// Strachey(1965) "An impossible program"
// CPL translated to C
// https://doi.org/10.1093/comjnl/7.4.313
void P(u32 x)
{ if (H(x, x))
HERE: goto HERE;
}

int main()
{ Output("Input_Halts = ", P((u32)P);
}

*Conventional Halt Deciding Axiom*
When the pure simulation of the machine description ⟨P⟩ of a machine P
on its input I never halts we know that P(I) never halts. // based on
UTM(⟨P⟩,I) ≡ P(I)

*Stipulated Definition of Halting*
An input to a halt decider is defined to halt if and only if this input
stops running while simulating halt decider H remains a pure simulator
of this input.

*Pathological Input* to a halt decider is defined as any input that was
defined to do the opposite of whatever its corresponding halt decider
decides.

It seems to me that the Stipulated Definition of Halting does not add
anything but clarity to the Conventional Halt Deciding Axiom. Others may
see this differently.

None-the-less the Stipulated Definition of Halting does provide the
means to correctly decide the halting status of Pathological Inputs.

int main() { P(P); } is defined to be a non-halting computation under
the stipulated definition.

The stipulated definition of halting defines the exact same set as the
conventional definition of halting with the possible exception that
pathological inputs are decided as non-halting inputs.

Because the stipulated definition of halting is merely a paraphrase of
the Conventional Halt Deciding Axiom I propose that this stipulated
definition of halting merely provides clarity and does not change the
conventional definition of halting at all.

https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation

--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein

On 7/17/2021 1:36 PM, olcott wrote:
> // Strachey(1965) "An impossible program"
> // CPL translated to C
> // https://doi.org/10.1093/comjnl/7.4.313
> void P(u32 x)
> {
>   if (H(x, x))
>     HERE: goto HERE;
> }
>
> int main()
> {
>   Output("Input_Halts = ", P((u32)P);
> }
>
> *Conventional Halt Deciding Axiom*
> When the pure simulation of the machine description ⟨P⟩ of a machine P
> on its input I never halts we know that P(I) never halts. // based on
> UTM(⟨P⟩,I) ≡ P(I)
>
> *Stipulated Definition of Halting*
> An input to a halt decider is defined to halt if and only if this input
> stops running while simulating halt decider H remains a pure simulator
> of this input.
>

A stipulative definition is a type of definition in which a new or
currently existing term is given a new specific meaning for the purposes
of argument or discussion in a given context.
https://en.wikipedia.org/wiki/Stipulative_definition

If it is stipulated that cats are all animals having DNA that exactly
matches cat DNA, then when a cat barks or gives birth to genetic puppies
it remains a cat by stipulative definition.

> *Pathological Input* to a halt decider is defined as any input that was
> defined to do the opposite of whatever its corresponding halt decider
> decides.
>
> It seems to me that the Stipulated Definition of Halting does not add
> anything but clarity to the Conventional Halt Deciding Axiom. Others may
> see this differently.
>
> None-the-less the Stipulated Definition of Halting does provide the
> means to correctly decide the halting status of Pathological Inputs.
>
> int main() { P(P); } is defined to be a non-halting computation under
> the stipulated definition.
>
> The stipulated definition of halting defines the exact same set as the
> conventional definition of halting with the possible exception that
> pathological inputs are decided as non-halting inputs.
>
> Because the stipulated definition of halting is merely a paraphrase of
> the Conventional Halt Deciding Axiom I propose that this stipulated
> definition of halting merely provides clarity and does not change the
> conventional definition of halting at all.
>
>
> https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation
>
>

--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein

On 7/17/21 12:36 PM, olcott wrote:
> // Strachey(1965) "An impossible program"
> // CPL translated to C
> // https://doi.org/10.1093/comjnl/7.4.313
> void P(u32 x)
> {
>   if (H(x, x))
>     HERE: goto HERE;
> }
>
> int main()
> {
>   Output("Input_Halts = ", P((u32)P);
> }
>
> *Conventional Halt Deciding Axiom*
> When the pure simulation of the machine description ⟨P⟩ of a machine P
> on its input I never halts we know that P(I) never halts. // based on
> UTM(⟨P⟩,I) ≡ P(I)
>
> *Stipulated Definition of Halting*
> An input to a halt decider is defined to halt if and only if this input
> stops running while simulating halt decider H remains a pure simulator
> of this input.

Which differs from the classical definition of the Halting Problem, so
you have just stipulated yourself out of the proof you want to make.

You CAN'T 'counter' a proof with a counter example that doesn't follow
the definitions of the original.

FAIL.

Since H switch from being a pure simulator, it ISN'T a pure simulator,
PERIOD. It doesn't matter that it acted like one for some period of time.

Please mail your trafic tickets for all the red lights you would have
run if you hadn't decided to stop for them.

>
> *Pathological Input* to a halt decider is defined as any input that was
> defined to do the opposite of whatever its corresponding halt decider
> decides.
>
> It seems to me that the Stipulated Definition of Halting does not add
> anything but clarity to the Conventional Halt Deciding Axiom. Others may
> see this differently.

FALSE. It adds inconsistency.
>
> None-the-less the Stipulated Definition of Halting does provide the
> means to correctly decide the halting status of Pathological Inputs.
>
> int main() { P(P); } is defined to be a non-halting computation under
> the stipulated definition.

But UNSOUND logic can 'prove' anything.

>
> The stipulated definition of halting defines the exact same set as the
> conventional definition of halting with the possible exception that
> pathological inputs are decided as non-halting inputs.
>
> Because the stipulated definition of halting is merely a paraphrase of
> the Conventional Halt Deciding Axiom I propose that this stipulated
> definition of halting merely provides clarity and does not change the
> conventional definition of halting at all.
>
>
> https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation
>
>

On 7/17/21 1:40 PM, olcott wrote:
> On 7/17/2021 1:36 PM, olcott wrote:
>> // Strachey(1965) "An impossible program"
>> // CPL translated to C
>> // https://doi.org/10.1093/comjnl/7.4.313
>> void P(u32 x)
>> {
>>    if (H(x, x))
>>      HERE: goto HERE;
>> }
>>
>> int main()
>> {
>>    Output("Input_Halts = ", P((u32)P);
>> }
>>
>> *Conventional Halt Deciding Axiom*
>> When the pure simulation of the machine description ⟨P⟩ of a machine P
>> on its input I never halts we know that P(I) never halts. // based on
>> UTM(⟨P⟩,I) ≡ P(I)
>>
>> *Stipulated Definition of Halting*
>> An input to a halt decider is defined to halt if and only if this
>> input stops running while simulating halt decider H remains a pure
>> simulator of this input.
>>
>
> A stipulative definition is a type of definition in which a new or
> currently existing term is given a new specific meaning for the purposes
> of argument or discussion in a given context.
> https://en.wikipedia.org/wiki/Stipulative_definition
>
> If it is stipulated that cats are all animals having DNA that exactly
> matches cat DNA, then when a cat barks or gives birth to genetic puppies
> it remains a cat by stipulative definition.

Note, that 'stipulating' a definition doesn't actually make it so. It
clearifies how YOU are going to use the term. If you stipulate a
definition that differs from conventional usage, then this makes your
argument irrelevant to things that use that conventional definition.

>