In general, the size of the DFA that recognizes the intersection of $n$ languages is exponential in $n$. However, in my case I am computing the intersection of a very restricted subset of possible languages. I am curious if this weakens the lower complexity bound.
With an alphabet $\Sigma$, for the $i$th language, I have set $A_i \subseteq \Sigma$ and another set $B_i \subseteq \Sigma$. The $i$th language $L_i$ consists of strings containing any number of symbols in $A_i$, followed by one symbol in $B_i$, followed by any number of symbols in $\Sigma$. I would like to find a DFA that recognizes $L = L_1 \cap L_2 \cap ... \cap L_n$.
Do these restrictions on the input languages eliminate the exponential lower bound?