Abstract
The generative capacity of combinatory categorial grammars (CCGs) as generators of tree languages is investigated. It is demonstrated that the tree languages generated by CCGs can also be generated by simple monadic context-free tree grammars. However, the important subclass of pure combinatory categorial grammars cannot even generate all regular tree languages. Additionally, the tree languages generated by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then these grammars can generate only a proper subset of the regular tree languages, whereas they can generate exactly the regular tree languages once first-degree composition rules are permitted.
Highlights
Categorial grammars [5] were introduced alongside the phrase-structure grammars of the Chomsky hierarchy [6] inspired by classical notions from proof theory [1, 3]
The mentioned equivalence result due to Vijay-Shanker and Weir [28] shows that Combinatory Categorial Grammar (CCG), Tree-Adjoining Grammar (TAG) [12] as well as linear indexed grammars [11] are equivalent in expressive power, which establishes that they generate the same string languages
We have shown that the tree languages accepted by CCGs with limited composition depth and rule restrictions are a subset of the tree languages generated by simple monadic contextfree tree grammars
Summary
Categorial grammars [5] were introduced alongside the phrase-structure grammars (regular, context-free, context-sensitive grammars, etc.) of the Chomsky hierarchy [6] inspired by classical notions from proof theory [1, 3]. 44:2 Tree-Generative Capacity of CCG construction depends on the ability to restrict the combination rules and to include entries for the empty word in the lexicon. Koller and Kuhlmann [13] show that CCG and TAG generate incomparable classes of dependency trees In this contribution, we answer the original question and characterize the tree languages accepted by CCGs, and relate them to the standard notions of regular [9, 10] and context-free tree languages [19, 20]. Our main result is that the tree languages accepted by CCGs can be generated by simple monadic context-free tree grammars (Theorem 20). We show that CCGs without composition operations, which are weakly equivalent to (ε-free) context-free grammars, generate a strict subclass of the regular tree languages that does not even include all local tree languages (Theorem 9). If we limit the permitted composition operators to first degree, exactly the regular tree languages are accepted (Theorem 14)
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