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

On 4/23/25 12:14 AM, olcott wrote:

On 4/22/2025 9:33 PM, Richard Damon wrote:

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.

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

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Except you are doing the logic backward.
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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.

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

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Which says if a machine exists, it is conputable.
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The machine does not need to exist.
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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.
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On Turing Machines inputs <are> finite strings, and
finite string transformation rules <are> applied to
these finite strings to derive corresponding outputs.

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Yes, so the results can only BE what is computable, but as pointed out, the correct answer need not be.
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 That would seem to indicate an error in the original
problem specification.

No, just in your understanding of it.

 Whatever can be derived by applying finite string
transformation to input finite strings <is> computable.

No, and the problem is you don't actually understand the meaning of the words you are using.
The point it that your phrase "Computations are finite string transformations" is not a definition, but just a description. All computations can be viewed as a finite string transformation, but not all finite string transformations are computations. The name for a process that maps finite strings to finite strings is a function, as a function is basically a mapping of (possibly a countably infinite set) strings to strings. Like in this case Descriptions of Turing Machines to their Halting Status. This can be specified in several manners, it could be specified by a finite algorithm (which makes it a computation), It could be an unbounded algorithm (run the machine specified and see if it ever halts) which by itself is NOT a computation (and thus doesn't make it computable) but there might be a finite algorithm that creates it, or it might just be an arbitrary mapping specified by an infinite number of finite string to finite string mappings. Note, that last case creates an UNCOUNTABLE INFINITE number of mappings.
A Computable Function, is a function where there exists a (finite) algorithm, that can produce each mapping in a finite number of steps.
Thus, not all "Functions" (which are the mappings) are by that definition computable, as some can be defined by non-finite algorithms or even be among the uncountable number of arbitrary mappings.

 

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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.
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The outputs DO need to correspond to the input, but not necessarily by a computable transform.
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 Yes necessarily by a computable transform.

Nope, see above. It can only DO a computable transform, but the problem isn't limited to such, but can be ANY transform, even one that needs an unbounded process to perform.

 

That only exists if the function is, in fact, computable.
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 Uncomputable functions are an incoherent idea when
computable functions are defined by deriving outputs
by applying finite string transformations to input
finite strings.
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Your mind is skipping over a term there. It is only a computable function if there exists the ALGORITMM that computes it, and the implications of that term in this context are finite description using finite time. Not all string transformations can so be processed.
This seems to be a common problem for you, if you don't understand a piece of something, you just ignore it but still think you know what it means.
I guess this comes from your failure to understand things like Rules, Reality, Truth, or Logic.