Path: ...!eternal-september.org!feeder3.eternal-september.org!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 14:46:39 -0400
Organization: A noiseless patient Spider
Lines: 59
Message-ID: <vuoied$3dd6e$7@dont-email.me>
References: <vu6lnf$39fls$2@dont-email.me> <vua9oi$2lub6$1@dont-email.me>
 <vudkah$1ona3$1@dont-email.me> <vufi61$3k099$1@dont-email.me>
 <vugddv$b21g$2@dont-email.me>
 <0a2eeee6cb4b6a737f6391c963386745a09c8a01@i2pn2.org>
 <vugvr3$pke9$8@dont-email.me>
 <4818688e0354f32267e3a5f3c60846ae7956bed2@i2pn2.org>
 <vuj18i$2lf64$6@dont-email.me>
 <f0d3f2e87d9a4e0b0f445f60a33d529f41a4fcf7@i2pn2.org>
 <vuj55m$2lf64$10@dont-email.me> <vuj8h3$2uahf$3@dont-email.me>
 <vujfuu$35hcg$1@dont-email.me>
 <65dddfad4c862e6593392eaf27876759b1ed0e69@i2pn2.org>
 <vujlj0$3a526$1@dont-email.me> <vujln7$32om9$8@dont-email.me>
 <vujmmm$3a526$2@dont-email.me> <vujmrj$32om9$9@dont-email.me>
 <vujtcb$3gsgr$1@dont-email.me> <vuju44$3hnda$1@dont-email.me>
 <vuk47o$3qkbb$1@dont-email.me> <vuk6b6$3l184$1@dont-email.me>
 <vuls34$1bf1j$4@dont-email.me> <vun87k$2m24h$2@dont-email.me>
 <vunb06$2fjjl$5@dont-email.me> <vuo57j$3h5l9$2@dont-email.me>
 <vuoath$3ljma$1@dont-email.me> <vuohgi$3td7u$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 7bit
Injection-Date: Mon, 28 Apr 2025 20:46:38 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="67af223ffcc413f8c29b457017b45374";
	logging-data="3585230"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18b/uRiObe8extwG+UTninG"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:D+1KwU9q6Cdqj1DdLl6giNCndh4=
Content-Language: en-US
In-Reply-To: <vuohgi$3td7u$1@dont-email.me>
Bytes: 4263

On 4/28/2025 2:30 PM, olcott wrote:
> On 4/28/2025 11:38 AM, Richard Heathfield wrote:
>> On 28/04/2025 16:01, 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
>>> this is the wrong behavior to measure.
>>>
>>> It is the behavior THAT IS derived by applying the finite
>>> string transformation rules specified by the x86 language
>>> to the input to HHH(DD) proves that THE EMULATED DD NEVER HALTS.
>>
>> The x86 language is neither here nor there. 
> 
> 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.
> *given an input of the function domain it*
> *can return the corresponding output*
> https://en.wikipedia.org/wiki/Computable_function
> 
> *Outputs must correspond to inputs*
> 
> *This stipulates how outputs must be derived*
> Every Turing Machine computable function is
> only allowed to derive outputs by applying
> finite string transformation rules to its inputs.
> 


And no turing machine exists that can derive the following mapping (i.e. 
the mapping is not a computable function), as proven by Linz and others:


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