Convergence and Diversity in the Control Hierarchy

Abstract

Weir has defined a hierarchy of language classes whose second member (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 L 2 is obtained using a contextfree 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 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.

Efficient Algorithms for Recognizing Weighted Tree-Adjoining Languages

“…In previous work (Butoi et al, 2023), we extended this mechanism to PDAs, both as controllers and controllees, and obtained four distinct formalisms by mixing a controller CFG or PDA with a controllee CFG or PDA. In this paper, we use semiringweighted versions of these formalisms.…”

“…Due to d-strong equivalence, this normal form also induces a normal form for TAG. But, in order to convert a TAG into the normal form, one needs to first extract the controller and the controllee as shown by Butoi et al (2023), then convert these to the Chomsky normal form, merge their rules, and convert them back to a TAG, also shown by Butoi et al (2023) (see App. E for a complexity analysis of these transformations).…”

“…A WLD-PDA can always be converted into an equivalent WLD-CFG (Butoi et al, 2023), converted into the normal form, then converted back into into a WLD-PDA.…”

“…1 Introduction Weir (1992) introduced a hierarchy of formal languages whose first level (L 1 ) is the class of contextfree languages and second level (L 2 ) is the class of tree-adjoining languages. Just as context-free languages can be characterized by both context-free grammars and pushdown automata, tree-adjoining languages are characterized by multiple formalisms as well, including tree-adjoining grammars (TAG;Joshi et al, 1975), linear indexed grammars (LIG; Gazdar, 1988), embedded pushdown automata (EPDA; Vijay-Shanker, 1987), and pushdownadjoining automata (PAA; Butoi et al, 2023).…”

“…Tree-adjoining languages can further be characterized through the mechanism of control (Weir, 1992), which yields various two-level formalisms. Specifically, we have shown that CFGs controlled by CFGs, CFGs controlled by PDAs, PDAs controlled by CFGs, and PDAs controlled by PDAs are equivalent to TAG, LIG, PAA, and EPDA, respectively, in a strict sense called d-strong equivalence (Butoi et al, 2023).…”