Penrose's Computing Pi Description?

Hello,

I have a question about Penrose's explanation on
Page 50 of "The Emperor's New Mind" (ENM)

"Likewise, _finite_ decimal expansions of unresticted length
give us nothing new, since they are just particular cases of
fractions. For example, the finite decimal approximation to
the irrational number Pi, given by 3.14159265, is simply
the fraction 314159265/100000000. However, _infinite_
decimal expressions, such as the _full_ non-terminating
expansion Pi = 3.14159265358979 . . .
present certain diffiuclties. Neither the input nor the output
of a Turing machine can, strictly speaking, be an infinite
decimel. One might think that we could find a Turing
machine to churn out _all_ the successive digits,
3,1,4,5,9, . . ., of the above expansion for Pi one after
another on the output tape, where we simply allow the
the machine to run on forever. But this is *not allowed*
for a Turing machine."

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?

Penrose continues: "We must wait for the machine to
halt (indicated by the bell ringing!) before we are
allowed to examine the output. So long as the machine
has not reached a STOP order, the output is subject
to possible change and so cannot be trusted. After
it has reached STOP, on the other hand, the output
is necessarily finite."

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

Penrose continues: "There is, however, a procedure
for _legitimately_ making a Turing machine produce
digits one after another, in a way very similar to this.
If we wish to generate an infinite decimal expansion,
say that of Pi, we could have a Turing machine
produce the whole-number part, 3, by making the
machine act on 0, then we could produce the first
decimal digit, 1, by making the machine act on 1,
then the second decimal digit, 4, by making it act
on 2, then the third, 1, by making the machine act
on 3, and so on. In fact, a Turing machine for
producing the entire decimal expansion of Pi,
in this _sense_ certainly _does_ exist though it
would be a little complicated to work it out
explicitly." ...

SH: I do understand that the value of the decimal
expansion has to be finitely completed, and that
there is an unending string of finite values which
make the the sequence. But I don't understand
why this needs to be accomplished with a STOP
order after every digit computed?

In the first example given by Penrose, isn't each
digit of Pi written after a finite process so that
a STOP order could be present after say 200
consecutive computations of Pi and this would
output the same 200 values in the expansion of
Pi as using method number two, in which the
TM receives a STOP order after every value
computed? Wouldn't they produce the exact
same sequence either way? No treacherous
value? I'm trying to find out why finitely computing
a value and continuing doesn't work and why only
finitely computing a value and printing that value
out, then appending that digit's value to the basis
before continuing, is how it must be done? (Chris)

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

What evil lurks in the innards of a TM in a hurry,
Stephen

What do white snow and white purses have in common?
You can't make an igloo out of an albino pig's ear.





.

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