Re: A computable function that reports on the behavior of its actual self is not allowed

On 5/12/2024 2:27 PM, olcott wrote:

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
 A computable function that reports on the behavior of its actual
self (or reports on the behavior of its caller) is not allowed.
 A decider must halt whereas simulating a pathological input
that would never halt unless aborted can only halt by aborting.
 This causes the direct execution of this input after it has been aborted
to have different behavior than the simulated input that cannot possibly
stop running unless aborted.
 

*MORE PRECISE WORDING* (this may take a few more rewrites)
When Ĥ is applied to ⟨Ĥ⟩
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
It is a verified fact that the directly executed Ĥ ⟨Ĥ⟩ cannot possibly
stop running unless simulating partial halt decider embedded_H aborts
its simulation of its input.
(a) The behavior of the directly executed Ĥ ⟨Ĥ⟩ is after
embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ has already aborted its simulation.
(b) The behavior of the simulated input to embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
is before embedded_H has aborted its simulation.
(c) These two behaviors (a) and (b) ARE NOT THE SAME. (a) will stop
running on its own (b) will never stop running unless aborted.
embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ is not allowed to report on the behavior of its
actual self, thus is not allowed to report on the behavior of the
directly executed Ĥ ⟨Ĥ⟩.

Since no one here understands that a simulating partial halt decider
is not even allowed to report on its own behavior or the behavior of its
caller they do not understand that the behavior of the directly executed
machine is irrelevant.
  

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer