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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string transformations to inputs
Date: Mon, 5 May 2025 11:21:02 +0300
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On 2025-05-05 00:35:13 +0000, olcott said:

> On 5/4/2025 5:34 PM, Mr Flibble wrote:
>> On Sun, 04 May 2025 23:30:54 +0100, Richard Heathfield wrote:
>> 
>>> On 04/05/2025 23:15, olcott wrote:
>>>> On 5/4/2025 2:21 PM, Richard Heathfield wrote:
>>>>> On 04/05/2025 18:55, olcott wrote:
>>>>>> Changing my words then rebutting these changed words is dishonest.
>>>>>> 
>>>>>> Functions computed by Turing Machines require INPUTS and produce
>>>>>> OUTPUTS DERIVED FROM THESE INPUTS.
>>>>> 
>>>>> Counter-example: a Turing Machine can calculate pi without any input
>>>>> whatsoever.
>>>>> 
>>>>> As Mikko rightly said: a Turing machine does not need to require an
>>>>> input.
>>>>> 
>>>>> 
>>>> IT IS NOT COMPUTING FUNCTION THEN
>>> 
>>> Quoth Alan Turing:
>>> 
>>> (viii) The limit of a computably convergent sequence is computable.
>>> 
>>> From (viii) and TT— 4(1—i-|--i—...) we deduce that TT is computable.
>>> 
>>> No input required.
>>> 
>>>> IT IS NOT COMPUTING FUNCTION THEN IT IS NOT COMPUTING FUNCTION THEN IT
>>>> IS NOT COMPUTING FUNCTION THEN
>>>> 
>>>> Computable functions are the basic objects of study in computability
>>>> theory. Computable functions are the formalized analogue of the
>>>> intuitive notion of algorithms, in the sense that 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
>>> 
>>> That's a very second-rate summary of computability. Turing was far more
>>> interested in whether a computation was possible than whether it needed
>>> inputs. Do most computations need inputs? Most useful ones that we care
>>> about, sure. But all? By no means.
>>> 
>>>> *Computer science is ONLY concerned with computable functions*
>>> 
>>> Computer science is concerned with the Halting Problem.
>>> The Halting Problem is concerned with an incomputable function.
>>> Therefore computer science is concerned with at least one incomputable
>>> function.
>> 
>> The function is neither computable nor incomputable because there is no
>> function at all, just a category error.
>> 
>> /Flibble
> 
> You can look at it that way or you can look
> at it as simulating termination analyzer HHH(DD)
> does correctly determine that DD cannot possibly
> reach its own final state, thus is correctly
> rejected as non-halting.

Or you can look at it as simulating termination analyzer HHH(DD)
does incorrectly determine that DD cannot possibly reach its own
final state, thus is incorrectly rejected as non-halting.

-- 
Mikko