Weir has defined a hierarchy of language classes whose second member ($\mathcal{L}_2$) is generated by tree-adjoining grammars (TAG), linear indexed grammars (LIG), combinatory categorial grammars, and head grammars. The hierarchy is obtained using the mechanism of control, and $\mathcal{L}_2$ is obtained using a context-free grammar (CFG) whose derivations are controlled by another CFG. We adapt Weir's definition of a controllable CFG to give a definition of controllable pushdown automata (PDAs). This yields three new characterizations of $\mathcal{L}_2$ as the class of languages generated by PDAs controlling PDAs, PDAs controlling CFGs, and CFGs controlling PDAs. We show that these four formalisms are not only weakly equivalent but equivalent in a stricter sense that we call d-weak equivalence. Furthermore, using an even stricter notion of equivalence called d-strong equivalence, we make precise the intuition that a CFG controlling a CFG is a TAG, a PDA controlling a PDA is an embedded PDA, and a PDA controlling a CFG is a LIG. The fourth member of this family, a CFG controlling a PDA, does not correspond to any formalism we know of, so we invent one and call it a Pushdown Adjoining Automaton.

翻译：Weir定义了一个语言类层次结构，其第二成员（$\mathcal{L}_2$）由树邻接文法（TAG）、线性索引文法（LIG）、组合范畴文法及头部文法生成。该层次通过控制机制获得，其中$\mathcal{L}_2$由受另一上下文无关文法（CFG）控制其推导过程的CFG生成。我们改进了Weir关于可控CFG的定义，提出了可控下推自动机（PDA）的定义。由此得到$\mathcal{L}_2$的三种新刻画：即由PDA控制PDA、PDA控制CFG、以及CFG控制PDA所生成的语言类。我们证明这四种形式化体系不仅弱等价，而且在更严格的等价意义下等价——我们称之为d-弱等价。进一步利用称为d-强等价的更严格等价概念，我们精确刻画了直觉认知：控制CFG的CFG对应TAG，控制PDA的PDA对应嵌入式PDA，控制CFG的PDA对应LIG。该家族第四成员——控制PDA的CFG——不匹配已知的任何形式化体系，因此我们发明了一种新体系并命名为"下推邻接自动机"。