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. In the most simple case $H$ would be real-valued with $H=0$ for memory-less automata*. But more complex definitions might be more appropriate, e.g. to distinguish automata that simply store the last input forever and automata that can recognize complex regular expressions.
What seems clear to me is that $H$ must be related to and reflect the complexity of a regular expression describing the automaton (in terms of size, nestedness, etc.)? And it must be related to the complexity of the state diagram (in terms of size, cycles, etc.). But it should not be identical with these complexities.
Is there already such a definition of hysteresis strength for finite automata?
*Memory-less automaton:
