Re: embedded_H applied to ⟨Ĥ⟩ ⟨Ĥ⟩ computes the mapping from its input to Ĥ.qn

On 8/1/2024 2:44 AM, Mikko wrote:

On 2024-07-31 17:27:33 +0000, olcott said:
 

On 7/31/2024 2:32 AM, Mikko wrote:

On 2024-07-30 14:16:20 +0000, olcott said:
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On 7/30/2024 1:37 AM, Mikko wrote:

On 2024-07-29 16:16:13 +0000, olcott said:
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On 7/28/2024 3:02 AM, Mikko wrote:

On 2024-07-27 14:08:10 +0000, olcott said:
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On 7/27/2024 2:21 AM, Mikko wrote:

On 2024-07-26 14:08:11 +0000, olcott said:
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When Ĥ is applied to ⟨Ĥ⟩
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

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The above is merely simplified syntax for the top of page 3
https://www.liarparadox.org/Linz_Proof.pdf
The above is the whole original Linz proof.

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And even more simplified semantics.
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(a) Ĥ copies its input ⟨Ĥ⟩
(b) Ĥ invokes embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
(c) embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩
(d) simulated ⟨Ĥ⟩ copies its input ⟨Ĥ⟩
(e) simulated ⟨Ĥ⟩ invokes simulated embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
(f) simulated embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩
(g) goto (d) with one more level of simulation
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You are supposed to evaluate the above as a contiguous
sequence of moves such that non-halting behavior is
identified.

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The above is an obvious tight loop of (d), (e), (f), and (g).
Its relevance (it any) to the topic of the discussion is not
obvious.
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When we compute the mapping from the input to embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
to the behavior specified by this input we know that embedded_H
is correct to transition to Ĥ.qn.

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The meaning of "correct" in this context is that if the transition of
embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qn is correct if H ⟨Ĥ⟩ ⟨Ĥ⟩ transitions to H.qn but
incorrect otherwise.

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No you are wrong.

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Which dictionary (or other authority) disagrees?

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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
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The common knowledge that a decider computes the mapping
from its input finite string...
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This is almost always the same as the direct execution of
the machine represented by this finite string.

 None of above indicates any disagreement by any authority.
 

Everyone (even Linz) has the wrong headed idea that a halt
decider must report on the behavior of the computation that
itself is contained within. This has always been wrong.
A halt decider must always report on the behavior that its
finite string specifies. This is different only when an
input invokes its own decider.

The one rare exception is shown above where Ĥ ⟨Ĥ⟩ halts
and the input to embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ cannot possibly reach
its own final state of ⟨Ĥ.qn⟩ when embedded_H acts as if
it was a UTM.

 That is not supported by any anuthority.
 

The authority says
*given an input of the function domain it*
*can return the corresponding output*
In other words all deciders compute the mapping from their
input (finite string) to an accept or reject state.
This means that they do not compute the mapping of the
executing process of themselves.
I am the first person in the world that noticed these
two could be different. Everyone that has disagreed with
me is disagreeing with the semantics of the x86 language.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer