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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: Tue, 6 May 2025 11:24:44 +0300
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On 2025-05-06 02:32:01 +0000, olcott said:

> On 5/5/2025 8:19 PM, Richard Damon wrote:
>> On 5/4/25 8:35 PM, olcott wrote:
>>> 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.
>>> 
>> 
>> Except that isn't the question that is being asked.
>> 
>> In fact, that question has a trivial answer, as we can make an H0 that 
>> just aborts its emulation and returns 0 and it is correct by your 
>> definition,
> 
> No that is stupidly wrong as I have said at least 100 times recently.
> The termination analyzer must compute the mapping from the input
> on the basis of the behavior that this input actually specifies.

You often say so and you often say otherwise. More specifically, you
have said that is correct to call the actual input non-halting if a
hypothetical non-decider would correctly decide that a hypthetical
input would not halt.

-- 
Mikko