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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 20:33:04 -0500
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On 3/24/2025 7:00 PM, André G. Isaak wrote:
> On 2025-03-24 17:42, olcott wrote:
>> On 3/24/2025 6:05 PM, André G. Isaak wrote:
>>> On 2025-03-24 17:04, olcott wrote:
> 
> 
>>>> _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]
>>>>
>>>> When III is emulated by pure emulator EEE for any finite
>>>> number of steps of emulation according to the semantics
>>>> of the x86 language it never reaches its own "ret"
>>>> instruction final halt state THUS DOES NOT HALT.
>>>>
>>>> When III is directly executed calls an EEE instance
>>>> that only emulates finite number of steps then this
>>>> directly executed III always reaches its own "ret"
>>>> instruction final halt state THUS HALTS.
>>>
>>> And that has what, exactly, to do with the post you are allegedly 
>>> responding to?
>>>
>>> André
>>>
>>
>>
>> THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.
>>
>> The behavior specified by the finite string input to a
>> computable function implemented on a model of computation
>>
>> does differ from the direct execution of this same finite
>> string because the direct execution avoids the pathological
>> self-reference that causes the recursive emulation.
>>
>> THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.
> 
> 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.

For example pure math functions don't have any specific
storage like a tape or machine registers.

This also would seem to mean that they would require
some actual input.


> The above copypasta doesn't address 
> this.
> 
> I pointed out that the domain of a computable function needn't be a 
> string. The above copypasta doesn't address this.
> 

When implemented using an actual model of computation
some concrete form or input seems required.

> I pointed out that there is no bijection natural numbers and strings, 

finite strings of decimal digits: [0123456789]

> but rather a one-to-many relation. The above copypasta doesn't address 
> this.

"12579" would seem to have a bijective mapping to
a single natural number.

> 
> I pointed out that the exact same sort of one-to-many relation exists 
> between computations and strings. The above copypasta doesn't address this.
> 

I pointed out above that the finite string of x86
machine code correctly emulated by EEE DOES
NOT MAP TO THE BEHAVIOR OF ITS DIRECT EXECUTION.


code of correctly
emulated by EEE machine code does not map to its direct
execution.

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