closed state/transition cluster: caught in a range of states
This decider does not correspond to a “family”, since it does describe the TMs future operation only by the states, and ignores the tape contents completely.
In mabu90 we have:
If there is a set S of states such that all transitions from elements of S are defined and their target state is also in S, (and the machine is in one of these states) it will never again leave S and thus not halt (closed state/transition cluster).
Currently (2022-07-10) I miss it from the collection of deciders. Since in 1990 it were important (and very efficient), I would think its omission should have an explicit explanation.
Decider examples and counterexamples
Decider code
Informally: we scan the states s of the TM downwards, fail if one of the transitions of s is not defined, compute the lowest target state occurring in the transitions of s, and succeed, when s <= (that minimum).
Decider tests
NYI
Results
That some of the TMs in question are (Translated) Cyclers, does not come as a surprise to me. That some of them are decided by “Backward reasoning” does come as a surprise.
Now, that I have thought about it, I suspect “Backward reasoning” to catch most of the TMs in question. But I am not sure about this… what do you think about it?
Database subset of application
NYI
Decider correctness
NYI