Re: Halting Problem: What Constitutes Pathological Input

On 5/5/2025 2:47 PM, olcott wrote:

On 5/5/2025 1:21 PM, dbush wrote:

On 5/5/2025 2:14 PM, olcott wrote:

On 5/5/2025 11:16 AM, dbush wrote:

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

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

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Which start with the assumption that the following mapping is computable
and that (in this case) HHH computes it:
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Given any algorithm (i.e. a fixed immutable sequence of instructions) X
described as <X> with input Y:
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A solution to the halting problem is an algorithm H that computes the
following mapping:
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(<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
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>
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int DD()
{
    int Halt_Status = HHH(DD);
    if (Halt_Status)
      HERE: goto HERE;
    return Halt_Status;
}
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https://github.com/plolcott/x86utm
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The x86utm operating system includes fully operational HHH and DD.
https://github.com/plolcott/x86utm/blob/master/Halt7.c
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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.
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Thus the Halting Problem's "impossible" input is correctly determined
to be non-halting.
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>

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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.

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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.
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/Flibble

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All algorithms either halt or do not halt when executed directly. Therefore the problem is not ill formed.
>

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When BOTH Boolean RETURN VALUES are the wrong answer
THEN THE PROBLEM IS ILL-FORMED. Self-contradiction must
be screened out as semantically incorrect.

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In other words, you're claiming that there exists an algorithm, i.e. a fixed immutable sequence of instructions, that neither halts nor does not halt when executed directly.
>

 That is not what I said.

Then there's no category error, and the halting function is well defined.  It's just that no algorithm can compute it.