Solution to Alan Turing’s 1936 Halting Problem Version(6) [ Decidability Decider Defined ]

[peteolcott]

https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States

If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

When-so-ever the potential halt decider cannot derive a formal proof from
its input strings to its final states of Halts or Loops, undecidability has
been decided.

A Halting Decidability Decider H could be defined as:
A David Hilbert formalist proof
where the inference steps are construed as an execution trace
in language of Turing machine descriptions
from the initial state of a Turing machine description H.q0 and
its finite string input pair: (Wm, W) to final states corresponding to
the mathematical logic predicates: Halts(Wm, W) and Loops(Wm, W) where
Loops(Wm, W) means that Wm contains a loop without a loop termination condition.

There does not exist any inference steps (execution trace) in the language of
Turing machine descriptions from H.q0 and input pair: (Wm, W) to any final state of
H corresponding to the logic predicates of Halts(Wm, W) or Loops(Wm, W).

~∃ H ∈ Turing_Machine_Descriptions
∀ Wm ∈ Turing_Machine_Descriptions
∀ W ∈ Finite_Strings
( H.Halts(Wm, W) ∨ H.Loops(Wm, W) )

[Shobe, Martin]

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there
are certain tuples that are undecidable.

Martin Shobe

[peteolcott]

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.
Martin Shobe

Definition of Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy // Wm is a TMD that would halt on its input W
H.q0 Wm W ⊢* H.qn // else

Definition of Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn

Linz proves that the H(Ĥ, Ĥ) is undecidable.

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf

[Shobe, Martin]

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there
are certain tuples that are undecidable.

Definition of Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy  // Wm is a TMD that would halt  on its input W
H.q0 Wm W ⊢* H.qn  // else
Definition of Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn
Linz proves that the H(Ĥ, Ĥ) is undecidable.

He proves that H doesn't exist and therefore there is no H(Ĥ, Ĥ) to decide.

Martin Shobe

[peteolcott]

Post by Shobe, Martin

Post by peteolcott

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.

Definition of Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy  // Wm is a TMD that would halt  on its input W
H.q0 Wm W ⊢* H.qn  // else
Definition of Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn
Linz proves that the H(Ĥ, Ĥ) is undecidable.

He proves that H doesn't exist and therefore there is no H(Ĥ, Ĥ) to decide.
Martin Shobe

First of all one cannot use a thing to prove that the thing itself
does not exist, that would be contradictory.

As it turns out (and I have shown) everyone in the world has been
wrong about halt deciders because they have always falsely assumed
that their specification requires fulfilling a self-contradictory
set of premises, like finding a number X such that X > 5 & X < 3.

[peteolcott]

Post by Shobe, Martin

Post by peteolcott

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.

Definition of Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy  // Wm is a TMD that would halt  on its input W
H.q0 Wm W ⊢* H.qn  // else
Definition of Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn
Linz proves that the H(Ĥ, Ĥ) is undecidable.

He proves that H doesn't exist and therefore there is no H(Ĥ, Ĥ) to decide.
Martin Shobe

AKA this tuple is undecidable: (H, Ĥ, Ĥ)

[Shobe, Martin]

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that
there are certain tuples that are undecidable.

Definition of Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy  // Wm is a TMD that would halt  on its input W
H.q0 Wm W ⊢* H.qn  // else
Definition of Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn
Linz proves that the H(Ĥ, Ĥ) is undecidable.

He proves that H doesn't exist and therefore there is no H(Ĥ, Ĥ) to decide.
Martin Shobe

AKA this tuple is undecidable: (H, Ĥ, Ĥ)

The tuple is nonexistent.

Martin Shobe

[peteolcott]

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.
Martin Shobe

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
Sure it is, the Linz proof is based on this tuple being undecidable
(H, Ĥ, Ĥ).

[Shobe, Martin]

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there
are certain tuples that are undecidable.

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
Sure it is, the Linz proof is based on this tuple being undecidable
(H, Ĥ, Ĥ).

It really isn't. Linz knew better than to base a proof off of something
as ridiculous as a single instance being undecidable.

Martin Shobe

[peteolcott]

Post by peteolcott

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
Sure it is, the Linz proof is based on this tuple being undecidable
(H, Ĥ, Ĥ).

It really isn't. Linz knew better than to base a proof off of something as ridiculous as a single instance being undecidable.
Martin Shobe

In case you didn't notice (because of its ⊢* wildcard state transition)
H specifies every element of the set of (at least hypothetical) halt
deciders that take exactly two inputs:

Turing Machine H (state transition sequence)
H.q0 Wm W ⊢* H.qy // Wm is a TMD that would halt on its input W
H.q0 Wm W ⊢* H.qn // else

Thus Ĥ specifies an adaptation to H specifying the infinite set of
conventional self-referential halting problem proof counter-examples.

Turing Machine Ĥ (state transition sequence)
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qy ∞
Ĥ.q0 Wm ⊢* Ĥ.qx Wm Wm ⊢* Ĥ.qn

Therefore the satisfaction of this sentence would refute Rice:
∃H ∈ Turing_Machine_Descriptions ∀ X ∈ Ĥ
( H.halting_decidable(X, X) ∨ H.halting_undecidable(X, X) )

[Shobe, Martin]

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott

Post by Shobe, Martin

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that
there are certain tuples that are undecidable.

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
Sure it is, the Linz proof is based on this tuple being undecidable
(H, Ĥ, Ĥ).

It really isn't. Linz knew better than to base a proof off of
something as ridiculous as a single instance being undecidable.

In case you didn't notice (because of its ⊢* wildcard state transition)
H specifies every element of the set of (at least hypothetical) halt

A set that is empty, in case you didn't notice. It still doesn't make
Linz's proof one that is based of off (H, Ĥ, Ĥ) being undecidable.
(Which you clearly have not noticed).

[snip remaining nonsense]

Martin Shobe

[peteolcott]

Post by Shobe, Martin

Post by peteolcott

Post by peteolcott

Post by peteolcott
https://www.researchgate.net/publication/323384939_Halting_Problem_Proof_from_Finite_Strings_to_Final_States
If there truly is a proof that shows that no universal halt decider exists
on the basis that certain tuples of {tm H, tm X, input I} are undecidable,
then this very same proof can be used by H to reject some of its inputs.

There isn't. The proof is not based on the nonsensical idea that there are certain tuples that are undecidable.

http://liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
Sure it is, the Linz proof is based on this tuple being undecidable
(H, Ĥ, Ĥ).

It really isn't. Linz knew better than to base a proof off of something as ridiculous as a single instance being undecidable.

In case you didn't notice (because of its ⊢* wildcard state transition)
H specifies every element of the set of (at least hypothetical) halt

A set that is empty, in case you didn't notice.

I did notice that both you and Ben take the conventional conclusion as a premise
thereby short-circuiting my correct refutation of the proof.

Post by Shobe, Martin
It still doesn't make Linz's proof one that is based of off (H, Ĥ, Ĥ) being undecidable.

Yes is damn well does. Linz proves that the Halting Problem is undecidable
entirely on the basis that H cannot decide input pair: (Ĥ, Ĥ).

Post by Shobe, Martin
(Which you clearly have not noticed).
[snip remaining nonsense]
Martin Shobe