Re: Turing computable functions

On 3/25/25 3:24 PM, olcott wrote:

Cannot possibly derive any outputs not computed from
their inputs.

Right, but the problem might require that.

 A Turing machine halt decider cannot possibly report
on the behavior of any directly executing process.
No Turing machine can every do this. This has always
been beyond what any Turing machine can ever do.

Sure it can for many inputs, because it can simulate some to completion and some to a repeat state, proving that they will not halt.
The problem is this doesn't

 The best that any Turing machine halt decider can
possibly do is determine the behavior that an input
finite string specifies.

But that behavior is specified to be the behavior of the actual machine it represents, or equivalently, the behavior of the UTM that interprets it.
Remember, a UTM, by its definition, exactly reproduces *ALL* the behavior of the Turing Machine described by its input.

 When we make these things 100% concrete in a language
that has been fully operational for many years...
 

Your problem is that those things HAVE been concrete for like 80 years, you just don;t

int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}

Which is NOT a "Program" since it doesn't include the definitioin of HHH.

 When an input finite string specifies a pathological
relationship with its simulating halt decider the actual
behavior that pathological relationship derives must
be reported because THAT IS THE BEHAVIOR THAT IS SPECIFIED
BY THIS INPUT FINITE STRING.
 

But the program doesn't HAVE a "it's decider", only a decider that it was designed to cofound by being contrary, and does so by purely legal methods.
Note, the behavior of YOUR finite string is "Categpry Error" as it isn't a program.
Add the code for HHH to it, and its behavior is determied by UTM(DD) which will be halting if HHH(DD) returns 0 or just aborts its execution, and non-halting if HHH(DD) return 1 or loops forver.
THus, it is clear that there can be no correct answer from however you had defined HHH (which must have been defined BEFORE we could create the DD) but there *IS* a correct answer that other deciders could return, they just need to answer the opposite of HHH(DD).
Sorry, you are just proving you don't understand what you are talking about.