In this paper we propose a generalization of the local internal contextual grammars, introduced by Ilie in 1997, namely the
-local internal contextual grammars. These grammars are, in fact, classical internal contextual grammars, but their only permitted derivations are those that can be described in a restricted manner (that depends on the number
). Using this formalism we define different classes of languages, and obtain a series of language theoretic results for them. Also, we show that the membership problem for
-local internal contextual grammars with polynomial choice can be solved in polynomial time. This seems interesting to us, as it shows that the descriptional restrictions on the derivations of a grammar reflect on the computational efficiency of accepting the language generated by that grammar.