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Unlike other mildly context-sensitive formalisms, Combinatory Categorial Grammar (CCG) cannot be parsed in polynomial time when the size of the grammar is taken into account. Refining this result, we show that the parsing complexity of CCG is exponential only in the maximum degree of composition. When that degree is fixed, parsing can be carried out in polynomial time. Our finding is interesting from a linguistic perspective because a bounded degree of composition has been suggested as a universal constraint on natural language grammar. Moreover, ours is the first complexity result for a version of CCG that includes substitution rules, which are used in practical grammars but have been ignored in theoretical work. [ABSTRACT FROM AUTHOR]

In this paper, we consider probabilistic context-free grammars, a class of generative devices that has been successfully exploited in several applications of syntactic pattern matching, especially in statistical natural language parsing. We investigate the problem of training probabilistic context-free grammars on the basis of distributions defined over an infinite set of trees or an infinite set of sentences by minimizing the cross-entropy. This problem has applications in cases of context-free approximation of distributions generated by more expressive statistical models. We show several interesting theoretical properties of probabilistic context-free grammars that are estimated in this way, including the previously unknown equivalence between the grammar cross-entropy with the input distribution and the so-called derivational entropy of the grammar itself. We discuss important consequences of these results involving the standard application of the maximum-likelihood estimator on finite tree and sentence samples, as well as other finite-state models such as Hidden Markov Models and probabilistic finite automata. Index Terms--Probabilistic context-free grammars, maximum-likelihood estimation, derivational entropy, cross-entropy, expectation-maximization methods, Hidden Markov Models.

We present algorithms for extracting Hyperedge Replacement Grammar (HRG) rules from a graph along with a vertex order. Our algorithms are based on finding a tree decomposition of smallest width, relative to the vertex order, and then extracting one rule for each node in this structure. The assumption of a fixed order for the vertices of the input graph makes it possible to solve the problem in polynomial time, in contrast to the fact that the problem of finding optimal tree decompositions for a graph is NP-hard. We also present polynomial-time algorithms for parsing based on our HRGs, where the input is a vertex sequence and the output is a graph structure. The intended application of our algorithms is grammar extraction and parsing for semantic representation of natural language. We apply our algorithms to data annotated with Abstract Meaning Representations and report on the characteristics of the resulting grammars. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Motivated by the task of semantic parsing, we describe a transition system that generalizes standard transition-based dependency parsing techniques to generate a graph rather than a tree. Our system includes a cache with fixed size m, and we characterize the relationship between the parameter m and the class of graphs that can be produced through the graph-theoretic concept of tree decomposition. We find empirically that small cache sizes cover a high percentage of sentences in existing semantic corpora. [ABSTRACT FROM AUTHOR]

Graphs have a variety of uses in natural language processing, particularly as representations of linguistic meaning. A deficit in this area of research is a formal framework for creating, combining, and using models involving graphs that parallels the frameworks of finite automata for strings and finite tree automata for trees. A possible starting point for such a framework is the formalism of directed acyclic graph (DAG) automata, defined by Kamimura and Slutzki and extended by Quernheim and Knight. In this article, we study the latter in depth, demonstrating several new results, including a practical recognition algorithm that can be used for inference and learning with models defined on DAG automata. We also propose an extension to graphs with unbounded node degree and show that our results carry over to the extended formalism. [ABSTRACT FROM AUTHOR]

The complexity of parsing with synchronous context-free grammars is polynomial in the sentence length for a fixed grammar, but the degree of the polynomial depends on the grammar. Specifically, the degree depends on the length of rules, the permutations represented by the rules, and the parsing strategy adopted to decompose the recognition of a rule into smaller steps. We address the problem of finding the best parsing strategy for a rule, in terms of space and time complexity. We show that it is NP-hard to find the binary strategy with the lowest space complexity. We also show that any algorithm for finding the strategy with the lowest time complexity would imply improved approximation algorithms for finding the tree width of general graphs. [ABSTRACT FROM AUTHOR]

The weak equivalence of Combinatory Categorial Grammar (CCG) and Tree-Adjoining Grammar (TAG) is a central result of the literature on mildly context-sensitive grammar formalisms. However, the categorial formalism for which this equivalence has been established differs significantly from the versions of CCG that are in use today. In particular, it allows restriction of combinatory rules on a per grammar basis, whereas modern CCG assumes a universal set of rules, isolating all cross-linguistic variation in the lexicon. In this article we investigate the formal significance of this difference. Our main result is that lexicalized versions of the classical CCG formalism are strictly less powerful than TAG. [ABSTRACT FROM AUTHOR]