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

(1) Regular control does not increase the generating power of 1-restricted [d]$K$-iteration grammars provided that $K\supseteq{\rm SYMBOL}$, and $K$ is closed under isomorphism and under union with ${\rm SYMBOL}$-languages. (2) Let $\Gamma$ be a prequasoid closed under the regular operations. If $K$ is a prequasoid [pseudoid], then $H(\Gamma)\subseteq H(\Gamma,K)$ [$\eta(\Gamma) \subseteq\eta(\Gamma,K)$. In particular we have $H(\Gamma) \subseteq (\Gamma){\rm ETOL}$ and $\eta(\Gamma) \subseteq (\Gamma){\rm EDTOL}$.(3) Underweak assumptions on $\Gamma$ and $K$, the decidability of the emptiness problem for $\Gamma$ and $K$ implies the decidability of the emptiness problem and the membership problem for the families $\eta(\Gamma,K)$ and $\eta(K)$

Iterating iterated substitution

By iterating iterated substitution not all regular languages can be copied. Hence the smallest full hyper (1)-AFL is properly contained in ETOL, the smallest full hyper-AFL. The number of iterations of iterated substitution gives rise to a proper hierarchy. Consequently the smallest full hyper (1)-AFL is not a full principal AFL

Hierarchies of hyper-AFLs

For a full semi-AFL K, B(K) is defined as the family of languages generated by all K-extended basic macro grammars, while H(K) B(K) is the smallest full hyper-AFL containing K; a full basic-AFL is a full AFL K such that B(K) = K (hence every full basic-AFL is a full hyper-AFL). For any full semi-AFL K, K is a full basic-AFL if and only if B(K) is substitution closed if and only if H(K) is a full basic-AFL. If K is not a full basic-AFL, then the smallest full basic-AFL containing K is the union of an infinite hierarchy of full hyper-AFLs. If K is a full principal basic-AFL (such as INDEX, the family of indexed languages), then the largest full AFL properly contained in K is a full basic-AFL. There is a full basic-AFL lying properly in between the smallest full basic-AFL and the largest full basic-AFL in INDEX