Re: Unpartial Halt Deciders

Mr Flibble <flibble@red-dwarf.jmc.corp> writes:

On Fri, 18 Apr 2025 12:32:41 -0700, Keith Thompson wrote:

Mr Flibble <flibble@red-dwarf.jmc.corp> writes:

On Fri, 18 Apr 2025 12:25:36 -0700, Keith Thompson wrote:

Mr Flibble <flibble@red-dwarf.jmc.corp> writes:

I, aka Mr Flibble, have created a new computer science term, the
"Unpartial Halt Decider".  It is a Halt Decider over the domain of
all program-input pairs excluding pathological input (a manifestation
of the self referencial category error).

[...]
 
Do you have a rigorous definition of "pathological input"?
 
Is there an algorithm to determine whether a given input is
"pathological" or not?
 
I could define an is_prime() function like this:
 
    bool is_prime(int n) {
        return n >= 3 && n % 2 == 1;
        // returns true for odd numbers >= 3, false for all others
    }
 
I'll just say that odd numbers that are not prime are pathological
input, so I don't have to deal with them.

>
Pathological input:
>
Self-referencial to the decider.

 
OK.
 
Do you have a *rigorous* definition of "pathological input"?
 
Is there an algorithm to determine whether a given input is
"pathological" or not?

>
In the general case pathological input is not computable as it is a
category/type error (ergo not logically sound) so there is no algorithm
that can detect it.  Specific forms of it can however be detected by a
Simulating Halt Decider given certain constraints - see Mr Olcott for
details.

So your "new computer science term" is based on a concept that is
not logically sound.  Got it.

Another point: The well known proof that the Halting Problem is
not solvable works by assuming that a halt decider exists and then
creating *one* input, based on the code of the decider, on which
the decider cannot give a correct answer.  You can call that input
"self-referential to the decider".

Producing just one input on which any halt decider must fail is all
that was required to prove that the Halting Problem is in general
not solvable.  But it doesn't imply that there aren't plenty of
other inputs on which any purported halt decider must fail.

Let's say you have a halt decider that works reliably for
non-pathological inputs.  And let's say you write a program
(or Turing machine) that halts if and only if it finds an even
natural number greater than 2 that is not the sum of two primes.
This program evetually halts if Goldbach's Conjecture is false, and
never halts if Goldbach's Conjecture is true.  Assume the program
was written without any knowledge of any halt decider to which it
might be fed.  (Of course it must work with arbitrary precision
integers, something that's entirely possible for a Turing machine,
which has unlimited storage on its tape.)

Now let's feed this program to your halt decider.  *Either* your halt
decider can resolve whether Goldbach's Conjecture is true or false
(something that mathematicians have been unable to do for centuries)
*or* that particular input is "pathological", even though it is
not in any way "self-referential to the decider".

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */