Re: Halting Problem: What Constitutes Pathological Input

On 5/5/2025 4:10 PM, olcott wrote:

On 5/5/2025 3:00 PM, dbush wrote:

On 5/5/2025 3:54 PM, olcott wrote:

On 5/5/2025 2:49 PM, dbush wrote:

On 5/5/2025 3:38 PM, olcott wrote:

On 5/5/2025 2:23 PM, Richard Heathfield wrote:

On 05/05/2025 20:20, olcott wrote:

Is "halts" the correct answer for H to return?  NO
Is "does not halt" the correct answer for H to return?  NO
Both Boolean return values are the wrong answer

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Or to put it another way, the answer is undecidable, QED.
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See? You got there in the end.
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Is this sentence true or false: "What time is it?"
is also "undecidable" because it is not a proposition
having a truth value.
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Is this sentence true or false: "This sentence is untrue."
is also "undecidable" because it is not a semantically sound
proposition having a truth value.
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Can Carol correctly answer “no” to this (yes/no) question?
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Both Yes and No are the wrong answer proving that
the question is incorrect when the context of who
is asked is understood to be a linguistically required
aspect of the full meaning of the question.

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And "does algorthm X with input Y halt when executed directly" has a single well defined answer.
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That is not even the actual question.
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In other words, you don't understand what the halting problem is about, because that is EXACTLY the question.
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 That question is in many textbooks yet is still
wrong because functions computed by models of
computation such as Turing Machines or RASP machines
are only allowed to use actual inputs as their basis.

And no Turing machine can compute the following mapping, as proven by Linz and other and as you have *explicitly* agreed is correct.
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

 When everyone here insists is that we simply ignore
the above fundamental rule of how functions must be
computed they are necessarily incorrect.

But not all functions can be computed, such as the above halting function, as proven by Linz and other and as you have *explicitly* agreed is correct.

 

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.
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Does an algorithm H exist that can tell me that or not?
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