Let $A_1=(Q,\Sigma,q_0,F_1,\delta _1)$ and $A_2=(P,\Sigma ,p_0,F_2,\delta _2)$ a finite non determinustic automaons.
Describe algorithm that deside if there are two words $x$ and $y$ in the same length such that $x\in L(A_1)$ and $y\in L(A_2)$
Hint: Intersection automaton
My try
To bulid the intersection automaton $L_1\cap L_2$ and check if $L_1\cap L_2 \neq \varnothing$ with $"\color{blue}{\text{emptiness problem algorithm} }"$ and if $L_1\cap L_2 \neq \varnothing$ so $\exists $ word $w$ such that $w \in \mathscr L(L_1\cap L_2)$