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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory,sci.logic
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Tue, 28 May 2024 21:23:56 -0500
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On 5/28/2024 9:04 PM, Richard Damon wrote:
> On 5/28/24 12:16 PM, olcott wrote:
>> 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)
> 
> But since for x being the description of the H^ built from that H and y 
> being the same, it turns out that no matter what answer H gives, it will 
> be wrong.
> 

We have not gotten to that point yet this post is so that
you can fully understand what templates are and how they work.

> (And I think you have an error in your reference to Halts, I think you 
> mean Halts(x,y) not Halts(x,x)
> 

Yes good catch. I was trying to model embedded_H / ⟨Ĥ⟩
and then changed my mind to make it more general.

>>
>> *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)
> 
> Not the same thing.
> ∃H ∈ C_Functions
> is not equivalent to
> ∃H  ∈ Turing_Machines
> 
> as there are many C_Functions that are not the equivalent of Turing 
> Machines.
> 

The whole purpose here is to get you to understand what
templates are and how they reference infinite sets.

> 
>>
>> In both cases infinite sets are examined to see
>> if any H exists with the required properties.
>>
> 
> Yes, but the logic of Turing Machines looks at them one at a time, and 
> the input is a FULL INDEPENDENT PROGRAM.
> 

∃H  ∈ Turing_Machines
That does not look at one machine it looks as an infinite set of
machines. I am very happy to find out that you were not playing head
games. Linz actually used the words that you referred to.

> I'm not sure what you can define your computation system to be actually 
> based on, and what its supposed use is, since your 'decider' and 'input' 
> are so intertwined.
> 

The whole purpose here is to get you to understand what
templates are and how they reference infinite sets.

> And your supposed algorithm just doesn't work when you try to make you 
> system "Turing Complete" by letting D have the ability to have a COPY of 
> H, and being able to make copies of its input, like real Turing machines 
> can.
> 

The whole purpose here is to get you to understand what
templates are and how they reference infinite sets.

All the other issues are for another different post.

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