y).
S21 (4, "x + y") = "4 + y".
The SMN just do some substitution, and might eliminate some "read x"
in the program given as input.
Ohh, nevermind (delete). Ref for Rogers, please?
ROGERS H., 1958, Gödel Numbering of the Partial Recursive Functions,
Journal of
Symbolic Logic, 23, pp. 331-341.
But it presupposes familiarity with theorems like the SMN theorem, so
you might buy the bible of recursion theory, written by the same Rogers:
ROGERS H.,1967, Theory of Recursive Functions and Effective
Computability, McGraw-
Hill, 1967. (2ed, MIT Press, Cambridge, Massachusetts 1987).
A good introductory book is the one by Cutland:
CUTLAND N. J., 1980, Computability An introduction to recursive
function theory,
Cambridge University Press.
Someday I will prove Kleene's second recursion theorem (which is the
math of the Dx = "xx" method) by using only the SMN (and the
diagonalization).
Bruno
http://iridia.ulb.ac.be/~marchal/
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