Abstract grammars based on transductions

We study an abstract grammatical model in which the effect (or application) of a production - determined by a so-called transduction - plays the main part rather than the notion of production itself. Under appropriately chosen assumptions on the underlying family T of transductions, we establish elementary, decidability, and complexity properties of the corresponding family L(T) of languages generated by T-grammars. These results are special instances of slightly more general properties of so-called Γ-controlled T-grammars, since regular control does not increase the generating power of T-grammars. In a Γ-controlled T-grammar we strict the iteration of T-transductions to those sequences of transductions that belong to a given control language, taken from a family Γ of control languages.