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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_A_simulating_halt_decider_applied_to_the_The_Peter_?=
 =?UTF-8?Q?Linz_Turing_Machine_description_=E2=9F=A8=C4=A4=E2=9F=A9?=
Date: Fri, 31 May 2024 10:35:18 -0500
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On 5/31/2024 8:00 AM, Mikko wrote:
> On 2024-05-30 13:20:09 +0000, olcott said:
> 
>> On 5/30/2024 2:06 AM, Mikko wrote:
>>> On 2024-05-29 13:13:13 +0000, olcott said:
>>>
>>>> On 5/29/2024 3:37 AM, Mikko wrote:
>>>>> On 2024-05-28 11:34:24 +0000, Richard Damon said:
>>>>>
>>>>>> On 5/27/24 10:59 PM, olcott wrote:
>>>>>>> On 5/27/2024 9:52 PM, Richard Damon wrote:
>>>>>>>> On 5/27/24 10:41 PM, olcott wrote:
>>>>>>>>> On 5/27/2024 9:23 PM, Richard Damon wrote:
>>>>>>>>>> On 5/27/24 10:01 PM, olcott wrote:
>>>>>>>>>>> On 5/27/2024 8:24 PM, Richard Damon wrote:
>>>>>>>>>>>> On 5/27/24 9:04 PM, olcott wrote:
>>>>>>>>>>>
>>>>>>>>>>>>>>> I totally do. Can you please write down the
>>>>>>>>>>>>>>> "completely specified state transition/tape operation 
>>>>>>>>>>>>>>> table."
>>>>>>>>>>>>>>> of this specific (thus uniquely identifiable) machine I 
>>>>>>>>>>>>>>> would
>>>>>>>>>>>>>>> really like to see it.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> But it was proven that no such machine exists!
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Remember, the proof starts with the hypothetical that such 
>>>>>>>>>>>>>> a machine exists. Such a machine WOULD HAVE a completely 
>>>>>>>>>>>>>> specified state transition/tape operation table.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> That is not what you said.
>>>>>>>>>>>>>  >>>>> There doesn't need to be a unique finite string, but 
>>>>>>>>>>>>> it is a 100%
>>>>>>>>>>>>>  >>>>> completely specified state transition/tape operation 
>>>>>>>>>>>>> table.
>>>>>>>>>>>>>
>>>>>>>>>>>>> "a 100% completely specified state transition/tape 
>>>>>>>>>>>>> operation table"
>>>>>>>>>>>>> of a non-existent machine.
>>>>>>>>>>>>
>>>>>>>>>>>> Right, by presuming that you have a Turing Machine, you have 
>>>>>>>>>>>> a completly specified state transition/tape operation table.
>>>>>>>>>>>>
>>>>>>>>>>>> You may not KNOW what that table is if you don't know what 
>>>>>>>>>>>> the exact machine is, but you know it exists.
>>>>>>>>>>>
>>>>>>>>>>>  >>> But it was proven that no such machine exists!
>>>>>>>>>>>  > ... but you know it exists.
>>>>>>>>>>>
>>>>>>>>>>>  >>> But it was proven that no such machine exists!
>>>>>>>>>>>  > ... but you know it exists.
>>>>>>>>>>>
>>>>>>>>>>>  >>> But it was proven that no such machine exists!
>>>>>>>>>>>  > ... but you know it exists.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Really, then show that one exists!
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>> *I am quoting your words. You did contradict yourself*
>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Really, where did I say that H exists?
>>>>>>>>
>>>>>>>> I said that if a Turing Machine exists, then its transition 
>>>>>>>> table does too.
>>>>>>>>
>>>>>>>
>>>>>>> OK my mistake this time. I did not take into account the full 
>>>>>>> context.
>>>>>>> I will go back an read the Linz proof and see if he said anything
>>>>>>> about a specific machine.
>>>>>>
>>>>>> Read the DEFINITION of the problem. He talks about "a" machine. 
>>>>>> Using a singular article means you are working with just one.
>>>>>>
>>>>>>
>>>>>> Taking stuff out of context is a common problem with you, when you 
>>>>>> don't understand something, you just make up what it must mean, 
>>>>>> and stick to that. That isn't the way to learn.
>>>>>>
>>>>>>
>>>>>>>
>>>>>>> None of the proofs ever try to show that there exists one machine 
>>>>>>> that
>>>>>>> gets the wrong answer. They are always at least trying to prove 
>>>>>>> that no
>>>>>>> machine of the infinite set of machine gets the right answer.
>>>>>>>
>>>>>>
>>>>>> What I see, is they always start with a prototypical single 
>>>>>> machine, and show that it gets the answer wrong, and then they use 
>>>>>> categorical logic to say that we can do this same thing for all of 
>>>>>> them.
>>>>>
>>>>> It is possible to formulate the claim and proof so that H is an 
>>>>> universally
>>>>> quantified variable. But the usual way is apparently equally good 
>>>>> for the
>>>>> target audience.
>>>>>
>>>>
>>>> *Formalizing the Linz Proof structure*
>>>> ∃H  ∈ Turing_Machines
>>>> ∀x  ∈ Turing_Machines_Descriptions
>>>> ∀y  ∈ Finite_Strings
>>>> such that H(x,y) = Halts(x,y)
>>>
>>> That is not a proof structure. That is the counter-hypothesis of 
>>> Linz' proof.
>>> Also note that both x and y are finite strings.
>>>
>>
>> The above is what Linz is claiming evaluates to false, he says
>> there is no such H.
> 
> Yes, and proves the claim.
> 
>> A decider computes the mapping from finite string inputs to
>> its own accept or reject state.
> 
> An existing decider.
> 
>> A decider does not and cannot compute the mapping from Turing_Machine
>> inputs to its own accept or reject state.
> 
> An exsiting decider.
> 

When Ĥ is applied to ⟨Ĥ⟩
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

*Formalizing the Linz Proof structure*
∃H  ∈ Turing_Machines
∀x  ∈ Turing_Machine_Descriptions
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,y)

That <is> what Linz is claiming is false.
*Here is the same claim with 100% complete specificity*
such that H(⟨Ĥ⟩, ⟨Ĥ⟩) != Halts(⟨Ĥ⟩, ⟨Ĥ⟩)

*A quick summary of the reasoning provided below*
The LHS is behavior that embedded_H is allowed to report on.
The RHS is behavior that embedded_H NOT is allowed to report on.
The LHS and the RHS specify different behaviors.

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