What are the main results and/or literature on the (self) halting problem for other machines than Turing machines? Alternatively, what would be the right keywords or tags to search for it.
I am considering of course the ability to determine halting on given input of automata in a family $F$ by one of the members of $F$, i.e. the $F$-decidability of the halting problem for $F$-automata.
I am not here interested in hypercomputation, but rather in "hypocomputation", i.e. computational models that are weaker than Turing machines.
One problem I looked at is the definition of halting for non-deterministic models of computation, since non-determinism can make a difference for PDA and possibly for LBA. I asked a question on that, Defining the halting problem for non-deterministic automata, but was a bit disappointed by answers, and ended up answering it myself (how well, I do not know).
Other issues I have in mind are:
what would be a proper definition of a family of automata defining a model of computation? Should it satisfy some closure property or other?
The reason is that one can contrive simple collections of automata such that one of them will decide halting for the whole family. But it does not seem very meaningful, if that is all the family has to offer.
when asking one automaton $H$ to decide on halting of automaton $A$ on input $I$, is one free to encode $A$ and $I$ in whatever way is deemed convenient?
I am not even sure whether stating that there is a standard encoding has any meaning.
I realize the question is open. But I do not know how much may exist as possible answer, possibly little. If it is too wide, suggestions to tighten it would be welcome.