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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Sun, 2 Jun 2024 09:04:14 -0500
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On 6/2/2024 3:24 AM, Mikko wrote:
> On 2024-06-01 14:52:54 +0000, olcott said:
> 
>> On 6/1/2024 3:20 AM, Mikko wrote:
>>> On 2024-05-31 15:44:22 +0000, olcott said:
>>>
>>>> On 5/31/2024 8:10 AM, Mikko wrote:
>>>>> On 2024-05-28 16:16:48 +0000, olcott said:
>>>>>
>>>>>> typedef int (*ptr)();  // ptr is pointer to int function in C
>>>>>> 00       int H(ptr p, ptr i);
>>>>>> 01       int D(ptr p)
>>>>>> 02       {
>>>>>> 03         int Halt_Status = H(p, p);
>>>>>> 04         if (Halt_Status)
>>>>>> 05           HERE: goto HERE;
>>>>>> 06         return Halt_Status;
>>>>>> 07       }
>>>>>> 08
>>>>>> 09       int main()
>>>>>> 10       {
>>>>>> 11         H(D,D);
>>>>>> 12         return 0;
>>>>>> 13       }
>>>>>>
>>>>>> 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_Machines_Descriptions
>>>>>> ∀y  ∈ Finite_Strings
>>>>>> such that H(x,y) = Halts(x,x)
>>>>>>
>>>>>> *Here is the same thing applied to H/D pairs*
>>>>>> ∃H ∈ C_Functions
>>>>>> ∀D ∈ x86_Machine_Code_of_C_Functions
>>>>>> such that H(D,D) = Halts(D,D)
>>>>>>
>>>>>> In both cases infinite sets are examined to see
>>>>>> if any H exists with the required properties.
>>>>>
>>>>> That says nothing about correct simulation. It says
>>>>> something abuout some D but not whether it is correctly
>>>>> simulated. Also nothing is said about templates or
>>>>> infinite sets. At the end is claimed that some
>>>>> infinite sets are examined but not who examined, nor
>>>>> how, nor what was found in the alleged examination.
>>>>>
>>>>
>>>> *Formalizing the Linz Proof structure*
>>>> ∃H  ∈ Turing_Machines
>>>> ∀x  ∈ Turing_Machines_Descriptions
>>>> ∀y  ∈ Finite_Strings
>>>> such that H(x,y) = Halts(x,x)
>>>
>>> The above is the counter hypothesis for the proof. Proof structore
>>> is that a contradiction is derived from the counter hypthesis.
>>>
>>>> The above disavows Richard's claim based on a misinterpretation of
>>>> Linz that the Linz proof is about a single specific Turing machine.
>>>
>>> Your ∃H declares H as a new symbol for a specific Turing machine.
>>> Therefore everything that follows refers to that specific Turing 
>>> machine.
>>> There may be others that could be discussed the same way but they 
>>> aren't.
>>>
>>
>> ∃H  ∈ Turing_Machines
>> There exists at least one H
>> from the infinite set of all Turing_Machines
>>
>> ∃!H  ∈ Turing_Machines
>> There exists a single unique H
>> from the infinite set of all Turing_Machines
> 
> The symbol ∃! is non-standard and should be defined when used.
> 
> And there is no point to present some formulas without saying
> anything about them.
> 
>>>>     The domain of this problem is to be taken as the set of
>>>>     all Turing machines and all w; that is, we are looking
>>>>     for a single Turing machine that, given the description
>>>>     of an arbitrary M and w, will predict whether or not the
>>>>     computation of M applied to w will halt.
>>>
>>> Note the words "a single Turing machine".
>>
>> I know that he said that yet he meant this
>> ∃H ∈ Turing_Machines *and didn't mean this* ∃!H ∈ Turing_Machines
>> or he would be contradicting every other HP proof.
> 
> He didn't mean either. Your false claim is merely an attempted
> preparation of strawman deception.
> 
>>>> Linz <IS NOT> looking for a single machine that gets the wrong answer.
>>>> Linz is looking for at least one Turing Machine that gets the right
>>>> answer: ∃H ∈ Turing_Machines
>>>
>>> Not at least one but exactly one. The Halting Problem asks for one
>>> or a proof that there is none.
>>
>> In other words when there are two machines that solve the halting
>> problem then the halting problem IS NOT SOLVED?
> 
> If there are two then one of them is one and the other is irrelevant.

∃!H  ∈ Turing_Machines
∀x  ∈ Turing_Machines_Descriptions
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,x)

https://en.wikipedia.org/wiki/Uniqueness_quantification

He was saying that Linz was saying that Linz was looking for exactly one 
unique Turing machine that solved the halting problem.

> When Linz says "we are looking for a single Turing machine that ..."
> he implies that when he (or someone) finds one he is not going to
> look for another one. When he finally proves that it is not possible
> to find one it is obvious that it is not possible to find more.
> 
> And if you need to ask that you don't need to as why you look stupid.
> 

Unless we attain 100% perfectly identical encoding and decoding of
meanings the truth of what I am saying with slip through the cracks of
ambiguity.

∃H  ∈ Turing_Machines
∀x  ∈ Turing_Machines_Descriptions
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,x)

Does there exist at least one H that solves the halting problem?

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