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.

中文翻译：

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加权迭代线性控制

我们结合了上下文无关文法的三个扩展：(a) 将其非终结符与存储配置相关联，(b) 为其规则配备权重，以及 (c) 控制其派生。对于可交换半环 K，我们引入了由 K 加权线性上下文无关文法生成的加权语言类，该文法具有存储 S 和由 (S, K) 可识别加权语言控制的派生。对派生的控制可以以自然的方式进行迭代。我们根据控制加权语言的存储 S 上的一圈下推算子的第 n 次迭代来表征控制的第 n 次迭代。此外，对于每个适当的半环，我们证明迭代控制会产生无限的、严格的加权语言类层次结构。