Iterated linear control and iterated one-turn pushdowns

Abstract

For a class of languages £, an £-controlled linear grammar K consists of a linear context-free grammar G and a control language H in £, where the terminals of H are interpreted as the labels of rules of G. The language generated by K is obtained by derivations of G such that the corresponding words of applied rules are control strings in H. The control of linear grammars can be iterated by starting with £ and by taking the result of the k-th step as class of control languages for the (k+1)-st step. The language class obtained by the k-th step is denoted by CTRL_k (£). Denote by £(S) the language class accepted by nondeterministic one-way S automata, where S is a storage type. We prove that for any S, CTRL_k(£(S))=£(P ^k_lt (S)), where P ^k_lt (S) is the storage type of which the configurations consist of k-iterated one-turn pushdowns of S-configurations, i.e., one-turn pushdowns of one-turn pushdowns of … of one-turn pushdowns of S-configurations (k times). Thereby we prove a strong connection between iterated linear control and iterated one-turn pushdowns. In particular, we characterize the members of the geometric language hierarchy (where £(S) is the class of context-free languages) by iterated one-turn pushdown automata in which the innermost pushdown is unrestricted.

The work of the author has been supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.)