Re: How computable functions actually work. (was Flibble)

On 4/22/25 6:19 PM, olcott wrote:

On 4/22/2025 4:58 PM, Andy Walker wrote:

On 22/04/2025 15:57, Mr Flibble wrote:

On Tue, 22 Apr 2025 15:43:27 +0100, Andy Walker wrote:

The "real" Mr Flibble is a malevolent penguin.  I wonder why
contributors take him so seriously?  If you want to debate with a
penguin, that's your prerogative, but to me it makes more sense to add
several pinches of salt and smile or groan as appropriate to everything
he writes.  He has a knack for writing things that are just about
plausible, which is enviable, but one response to anything interesting
is surely enough?

Mr Flibble is very cross.

>
     He shouldn't be.  As hinted above, being able to write successful
satire is a rare skill.  But it loses its point if too many people take
it seriously.
>

 Flibble <is> factually correct.
 All computation is defined to be represented as finite string
transformations to finite strings.

Except you are doing the logic backward.

 This <is> how Turing Machine computable functions actually work.
Outputs are forced to correspond to inputs when finite string
transformation rules are applied to inputs to derive outputs.

And you need to know that the function *IS* computable to use that.
What the machine actually produces will be computable.
What the machine is SUPPOSED to produce might not be.

     a function is computable if there exists an algorithm
    that can do the job of the function, i.e. given an input
    of the function domain it can return the corresponding
    output. https://en.wikipedia.org/wiki/Computable_function

Which says if a machine exists, it is conputable.
The machine does not need to exist.
This seems to be a flaw in your logic, you seem to think there is a Truth Faerie that can magically make the impossible happen.

 On Turing Machines inputs <are> finite strings, and
finite string transformation rules <are> applied to
these finite strings to derive corresponding outputs.

Yes, so the results can only BE what is computable, but as pointed out, the correct answer need not be.

 People here stupidly assume that the outputs are not
required to correspond to the inputs. That comes from
learn-by-rote with zero depth of understanding.
 

The outputs DO need to correspond to the input, but not necessarily by a computable transform.
That only exists if the function is, in fact, computable.
Nearly all "random" functions will not be, by a simple counting argument, There are more functions/mapping (Aleph-1) that can be asked for than machines (Aleph-0) to do the mapping. Therefore only a infintisimal fraction of all possible mappings are in fact computable.
Now, when we limit ourselves to mapping that have a use, it seems we find a lot that do have a mapping, but not all.
There is no computation that can compute for *ALL* possible programs, if that progran will halt when run. Turing proves that by showing for every possible decider that we might think can do it, that we can make a program for it to decide on that it will get wrong.
Thus the mapping must be uncomputable.