Insertion operations: closure properties.dvi

The basic notions used for specifying languages are of algorithmic or operational character: automata (accepting devices) or grammars (generating devices). This duality reflects the original motivations coming from computer science or linguistics. A deeper theory and more involved proofs called for alternative definitional devices, where the operations have a classical mathematical character. Early examples of such definitional devices are the sequential functions for automata (see [14], pp.48-53) or substitutions and morphisms for grammars. This paper goes into the fundamentals of the substitution operation, aiming thus to a better understanding of the mechanisms of generating languages. So far in the literature a substitution has been defined as an operation on an alphabet. A substitution is never applied to λ (except for the convention that λ is always mapped into λ). Our work can be viewed as an attempt to understand the substitution on the empty word. Let L1, L2 be two languages over an alphabet Σ. The operation of sequential insertion of a language L2 into a language L1, can be viewed as a nonstandard modification of the notion of substitution. It maps all letters of Σ into themselves and the empty letter into L2, with the following additional convention. For each word w, only one of the empty words occurring in it is substituted. The result of applying this ”substitution” to L1 consists of words obtained from words in L1 in which an arbitrary word from L2 has been inserted. Note that the sequential insertion is also a generalization of the catenation operation. The sequential insertion operation can be viewed also as a one-step rewriting rule of a Thue system (see [4], [3]). Consequently, some closure properties of the families in the Chomsky hierarchy under iterated sequential insertion can be obtained using some results about Thue systems.