Re: Halting Problem: What Constitutes Pathological Input

On 2025-05-05 17:45:07 +0000, dbush said:

On 5/5/2025 1:37 PM, olcott wrote:

On 5/5/2025 11:13 AM, 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

 The above example is category error because it asks
HHH(DD) to report on the direct execution of DD() and
the input to HHH specifies a different sequence of steps.
 

 In other words, you're demonstrating that you don't understand proof by contradiction, a concept taught to and understood by high school students more than 50 years your junior.

Every theorem that can be proven with an indirect proof can also be proven
with a direct proof. But I have observed small children using indirect
proofs so apparently it is easier or more intuitive for them.
--
Mikko