I have an algorithmic problem, where I need to build an algorithm and say if the problem is decidable. Here it is:
Regular languages $L_1$, $L_2$, and $L_3$ are given by finite automata. Is the problem decidable? $$\exists v \in L_1, \forall w \in L_2, \quad vw \in L_3.$$
I'm new to regular languages and finite automatons, so I don't understand how to do it. Can you help me to build this algorithm and give some useful links to read more about these themes?
It suffices to (decidably) construct a finite automaton for the language $$L = \{v \mid vL_2 \subset L_3\}$$ since then the problem reduces to deciding if the regular language $L \cap L_1 \ne \varnothing$, which amounts to reachability in a DFA/NFA. Consider a DFA $\mathcal{A}$ for $L_3$ with states $Q$. Now, given $q \in Q$, let $L_q$ be the regular language recognized by the DFA $\mathcal{A}_q$ which is the same as $\mathcal{A}$ except with start state $q$. Let $F = \{q \in Q \mid L_2 \subset L_q\}$. Note that $F$ is decidable by the same reachability argument, since $L_2 \subset L_q \iff L_2 \cap L_q^c = \varnothing$. But now $L$ is recognized by the DFA which is the same as $\mathcal{A}$ but with accepting states $F$.