Let $A,B$ be two automata over the same alphabet $\Sigma$; they are supposed to be complete, strongly connected DFAs. We denote by $._A$ (resp. $._B$) the action induced by $\Sigma^*$ over $Q(A)$, i.e. $u ._A s$ designates the state reached from $s$ upon reading $u$ - note that this is more commonly denoted by $\delta_A(u,s)$ but we adopt the first notation for conciseness. Say that an "automata relation" between $A$ and $B$ is a relation $R \subseteq Q(A) \times Q(B)$ such that:
(*) if $R(s,t)$ holds, then for any $u,v \in \Sigma^*$ we have: $u ._A s = v ._A s \Rightarrow u ._B t = v ._B t$.
Given $R$ and $t \in Q(B)$, we then define $R^{-1}(t) = \{ s \in Q(A) : R(s,t) \text{ holds} \}$.
Given such a relation, it is possible to define an action of $Q(A)$ over $Q(B)$ as follows: given $t \in Q(B)$ and $s \in Q(A)$, we define $s^{-1} t$ as the unique state $t' \in Q(B)$ such that for every $s' \in R^{-1}(t)$, $L_A(s',s).t = \{t'\}$. Note that the existence of $t'$ follows from the strong connectedness of $A$, and that the unicity follows by observing that if $u,v \in L_A(s',s)$, we have $u ._A s' = v ._A s' = s$ and thus $u ._B t = v ._B t = t'$.
Now, given the tuple $T = (A,B,R)$ and given two weight functions $f,g$ over $Q(B)$, let us define the "theta function" $\Theta_{T,f,g}(s) = \sum_{t \in Q(B)} f(t) g(s^{-1} t)$.
Question: what are the interesting questions to ask about this notion?
(Some examples and suggestions should follow, I'm still trying to clear my thoughts on this problem.)