Modest successes have been achieved in learning grammars in this way. The complexity is high (cubic-time in the size of the training corpus as well as in the number of nonterminals), and as noted, EM does not in general find globally optimal models.  Thus it is important to place some constraints on the initial grammar, e.g., allowing nonterminals to generate either pairs of nonterminals or words, but not both, and also severely limiting the number of allowed nonterminals. A method of preferring small rule sets over large ones, without setting a fixed upper bound, is the use of a Dirichlet process that supplies a probability distribution over the probabilities of an unbounded number of rules. (This method is nonparametric, in the sense that it does not commit to any fixed number of building blocks or parameters in the modeling.) Whatever method of bounding the rules is used, the initial PCFG must be carefully chosen if a reasonably good, meaningful rule set is to be learned. One method is to start with a linguistically motivated grammar and to use “symbol splitting” (also called “state splitting”) to generate variants of nonterminals that differ in their expansion rules and probabilities. Recent spectral algorithms offer a relatively efficient, and globally optimal, alternative to EM (Cohen et al. 2013), and they can be combined with symbol splitting.  