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

Kanazawa, Manabu

doi:10.1007/978-3-662-44121-3_6

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…, u k is the alphabetical listing of the elements of P . 18 Example 9. Derivation trees of a simple context-free tree grammar G = (N, Σ, P, S) can be represented as 3-dimensional trees over the alphabet N ∪ Σ.…”

Section: Multi-dimensional Treesmentioning

confidence: 99%

“…The Simple context-free tree grammars hold interest for theoretical computational linguists because of their many attractive formal properties [15,14,19] and their ability to lexicalize tree-adjoining grammars without changing the set of derived trees [27]. The notion of a Dyck tree language I introduce in this paper also leads to a generalization of linear indexed grammars that is equivalent to simple context-free tree grammars in the same way that linear indexed grammars are equivalent to tree-adjoining grammars [18]. I believe these connections make it particularly interesting that the classic result of Chomsky and Schützenberger naturally extends to the level of simple context-free tree languages.…”

Section: Introductionmentioning

confidence: 99%

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Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Kanazawa

2014

J Logic Computation

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Weir [43] proved a Chomsky-Schützenberger-like representation theorem for the string languages of tree-adjoining grammars, where the Dyck language Dn in the Chomsky-Schützenberger characterization is replaced by the intersection D2n ∩ g −1 (D2n), where g is a certain bijection on the alphabet consisting of 2n pairs of brackets. This paper presents a generalization of this theorem to the string languages generated by simple (i.e., linear and non-deleting) context-free tree grammars. This result is obtained through a natural generalization of the original Chomsky-Schützenberger theorem to the tree languages of simple context-free tree grammars. I use Baldwin and Strawn's [2] notion of multi-dimensional trees to state this latter theorem in a very general, abstract form.

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Section: Multi-dimensional Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Kanazawa

2014

J Logic Computation

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“…Existing definitions of LIG (Vijay-Shanker and Weir, 1989;Joshi et al, 1991;Kanazawa, 2014) are equivalent to CFGs controlled by simple PDAs, and existing recognition algorithms (e.g., Vijay-Shanker and Weir, 1989) have the same limitation that makes Lang's algorithm suboptimal. By using a top-down PDA as a controller instead, we obtain a new stringsum algorithm that is more space-efficient and runs asymptotically faster.…”

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confidence: 99%

Efficient Algorithms for Recognizing Weighted Tree-Adjoining Languages

Butoi,

Vieira,

Cotterell

et al. 2023

Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing

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The class of tree-adjoining languages can be characterized by various two-level formalisms, consisting of a context-free grammar (CFG) or pushdown automaton (PDA) controlling another CFG or PDA. These four formalisms are equivalent to tree-adjoining grammars (TAG), linear indexed grammars (LIG), pushdownadjoining automata (PAA), and embedded pushdown automata (EPDA). We define semiringweighted versions of the above two-level formalisms, and we design new algorithms for computing their stringsums (the weight of all derivations of a string) and allsums (the weight of all derivations). From these, we also immediately obtain stringsum and allsum algorithms for TAG, LIG, PAA, and EPDA. For LIG, our algorithm is more time-efficient by a factor of O(n|N |) (where n is the string length and |N | is the size of the nonterminal set) and more space-efficient by a factor of O(|Γ|) (where Γ is the size of the stack alphabet) than the algorithm of Vijay-Shanker and Weir (1989). For EPDA, our algorithm is both more spaceefficient and time-efficient than the algorithm of Alonso et al. ( 2001) by factors of O(|Γ| 2 ) and O(|Γ| 3 ), respectively. Finally, we give the first PAA stringsum and allsum algorithms.

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“… Most LIG definitions (Vijay-Shanker, 1987;Joshi et al, 1991;Kanazawa, 2014) allow only productions that pop/push at most one symbol from/to a nonterminal's stack.2 The terms variable and constant come fromLang (1994) and are equivalent to nonterminals with obligatory and null adjunction constraints, respectively.…”

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confidence: 99%

Convergence and Diversity in the Control Hierarchy

Butoi¹,

Cotterell²,

Chiang³

2023

Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)

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Weir has defined a hierarchy of language classes whose second member (L 2 ) is generated by tree-adjoining grammars (TAG), linear indexed grammars (LIG), combinatory categorial grammars, and head grammars. The hierarchy is obtained using the mechanism of control, and L 2 is obtained using a contextfree grammar (CFG) whose derivations are controlled by another CFG. We adapt Weir's definition of a controllable CFG to give a definition of controllable pushdown automata (PDAs). This yields three new characterizations of L 2 as the class of languages generated by PDAs controlling PDAs, PDAs controlling CFGs, and CFGs controlling PDAs. We show that these four formalisms are not only weakly equivalent but equivalent in a stricter sense that we call d-weak equivalence. Furthermore, using an even stricter notion of equivalence called d-strong equivalence, we make precise the intuition that a CFG controlling a CFG is a TAG, a PDA controlling a PDA is an embedded PDA, and a PDA controlling a CFG is a LIG. The fourth member of this family, a CFG controlling a PDA, does not correspond to any formalism we know of, so we invent one and call it a Pushdown Adjoining Automaton.

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

Abstract: Abstract. I define a generalization of linear indexed grammars that is equivalent to simple context-free tree grammars in the same way that linear indexed grammars are equivalent to tree-adjoining grammars.

Cited by 4 publications

References 23 publications

Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Multidimensional trees and a Chomsky–Schützenberger–Weir representation theorem for simple context-free tree grammars

Efficient Algorithms for Recognizing Weighted Tree-Adjoining Languages

Convergence and Diversity in the Control Hierarchy

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