Morphic characterizations of language families in terms of insertion systems and star languages

Insertion systems have a unique feature in that only string insertions are allowed, which is in marked contrast to a variety of the conventional computing devices based on string rewriting. This paper will mainly focus on those systems whose insertion operations are performed in a context-free fashion, called context-free insertion systems, and obtain several characterizations of language families with the help of other primitive languages (like star languages) as well as simple operations (like projections, weak-codings). For each k < 1, a language L is a k-star language if L = F⁺ for some finite set F with the length of each string in F is no more than k. The results of this kind have already been presented in [10] by Pun et al., while the purpose of this paper is to prove enhanced versions of them. Specifically, we show that each context-free language L can be represented in the form L = h(L(γ) ∩F⁺), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context-free manner), h is a projection, and F⁺ is a 2-star language. A similar characterization can be obtained for recursively enumerable languages, where insertion systems of weight (3, 3) and 2-star languages are involved.