On 7/28/2021 6:36 PM, André G. Isaak wrote:
> On 2021-07-28 14:06, olcott wrote:
>> On 7/28/2021 1:40 PM, André G. Isaak wrote:
>>> On 2021-07-28 10:38, olcott wrote:
>>>> On 7/28/2021 10:31 AM, André G. Isaak wrote:
>>>>> On 2021-07-28 08:09, olcott wrote:
>>>>>> On 7/27/2021 10:39 PM, André G. Isaak wrote:
>>>>>
>>>>>>> So does P(P) halt or not?
>>>>>>>
>>>>>>
>>>>>> Even though the first P in the invocation sequence reaches its
>>>>>> final state the fact that it only reaches its final state because
>>>>>> the second P in the invocation sequence was aborted proves that
>>>>>> H(P,P)==0 is correct.
>>>>>
>>>>> If a car only crashes because its brakes failed, that does not
>>>>> imply that it didn't crash.
>>>>>
>>>>> If a program returns the wrong result only because it has a bug,
>>>>> that does not imply that it didn't return the right answer.
>>>>>
>>>>> If a program only halts because the second P was aborted, that does
>>>>> not imply that it didn't halt.
>>>>>
>>>>
>>>> If an infinitely recursive sequence of function calls is aborted on
>>>> the second call instead of the first call that does not mean that it
>>>> was not an infinitely recursive sequence.
>>>
>>> But the definition of halting makes no mention of infinitely
>>> recursive sequences or aborted function calls. It only requires that
>>> P(P) reach a final state in a finite amount of time. P(P) meets this
>>> definition.
>>>
>>
>> If we base the decision on whether or not P halts entirely on the fact
>> that it reaches its final state
>
> That's the DEFINITION of halting, so of course that's the only thing we
> should base the decision of whether P halts on.
>

This is equally the definition of not halting:
Every input to a simulating halt decider that only stops running when
its simulation is aborted unequivocally specifies a computation that
never halts.

>> then we have the same situation as "This sentence is not true." It is
>> indeed not true and the definition of a true sentence is whether or
>> not its assertion is satisfied.
>
> I fail to see any parallels between P and the liar.
>
>> When we explicitly take into account the self-contradictory nature of
>> these two cases things are not as cut-and-dried.
>
> There's nothing self-contradictory about it.
>
>> "This sentence is not true." is indeed not true yet when we apply this
>> to the satisfaction of the whole sentence and not just its assertion
>
> That is gibberish.
>
>> then we get a contradiction. If it is true that it is not true then
>> that makes is not true.
>>
>> It is an easily verifiable fact that the input to H(P,P) cannot
>> possibly reach its final state while H remains a pure simulator.
>> Because H
>
> But the definition of halting makes no reference to what H does at all,
> nor to pure simulators. whether P(P) halts is based solely on the
> behaviour of P(P).
>

The definition of UTM does make reference to computational equivalence
forcing this statement to be necessarily true:

Every input to a simulating halt decider that only stops running when
its simulation is aborted unequivocally specifies a computation that
never halts.

>> remains a pure simulator until after it makes its halt status decision
>> then its decision that its input never halts is necessarily correct.
>>
>> That the first P in the infinitely recursive sequence reaches its
>> final state after H has made its correct halt status decision is just
>> like saying the liar paradox is true on the basis that its assertion
>> is satisfied.
>
> No. That the first P reaches its final state means that P(P) halts. The
> definition of halting doesn't care how or why it reaches this state.
> Only that it does.

So then we must conclude the the self contradictory input causes the
logic of the system to be inconsistent because it proves that P halts
and it proves that P never halts.

>>>>>> Because this is too difficult to understand and accept I have
>>>>>> temporarily changed the subject to refuting Rice's theorem. The
>>>>>> fact that the first P reaches its final state and the second P is
>>>>>> aborted can   be used as the criterion measure to consistently
>>>>>> reject all and only self-contradictory inputs. This does refute
>>>>>> Rices theorem.
>>>>>>
>>>>>>> You have on the one hand acknowledged that it does, while at the
>>>>>>> same time claimed that it doesn't halt in a 'pure simulator'. So
>>>>>>> if your 'global simulator' is not a pure simulator, what kind of
>>>>>>> simulator is it?
>>>>>
>>>>> You didn't answer the above. In the past you've claimed (falsely)
>>>>> that in a pure simulator, P(P) doesn't halt.
>>>>>
>>>>
>>>> While H remains a pure simulator of its input H(P,P) its input never
>>>> halts thus proving that its input never halts.
>>>
>>> But the definition of halting makes no mention of what happens inside
>>> H, regardless of whether it remains a 'pure simulator'. It only
>>> requires that the actual computation P(P) reach a final state in a
>>> finite amount of time. P(P) meets this definition.
>>>
>>>>> Now you appear to be using your 'global simulator' to recognise
>>>>> that P(P) does halt so that you can compare this with the results
>>>>> of H(P, P).
>>>>
>>>> It is still true that H(P,P) did correctly decide that its input
>>>> never halts. Because this is difficult to understand I am
>>>> temporarily changing the subject to Rice's theorem.
>>>
>>> No, it is not. The definition of halting is clearly defined, and P(P)
>>> clearly meets the definition of halting. Rice's theorem has no
>>> bearing on the fact that P(P) is halting computation.
>>>
>>
>> In this exact same way we would have to say that the liar paradox is
>> true because its assertion that it is not true is fully satisfied.
>
> The LP's "assertion that it is not true" is *not* satisfied, fully or
> otherwise.

The Liar Paradox: "This sentence is not true."
is indeed provably not true on the basis that it leads to a
contradiction and on the basis that it specifies and infinite cycle.

Only my own unique work finally resolves this:

Apparently in all of the years prior to my original (2016) paper no one
ever separated the analysis of propositions into their atomic units of
semantic compositionality:
(a) Assertion (Boolean) // What the expression of language claims to
be True
(b) Satisfied (Boolean) // Whether or not the expression of language
is True

https://www.researchgate.net/publication/323866366_The_Notion_of_Truth_in_Natural_and_Formal_Languages

> P(P), on the other hand, fully meets the definition of halting.
>
>> Whenever the assertion of a declarative sentence is satisfied we know
>> that this declarative sentence is true unless this declarative
>> sentence is self-contradictory.
>>
>> Whenever H decides that its input never halts its input never reaches
>> a final state. When its input is self-contradictory then another
>> different
>
> The input to H is simply *data*. Halting applies to *computations*.
>

Not exactly: UTM(P,I) ≡ P(I)

>> instance of the program that is not an input to H may halt.
>>
>>>>> But if P(P) doesn't halt in a 'pure simulator' then what kind of
>>>>
>>>> I did not say that P(P) does not halt in a pure simulator, you must
>>>> pay careful attention to every single word that I say. When you skip
>>>> a single word that reverses the meaning of what I say.
>>>>
>>>> The input to H(P,P) never halts while H remains in pure simulator mode.
>>>
>>> So what's the difference between a 'pure simulator' and H running in
>>> 'pure simulator mode'? One would have assumed that the latter meant
>>> that H was acting as a pure simulator.
>>>
>>
>> H is evaluating the halt status of its input on the basis of what the
>> behavior of this input would be if H never aborts the simulation of
>> this input.
>
> Which is the wrong criterion for evaluating this. The correct criterion
> is simply whether P(P) reaches a final state.
>
> <snippage>
>
>> The bottom line is that self-contradiction must be treated
>> differently, the conventional rules do not apply to self-contradiction.
>
> Where does the definition of halting entail some class which must be
> treated differently? All computations either reach a final state in a
> finite amount of time or they do not. There is no third option.
>

It is generally always the case that self-contradictory expressions of
language must always be treated differently than non self-contradictory
expressions of language.

That some sub-fields never noticed this is their fault and limitation.

>> When the assertion of a declarative sentence is satisfied this makes
>> the sentence true unless the sentence is self-contradictory.
>>
>> The fact that "This sentence is not true." is not true does not make
>> the sentence true because the sentence is self-contradictory.
>>
>> When a simulating halt decider reports that its input does not halt
>> then no instance of the same program ever reaches a final state unless
>> the input program specifies self-contradiction.
>
> There are no 'instances of the same program' when talking in terms of
> Turing Machines.
>
> André
>

Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qy ∞
if M applied to wM halts, and

Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
if M applied to wM does not halt

Ĥ( ⟨Ĥ⟩ ) specifies an infinite cycle from Ĥ.qx to Ĥ.q0 all the time that
Ĥ.qx remains a pure simulator of its input.

Every simulation that would never stop unless its simulating halt
decider stops it at some point specifies infinite execution. This
remains true for: Ĥ.qx(⟨Ĥ⟩, ⟨Ĥ⟩)

Ĥ( ⟨Ĥ⟩ ) only stops because its simulating halt decider stops it at some
point.

--
Copyright 2021 Pete Olcott

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