olcott <NoOne@NoWhere.com> writes:
On 7/31/2021 5:08 PM, Ben Bacarisse wrote:
olcott <NoOne@NoWhere.com> writes:
On 7/30/2021 2:58 PM, Ben Bacarisse wrote:Distraction. Everything you ignore below is about the proof and refers
olcott <NoOne@NoWhere.com> writes:
On 7/30/2021 7:39 AM, Ben Bacarisse wrote:
Any chance you will now say if
Ĥ.qx(⟨Ĥ⟩, ⟨Ĥ⟩)
transitions to Ĥ.qn or Ĥ.qy? If you find this question difficult, >>>>>>> please ask for some help in understanding it.
Ĥ.qx(⟨Ĥ⟩, ⟨Ĥ⟩) transitions to Ĥ.qn
An answer. Thank you.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
For Ĥ to be "exactly and precisely as in Linz" this, then, is the clause >>>>> that applies to your H and Ĥ:
There is no H in the relevant last paragraph of the Linz proof that
forms the basis for the Linz conclusion.
only to Ĥ.
This is an abuse of the notation (but I know what you mean). There isĤ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt
so Ĥ (M) applied to ⟨Ĥ⟩ (wM) does not halt, but you have just told me
that it does. That is what this full (but abbreviated) state transition >>>>> sequence means:
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
Which is it?
Ĥ0.q0 copies its input ⟨Ĥ1⟩ to ⟨Ĥ2⟩ then Ĥ0.qx simulates Ĥ1 with the
⟨Ĥ2⟩ copy then
Ĥ1.q0 copies its input ⟨Ĥ2⟩ to ⟨Ĥ3⟩ then Ĥ1.qx simulates Ĥ2 with the
⟨Ĥ3⟩ copy then
Ĥ2.q0 copies its input ⟨Ĥ3⟩ to ⟨Ĥ4⟩ then Ĥ2.qx simulates Ĥ3 with the
⟨Ĥ4⟩ copy then ...
no Ĥ1 or Ĥ2. If you think it helps to show which copy of ⟨Ĥ⟩ your >>> simulating "decider" is either running and/or currently looking at, you
need to come up with a notation that does that.
A better notation is what I have in my PDF actual subscripts but
people here tel me that their newsreader makes sure to totally ignore
posts with HTML so that do even see the post at all.
I am happy you have a notation you like. Are you prepared to address
that fact that your H^ is not "as in Linz"?
At least I know what
this "math poem" means, because you've been saying this "it's a
simulator until" stuff for years.
The outermost Ĥ0.qx correctly decides that its input: (⟨Ĥ1⟩, ⟨Ĥ2⟩)Apart from the bad notation, yes. All those copies and tests and
can't possibly ever reach its final state. Then it transitions to
Ĥ0.qn causing the outermost Ĥ0 to halt.
eventual deciding are neatly summed up in the last ⊢* Ĥ.qn of
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
Because the outermost Ĥ0.qx did not decide that Ĥ0 would never haltYou are free to define "decide correctly" in any way you like provided
and it is self evident that its input: (⟨Ĥ1⟩, ⟨Ĥ2⟩) can't possibly
ever reach its final state there is no contradiction or paradox and it >>>> decided correctly.
you are honest about it. But you hooked people in by saying that your Ĥ >>> is "exactly and precisely as in Linz", and you quoted, even now, what
Linz has to say about such TMs:
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt
This is your quote. You brought it up. You claimed your Ĥ was as Linz >>> states -- that Ĥ.q0 wM ⊢* Ĥ.qn if and only if M applied to wM does not >>> halt. Linz makes no exceptions based on why the transitions from Ĥ.q0
wM to Ĥ.qn occur. Linz does not say
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt or if M applied wM only halts
because...
Are you now saying that your TM was not "as in Linz"? (You should,
because you've admitted that elsewhere.)
Ĥ[0] is to be interpreted to mean Ĥ<sub>0</sub>
[0] Means the actual Turing machine and not a TM description.
[1] Means the first TM description parameter
[2] Means a copy of the the first TM description parameter
Now I am saying that when the actual unmodified Linz Ĥ is understood
to have a UTM/Halt-Decider at Ĥ[0].qx that this Ĥ[0].qx does correctly
decide that its input: (⟨Ĥ[1]⟩, ⟨Ĥ[2]⟩) can't possibly ever reach its
final state of Ĥ[1].qn or Ĥ[2].qn, therefore we know that its input
never halts therefore we know that a state transition from Ĥ[0].qx to
Ĥ0.qn is necessarily correct.
We all know you are declaring that to be correct. Here's why your Ĥ is
not "as in Linz". Linz requires that
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt
Any Ĥ that eventually transitions to Ĥ.qn on input wM must do so
if, and only if, the encoded M applied to wM does not halt. But you've
given us a case where your Ĥ is not like this:
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
Here we can see that Ĥ applied to ⟨Ĥ⟩ halts. You can call your Ĥ's behaviour "correct". You can call it anything you like. But it's not
"as in Linz". It does not say anything about Linz's proof. It does not
do anything people would call impossible or even interesting.
Presumably, you will simply explain, yet again, why you choose to call
it correct. You might even, yet again, quote the symbols from Linz that don't apply to your Ĥ in order to make you posts seem relevant.
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