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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Thu, 30 May 2024 21:37:34 -0400
Organization: i2pn2 (i2pn.org)
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On 5/30/24 9:11 AM, olcott wrote:
> On 5/30/2024 4:11 AM, joes wrote:
>> Am Wed, 29 May 2024 22:48:45 -0500 schrieb olcott:
>>> On 5/29/2024 9:55 PM, Richard Damon wrote:
>>>> On 5/29/24 10:36 PM, olcott wrote:
>>>>> On 5/29/2024 9:25 PM, Richard Damon wrote:
>>>>>> On 5/29/24 9:55 PM, olcott wrote:
>>>>>>> When the category is examined all at once then there is no need
>>>>>>> to look at each individual element.
>>>>>> So, which one or ones gave the correct answer for their input?
>>
>>>>> *Formalizing the Linz Proof structure*
>>>>> ∃H  ∈ Turing_Machines
>>>>> ∀x  ∈ *Turing_Machines_Descriptions*
>>>>> ∀y  ∈ Finite_Strings
>>>>> such that H(x,y) = Halts(x,y)
>>>>>
>>>>> When we formalize it that way then some simulating halt deciders
>>>>> get the correct answer.
>>>>>
>>>>> *Everyone else implicitly assumes this incorrect formalization*
>>>>> ∃H  ∈ Turing_Machines
>>>>> ∀x  ∈ *Turing_Machines*
>>>>> ∀y  ∈ Finite_Strings
>>>>> such that H(x,y) = Halts(x,y)
>>>>>
>>>> Nope.
>>>> You just don't understand the meaning of a "Description" in the 
>>>> problem.
>>>>
>>> I have an OCD/Aspergers degree of single-minded focus.
>> Checks out.
>>
>>> *A deciders compute the mapping*
>>> FROM ITS INPUTS
>>> *to it own accept or reject state*
>>>
>>> *Deciders cannot take*
>>> ACTUAL TURING MACHINES AS INPUTS
>>>
>>> *Deciders can only take*
>>> FINITE STRINGS AS INPUTS
>> Poetic.
>> What is an „actual Turing machine”?
>>
> 
> *Formalizing the Linz Proof structure*
> ∃H  ∈ Turing_Machines
> ∀x  ∈ Turing_Machine_Descriptions
> ∀y  ∈ Finite_Strings
> such that H(x,y) = Halts(x,y)
> 
> Every H is an actual Turing_Machine
> 
> Every x is a Turing_Machine_Description
> thus not an actual Turing_Machine
> 

But represents one, and thus we can determine the actual behavior of 
that Turing Machine so desceribed.

And, if you actually READ what Linz says, his formalation would be:

For Halting to be computable we must have:
∃H  ∈ Turing_Machines such that
∀M  ∈ Turing_Machines, with a description Wm, and
∀w  ∈ Finite_Strings
such that H(Wm,w) = Halts(M,w)

Try to read his work a bit betetr.

He describes FIRST having a Turing Machine M to be decided on, and THEN 
converts it to Wm to give it to the decider.

SKippig steps is not allowed.