Re: Penrose's Computing Pi Description?

Stephen Harris wrote:

<snip penrose>

SH: I realize that computing infinity does not come to
an end or reach a completion. What does this have to
do with a condition of *not being allowed* for a TM?


In the standard definition of a TM, it can only output things
after it has finished. The "output" of a non-terminating TM
is usually taken to be undefined.

<snip penrose>

SH: I realize that expansions made earlier in the
computation will not have the same number of places
as some value reached later in the calculation, but I
wouldn't think this would be described as "subject
to possible change and so cannot be trusted."
Why can't it be trusted, how can it be *changed*?


The way a TM output things, is by leaving the output on the
tape. However, the tape is also where the TM stores
intermediate results. Therefore, in principle, while the TM
is running there is no way (in general) to distinguish the
output from the intermediate results.

Of course, for some particular TM, you could prove that
some symbol on the tape will never change again. But in
general, that is not true.

I also don't fathom how his next method is
legitimate and excludes the possibility of "change"?


The difference between the two methods is that the second
stops for each digit of Pi, and therefore we can take the
digit as output. The first never stops, and therefore, by
definition, never outputs anything.

Of course, in the first case, you have to "manually" run
the TM for any natural number, so it is not really
"computing Pi" on its own, IMHO.

<snip>

Is there a  flaw in how I understand this theory works?
Could someone please clarify a bit, Penrose's thinking.


I hope this helps.

groente
-- Sander
.

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