Context-Free Recognition via Shortest Paths Computation: A Version of Valiant's Algorithm

We present a new algorithm recognizing general context-free languages in O(BM(n)) time, where BM(n) is the time to multiply two n by n boolean matrices. The only known algorithm for this problem with the same asymptotic complexity is the Valiant's algorithm [Va75], which is quite sofisticated. The probem related to texts is reduced to arithmetics of matrices whose elements are in a semiring of a constant size. The main difficulty in Valiant’s algorthm is nonassociativity of considered semirings, the main point in our algorithm is that nonassociative semirings are replaced by associative ones. This simplifies the algorithm considerably. Our algorithm is more structured, its main part is a computation of shortest paths in a special graph called here the lattice graph. The Valiant's lemma, see [Ha78], is replaced here by a shortest paths lemma. The shortest paths problem for lattice graphs is also interested in its own.