We present a machine translation framework that can incorporate arbitrary features of both input and output sentences. The core of the approach is a novel decoder based on lattice parsing with quasisynchronous grammar (Smith and Eisner, 2006), a syntactic formalism that does not require source and target trees to be isomorphic. Using generic approximate dynamic programming techniques, this decoder can handle “non-local ” features. Similar approximate inference techniques support efficient parameter estimation with hidden variables. We use the decoder to conduct controlled experiments on a German-to-English translation task, to compare lexical phrase, syntax, and combined models, and to measure effects of various restrictions on nonisomorphism. 1