We consider the following restriction of internal contextual grammars,
called local: in any derivation in a grammar, after applying a contex
t, further contexts can be added only inside of or at most adjacent to
the previous ones. We further consider a natural restriction of this
derivation mode by requiring that no superword of the word considered
as selector can be used as selector. We investigate the relevance of t
he latter type of grammars for natural. language study. In this aim, w
e show that all the three basic non-context-free constructions in natu
ral languages, that is, multiple agreements, crossed agreements, and d
uplication, can be realized using this type of grammars. Our main resu
lt is that these languages are parsable in polynomial time. The proble
m of semilinearity remains open.