On 10/14/2025 10:34 PM, Kaz Kylheku wrote:On 2025-10-15, olcott <polcott333@gmail.com> wrote:>On 10/14/2025 9:46 PM, Kaz Kylheku wrote:On 2025-10-15, olcott <polcott333@gmail.com> wrote:>5. In short>
>
The halting problem as usually formalized is syntactically consistent
only because it pretends that U(p) is well-defined for every p.
>
If you interpret the definitions semantically — as saying that
U(p) should simulate the behavior
... then you're making a grievous mistake. The halting function doesn't
stipulate simulation.
>
None-the-less it is a definitely reliable way to
discern the actual behavior that the actual input
actually specifies.
No, it isn't. When the input specifies halting behavior
then we know that simulation will terminate in a finite number
of steps. In that case we discern that the input has terminated.
When the semantics of the language specify
that when DD calls HHH(DD) that HHH must
simulate an instance of itself simulating
DD ChatGPT knows that this cannot be simply
ignored.
It is obvious that when H denotes a simulator, then its diagonal program
D ends up infinite regress, and is nonterminating.
H(D) doesn't terminate, and fails to be a decider that way, not
on account of returning an incorrect value.
This situation is of no particular significance.
When H is a simulator equipped with some break condition by which it
stops simulating and returns a value, that H's diagonal program D
ensures that the return value is wrong; if the value is 0, D is
terminating. It is necessarily always the case that H will never
simulate D far enough to reproduce the situation where the
simulated H(D) returns a value to D. That is always out of reach
of H for one reason or another.
These observations are interesting, but ultimately of no significance;
there is no deep truth within.
When D is based on a breaking decider H, the "opposite behavior" of D
/is/ reached in a bona fide simulation (i.e. one not conducted by
a procedure other than H).
** Whether or not a calculation maps to a halting state is not
** determined by whether given simulations of it /demonstrate/
** that state or not.
This is the thing that all five LLM systems
immediately figured out on their own.
All five LLM systems, and throngs of CS undergraduates
during their first lecture on halting.
--
TXR Programming Language: http://nongnu.org/txr
Cygnal: Cygwin Native Application Library: http://kylheku.com/cygnal
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