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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Tue, 28 May 2024 22:04:14 -0400
Organization: i2pn2 (i2pn.org)
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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.

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

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.



> 
> 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.

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.

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.