Re: Halting Problem: What Constitutes Pathological Input

On 05/05/2025 22:35, Mr Flibble wrote:

On Mon, 05 May 2025 21:57:42 +0100, Richard Heathfield wrote:
 

On 05/05/2025 21:03, Mr Flibble wrote:

On Mon, 05 May 2025 16:00:11 -0400, dbush wrote:

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<snip>
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I want to know if any arbitrary algorithm X with input Y will halt
when executed directly.  It would be *very* useful to me if I had an
algorithm H that could tell me that in *all* possible cases.  If so, I
could solve the Goldbach conjecture, among many other unsolved
problems.
>
Does an algorithm H exist that can tell me that or not?

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That isn't what the halting problem is about at all:

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Yes, it is.
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the halting problem is about pathological input being undecidable but
not for the reason claimed in any halting problem proof.

>
By "pathological input", you mean a program the halt behaviour of which
is undecidable. So we are in agreement, even if you don't yet realise
it.

 No we are not in agreement

You now acknowledge the existence of programs whose fate is undecidable. You give it a different name, sure, but it's the same thing.

and I suspect you are being dishonest and know
this already.

On the contrary, when you talk about 'pathological input' you use the term to describe uncomputable mappings between programs and termination statuses, so you're rather closer to the truth than you perhaps intended.
To put it in terms you might be able to understand better:
Turing hypothesised the existence of a universal halt decider, but then showed that were such a decider to exist it would be possible to use it to create a 'pathological input' that it couldn't decide, so it follows that no decider can possibly be universal. /At best/, it can decide for all non-pathological inputs.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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