Three Notes on Controlled Hyper-Algebraic and Dhyper-Algebraic Extensions

(1) Regular control does not increase the generating power of 1restricted [d]K-iteration grammars provided that K ⊇ SYMBOL, and K is closed under isomorphism and under union with SYMBOL-languages. (2) Let Γ be a prequasoid closed under the regular operations. If K is a prequasoid [pseudoid], then H(Γ) ⊆ H(Γ,K) [η(Γ) ⊆ η(Γ,K)]. In particular we have H(Γ) ⊆ (Γ)ETOL and η(Γ) ⊆ (Γ)EDTOL. (3) Under weak conditions on Γ and K, the decidability of the emptiness problem for Γ and K implies the decidability of the emptiness problem and the membership problem for the families η(Γ,K) and η(K). ∗This research has been supported by the Netherlands Organization for the Advancement of Pure Research (ZWO).