Controlled Iteration Grammars and Full Hyper-AFL's

We investigate derivation-controlled $K$-iteration grammars, called $(\Gamma,K)$-iteration grammars, where $\Gamma$ can be any family of control languages. We prove that already under very weak restrictions on $\Gamma$ and $K$ the following hold: (i) Regular control does not increase the generating power of $K$-iteration grammars, (ii) for each $(\Gamma,K)$-iteration grammar there exists an equivalent propagating $(\Gamma,K)$-iteration grammar, (iii) the family of $(\Gamma,K)$-iteration languages is a full hyper-AFL, (iv) for each $(\Gamma,K)$-iteration grammar there exists an equivalent $(\Gamma,K)$-iteration grammar with exactly two substitutions. We also discuss some additional properties and applications of (uncontrolled) $K$-iteration grammars and controlled (deterministic) ET0L systems and their languages in a wider context.