The concept of hysteresis seems well suited to describe and distinguish finite automata:
"Hysteresis is the dependence of the state of a system on its history." (Wikipedia, Hysteresis)
"[The state of a finite state machine] is determined by its history. (Minsky, Computation)
What I am looking for is a definition of hysteresis strength $H$ of finite automata distinguishing finite automata with a stronger dependence on history from those with a weaker one.
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Thanks to domotorp's hint, I found that the Nerode index is a good such measure: finite automata (equivalent to finite regular expressions) have a finite Nerode index, and the smaller the Nerode index is, the less "memory" (or hysteresis strength) the automaton has.
But: The Nerode index is defined rather abstractly, as the number of equivalence classes of a language with respect to the Nerode equivalence relation.
What I would like to know:
How is the Nerode index related to the structure/complexity/size of a regular expression?
How is the Nerode index related to the structure/complexity/size of an equivalent state diagram?
Is there more to say than:
The lesser complex the regular expression describing an automaton is, the smaller its Nerode index is.
The Nerode index is correlated to the number of (accessible) states.