Re: A computable function that reports on the behavior of its actual self *is* allowed +++

On 5/13/24 9:22 PM, olcott wrote:

On 5/13/2024 7:41 PM, immibis wrote:

On 14/05/24 00:57, olcott wrote:

On 5/13/2024 4:50 PM, immibis wrote:

On 13/05/24 15:39, olcott wrote:

On 5/13/2024 4:34 AM, Fred. Zwarts wrote:

Op 12.mei.2024 om 21:27 schreef olcott:

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
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A computable function that reports on the behavior of its actual
self (or reports on the behavior of its caller) is not allowed.

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So, olcott uses his authority to create a new problem. Why would anybody be interested in such limitation?
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The definition of computable function is an axiomatic basis
not any mere authority.
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There's no axiom that says computable functions aren't allowed to have themselves as input.

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If you are 100% precise with the meaning of your words you
already know that no executed embedded_H can possibly report
on its own behavior because no TM can take another TM as input.

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If you are 90% precise with the meaning of your words you know that the halting problem is about the machine description of a Turing Machine and that is OBVIOUSLY what I meant.

 Please be much more careful in the future.
 The halting problem is not allowed to alter the fundamental notion of
a decider and require this decider do anything besides compute the
mapping from this input on the basis of the behavior that this input
specifies.

Which, by the defintion of the Halting problem, is the behavior of the Turing Machine and input specified / described / represented by the input to the decider.

 When Ĥ is applied to ⟨Ĥ⟩
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
 THE INPUT TO SIMULATING PARTIAL HALT DECIDER embedded_H
SPECIFIES THAT IT CANNOT POSSIBLY REACH ITS OWN SIMULATED
FINAL STATE ⟨Ĥ.qn⟩

So? That isn't the question, so you are working on a straw man,

 

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If you are 110% precise with the meaning of your words you know that you said computable functions aren't the same as Turing machines and that a function which takes a function as input can exist in some theories (more commonly in computer science rather than mathematics) and is called a higher-order function.
https://en.wikipedia.org/wiki/Higher-order_function
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