Children are facile at both discovering word boundaries and using those words to build higher-level structures in tandem. Current research treats lexical acquisition and grammar induction as two distinct tasks. Doing so has led to unreasonable assumptions. Existing work in grammar induction presupposes a perfectly segmented, noise-free lexicon, while lexical learning approaches largely ignore how the lexicon is used. This paper combines both tasks in a novel framework for bootstrapping lexical acquisition and grammar induction. We present an algorithm that iteratively learns a lexicon and a grammar for a class of regular languages in polynomial time, and we report experimental results for benchmark languages. The ease with which children learn to discover boundaries in their environments while building grounded high-level structures belies the complexity and computational challenges of the task. We address these two disparate problems by proposing a bootstrap between lexical and gramm...