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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 11:56:46 -0500
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On 3/25/2025 11:46 AM, dbush wrote:
> On 3/25/2025 12:40 PM, olcott wrote:
>> 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 
> 
> Is called by III makes the code of EEE part of the fixed input, as well 
> as everything that EEE calls down to the OS level.
> 

Which is not relevant to whether or not III emulated
by EEE reaches its own final halt state.

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