Second-Order Abstract Categorial Grammars as Hyperedge Replacement Grammars

Abstract

Abstract. Second-order abstract categorial grammars (de Groote 2001) and hyperedge replacement grammars (see Engelfriet 1997) are two natural ways of generalizing "context-free" grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. 1991) or linear context-free rewriting systems (Weir 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars.

Multiple context-free tree grammars: Lexicalization and characterization

“…✷ Third, MCFT and MCF can be characterized in terms of second-order abstract categorial grammars. It is shown in [57] that such grammars have the same tree and string generating power as hyperedgereplacement context-free graph grammars, which was already known for strings from earlier results as discussed in [57].…”

“…They are of interest to computational linguists because they can model cross-serial dependencies, whereas they can still be parsed in polynomial time and generate semi-linear languages. Multiple context-free tree grammars were introduced in [57], in the sense that it is suggested in [57,Section 5] that they are the hyperedgereplacement context-free graph grammars in tree generating normal form, as defined in [27]. Such graph grammars generate the same string languages as MCFGs [21,94].…”

“…Such graph grammars generate the same string languages as MCFGs [21,94]. It is shown in [57] that they generate the same tree languages as second-order abstract categorial grammars (2ACG), generalizing the fact that MCFGs generate the same string languages as 2ACGs [82]. It is also observed in [57] that the set-local multi-component tree adjoining grammar (MC-TAG, see [53,93]), well-known to computational linguists, is roughly the monadic restriction of the multiple context-free tree grammar, just as the tree adjoining grammar (TAG, see [49,51]) is roughly the monadic restriction of the (linear and nondeleting) contextfree tree grammar, see [37,61,71].…”

A syntax-semantics interface for Tree-Adjoining Grammars through Abstract Categorial Grammars

“…When considering strings as the object language, the generated languages coincide with multiple context-free languages (Salvati 2006). When considering trees, the generated languages coincide with the tree languages generated by hyperedge replacement grammars (Kanazawa 2009). A further refinement on the ACG hierarchy provides a fine-grained correspondence with regular (string or tree) languages, context-free string and linear context-free tree languages, or well-nested multiple context-free languages (string), in particular tree-adjoining languages.…”

A Generalization of Linear Indexed Grammars Equivalent to Simple Context-Free Tree Grammars

“…Elsewhere [12], I have argued that a class of string languages that falls in between these two classes, namely, the class equivalently captured by well-nested multiple context-free grammars [13], 2 coupled-context-free grammars [10], nonduplicating macro grammars [25], simple (i.e., linear and non-deleting) contextfree tree grammars [16], and second-order abstract categorial grammars of lexicon complexity 3 (see [14]), may be more attractive than the broader class as a formalization of mild context-sensitivity. I will not repeat the arguments here, 3 but one counterargument might be that this intermediate class (the class of well-nested multiple context-free languages) does not look as robust as the other two.…”