On 3/9/2022 8:50 PM, André G. Isaak wrote:
> On 2022-03-09 06:03, olcott wrote:
>> On 3/8/2022 11:32 PM, André G. Isaak wrote:
>>> On 2022-03-08 21:20, olcott wrote:
>>>> On 3/8/2022 9:41 PM, Ben Bacarisse wrote:
>>>
>>>>> No.  Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn, on its own, is neither
>>>>> "correct" nor "incorrect".  It is simply a statement about what a TM
>>>>> does when presented with some input.  There has to be an associated
>>>>> condition by which the correctness can be assessed, and you
>>>>> steadfastly
>>>>> remove the conditions that Linz so clearly gives you:
>>>>>
>>>>
>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
>>>> If the pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H would reach its
>>>> final state.
>>>>
>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>>>> If the pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H would never reach
>>>> its final state.
>>>
>>> The above doesn't jive with the nonsense you keep asserting
>>> elsewhere. You keep claiming that TMs can only deal with the finite
>>> strings which are their inputs.
>>>
>>
>>  From the above:
>> ⟨Ĥ⟩ is the common convention for a finite string Turing machine
>> description.
>>
>> Ĥ is the convention for Turing machine Ĥ.
>>
>> ⟨Ĥ⟩ ⟨Ĥ⟩ is the convention for the input to the copy of Linz H at Ĥ.qx.
>>
>> embedded_H is the convention for referring to the copy of Linz H
>> embedded at Ĥ.qx.
>
> I'm not sure why you're responding with all the above. I've read Linz
> and I've read your previous posts so I already know how you are using
> these terms.
>
>>> But the pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H is neither a finite
>>> string nor an input to embedded_H. So why do you think that non-input
>>> non-finite string should be a valid basis for a decision criterion
>>> whereas Ĥ applied to ⟨Ĥ⟩ should not be?
>>>
>>
>> It it always that case that every decider computes the mapping from
>> its finite string input to accept or reject state.
>>
>> In the case of a simulating halt decider this mapping is computed on
>> the basis of simulating the finite string and examining the simulated
>> behavior for any infinite behavior patterns.
>
> But the point is that in your criteria above you keep talking about the
> "pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H"
>
> Your embedded_H is *not* a pure simulator; it is a modified copy of what
> you claim to be a simulating halt decider but with its accepting state
> replaced by an infinite loop.
>
> So you are asking it to base its decision on the behaviour of some
> hypothetical "pure simulator" which is *not* a string nor the input to
> embedded_H.

embedded_H contains all of the functionality of a UTM as well as
additional halt deciding functionality. The halt deciding functionality
bases its halt status decision on matching infinite behavior patterns,
or failing to match any of these patterns.

> Why is that allowed but it is not allowed to base a decision
> on the behaviour of Ĥ (which while not a string nor an actual input is
> at least a TM which embedded_H has a complete description of) applied to
> ⟨Ĥ⟩?
>

All deciders map finite string inputs to an accept or reject state.
⟨Ĥ⟩ ⟨Ĥ⟩ are finite string inputs to embedded_H. Ĥ is neither a finite
string nor an input to to embedded_H.

>>>>>    Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qn  if (and only if) Ĥ does not halt on input ⟨Ĥ⟩.
>>>>>
>>>>> For Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn to be correct, that must
>>>>> occur if, and only if, Ĥ does not halt on input ⟨Ĥ⟩.
>>>>>
>>>>> The whole world knows why you keep omitting the condition and,
>>>>> instead,
>>>>> claim that transitioning to Ĥ.qn "correct" with no reference to the
>>>>> actual condition for correctness.  When pushed, you replace the actual
>>>>> condition with your "it had to be aborted, honest, guv" waffle in
>>>>> place
>>>>> of the proper condition derived from what Linz asserts about H to
>>>>> start
>>>>> with.
>>>>>
>>>>> It's all so tediously obvious.  You won't be able to fool anyone with
>>>>> it.
>>>>>
>>>>>> You (and Linz) continue to be under the mistaken notion that a halt
>>>>>> decider must determine the halt status of the computation that
>>>>>> contains itself.
>>>>>
>>>>> A halt decider, H, must be correct about every input -- even those
>>>>> strings that include something very like an encoding of H (as is the
>>>>> case for Ĥ).
>>>>>
>>>>
>>>> You were much sloppier in your first comment that I reponded to.
>>>> Halt deciders do not compute the halt status of non-finite string
>>>> non-inputs.
>>>>
>>>>> It almost sounds like are preparing to give up on the ruse you've been
>>>>> trying to pull for the last few years, and that you are going to go
>>>>> back
>>>>> to the "it's an invalid question nonsense" from many years ago.
>>>>>
>>>>>> This mistake would require a decider to base its accept or reject
>>>>>> decision on a non-string, non-input.
>>>>>
>>>>> A decider must accept or reject every string and which one of those is
>>>>> the case is determined by the input string and nothing else.
>>>>> Neither I,
>>>>> nor Linz, nor anyone else, has suggested otherwise.
>>>>
>>>> Both you and Linz incorrectly believe that embedded_H must report on
>>>> the non-input non-finite string of Ĥ ⟨Ĥ⟩.
>>>
>>> It's not just Ben and Linz, but every single author who has ever
>>> written about this particular proof. That's because it *isn't* a
>>> mistake.
>>>
>>
>> As it turns out to actually be even an actual universal consensus does
>> not guarantee truth. Prior to Pythagoras there was a universal
>> consensus that the Earth was flat.
>
> <pedantry> Your facts here are incorrect. There is no evidence to
> suggest that there was a universal consensus that the earth was flat
> prior to Pythagoras. </pedantry>
>
>>> Don't you think that it's a bit telling that everyone in the universe
>>> who actually *understands* how Turing Machine agrees with Linz,
>>> whereas you, who have consistently demonstrated that you've never
>>> once actually designed or worked with actual Turing Machines, are the
>>> only one who thinks otherwise?
>>>
>>
>> Especially not when every rebuttal to my work was like yours and
>> entirely anchored in a misunderstanding of what I am even saying.
>>
>>> Maybe you start by learning the basics and leave talking about Turing
>>> Machines to people who have actually dealt with Turing Machines.
>>>
>>> André
>>>
>>
>> I am actually starting with the establishing the fundamental
>> philosophical underpinnings of the notion of truth itself.
>
> But the issue here has nothing to do with "truth" but with how Turing
> Machines work.
>
> If you were to actually look at some examples of real Turing Machines
> maybe you would come to understand this. When we try to construct a TM
> to compute some function one of the first steps involved is to figure
> out how to encode elements of the domain of that function as strings
> which can actually be passed to a TM. But the TM is still expected to
> process those strings in a way which actually conforms to the function
> being computed.
>
> Consider a TM which decides whether a given integer x is even or not.
> There is no way in which we can pass an actual integer to a TM, since an
> integer is not a string. But we can easily come up with a way of
> encoding the integer x as a string ⟨x⟩ which *can* be passed to an integer.
>
> Whether the above TM accepts or rejects ⟨x⟩ *must* depend on whether x
> is even or odd. Otherwise it isn't computing the function it purports to
> compute. And if you claim it is only responsible for determining the
> mapping from the string ⟨x⟩ which is its input rather than the actual
> integer x, which is not its input, then any reasonable person is going
> to respond with a look of utter confusion. What does it even *mean* to
> talk about the evenness of ⟨x⟩ if it is not related to the evenness of
> the actual integer x? Strings aren't 'even' anymore than they are
> 'halting'.
>

embedded_H is only responsible for computing the mapping from its inputs
⟨Ĥ⟩ ⟨Ĥ⟩ to an accept or reject state based on the behavior that ⟨Ĥ⟩ ⟨Ĥ⟩
specifies. A pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H is the ultimate
measure of the behavior that ⟨Ĥ⟩ ⟨Ĥ⟩ specifies.

>> The majority of philosophers agree that this cannot be done because
>> Willard Van Orman Quine has convinced them that they cannot really
>> know that all bachelors are unmarried.
>
> Yes, you've already demonstrated that you are unable to appreciate the
> nuance of Quine's article and have instead settled on this ridiculous
> interpretation. Nobody cares.
>
> André

Rudolf Carnap wrote a meaning postulate in his debate with Quine that
specifies ∀x (bachelor(x) → ¬married(x)) and Quine still did not get it.
To the best of my knowledge this was the first step towards the Richard
Montague grammar of natural language semantics.
https://liarparadox.org/Meaning_Postulates_Rudolf_Carnap_1952.pdf

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer