Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string
 transformations to inputs
Date: Mon, 28 Apr 2025 12:54:45 -0400
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In-Reply-To: <vuoatj$3jn5n$6@dont-email.me>

On 4/28/2025 12:38 PM, olcott wrote:
> On 4/28/2025 10:41 AM, dbush wrote:
>> On 4/28/2025 11:35 AM, olcott wrote:
>>> On 4/28/2025 10:18 AM, dbush wrote:
>>>> On 4/28/2025 11:01 AM, olcott wrote:
>>>>> On 4/28/2025 2:33 AM, Richard Heathfield wrote:
>>>>>> On 28/04/2025 07:46, Fred. Zwarts wrote:
>>>>>>
>>>>>> <snip>
>>>>>>
>>>>>>> So we agree that no algorithm exists that can determine for all 
>>>>>>> possible inputs whether the input specifies a program that 
>>>>>>> (according to the semantics of the machine language) halts when 
>>>>>>> directly executed.
>>>>>>> Correct?
>>>>>>
>>>>>> Correct. We can, however, construct such an algorithm just as long 
>>>>>> as we can ignore any input we don't like the look of.
>>>>>>
>>>>>
>>>>> The behavior of the direct execution of DD cannot be derived
>>>>> by applying the finite string transformation rules specified
>>>>> by the x86 language to the input to HHH(DD). This proves that
>>>>
>>>> The assumption that an H exists that meets the below requirements is 
>>>> false, as shown by Linz and others:
>>>>
>>>
>>> I have just proved that those requirements are stupidly wrong
>>
>> Category error.  The mapping exists 
> 
> 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.

i.e. a computable function is a mathematical mapping for which an 
algorithm exists to compute in.

And the halting function below is not a computable function:


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


> *given an input of the function domain*
> *it can return the corresponding output*
> https://en.wikipedia.org/wiki/Computable_function
> 

In other words, the algorithm, given an input in the domain of the 
mathematical function, returns the corresponding output of the 
mathematical function.


> There is a mapping from the input to HHH(DD) by applying
> the finite string transformation rules specified by the
> x86 language to this DD input that derives:
> *no correctly emulated DD ever reaches its final halt state*
> 

In other words, the mathematical function that algorithm HHH is mapping 
is not the halting function, and therefore isn't a solution to the 
halting problem.