Context-free grammars and cubic time parsing are so related in people's minds that they often think that parsing any extension of context-free grammars must need some extra time. Of course, this is not true and this paper presents a generalization of context-free grammars which keeps a cubic time complexity. This extension, which defines a sub-class of context-sensitive languages, has both a theoretical and a practical interest. The class of languages defined by these grammars is closed under both intersection and complementation (in fact it is the class containing the intersection and the complementation of context-free languages). On the other hand, these grammars can be considered as being mildly context-sensitive and can therefore be used in natural language processing.