It has long been appreciated that monolingual parsing computes the intersection of an FSA and a CFG (Bar-Hillel et al., 1961;van Noord, 1995).
if S is an FSA encoding some sentence s, intersection of S with a CFG, G, results in a parse forest which contains all and only derivations of s, that is L(S) ∩ L(G) ∈ {{s}, ∅}.