I've been exploring the idea of a nondeterministic continuous automaton based on germs:
Two functions $f,g: \mathbb{R} \to S$ have the same right germ at $x$ if there is some interval $[x,a)$ on which $f$ and $g$ agree.
Given an input word $i: \mathbb{R} \to \Sigma$, where $\Sigma$ is some alphabet, a run of an automaton $r: \mathbb{R} \to Q$, where $Q$ is some set of states where for any $x$, the collection of:
- The input at time $x$, $i(x)$
- Possibly also the left/right germs of $i$ at $x$
- The left germ of the run at $x$ (this corresponds to the "previous state")
- The value of the run at $x$, $r(x)$
- Possibly also the right germ of the run at $x$
Taken together fall into some set of "allowable transitions".
There are several parameters that I suspect need to be tweaked to get a nice notion of automaton, and before I try all the possibilities, I'd like to know if these sorts of things have been studied before, or picked up and discarded as uninteresting. Have they?
