Re: Hey CANTORIANS!! HOW MANY DIGITS?

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 01/09/05 

Date: Sun, 09 Jan 2005 02:00:16 GMT

In sci.math, |-|erc
<h@r.c>
 wrote
on Sun, 9 Jan 2005 10:55:42 +1000
<34bdo1F480lkbU1@individual.net>:
> "The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in
>> In sci.logic, |-|erc
>> <h@r.c>
>> wrote
>> on Sat, 8 Jan 2005 17:59:06 +1000
>> <349i5tF46q2ljU1@individual.net>:
>> > (A)
>> > SEQUENCE = <314159265..........................................................................>>
>> >
>> > <--- HOW MANY DIGITS???--->
>> >
>> >
>> >
>> > (B)
>> > COMPUTABLES
>> > 1 <398498498.................>
>> > 2 <484849848.................>
>> > 3 <383873838.................>
>> > ..
>> >
>> > How many digits of (A) appear in correct sequence in (B), guaranteed?
>>
>> (a) presumably is pi, though you've not given an explicit computation
>> therefor (however, that's a minor detail, as almost everyone,
>> with the possible exception of S. Enterprize, has memorized
>> at least 5, and probably more, digits thereof).
>>
>> (b) looks like FumbleFingered Typing(tm). Did you have an explicit
>> construction method, or is this generated by, say, the
>> number of muons falling through a certain region of a
>> detector unit?
>>
>> (c) I should mention that the digit sequence of pi is computable
>> using a Turing machine; the simplest method I can think of
>> involves a modified Maclaurin sequence based on
>> 16 * atan(1/5) - 4 * atan(1/239); since the tape is infinite
>> the primary problems are technical ones, plus the issue
>> that, when extending the accumulator, the entire computation
>> may have to be redone over that accumulator. There are
>> probably more elegant methods.
>>
>> Therefore, pi may be included in your computables list.
>> But without (b) it's hard to say.
>>
>
> B is the output of a UTM(number, digit) mod 10
>
> Yes pi appears on a UTM, but you have to *guarantee*
> <314159265.................>
> is on there, (how many digits are guaranteed to be on the list)
> Ambiguous but you can't assume it is pi, (here).
>
> Herc
>
>

I can guarantee that there's a UTM that can compute pi's digits.
Of course, the more digits, the slower each cycle, but
pi = 16 * atan(1/5) - 4 * atan(1/239), and
atan(x) = x - x^3/3 + x^5/5 ..., which is computable;
hence pi is computable.

There's even a method by which one can compute hex digits
of pi. Pi may be transcendental, but it is computable.

Chaitin's Constant, however, is not. 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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