It has long been appreciated that parsing computes the intersection of an FSA and a CFG (Bar-Hillel et al., 1961;van Noord, 1995;Grune and Jacobs, 2008).
parsing an FSA, S, with a CFG, G, results in a parse forest which contains derivations of strings in I = L(S) ∩ L(G), 1 and which may be ∅.