Efficient Algorithms for Parsing the DOP Model? A Reply to Joshua Goodman

In the introduction of his paper, Goodman observes that the DOP model ".. can be summarized as a special kind of Stochastic Tree Substitution Grammar (STSG): given a bracketed, labeled training corpus, let every subtree of that corpus be an elementary tree, with a probability proportional to the number of occurrences of that subtree in the training corpus." Goodman then neglects to add that according to the DOP model, the "preferred" or "best" parse tree of a sentence is the most probable parse tree of that sentence. This definition is found in all my publications about DOP (e.g. Bod, 1992-96; van den Berg, Bod and Scha, 1994; Bod and Scha, 1994; Bod, Krauwer and Sima'an, 1994; Bod, Bonnema and Scha, 1996; Sima'an, Bod, Krauwer and Scha, 1994). Since in DOP, the probability of a tree is the sum of the probabilities of all distinct derivations that produce that tree, the computation of the most probable tree is very expensive. In Sima'an (1996b), a proof is given that the problem of computing the most probable tree of a sentence in DOP is NP-hard. This proof does not mean that there is no algorithm that can estimate the most probable tree of a sentence with an error that can be made arbitrarily small (cf. Bod, 1993b), but it does mean that there is no deterministic polynomial time algorithm for finding the most probable tree.