Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Mon, 5 May 2025 13:08:20 -0500
Organization: A noiseless patient Spider
Lines: 89
Message-ID: <vvauqk$vtiu$3@dont-email.me>
References: <GE4SP.47558$VBab.42930@fx08.ams4> <vvamqc$o6v5$4@dont-email.me>
 <vvan7q$o4v0$1@dont-email.me> <ts5SP.113145$_Npd.41800@fx01.ams4>
 <vvat0g$vtiu$1@dont-email.me> <vvatf3$o4v0$3@dont-email.me>
 <X67SP.203973$wBVe.8502@fx06.ams4>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Mon, 05 May 2025 20:08:20 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="0a6320ec149f030cd98ca15e4d2d5e5f";
	logging-data="1046110"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX199MgRvaGplOOOqu20dBdwN"
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:SW1WBaD6XKXCt2p7PILFjlI7PhU=
X-Antivirus-Status: Clean
X-Antivirus: Norton (VPS 250505-4, 5/5/2025), Outbound message
In-Reply-To: <X67SP.203973$wBVe.8502@fx06.ams4>
Content-Language: en-US
Bytes: 4541

On 5/5/2025 1:06 PM, Mr Flibble wrote:
> On Mon, 05 May 2025 13:45:07 -0400, dbush wrote:
> 
>> On 5/5/2025 1:37 PM, olcott wrote:
>>> On 5/5/2025 11:13 AM, Mr Flibble wrote:
>>>> On Mon, 05 May 2025 11:58:50 -0400, dbush wrote:
>>>>
>>>>> On 5/5/2025 11:51 AM, olcott wrote:
>>>>>> On 5/5/2025 10:17 AM, Mr Flibble wrote:
>>>>>>> What constitutes halting problem pathological input:
>>>>>>>
>>>>>>> Input that would cause infinite recursion when using a decider of
>>>>>>> the simulating kind.
>>>>>>>
>>>>>>> Such input forms a category error which results in the halting
>>>>>>> problem being ill-formed as currently defined.
>>>>>>>
>>>>>>> /Flibble
>>>>>>
>>>>>> I prefer to look at it as a counter-example that refutes all of the
>>>>>> halting problem proofs.
>>>>>
>>>>> Which start with the assumption that the following mapping is
>>>>> computable and that (in this case) HHH computes it:
>>>>>
>>>>>
>>>>> 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
>>>>>
>>>>>
>>>>>
>>>>>> int DD()
>>>>>> {
>>>>>>      int Halt_Status = HHH(DD);
>>>>>>      if (Halt_Status)
>>>>>>        HERE: goto HERE;
>>>>>>      return Halt_Status;
>>>>>> }
>>>>>>
>>>>>> https://github.com/plolcott/x86utm
>>>>>>
>>>>>> The x86utm operating system includes fully operational HHH and DD.
>>>>>> https://github.com/plolcott/x86utm/blob/master/Halt7.c
>>>>>>
>>>>>> When HHH computes the mapping from *its input* to the behavior of DD
>>>>>> emulated by HHH this includes HHH emulating itself emulating DD.
>>>>>> This matches the infinite recursion behavior pattern.
>>>>>>
>>>>>> Thus the Halting Problem's "impossible" input is correctly
>>>>>> determined to be non-halting.
>>>>>>
>>>>>>
>>>>>>
>>>>> Which is a contradiction.  Therefore the assumption that the above
>>>>> mapping is computable is proven false, as Linz and others have proved
>>>>> and as you have *explicitly* agreed is correct.
>>>>
>>>> The category (type) error manifests in all extant halting problem
>>>> proofs including Linz.  It is impossible to prove something which is
>>>> ill-formed in the first place.
>>>>
>>>> /Flibble
>>>
>>> The above example is category error because it asks HHH(DD) to report
>>> on the direct execution of DD() and the input to HHH specifies a
>>> different sequence of steps.
>>>
>>>
>> In other words, you're demonstrating that you don't understand proof by
>> contradiction, a concept taught to and understood by high school
>> students more than 50 years your junior.
> 
> For proof by contradiction to be valid the contradition has to be well-
> formed which in the case of the Halting Problem it is not.
> 
> /Flibble

Yes. Self-contradiction is always ill-formed.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer