Convergence and Diversity in the Control Hierarchy
This work provides incremental theoretical insights for computational linguistics by refining equivalence notions in formal language hierarchies.
The paper tackles the problem of characterizing the language class L2 by adapting Weir's control hierarchy to pushdown automata, resulting in three new characterizations and establishing strict equivalence notions. It shows that these formalisms are d-weakly equivalent and uses d-strong equivalence to link them to known formalisms like TAG and LIG, while introducing a new formalism called Pushdown Adjoining Automaton.
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.