On 13 Aug 2012, at 06:42, meekerdb wrote:

On 8/12/2012 1:57 AM, Bruno Marchal wrote:
Let phi_i be an enumeration of the (partial) computable function.

u is universal if phi_u(x, y) = phi_x(y). (x,y) = some number code for the couple (x, y)

So can y be some number code for a pair (a,b) and b a code for a pair (c,d),...?

Why not? It can make sense in some circumstance, but by (x,y) I meant just a bijection between NxN and N. What the machine will do with the numbers is up to them.

It is indeed frequent to code list of numbers like (x, y, z) by the couple (x, (y, z)). Similarly for longer sequence, by iterating that procedure.

You wrote in another post:

Evgenii: What does intelligence means in this context that life is unintelligent? Let us compare for example a bacterium and a rock. Where there is more intelligence?

Bruno: Bacteria are provably Turing complete, rocks are not.

Brent: Bacteria a certainly smarter than rocks by any reasonable measure. But I don't think a bacterium has a semi-infinite tape.

All machines, including the universal one, are finite object. Turing discovery is really the discovery of a finite Turing machine u such that phi_u(x, y) = phi_x(y).

Wolfram's problem consisted in finding the smallest possible Turing- universal machine coded in cellular automata language. Again he asked for something finite (even if needing an infinite plan to do any of its possible work).

The infinite tape is only a rather misleading pedagogical folklore. Example of universal "number" are brain, computer, programming language interpreters, etc. Universal pattern in the game of life are finite pattern. The infinite tape here is the infinite plan. The infinite tape of the human has been provided by the wall of the cavern, the pebble, knots, the papers, the books and diaries, the magnetic tapes, the physical reality itself, etc. The infinite tape plays the role of a potential infinite neighborhood in which the memory of the machine can extend itself. It is not part of any machine, as the notion of machine requires finiteness. And in that sense bacteria have infinite tapes: the soil, or liquid, or gel in which they multiply.

That is why I use the label "universal *number*", to always keep in mind that those Turing universal being are finite entities. (By number I always mean natural number 0, 1, 2, ...). The universal machines are not God. In the arithmetical translation of Plotinus, they play the role of "man", or "discursive reasoner" in Plotinus. yet, their canonical first person person attached with them (by incompleteness; or by the distinction between Bp and Bp&p) is a sort of God, at least from the machine's point of view (it is infinite, in some sense, and not arithmetically definable).

Bruno




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



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