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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 11:40:51 -0500
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On 3/25/2025 10:17 AM, dbush wrote:
> On 3/25/2025 11:13 AM, olcott wrote:
>> On 3/25/2025 10:02 AM, dbush wrote:
>>> On 3/25/2025 10:53 AM, olcott wrote:
>>>> On 3/25/2025 9:45 AM, dbush wrote:
>>>>> On 3/24/2025 11:29 PM, olcott wrote:
>>>>>> On 3/24/2025 10:12 PM, dbush wrote:
>>>>>>> On 3/24/2025 10:07 PM, olcott wrote:
>>>>>>>> On 3/24/2025 8:46 PM, André G. Isaak wrote:
>>>>>>>>> On 2025-03-24 19:33, olcott wrote:
>>>>>>>>>> On 3/24/2025 7:00 PM, André G. Isaak wrote:
>>>>>>>>>
>>>>>>>>>>> In the post you were responding to I pointed out that 
>>>>>>>>>>> computable functions are mathematical objects.
>>>>>>>>>>
>>>>>>>>>> https://en.wikipedia.org/wiki/Computable_function
>>>>>>>>>>
>>>>>>>>>> Computable functions implemented using models of computation
>>>>>>>>>> would seem to be more concrete than pure math functions.
>>>>>>>>>
>>>>>>>>> Those are called computations or algorithms, not computable 
>>>>>>>>> functions.
>>>>>>>>>
>>>>>>>>
>>>>>>>> https://en.wikipedia.org/wiki/Pure_function
>>>>>>>> Is another way to look at computable functions implemented
>>>>>>>> by some concrete model of computation.
>>>>>>>>
>>>>>>>
>>>>>>> And not all mathematical functions are computable, such as the 
>>>>>>> halting function.
>>>>>>>
>>>>>>>>> The halting problems asks whether there *is* an algorithm which 
>>>>>>>>> can compute the halting function, but the halting function 
>>>>>>>>> itself is a purely mathematical object which exists prior to, 
>>>>>>>>> and independent of, any such algorithm (if one existed).
>>>>>>>>>
>>>>>>>>
>>>>>>>> None-the-less it only has specific elements of its domain
>>>>>>>> as its entire basis. For Turing machines this always means
>>>>>>>> a finite string that (for example) encodes a specific
>>>>>>>> sequence of moves.
>>>>>>>
>>>>>>> False.  *All* turing machine are the domain of the halting 
>>>>>>> function, and the existence of UTMs show that all turning 
>>>>>>> machines can be described by a finite string.
>>>>>>>
>>>>>>
>>>>>> You just aren't paying enough attention. Turing machines
>>>>>> are never in the domain of any computable function.
>>>>>> <snip>
>>>>>>
>>>>>
>>>>> False.  The mathematical function that counts the number of 
>>>>> instructions in a turing machine is computable.
>>>>>
>>>>
>>>> It is impossible for an actual Turing machine to
>>>> be input to any other TM.
>>>>
>>>
>>> But a description of a turing machine can be, for example in the form 
>>> of source code or a binary.  And a turing machine by definition 
>>> *always* behaves the same for a given input when executing directly.
>>
>> IT IS COUNTER-FACTUAL THAT A MACHINE DESCRIPTION ALWAYS
>> SPECIFIES
>> BEHAVIOR IDENTICAL TO THE DIRECTLY EXECUTED MACHINE.
>>
>> _III()
>> [00002172] 55         push ebp      ; housekeeping
>> [00002173] 8bec       mov  ebp,esp  ; housekeeping
>> [00002175] 6872210000 push 00002172 ; push III
>> [0000217a] e853f4ffff call 000015d2 ; call EEE(III)
>> [0000217f] 83c404     add  esp,+04
>> [00002182] 5d         pop  ebp
>> [00002183] c3         ret
>> Size in bytes:(0018) [00002183]
> 
> That is not the complete description.  The complete description consists 
> of the code of III

and the fact that EEE emulates III according to the semantics
of the x86 language.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer