Contexts and the Concept of Mild Context-Sensitivity

We introduce and study a natural extension ofMarcus external contextual grammars. This mathematically simple mechanism whichgenerates a proper subclass of simple matrix languages,known to be mildly context-sensitive ones, is still mildlycontext-sensitive. Furthermore, we get an infinite hierarchy ofmildly context-sensitive families of languages.Then we attempt to fill a gap regarding the linguistic relevanceof these mechanisms which consists in defining a tree structure on thestrings generated by many-dimensional external contextual grammars,and investigate some related issues. Several open problemsare finally discussed.