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). 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.

If you prove "1)" and "2)" for the CY machines, you are done.

Of course another way to prove that would be to directly construct one universal CY machines, emulating for example one universal Turing machine, or the SK combinators.

Probably the paper by Schmidhuber on formal strings, that I refer to you some times ago, should help.

Bruno



http://iridia.ulb.ac.be/~marchal/



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