Re: Linz's proofs. --- Good catch !

On 3/7/2024 12:46 PM, Richard Damon wrote:

On 3/7/24 10:30 AM, olcott wrote:

On 3/7/2024 10:44 AM, immibis wrote:

On 7/03/24 17:16, Ben Bacarisse wrote:

immibis <news@immibis.com> writes:
>

On 7/03/24 12:32, Ben Bacarisse wrote:

The students I taught seemed to have no problem with this sort of case
analysis.  But the "assume H does X" argument lead to lots of "but H1
could be better" arguments.

>
They aren't satisfied with "we can do the exact same thing with H1 to prove
that H1 doesn't work either"?

>
In the vast majority of cases, yes, but even then there is a logical
problem with going down that route -- there is no H so there can't be an
H1 that does better.  Once this objection is properly examined, it turns
out to be the argument I ended up preferring anyway.  H isn't a halt
decider, it's just any old TM and we show it can't be halt decider for
one reason or another.
>

>
Unless your students are extremely pedantic... maybe they are... I don't see what's illogical with:
>
"I think H is a halt decider."
"But it doesn't: see this proof."
"Oh. Well, even though H isn't a halt decider, how do we know there isn't a program H1 which is a halt decider?"
"The proof would still work for H1, or H2, or any other program you think is a halt decider."
>
>

>
It is an easily verified fact that:
H(D,D) Sees that D(D) is calling H(D,D) at machine address 00001522
H1(D,D) Sees that D(D) is NOT calling H1(D,D) at machine address 00001422
*different machine addresses is the reason for different return values*
>
>

 Which proves that H1 and H are different computation and thus different Turing Machines, so H1 getting the right answer doesn't "fix" H's getting the wrong answer.
 

Good catch !!!
For Olcott machines Linz H ⟨Ĥ⟩ ⟨Ĥ⟩ would map its input with
its own TMD concatenated to this input to its Boolean result.
Linz Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ would map its input with its own TMD
concatenated to this input to its Boolean result.
thus finally explaining how Linz H ⟨Ĥ⟩ ⟨Ĥ⟩ can correctly
determine the halt status of its input while Linz Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
cannot.

H1^ will confound H1, thus showing it isn't a halt decider either.

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