On 5/27/2022 6:11 PM, Richard Damon wrote:
> On 5/27/22 6:52 PM, olcott wrote:
>> On 5/27/2022 5:31 PM, André G. Isaak wrote:
>>> On 2022-05-27 16:20, olcott wrote:
>>>> On 5/27/2022 5:15 PM, André G. Isaak wrote:
>>>>> On 2022-05-27 16:01, olcott wrote:
>>>>>> On 5/27/2022 4:34 PM, André G. Isaak wrote:
>>>>>>> On 2022-05-27 15:27, olcott wrote:
>>>>>>>> On 5/27/2022 4:19 PM, André G. Isaak wrote:
>>>>>>>>> On 2022-05-27 15:04, olcott wrote:
>>>>>>>>>> On 5/27/2022 3:19 PM, André G. Isaak wrote:
>>>>>>>>>>> On 2022-05-27 13:58, olcott wrote:
>>>>>>>>>>>> On 5/27/2022 2:46 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 5/27/22 2:59 PM, olcott wrote:
>>>>>>>>>>>>>> On 5/27/2022 1:45 PM, André G. Isaak wrote:
>>>>>>>>>>>>>>> On 2022-05-27 12:37, Andy Walker wrote:
>>>>>>>>>>>>>>>> On 27/05/2022 18:57, André G. Isaak wrote:
>>>>>>>>>>>>>>>>> The (positive) square root function you talk about maps
>>>>>>>>>>>>>>>>> real numbers
>>>>>>>>>>>>>>>>> (not scrambled eggs and not finite strings) to real
>>>>>>>>>>>>>>>>> numbers (again,
>>>>>>>>>>>>>>>>> not finite string). Unlike the prime() function,
>>>>>>>>>>>>>>>>> however, the
>>>>>>>>>>>>>>>>> positive square root function is NOT computable.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>      Um.  This is technically true, but [IMO]
>>>>>>>>>>>>>>>> misleading. The reason
>>>>>>>>>>>>>>>> for the failure is that most [indeed, almost all] real
>>>>>>>>>>>>>>>> numbers are not
>>>>>>>>>>>>>>>> computable.  But non-computable reals are of [almost] no
>>>>>>>>>>>>>>>> interest for
>>>>>>>>>>>>>>>> practical applied maths and theoretical physics, and are
>>>>>>>>>>>>>>>> the sorts of
>>>>>>>>>>>>>>>> object that give maths a bad name in the outside world.
>>>>>>>>>>>>>>>> If "x" is a
>>>>>>>>>>>>>>>> computable positive real, then "sqrt(x)" is also a
>>>>>>>>>>>>>>>> computable real [eg
>>>>>>>>>>>>>>>> by using Newton-Raphson], which is all you really have
>>>>>>>>>>>>>>>> any right to
>>>>>>>>>>>>>>>> expect.  If you can't compute "x", then what does it
>>>>>>>>>>>>>>>> even mean to talk
>>>>>>>>>>>>>>>> about its "sqrt"?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> All I was really trying to get Olcott to see was a case
>>>>>>>>>>>>>>> where it really *isn't* possible to encode all elements
>>>>>>>>>>>>>>> of the domain or codomain as finite strings, which is
>>>>>>>>>>>>>>> rather different from his strange claim that computations
>>>>>>>>>>>>>>> like P(P) cannot be encoded as finite strings.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Computations like P(P) can be encoded as finite string
>>>>>>>>>>>>>> inputs to H1, they cannot be encoded as finite string
>>>>>>>>>>>>>> inputs to H simply because the behavior specified by the
>>>>>>>>>>>>>> correctly emulated input to H(P,P) is entirely different
>>>>>>>>>>>>>> behavior than the correctly emulated input to H1(P,P).
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> We don't even need to understand why this is the case we
>>>>>>>>>>>>>> only need to understand that it is a verified fact.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> If P(P) can't be encoded to give to H, then H fails to be a
>>>>>>>>>>>>> (Universal) Halt Decider from the begining, and can't be a
>>>>>>>>>>>>> counter example.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Not at all. Halt deciders have sequences of configurations
>>>>>>>>>>>> encoded as finite strings as the domain of their computable
>>>>>>>>>>>> function.
>>>>>>>>>>>
>>>>>>>>>>> This is an entirely mangled sentence. You really need to go
>>>>>>>>>>> back to the basics:
>>>>>>>>>>>
>>>>>>>>>>> First, a Turing Machine is *NOT* a computable function.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> A Turing machine would compute only the inputs to a its
>>>>>>>>>> corresponding computable function that can be encoded as
>>>>>>>>>> finite strings.
>>>>>>>>>
>>>>>>>>> Computable functions don't have inputs, they have domains. And
>>>>>>>>> *all* of the elements in the domain of a computable function
>>>>>>>>> can be encoded as finite string. Otherwise it wouldn't be a
>>>>>>>>> computable function.
>>>>>>>>>
>>>>>>>>
>>>>>>>>       Computable functions are the basic objects of study in
>>>>>>>> computability theory.
>>>>>>>>       Computable functions are the formalized analogue of the
>>>>>>>> intuitive notion of
>>>>>>>>       algorithms, in the sense that a function is computable if
>>>>>>>> there exists an algorithm
>>>>>>>>       that can do the job of the function, i.e. given an input
>>>>>>>> of the function domain it
>>>>>>>>       can return the corresponding output.
>>>>>>>>       https://en.wikipedia.org/wiki/Computable_function
>>>>>>>> *I am going by the above*
>>>>>>>
>>>>>>> No. You're going by your *flawed* reading of the above. In the
>>>>>>> above paragraph it is the algorithm which has an input,
>>>>>>
>>>>>> *input of the function domain*
>>>>>
>>>>> What that means is "input [to the algorithm] of [i.e. taken from]
>>>>> the function domain".
>>>> Thus mandating a bijection to finite strings.
>>>
>>> There is no bijection (and bijections hold between things, not to
>>> things). Every element of the function's domain can potentially be
>>> encoded in an infinite number of different ways. And this has no
>>> relevance to the particular confusion of yours that I was pointing out.
>>>
>>> André
>>>
>>>
>>
>> The simpler way around all this is to deduce that set of possible
>> inputs to a TM halt decider is the set of Turing machine descriptions.
>>
>> This is fairly widely known as an aspect of the definition of a halt
>> decider.
>>
>
> Right, so H can be given the description of ANY Turing Machine (even H^)
> and an input for that, and needs to decide what that the Turing Machine
> that input describes, applied to that input would do (Halt or Not) and
> give the right answer to be correct.
>

The ultimate measure of the behavior of an input its its correct
emulation. If the input to H(P,P) calls H(P,P) then P must actually call
H(P,P). If the fact that the input calls H(P,P) makes its correct x86
emulation never reach its "ret" instruction then this is the behavior
that H must report.

> THus H applied to <H^> <H^> has been given a VALID input, and needs to
> output if H^ applied to <H^> would halt or not.  (here <> means a
> description of, being a finite string representation of the machine
> within the <>s)

--
Copyright 2022 Pete Olcott

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