As noted at the beginning of this section, traditional formal grammars proved too limited in coverage and too rigid in their grammaticality criteria to provide a basis for robust coverage of natural languages as actually used, and this situation persisted until the advent of probabilistic grammars derived from sizable phrase-bracketed corpora (notably the Penn Treebank). The simplest example of this type of grammar is a probabilistic context-free grammar or PCFG. In a PCFG, each phrase structure rule X → Y1 … Yk is assigned a probability, viewed as the probability that a constituent of type X will be expanded into a sequence of (immediate) constituents of types Y1, …, Yk. At the lowest level, the expansion probabilities specify how frequently a given part of speech (such as Det, N, or V) will be realized as a particular word. Such a grammar provides not only a structural but also a distributional model of language, predicting the frequency of occurrence of various phrase sequences and, at the lowest level, word sequences. 