Learnability theory involves the construction of formal mathematical proofs whose goal is to demonstrate how the child can successfully induce a mature grammar. An empirically adequate learnability proof constitutes a detailed hypothesis concerning the boundary conditions within which acquisition proceeds and can provide a general framework for developmental accounts of specific aspects of language acquisition. A central difficulty in constructing a feasible learnability proof for natural languages is that the child's language input is strictly limited. One line of inquiry has been concerned with demonstrating how transformational grammars might in principle be learned, given strict constraints on the complexity of available input. Degree 1 learnability involves learning from constructions with one level of embedding, less complex constructions than have been used for Degree 2 learnability proofs. The Degree 1 proof avoids some of the serious problems encountered in Degree 2 proofs, and in the result outlined it is shown that in the acquisition of complex grammars such as those of natural languages, the required complexity of input can be reduced and constraints on possible grammatical hypotheses can simultaneously be relaxed if the input available to the language learner is rich in structural information. (Author/MSE)