AbstractWeir has defined a hierarchy of language classes whose second member (L2) 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 L2 is obtained using a context-free grammar (CFG) whose derivations are controlled by another CFG. We adapt Weir’s definition of a controllable CFG (called a labeled distinguished CFG) to give a definition of controllable pushdown automata (PDAs), called labeled distinguished PDAs. This yields three new characterizations of L2 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 kind of automaton we know of, so we invent one and call it a Pushdown Adjoining Automaton (PAA).