Disclaimer: I'm studying for an exam, but I don't have to hand in anything.
I have the state transition diagram of the FSM and its states encoding:
I want to find the sum-of-products (SOP) Boolean expression of $S_0^1$, $S_1^1$ and output $y$. I've drawn tables for each next state, based on all current states ($S_0$ and $S_1$) and inputs ($A$ and $B$):
After this, I don't have very clear how to proceed with drawing the output and state table truth / k-map to get the final Boolean expression.