The problemAll languages have distributional regularities: patterns which restrict what sounds can appear where, including nowhere, as determined by local syntagmatic factors independent of any particular morphemic alternations. Early generative phonology tended to slight the study of distributional relations in favor of morphophonemics, perhaps because word-relatedness phonology was thought to be more productive of theoretical depth, reliably leading the analyst beyond the merely observable. But over the last few decades it has become clear that much morphophonemics can be understood as accommodation to phonotactic requirements, e.g. Kisseberth (1970), Sommerstein (1974), Kiparsky (1980), Goldsmith (1993), etc. A German-like voice-neutralizing alternation system resolves rapidly when the phonotactics of obstruent voicing is recognized. And even as celebrated a problem in abstractness and opacity as Yawelmani Yokuts vocalic phonology turns on a surface-visible asymmetry in height-contrasts between long and short vowels. Distributions require nontrivial learning: the data itself does not explicitly indicate the nature, or even the presence, of distributional regularities, and every distributional statement goes beyond what can be observed as fact, the 'positive evidence'. From seeing X in this or that environment the learner must somehow conclude 'X can only appear under these conditions and never anywhere else' -when such a conclusion is warranted.A familiar learning hazard is immediately encountered. Multiple grammars can be consistent with the same data, grammars which are empirically distinct in that they make different predictions about other forms not represented in the data. If learning is based upon only positive evidence, then the simple consistency of a grammatical hypothesis with all the observed data will not guarantee that the hypothesis is correct, even when adequate data are provided. This problem is typically characterized in terms of relationships among grammars. If one grammar generates all the observable forms licensed by another grammar along with yet other forms, then it is tricky to tell them apart, though crucial to do so. All positive data that support the less-inclusive grammar 1 The authors names are given alphabetically. Prince would like to thank the John Simon Guggenheim Memorial Foundation for support. Thanks to Jane Grimshaw, John McCarthy, Paul Smolensky, and the Rutgers Optimality Research Group for helpful discussion.Hayes (1999, this volume) independently arrives at the same strategy for approaching the learning problems dealt with here, and he offers similar arguments for the utility and interest of the approach. We have tried to indicate by references in the text where the identities and similarities lie. Along with the overlap in basics, there are significant divergences in emphasis, analytical procedure, and development of the shared ideas; we suggest that readers interested in the issues discussed below should consult both papers.2 The nonexistence of long high vowels on t...