  Most categorical models of meaning use a functor from the syntactic category
to the semantic category. When semantic information is available, the problem
of grammar induction can therefore be defined as finding preimages of the
semantic types under this forgetful functor, lifting the information flow from
the semantic level to a valid reduction at the syntactic level. We study the
complexity of grammar induction, and show that for a variety of type systems,
including pivotal and compact closed categories, the grammar induction problem
is NP-complete. Our approach could be extended to linguistic type systems such
as autonomous or bi-closed categories.
