We present a polynomial update time algorithm for the inductive inference of a large class of context-free languages using the paradigm of positive data and a membership oracle. We achieve this result by moving to a novel representation, called Contextual Binary Feature Grammars (CBFGs), which are capable of representing richly structured context-free languages as well as some context sensitive languages. These representations explicitly model the lattice structure of the distribution of a set of substrings and can be inferred using a generalisation of distributional learning. This formalism is an attempt to bridge the gap between simple learnable classes and the sorts of highly expressive representations necessary for linguistic representation: it allows the learnability of a large class of context-free languages, that includes all regular languages and those context-free languages that satisfy two simple constraints. The formalism and the algorithm seem well suited to natural langua...