Extensions of Language Families and Canonical Forms for Full AFL-structures∗

We consider the following ways of extending a family of languages K to an “enriched ” family X(K): (i) hyper-algebraic extension (X = H) based on iterated parallel substitution, (ii) algebraic extension (X = A) obtained by nested iterated substitution, (iii) rational extension (X = R) achieved by not self-embedding nested iterated substitution, and (iv) a few subrational extensions (X = M,C, S, P, F ) based on several kinds of substitution. We introduce full X-AFL’s, i.e., nontrivial families closed under finite substitution, intersection with regular sets and under X, which turn out to be equivalent to well-known full AFL-structures such as full hyperAFL (X = H), full super-AFL (X = A), full substitution-closed AFL (X = R), full semi-AFL (X = S), etc. Then we establish Canonical Forms for the smallest full XAFL X̂ (K) containing K, i.e., we decompose the operator X̂ into simpler operators. Using Canonical Forms for full X-AFL’s we obtain expressions for the smallest full X-AFL containing the result of substituting a family of languages into another family.