Re: Halting Problem: What Constitutes Pathological Input

On 5/5/2025 12:13 PM, Mr Flibble wrote:

On Mon, 05 May 2025 11:58:50 -0400, dbush wrote:
 

On 5/5/2025 11:51 AM, olcott wrote:

On 5/5/2025 10:17 AM, Mr Flibble wrote:

What constitutes halting problem pathological input:
>
Input that would cause infinite recursion when using a decider of the
simulating kind.
>
Such input forms a category error which results in the halting problem
being ill-formed as currently defined.
>
/Flibble

>
I prefer to look at it as a counter-example that refutes all of the
halting problem proofs.

>
Which start with the assumption that the following mapping is computable
and that (in this case) HHH computes it:
>
>
Given any algorithm (i.e. a fixed immutable sequence of instructions) X
described as <X> with input Y:
>
A solution to the halting problem is an algorithm H that computes the
following mapping:
>
(<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
(<X>,Y) maps to 0 if and only if X(Y) does not halt when executed
directly
>
>
>

int DD()
{
    int Halt_Status = HHH(DD);
    if (Halt_Status)
      HERE: goto HERE;
    return Halt_Status;
}
>
https://github.com/plolcott/x86utm
>
The x86utm operating system includes fully operational HHH and DD.
https://github.com/plolcott/x86utm/blob/master/Halt7.c
>
When HHH computes the mapping from *its input* to the behavior of DD
emulated by HHH this includes HHH emulating itself emulating DD. This
matches the infinite recursion behavior pattern.
>
Thus the Halting Problem's "impossible" input is correctly determined
to be non-halting.
>
>

>
Which is a contradiction.  Therefore the assumption that the above
mapping is computable is proven false, as Linz and others have proved
and as you have *explicitly* agreed is correct.

 The category (type) error manifests in all extant halting problem proofs
including Linz.  It is impossible to prove something which is ill-formed
in the first place.
 /Flibble

All algorithms either halt or do not halt when executed directly. Therefore the problem is not ill formed.
You only get something that appears that way when a false assumption is made, namely that the halting function is computable.