Re: That P(P) of main() halts does not contradict H(P,P)==0 [ pure functions ][ Rice's Theorem ]

[olcott]

On 9/7/2021 10:39 AM, André G. Isaak wrote:
> On 2021-09-07 08:26, olcott wrote:
>> On 9/7/2021 9:14 AM, André G. Isaak wrote:
>>> On 2021-09-07 07:51, olcott wrote:
>>>> On 9/7/2021 12:36 AM, André G. Isaak wrote:
>>>
>>>>> The input to H and to Ĥ describe the *exact* same computation. You
>>>>> can't claim that it must be aborted in one instance but that it
>>>>> halts without being aborted in the other.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> So in other words you "believe" that a TM description simulated by a
>>>> UTM is completely aware of every step that the UTM makes in
>>>> simulating itself. I would say that you are alone in this belief.
>>>
>>> I'm saying that a given string specifying a TM description and an
>>> input to that TM specifies a single, unique computation. That
>>> computation is either a halting computation or it isn't. What some
>>> putative UTM does has no bearing on this fact.
>>>
>>
>> What the putative halt decider / UTM does is an integral aspect of how
>> the input has been defined making this input pathological:
>>
>>       Now we construct a new Turing machine D with H as a subroutine.
>>       This new TM calls H to determine what M does when the input to M
>>       is its own description ⟨M⟩. Once D has determined this information,
>>       it does the opposite.  (Sipser:1997:165)
>>
>>> If that computation halts when run on its own but doesn't halt when
>>> emulated by some "UTM", then the "UTM" is broken.
>>>
>>> André
>>>
>>
>> When H simulates ⟨Ĥ⟩ ⟨Ĥ⟩ it can see that Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ transitions to Ĥ.qn
>>
>> When Ĥ.qx simulates ⟨Ĥ⟩ ⟨Ĥ⟩ it sees that none of its slaves ever
>> transition to any final state.
>>
>> The law of computer science that says that a function must derive that
>> same results with the same input does not seem to apply to some cases
>> of identical functions having the same input.
>>
>> Any idiot can see that H has an entirely different halt deciding basis
>> than Ĥ.qx thus providing the means for the behavior of H to diverge
>> from the behavior of Ĥ.qx.
>
> You are really very confused here.
>
> So let's review some basic facts here.
>
> First, as I said, a string specifying a TM description and an input to
> that TM specifies a single, unique computation.
>
> Halting is an *intrinsic* property of a computation.
>
> Do you understand what 'instrinsic' means?
>
> Remember that computations, as defined in computational theory, are
> entirely self-contained objects. When you perform a computation, there
> is no hardware, or operating system, or software environment involved.
> Every computation either halts or doesn't halt, and the halting status
> of a particular computation is a fixed, eternal value.
>

int main()
{
if (H1((u32)P, (u32)P) != H((u32)P, (u32)P))
OutputString("Pathological self-reference error!");
}

if (H applied to ⟨Ĥ⟩ ⟨Ĥ⟩ != Ĥ applied to ⟨Ĥ⟩)
OutputString("Pathological self-reference error!");

> Sure, you can define different "simulator" functions H and H1 in which
> H(P, P) behaves differently from H1(P, P), but that's because H(P, P)
> and H1(P, P) are different computations[1] which means they are distinct
> from one another and are also distinct from P(P).
>

The master slave relationship from H to Ĥ.qx makes them distinctly
different computations even though they are otherwise identical and have
the same input.

> But when we ask about the halting behaviour of the computation P(P), the
> halting problem is concerned *only* with the behaviour of P(P). Whatever
> H(P, P) or H1(P, P) do has no relevance to the halting behaviour of
> P(P). If H and H1 partially emulate their input, whatever might happen
> during that emulation is also entirely irrelevant to the halting
> behaviour of P(P). If H and H1 "see" different things and then behave
> differently that may be relevant to the computations H(P, P) and H1(P,
> P) but it is irrelevant to the computation P(P).
>
> You don't seem to get this very simple fact.
>
> André
>
> [1] Well, you claim they are different but also claim they have
> identical code. I won't address that mysterious claim here.
>
>


--
Copyright 2021 Pete Olcott

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