  Children learn their native language by exposure to their linguistic and
communicative environment, but apparently without requiring that their mistakes
are corrected. Such learning from positive evidence has been viewed as raising
logical problems for language acquisition. In particular, without correction,
how is the child to recover from conjecturing an over-general grammar, which
will be consistent with any sentence that the child hears? There have been many
proposals concerning how this logical problem can be dissolved. Here, we review
recent formal results showing that the learner has sufficient data to learn
successfully from positive evidence, if it favours the simplest encoding of the
linguistic input. Results include the ability to learn a linguistic prediction,
grammaticality judgements, language production, and form-meaning mappings. The
simplicity approach can also be scaled-down to analyse the ability to learn a
specific linguistic constructions, and is amenable to empirical test as a
framework for describing human language acquisition.
