Re: Defining a correct halt decider

Am Tue, 03 Sep 2024 08:17:56 -0500 schrieb olcott:

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.

What would those assumptions be?

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

How can the same code have different semantics?

HHH is required to report on the behavior tat its finite string input
specifies even when this requires HHH to emulate itself emulating DDD.

The input specifies an aborting HHH - which you don’t simulate.

DDD never halts unless it reaches its own final halt state. The fact
that the executed HHH halts has nothing to do with this.

Other than that DDD calls HHH?

HHH is not allowed to report on the computation that itself is contained
within.

Then it is only partial, and doesn’t even solve the case it was built for.

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 sure sounds like a mistake to me.

That no one has noticed that they can differ does not create an axiom
where they are not allowed to differ.

They were never allowed, that was the definition.

No one noticed that they differ only because everyone rejected the idea
of a simulating halt decider out-of-hand without review.

I think after 3 years that excuse has grown a bit stale.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.