Re: Defining a correct halt decider

On 9/3/2024 3:44 AM, Mikko wrote:

On 2024-09-02 16:06:11 +0000, olcott said:
 

A correct halt decider is a Turing machine T with one accept state and one reject state such that:
>
If T is executed with initial tape contents equal to an encoding of Turing machine X and its initial tape contents Y, and execution of a real machine X with initial tape contents Y eventually halts, the execution of T eventually ends up in the accept state and then stops.
>
If T is executed with initial tape contents equal to an encoding of Turing machine X and its initial tape contents Y, and execution of a real machine X with initial tape contents Y does not eventually halt, the execution of T eventually ends up in the reject state and then stops.

 Your "definition" fails to specify "encoding". There is no standard
encoding of Turing machines and tape contents.
 

That is why I made the isomorphic x86utm system.
By failing to have such a concrete system all kinds
of false assumptions cannot be refuted.
The behavior of DDD emulated by HHH** <is> different
than the behavior of the directly executed DDD**
**according to the semantics of the x86 language
HHH is required to report on the behavior tat its finite
string input specifies even when this requires HHH
to emulate itself emulating DDD.
DDD never halts unless it reaches its own final
halt state. The fact that the executed HHH halts
has nothing to do with this.
HHH is not allowed to report on the computation that
itself is contained within.
Except for the case of pathological self-reference the
behavior of the directly executed machine M is always
the same as the correctly simulated finite string ⟨M⟩.
That no one has noticed that they can differ does not
create an axiom where they are not allowed to differ.
No one noticed that they differ only because everyone
rejected the idea of a simulating halt decider out-of-hand
without review.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer