On 02 Apr 2015, at 15:04, Bruce Kellett wrote:
Bruno Marchal wrote:
On 02 Apr 2015, at 04:08, Bruce Kellett wrote:
Emulation is a dynamical process in time. I wonder where you get a
time variable for your UTM.
By a variable on the computational steps. It has nothing to do with
a physical time a priori.
What variable? A simple numbering of steps? But that will not work,
or at least, you have hidden an assumption of an external time in
your notation.
The external time is given by the universal machine running the
computation. It can be the basic level (elementary arithmetic, or the
universal dovetailer), or it can be some other universal layer,
running on some other universal layer, running on some other ....
running on the basic level.
At the basic level, we have the block mindscape, like the UD* (the
infinite "cone" of all computations).
"At the end of step 27, move to step 28." That contains an implicit
notion of time -- 'ending' and 'moving' are temporal concepts. I do
not see that you can remove all traces of the idea of an external
temporal parameter. Otherwise the machine could just halt
arbitrarily at some point and never know that it had halted.
I think you ned to flesh your ideas here out a great deal more.
Well, this is a forum, and I explain things already explained with all
details in much longer text (which sometime does not help, because
busy people tend to skip even more the long texts nowadays).
Let me try to help a bit though. Fix some universal programming
language, like Fortran, say. Enumerate all programs computing function
with 1 argument, p_0, p_1, p_2, ....
Let us denote by [p_i(j)^k] the kième step of the execution of the
ième program on argument j, by the universal dovetailer (which
dovetails then on all such [p_i(j)^k] .
Then we can define, indeed already in Robinson arithmetic, a
computation by a sequence of such steps, when i and j are fixed. So a
computation is given by the sequence
[p_456(666)^0]
[p_456(666)^1]
[p_456(666)^2]
[p_456(666)^3]
[p_456(666)^4]
etc.
This sequence is a subsequence of the general universal dovetailing,
which dovetails on all [p_i(j)^k].
It is a computation, only in virtue of the universal dovetailing, and
the universal dovetailing can be defined in arithmetic. I can
translate the proposition "the UD access to [p_345(898786)^89]"
entirely in term of arithmetic, using only the notion of addition,
multiplication, successor (of natural numbers) and 0, and predicate
logic.
The only external time used is the ordering of the natural number,
which is easily translated in arithmetic: x < y means Ez(x + z) = y).
OK?
Bruno
Bruce
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