Re: Halting Problem: What Constitutes Pathological Input

On 5/5/2025 1:06 PM, Mr Flibble wrote:

On Mon, 05 May 2025 13:45:07 -0400, dbush wrote:
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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:
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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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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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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.
>
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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.

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For proof by contradiction to be valid the contradition has to be well-
formed which in the case of the Halting Problem it is not.
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/Flibble

 Yes. Self-contradiction is always ill-formed.