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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Flibble's Law
Date: Wed, 23 Apr 2025 07:26:14 -0400
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On 4/22/25 11:23 PM, Keith Thompson wrote:
> Richard Damon <richard@damon-family.org> writes:
> [...]
>> Yes, there is a nuance, but there ARE some processes that aren't just
>> unbounded by go infinite. Unbounded can only reach countable infinity,
>> but not uncountable infinity. Thus a process that was performing the
>> super-task of writing out all the real numbers between 0 and 1 would
>> use TRULY infinite tape (needing at least Aleph-1 cells) and infinite
>> time to complete.
> 
> Hmm.  It seems to me that that would be beyond the scope of any
> Turing machine.

Yes, I was discussing beyond the domain of Turing Machines, which is why 
I used process and not comutation, and used the term super-task.

> 
> A trivial Turing machine could write out an unending sequence of
> 1s to its tape, never halting.  Such a machine would need a truly
> infinite tape (Aleph-0 cells) if you let it run for an infinite
> time.  I suppose you could loosely say that it "finishes" its task
> of writing infinite 1s after an infinite time.  Another TM could
> repeatedly write 1s to the same location without moving the tape;
> such a machine could be left running for an infinite type but would
> only consume finite tape.
> 
> I suggest that something that can potentially execute Aleph-1 steps
> or consume Aleph-1 tape cells is not a TM.

I agree.

> 
>> In the same way, an unbounded process, could be thought of, as
>> "finishing" after a countable infinite number of steps, while another
>> process might just never stop as in a countable infinite number of
>> steps it never reaches its end (because it takes an uncountable
>> infinite number of steps).
> 
> I'd say that a TM that takes a countably infinite number of steps
> just never completes (though we can discuss the *limit* of its
> output), while a machine that takes an uncountably infinite number
> of steps isn't a TM.
> 
> Actual experts, please jump in and tell me where I'm wrong.
> 
> [...]
> 

I am just going into the realm of hyper-computations, which seems to be 
where Flibble is headed (even if he doesn't realize it).

This might be a Turing Machine with an unbounnded number of tapes, or 
one that can process actual Real numbers like Pi as fundamental elements.

A hyper-process might as a single hyper-step sum up all the numbers on 
an unbounded tape. An operation that would be beyond a Turing Machine to 
perform unless given unbounded time.