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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Unpartial Halt Deciders --- category error
Date: Sat, 19 Apr 2025 19:23:16 -0400
Organization: i2pn2 (i2pn.org)
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On 4/19/25 4:44 PM, olcott wrote:
> On 4/19/2025 1:06 PM, Mr Flibble wrote:
>> On Sat, 19 Apr 2025 13:34:40 -0400, Richard Damon wrote:
>>
>>> On 4/19/25 8:05 AM, Mr Flibble wrote:
>>>> On Sat, 19 Apr 2025 07:55:55 -0400, Richard Damon wrote:
>>>>
>>>>> On 4/18/25 11:52 PM, olcott wrote:
>>>>>> On 4/18/2025 2:32 PM, Keith Thompson wrote:
>>>>>>> Mr Flibble <flibble@red-dwarf.jmc.corp> writes:
>>>>>>>> On Fri, 18 Apr 2025 12:25:36 -0700, Keith Thompson wrote:
>>>>>>>>> Mr Flibble <flibble@red-dwarf.jmc.corp> writes:
>>>>>>>>>> I, aka Mr Flibble, have created a new computer science term, the
>>>>>>>>>> "Unpartial Halt Decider".  It is a Halt Decider over the domain
>>>>>>>>>> of all program-input pairs excluding pathological input (a
>>>>>>>>>> manifestation of the self referencial category error).
>>>>>>>>> [...]
>>>>>>>>>
>>>>>>>>> Do you have a rigorous definition of "pathological input"?
>>>>>>>>>
>>>>>>>>> Is there an algorithm to determine whether a given input is
>>>>>>>>> "pathological" or not?
>>>>>>>>>
>>>>>>>>> I could define an is_prime() function like this:
>>>>>>>>>
>>>>>>>>>        bool is_prime(int n) {
>>>>>>>>>            return n >= 3 && n % 2 == 1;
>>>>>>>>>            // returns true for odd numbers >= 3, false for
>>>>>>>>>            all others
>>>>>>>>>        }
>>>>>>>>>
>>>>>>>>> I'll just say that odd numbers that are not prime are pathological
>>>>>>>>> input, so I don't have to deal with them.
>>>>>>>>
>>>>>>>> Pathological input:
>>>>>>>>
>>>>>>>> Self-referencial to the decider.
>>>>>>>
>>>>>>> OK.
>>>>>>>
>>>>>>> Do you have a *rigorous* definition of "pathological input"?
>>>>>>>
>>>>>>> Is there an algorithm to determine whether a given input is
>>>>>>> "pathological" or not?
>>>>>>>
>>>>>>>
>>>>>> int DD()
>>>>>> {
>>>>>>      int Halt_Status = HHH(DD);
>>>>>>      if (Halt_Status)
>>>>>>        HERE: goto HERE;
>>>>>>      return Halt_Status;
>>>>>> }
>>>>>>
>>>>>> Patterns isomorphic to the above when simulated by HHH.
>>>>>>
>>>>>>
>>>>>>
>>>>> Examples are not definitions.
>>>>>
>>>>> And the problem is that the above example is itself a category error
>>>>> for the problem, as the DD provided above isn't a complete program, as
>>>>> it doesn't include the code for HHH as required, and when you include
>>>>> Halt7.c as part of the input, your HHH isn't a seperate program of its
>>>>> own, and thus doesn't have a Turing Complete range of inputs it can
>>>>> accept.
>>>>>
>>>>> Sorry, you are just showing you don't understand what it means to
>>>>> DEFINE something.
>>>>
>>>> Ah, the fundamental mistake you have been making all this time, Damon!
>>>> The self-referencial category error doesn't magically disappear by
>>>> providing source code rather than a run-time function address to the
>>>> decider; you are simply transforming the same input without affecting
>>>> the result.
>>>>
>>>> /Flibble
>>>
>>> And WHAT is the category error?  You stil can't show the difference in
>>> CATEGORY between what is allowed and what isn't, and in fact, you can't
>>> even precisely define what is and isn't allowed.
>>>
>>> Now, you also run into the issue that the "Olcott System" begins with an
>>> actual category error as we do not have the required two seperate
>>> programs of the "Decider" and the "Program to be decided on" given via
>>> representation as the input, as what you want to call that program to be
>>> decided isn't one without including the code of the decider it is using,
>>> and when you do include it, the arguments about no version of the
>>> decider being able to succeed is improper as it must always be that
>>> exact program that we started with, and thus it just FAILS to do a
>>> correct simulation, while a correct simulation of this exact input
>>> (which includes the ORIGINAL decider) will halt.
>>>
>>> Sorry, YOU are the one stuck with the fundamental mistake, or is it a
>>> funny mental mistake because you don't understand what you are talking
>>> about.
>>
>> The category error is extant over the domain of pathological inputs, no
>> matter what form those inputs take.
>>
>> /Flibble
> 
> The category error in the halting problem proof is
> to define an input D that is able to actually do the
> opposite of whatever value that H reports.


And what was done WRONG when we did that?

> 
> Now the question: Does the input D halt becomes
> self-contradictory for H.

No it doesn't, as the code for D is fully defined because before we 
could make D, the code for H had to be fully define.

And thus the actual behavior of the input is a fixed value, based on 
that prior decision made in the design of H.

> 
> So it is asking a yes/no question where both yes and
> no are the wrong answer that is the category error.

No, the right answer exists, it just isn't the answer that H gives.

Since you H returns 0, non-halting, the correct answer to the quesiton 
does the program described by the input halt, is Yes,

Now, if you change your H to be H1, a program that returns 1 when given 
the D1 built on it, then that D1, a different program than D, will end 
up being non-halting, so H1 is wrong for that input.

H(D1) might be correct, as might H1(D), it is just that for ANY Hn you 
want to try to make, there is a Dn that Hn(Dn) will give the wrong 
answer too.

But Dn still has defined behavior, and isn't itself 
"self-contardictory", it is just that Dn is designed to be 
Hn-contradictory, something fully allowed by the rules of construction 
of Turing Conplete Computation systems.

> 
> Objective and Subjective Specifications
> Eric C.R. Hehner
> Department of Computer Science, University of Toronto
> 
> (6) Can Carol correctly answer “no” to this (yes/no) question?
> https://www.cs.toronto.edu/~hehner/OSS.pdf
> 
> Richard Damon found a loophole in the original question.
> I inserted (yes/no) to close the loophole.
> 

And the problem is you switch from the actual question of what is the 
bahivor of the actual program described by the input, to the subjective 
question of what can the decider return to be right, which also makes 
the decider NOT A PROGRAM, as you are trying to define its behavior by 
something other than its own code, which isn't valid.

Sorry, you are just showing you don't know what yo are talking about.