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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: Re: A computable function that reports on the behavior of its actual
 self is not allowed
Date: Sun, 12 May 2024 18:14:37 -0500
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On 5/12/2024 5:40 PM, Richard Damon wrote:
> On 5/12/24 6:21 PM, olcott wrote:
>> On 5/12/2024 4:40 PM, Richard Damon wrote:
>>> On 5/12/24 5:18 PM, olcott wrote:
>>>> On 5/12/2024 2:27 PM, olcott wrote:

>>>>> Computable functions are the basic objects of study in computability
>>>>> theory. 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
>>>>>
>>>>> A computable function that reports on the behavior of its actual
>>>>> self (or reports on the behavior of its caller) is not allowed.
>>>>>
>>>>> A decider must halt whereas simulating a pathological input
>>>>> that would never halt unless aborted can only halt by aborting.
>>>>>
>>>>> This causes the direct execution of this input after it has been 
>>>>> aborted
>>>>> to have different behavior than the simulated input that cannot 
>>>>> possibly
>>>>> stop running unless aborted.
>>>>>
>>>>
>>>> *MORE PRECISE WORDING* (this may take a few more rewrites)
>>>> When Ĥ is applied to ⟨Ĥ⟩
>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>>>>
>>>> It is a verified fact that the directly executed Ĥ ⟨Ĥ⟩ cannot possibly
>>>> stop running unless simulating partial halt decider embedded_H aborts
>>>> its simulation of its input.
>>>
>>> But since embedded_H implements a specific algorithm, either it will 
>>> or it won't. "unless" is a meaningless word here, it implies a case 
>>> that can't happen.
>>>
>>> We can look at the two possible cases.
>>>
>>> First, if embedded_H doesn't ever abort its simulation, then, as you 
>>> have desceribed, THAT embedded_H creates a H^ that will never halt, 
>>> but the H that was based on will also never abort its simulation (or 
>>> you lied that embedded_H is the needed copy of H) and thus never 
>>> answer and fail to be a decider.
>>>
>>
>> It can answer without halting by transitioning to its own internal
>> non-final state of embedded_H.qn without ever reaching Ĥ.qn. Every
>> simulated instance of embedded_H would do this same thing and then
>> continue simulating its input.
> So, you just don't understand how algorithms work, and how compuations 
> are DEFINED.
> 
> 
> If you want to try to define a new system of compuation that allows 
> giving answer without the algorithm ending, but still allows all the 
> composition operations that are included in computation theory, go ahead 
> and try.
> 
> You then need to show that it is Turing Complete, which means that you 
> can't outlaw any computation allowed in a Turing Machine, like H^.
> 

*It <is> a way for embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ to get the correct answer*
*It <is> a way for embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ to get the correct answer*
*It <is> a way for embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ to get the correct answer*

Every prior work that I have ever seen and probably every prior
work that exists essentially concludes that both YES and NO are the
wrong answer for H to provide for every H/D pair where H and D have
the HP pathological relationship.

I have shown a weird yet not impossible way for H say say NO
correctly.

>>
>> In this case embedded_H <is> an actual UTM that has the extra
>> feature of examining all of the state transitions of its input
>> to see what we can all see that Ĥ ⟨Ĥ⟩ remains stuck in recursive
>> simulation.
>>
> 
> And a UTM doesn't reveal its answer until it come to a final state, just 
> like ALL Turing Machines or equivalent computation.
> 
>> *Or we can get an actual partial halt decider as follows*
>> *Or we can get an actual partial halt decider as follows*
>> *Or we can get an actual partial halt decider as follows*
>>
>> No decider is ever allowed to report on its own behavior thus embedded_H
>> as a simulating partial halt decider is NOT ALLOWED to report on the
>> direct execution of Ĥ ⟨Ĥ⟩ because this IS REPORTING ON ITS OWN BEHAVIOR.
> 
> WHO SAYS THIS?
> 


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