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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions ---HHH(DD)
Date: Tue, 25 Mar 2025 13:27:11 -0500
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On 3/25/2025 12:24 PM, dbush wrote:
> On 3/25/2025 1:18 PM, olcott wrote:
>> On 3/25/2025 12:04 PM, dbush wrote:
>>> On 3/25/2025 12:56 PM, olcott wrote:
>>>> 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.
>>>>
>>>
>>> Which is why III emulated by EEE is not relevant.
>>>
>>
>> When the question is:
>> Does III emulated by EEE reach its final halt state
>> when III defines a pathological relationship with its emulator?
>>
> 
> 
> But that's not the question.  The question is whether or not an H exists 
> that behaves as described below:
> 

Did you ever bother to notice the title of this thread?

Turing machines are only capable operating on input
finite strings. Turing machine computable functions cannot
compute anything that their input doesn't specify.

THEIR INPUT NEVER SPECIFIES THE ACTUAL BEHAVIOR OF ANY
OTHER TURING MACHINE

> 
> Given any algorithm (i.e. a fixed immutable sequence of instructions) X 
> described as <X> with input Y:
> 
> A solution to the halting problem is an algorithm H that computes the 
> following mapping:
> 
> (<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
> (<X>,Y) maps to 0 if and only if X(Y) does not halt when executed directly
> 
> 


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