## Grammar Factorization by Tree Decomposition

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@MISC{Gildea_grammarfactorization,

author = {Daniel Gildea},

title = {Grammar Factorization by Tree Decomposition},

year = {}

}

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### Abstract

We describe the application of the graph-theoretic property known as treewidth to the problem of finding efficient parsing algorithms. This method, similar to the junction tree algorithm used in graphical models for machine learning, allows automatic discovery of efficient algorithms such as the O(n 4) algorithm for bilexical grammars of Eisner and Satta. We examine the complexity of applying this method to parsing algorithms for general Linear Context-Free Rewriting Systems. We show that any polynomial-time algorithm for this problem would imply an improved approximation algorithm for the well-studied treewidth problem on general graphs. 1.

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Citation Context ...st node in the tree decomposition of Figure 8c includes eight position variables. This result is intermediate between the O(n 6 ) for binarizable SCFGs, also known as Inversion Transduction Grammars (=-=Wu 1997-=-), and the O(n 10 ) that we would achieve by recognizing the rule in a single deduction step. Gildea and ˇStefankovič (2007) use a combinatorial argument to show that as the number of nonterminals r i... |

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Citation Context ...e languages of string pairs generated by a synchronous grammar can be arranged in an infinite hierarchy, with each rank ≥ 4 producing languages not possible with grammars restricted to smaller rules (=-=Aho and Ullman 1972-=-). For any grammar with maximum rank r, converting each rule into a single deduction rule yields an O(n 2r+2 )parsing algorithm, because there are r + 1 boundary variables in each language. More effic... |

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Citation Context ...ependently of one another. In order to have a common representation for our algorithms to work with, we represent parsing algorithms as weighted deduction systems (Shieber, Schabes, and Pereira 1995; =-=Goodman 1999-=-; Nederhof 2003). Weighted deduction systems consist of axioms and rules for building items or partial results. Items are identified by square brackets, with their weights written to the left. Figure ... |

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Citation Context ...ions contain just one string. As CFG is equivalent to LCFRS with fanout = 1, SCFG and TAG can be represented as LCFRS with fan-out = 2. Higher values of fan-out allow strictly more powerful grammars (=-=Rambow and Satta 1999-=-). Polynomialtime parsing is possible for any fixed LCFRS grammar, but the degree of the polynomial depends on the grammar. Parsing general LCFRS grammars, where the grammar is considered part of the ... |

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Citation Context ...rd the remainder, for example, when factorizing rules automatically 236Gildea Grammar Factorization by Tree Decomposition extracted from word-aligned bitext (Wellington, Waxmonsky, and Melamed 2006; =-=Huang et al. 2009-=-) or from dependency treebanks (Kuhlmann and Nivre 2006; Gildea 2010). In this setting, the rules having treewidth k can be identified in time O(ℓ k+2 )using the simple algorithm of Arnborg, Corneil, ... |

21 | Recognition of linear context-free rewriting systems
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Citation Context ...ng is possible for any fixed LCFRS grammar, but the degree of the polynomial depends on the grammar. Parsing general LCFRS grammars, where the grammar is considered part of the input, is NP-complete (=-=Satta 1992-=-). 4.1 Graphs Derived from LCFRS Rules Given an LCFRS rule as defined previously, a weighted deduction rule for a bottom– up parser can be derived by creating an antecedent for each righthand nontermi... |

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(Show Context)
Citation Context ...ules automatically 236Gildea Grammar Factorization by Tree Decomposition extracted from word-aligned bitext (Wellington, Waxmonsky, and Melamed 2006; Huang et al. 2009) or from dependency treebanks (=-=Kuhlmann and Nivre 2006-=-; Gildea 2010). In this setting, the rules having treewidth k can be identified in time O(ℓ k+2 )using the simple algorithm of Arnborg, Corneil, and Proskurowski (1987), (where again ℓ is the number o... |

15 | Weighted deductive parsing and Knuth’s algorithm - Nederhof - 2003 |

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1 | Worst-case synchronous grammar rules - Gómez-Rodríguez, Kuhlmann, et al. |