Generalized Categorial Dependency Grammars

Dekhtyar, Michael I.; Dikovsky, Alexander

doi:10.1007/978-3-540-78127-1_13

Cited by 13 publications

(4 citation statements)

References 9 publications

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“…From the modern logical point of view, the calculus of Lambek grammars (the Lambek calculus) is a variant of Girard's linear logic (Girard, 1987) in its non-commutative intuitionistic version (Abrusci, 1990). Nowadays Lambek-style categorial grammars form one framework in a family of closely related formalisms, including combinatory categorial grammars (Steedman, 2000), categorial dependency grammars (Dekhtyar and Dikovsky, 2008), and others.…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

“…The Lambek calculus [23] is a substructural, non-commutative logical system (a variant of linear logic [15] in its intuitionistic non-commutative version [1]) that serves as the logical base for categorial grammars, a formalism that aims to describe natural language by means of logical derivability (see Buszkowski [9], Carpenter [11], Morrill [30], Moot and Retoré [28], etc). The idea of categorial grammar goes back to works of Ajdukiewicz [2] and Bar-Hillel [3], and afterwards it developed into several closely related frameworks, including combinatory categorial grammars (CCG, Steedman [39]), categorial dependency grammars (CDG, Dikovsky and Dekhtyar [12]), and Lambek categorial grammars. A categorial grammar assigns syntactic categories (types) to words of the language.…”

Section: Linguistic Introductionmentioning

confidence: 99%

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Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

Kanovich

Kuznetsov

Scedrov

2017

Fundamentals of Computation Theory

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Abstract. The Lambek calculus is a well-known logical formalism for modelling natural language syntax. The original calculus covered a substantial number of intricate natural language phenomena, but only those restricted to the context-free setting. In order to address more subtle linguistic issues, the Lambek calculus has been extended in various ways. In particular, Morrill and Valentín (2015) introduce an extension with so-called exponential and bracket modalities. Their extension is based on a non-standard contraction rule for the exponential that interacts with the bracket structure in an intricate way. The standard contraction rule is not admissible in this calculus. In this paper we prove undecidability of the derivability problem in their calculus. We also investigate restricted decidable fragments considered by Morrill and Valentin and we show that these fragments belong to the NP class. (CDG, Dikovsky and Dekhtyar [12]), and Lambek categorial grammars. A categorial grammar assigns syntactic categories (types) to words of the language. In the Lambek setting, types are constructed using two division operations, \ and /, and the product, ·. Intuitively, A \ B denotes the type of a syntactic object that lacks something of type A on the left side to become an object of type B; B / A is symmetric; the product stands for concatenation. The Lambek calculus provides a system of rules for reasoning about syntactic types. Linguistic Introduction

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Section: Mathematics Subject Classificationmentioning

confidence: 99%

Section: Linguistic Introductionmentioning

confidence: 99%

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

Kanovich

Kuznetsov

Scedrov

2017

Fundamentals of Computation Theory

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“…Now we list some properties of L(P DAC) which follows from theorem 2 and the results obtained in the papers [2,3,21,15]:…”

Section: Definition 12mentioning

confidence: 99%

PDA with Independent Counters

Дехтярь¹,

Karlov²

2015

Model. anal. inf. sist.

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-down automata with independent counters, projective and non-projective dependency, categorial dependency grammar, multimodal categorial dependency grammar, push-down automata with stacks of independent counters.Is dedicated to the memory of Alexander Dikovsky (1945Dikovsky ( -2014 Push-down automata with independent counters (PDACs) combine the power of PDAs and Petri Nets. They were developed in [21,15], as a tool of recognition of languages generated by Categorial Dependency Grammars (CDGs). CDGs are classical categorial grammars extended by oriented polarized valencies. They express both projective and non-projective dependencies between the words of a sentence. PDAC is a usual PDA equipped with a finite number of counters. The independence of counters means that their state has no effect on the choice of an automaton move. In the first part of the paper we compare some variants of PDACs and prove the equivalence of two variants of PDAs with independent counters: without syntactic and without semantic ε-loops. Some connections between PDAC-languages and Petri Net languages are noticed. Then we show that PDACs are equivalent to stack+bag push-down automata (SBPA) independently introduced by Søgaard and that ε-acyclic SBPAs recognize exactly CDG-languages.Multimodal Categorial Dependency Grammars (mmCDGs) were introduced in [4] as an extension of GDGs that allows control of some intersections of dependencies. The class of mmCDG-languages is rich enough and has good closure properties, that it forms AFL. In the second part of the paper we extend PDACs and introduce push-down automata with stacks of independent counters (PDASC). PDASCs extend PDACs twofold: (i) each counter is a stack of integers and (ii) there is a restriction function which allows to diminish a head of a counter only if the heads of all dependent counters are zeros. Our main result says that these PDASCs accept exactly the class of mmCDG-languages.

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“…For the special case of unidirectional Lambek grammars, which use only one kind of division operators (\, /), but not both, Kuznetsov (2016), using the ideas of Savateev (2009), presents a polynomial translation into context-free grammars. Other generalizations of categorial grammars include combinatory categorial grammars by Steedman (1996), categorial dependency grammars by Dekhtyar and Dikovsky (2008), and others.…”

Section: Introductionmentioning

confidence: 99%

Conjunctive Categorial Grammars

Kuznetsov

Okhotin

2017

Proceedings of the 15th Meeting on the Mathematics of Language

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Basic categorial grammars are enriched with a conjunction operation, and it is proved that the formalism obtained in this way has the same expressive power as conjunctive grammars, that is, context-free grammars enhanced with conjunction. It is also shown that categorial grammars with conjunction can be naturally embedded into the Lambek calculus with conjunction and disjunction operations. This further implies that a certain NP-complete set can be defined in the Lambek calculus with conjunction.

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Generalized Categorial Dependency Grammars

Cited by 13 publications

References 9 publications

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

PDA with Independent Counters

Conjunctive Categorial Grammars

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