I like to think that I influenced your choice of subject.
Yup, you did.
it seems that “head-state” is what would usually called “state” in TMs.
Correct. Really, the “state” of a TM (as the word is used most often in other math/engineering contexts) is both the head-state and whatever’s on the tape.
In a technical sense, the “state” of a system is usually whatever information forms a Markov blanket between future and past—i.e. the interaction between everything in the future and everything in the past should be completely mediated by the system state. There are lots of exceptions to this, and the word isn’t used consistently everywhere, but that’s probably the most useful heuristic.
Yup, you did.
Correct. Really, the “state” of a TM (as the word is used most often in other math/engineering contexts) is both the head-state and whatever’s on the tape.
In a technical sense, the “state” of a system is usually whatever information forms a Markov blanket between future and past—i.e. the interaction between everything in the future and everything in the past should be completely mediated by the system state. There are lots of exceptions to this, and the word isn’t used consistently everywhere, but that’s probably the most useful heuristic.