# An algorithm to decide if a context-free language like $L_1$ and a regular language like $L_2$ have common members

A context-free language (CFL) is a language generated by some context-free grammar (CFG).

A regular language (also called a rational language) is a formal language that can be expressed using a regular expression.

Question : Assume that $L_1$ is a context-free language and $L_2$ is a regular language. Is there any algorithm to decide that $L_1 \cap L_2$ is empty or not?

The answer is yes. Given a context-free grammar generating a language $L_1$ and a regular language $L_2$ given by some finite automaton, one can effectively construct a context-free grammar generating the language $L_1 \cap L_2$. Now, given a context-free grammar, one can decide whether the language it generates is empty or not.