Equivalent halting definitions

[olcott]

The set of Turing machine's that would not halt on their input is
identical to the set of Turing machines that must have their simulation
aborted to prevent their otherwise infinite execution because:

(a) Every input having infinite execution would have to have its
simulation aborted to prevent its otherwise infinite execution.

(b) Every input not having infinite execution not need to have its
simulation aborted to prevent infinite execution.

Thus the set of inputs that must be aborted to prevent their infinite
execution is the exact same set of inputs that have infinite execution
and their complementary set is the set of halting computations.

--
Copyright 2021 Pete Olcott

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

[olcott]

On 4/20/2021 4:31 PM, Richard Damon wrote:
> On 4/20/21 4:54 PM, olcott wrote:
>> On 4/20/2021 3:32 PM, Richard Damon wrote:
>>> On 4/20/21 3:23 PM, olcott wrote:
>>>> On 4/20/2021 1:57 PM, Richard Damon wrote:
>>>>> On 4/20/21 1:44 PM, olcott wrote:
>>>>>> On 4/20/2021 11:48 AM, Richard Damon wrote:
>>>>>>> On 4/20/21 12:38 PM, olcott wrote:
>>>>>>>> On 4/20/2021 11:18 AM, Richard Damon wrote:
>>>>>>>>> On 4/20/21 12:03 PM, olcott wrote:
>>>>>>>>>> On 4/20/2021 10:54 AM, Richard Damon wrote:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> As long as the first part is true
>>>>>>>>>>>> All the computations that would not halt unless the halt decider
>>>>>>>>>>>> aborts
>>>>>>>>>>>> their simulation
>>>>>>>>>>
>>>>>>>>>> then the second part is impossibly false.
>>>>>>>>>
>>>>>>>>> But, it is demonstartably true for H_Hat.
>>>>>>>>
>>>>>>>> That isn't even a complete thought.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> H_Hat is demonstrably a Halting Computation given the stipulation
>>>>>>> that
>>>>>>> Halts is a Computation and that Halts answers non-halting when asked
>>>>>>> about the halting of H_Hat(H_Hat).
>>>>>>>
>>>>>>> Thus the second case is demonstrably True, and thus the first MUST be
>>>>>>> False, or you admit your logic system is inconsistent.
>>>>>>>
>>>>>>
>>>>>> The set of computations that only halt when their simulation is
>>>>>> aborted
>>>>>> is an identical set to the set of computations that never halt.
>>>>>>
>>>>>
>>>>> WRONG. That is only true IF you have a perfect Halt Decider. Prove that
>>>>> you have that first before you use that fact to prove it.
>>>>>
>>>>
>>>> When it is proven that set Q has the exactly same elements as set R then
>>>> this proves that the definition of set Q is equivalent to the definition
>>>> of set R.
>>>>
>>>
>>> Except it hasn't be provem, in fact with your definition it has been
>>> shown that there IS a difference.
>>>
>>
>> Yes it has been shown that the definitions are different.
>> As long as these different definitions define the same set of elements
>> then these different definition are equivalent.
>>
>
> It has been shown that the sets are different. At least one element has
> moved from one set to the other
>
>
> Boy are you dense.
>

Try and find the differing element:

Whether or not a simulated input must have its simulation aborted to
prevent its infinite execution <is> equivalent to deciding that an input
has infinite execution because (a) and (b)

(a) Every input having infinite execution would have to have its
simulation aborted to prevent its otherwise infinite execution.

(b) Every input not having infinite execution need not have its

simulation aborted to prevent infinite execution.

Thus the set of inputs that must be aborted to prevent their infinite

execution is the exact same set of inputs that have infinite execution.

[Kaz Kylheku]

On 2021-04-20, olcott <No...@NoWhere.com> wrote:
> Whether or not a simulated input must have its simulation aborted to
> prevent its infinite execution <is> equivalent to deciding that an input
> has infinite execution because (a) and (b)

Yeah yeah; whether or not an integer must be incremented by one
to make it divisible by two <is> equivalent to whether it is odd.

[olcott]

You know that I am correct about this that is why you removed the
context. I am going to keep bugging you again and again on this until
you give it an accurate critique

Try and find the differing element:

Whether or not a simulated input must have its simulation aborted to
prevent its infinite execution <is> equivalent to deciding that an input
has infinite execution because (a) and (b)

[Richard Damon]

H_Hat, by the fact that when run as an independent machine (assuming
Halts behaves as defined) will be a Halting Computation (yes, it halts
because the Halts it used to emulate a H_Hat terminated that emulation
and returned a non-halting answer), and thus fall into category (b).

By YOUR definition, you claim that Halts needed to abort it to prevent
its otherwise infinite execution, so you put it in category (a).

(b) != (a) so we have a difference.

That, or your function Halts doesn't meet the requirements it needs to
in order to be called a Halt Decider.

[olcott]

Is an off topic dodge of this point:

Computations that only halted because the halt decider correctly decided
that their simulation must be aborted to prevent their otherwise
infinite execution define the exact same set of elements as the set of
non-halting computations.