We study the parsing complexity of Combinatory Categorial Grammar (CCG) in the formalism of Vijay-Shanker and Weir (1994). As our main result, we prove that any parsing algorithm for this formalism will take in the worst case exponential time when the size of the grammar, and not only the length of the input sentence, is included in the analysis. This sets the formalism of Vijay-Shanker and Weir (1994) apart from weakly equivalent formalisms such as Tree Adjoining Grammar, for which parsing can be performed in time polynomial in the combined size of grammar and input sentence. Our results contribute to a refined understanding of the class of mildly context-sensitive grammars, and inform the search for new, mildly context-sensitive versions of CCG.

We study the generalization of maximum spanning tree dependency parsing to maximum acyclic subgraphs. Because the underlying optimization problem is intractable even under an arc-factored model, we consider the restriction to noncrossing dependency graphs. Our main contribution is a cubic-time exact inference algorithm for this class. We extend this algorithm into a practical parser and evaluate its performance on four linguistic data sets used in semantic dependency parsing. We also explore a generalization of our parsing framework to dependency graphs with pagenumber at most k and show that the resulting optimization problem is NP-hard for k ≥ 2.