Approximate matching between a context-free grammar and a finite-state automaton

For a given context-free grammar (CFG) and a finite-state automaton (FA), we tackle the edit-distance problem—the problem of computing the most similar pair of strings in the two respective languages. In particular, we consider three different gap cost models for the edit-distance that are crucial for finding a proper alignment between two bio sequences: the linear, affine and concave models. We design efficient algorithms for the edit-distance between a CFG and an FA under these gap cost models. The time complexity of our algorithm for computing the linear or affine gap distance is polynomial and the time complexity for the concave gap distance is exponential.