On 11/29/22 8:15 PM, olcott wrote:
> Technically a halt decider must be all knowing and a halt determiner is
> too much of an unfamiliar term so I use the term halt decider and
> restrict its domain to a well defined set of inputs.
Programs don't "Know", they "Compute", and YES, a correct Halt Decider
by the Theory needs to be able to COMPUTE the answer for EVERY problem.
Note, the term "Halt Determiner" is WELL DEFINED, and in fact, is know
to exist. In computation Theory, a "Determiner" is a machine that will
ACCEPT all input that match its requirements, and might reject some
inputs that don't, but is allowed to no answer some for some (or all) of
the input that don't match the requirements.
Restricting your Halt Decider Definition to just select inputs means it
isn't actually a Halt Decider, so technically your whole work is worthless.
If your goal it to just decide the one particular input from templates
like the Linz proof, then you really need to add a qualifier to the
name, or you are just LYING. Perhaps you should call it a Partial Halt
Decider.
To make you point, you do still need to get it to answer the EXACT
program from that template. H needs to be able to answer about the
H^/P/D that is built from the exact H that is claimed to correctly be
deciding it.
Note, this means your set theory is not applicable, or perhaps more
precisely doesn't prove what you want to prove, as it just shows that no
H in your set can correct predict that its input is Halting, not that it
isn't, as for every H that does answer, the H^/P/D built on it is NEVER
simulated to the point that proves that it doesn't halt. The only ones
you do are those that are built on an H that never answers.
These are two disjoint sets, so the set of machines that have given the
correct answer is the empty set,