Weighted iterated linear control

Abstract

We combine three extensions of context-free grammars: (a) associating its nonterminals with storage configurations, (b) equipping its rules with weights, and (c) controlling its derivations. For a commutative semiring K, we introduce the class of weighted languages generated by K-weighted linear context-free grammars with storage S and with derivations controlled by (S, K)-recognizable weighted languages. The control on the derivations can be iterated in a natural way. We characterize the n-th iteration of the control in terms of the n-th iteration of the one-turn pushdown operator on the storage S of the control weighted language. Moreover, for each proper semiring we prove that iterating the control yields an infinite, strict hierarchy of classes of weighted languages.