Full AFLS and nested iterated substitution

Abstract

A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then S∞(L, the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in S∞ (L). The index languages form a superAFL which is not the least superAFL containing the one way stack languages. If L has a decidable emptiness problem, so does S∞ (L). If Ds is an AFA, L=L (Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L (Dw) = S∞(L) if and only if Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested.