Conjunctive Categorial Grammars

Kuznetsov, Stepan L.; Okhotin, Alexander

doi:10.18653/v1/w17-3414

Cited by 9 publications

(4 citation statements)

References 33 publications

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“…The original Lambek calculus includes only multiplicative connectives (multiplication and two implications, called divisions). It is quite natural, however, to equip the Lambek calculus also with additive connectives (conjunction and disjunction), as in linear logic [van Benthem, 1991, Kanazawa, 1992, Buszkowski, 2010, Kuznetsov and Okhotin, 2017. We'll call this bigger system the multiplicative-additive Lambek calculus (MALC).…”

Section: The Multiplicative-additive Lambek Calculus With Subexponent...mentioning

confidence: 99%

Subexponentials in non-commutative linear logic

Kanovich¹,

Kuznetsov²,

Nigam

et al. 2018

Math. Struct. Comp. Sci.

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Linear logical frameworks with subexponentials have been used for the specification of among other systems, proof systems, concurrent programming languages and linear authorization logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can always be applied. This means that formulae in such frameworks can only be organized as sets and multisets of formulae not being possible to organize formulae as lists of formulae. This paper investigates the proof theory of linear logic proof systems in the non-commutative variant. These systems can disallow the application of exchange rule on some subexponentials. We investigate conditions for when cut-elimination is admissible in the presence of non-commutative subexponentials, investigating the interaction of the exchange rule with local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

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Section: The Multiplicative-additive Lambek Calculus With Subexponent...mentioning

confidence: 99%

Subexponentials in non-commutative linear logic

Kanovich¹,

Kuznetsov²,

Nigam

et al. 2018

Math. Struct. Comp. Sci.

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“…Namely, as noticed by Kanazawa (1992), MALC * -grammars an generate finite intersections of context-free languages and, moreover, images of such intersections under symbol-to-symbol homomorphisms (that is, homomorphisms h : Σ * → Σ * that map Σ to Σ). Furthermore, as shown by Kuznetsov (2013) and Kuznetsov and Okhotin (2017), the class of MALC *languages includes the class of languages generated by conjunctive grammars Okhotin (2013). This latter class is strictly greater than the class of intersections of context-free languages, but, unless P = NP, is still not closed under symbol-to-symbol homomorphisms (Kuznetsov and Okhotin, 2017).…”

Section: Generative Power Of Categorial Grammarsmentioning

confidence: 99%

“…Furthermore, as shown by Kuznetsov (2013) and Kuznetsov and Okhotin (2017), the class of MALC *languages includes the class of languages generated by conjunctive grammars Okhotin (2013). This latter class is strictly greater than the class of intersections of context-free languages, but, unless P = NP, is still not closed under symbol-to-symbol homomorphisms (Kuznetsov and Okhotin, 2017).…”

Section: Undecidabilitymentioning

confidence: 99%

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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“…The first natural extension of the Lambek calculus is the so-called "full," or multiplicativeadditive Lambek calculus obtained by adding additive conjunction and disjunction, which correspond to intersection and union. This increases the expressive power of Lambek grammars: with additives, they can describe finite intersections of context-free languages [Kanazawa 1992] and even a broader class of languages generated by conjunctive context-free grammars [Kuznetsov 2013, Kuznetsov andOkhotin 2017]. No non-trivial upper bounds are known for the class of languages generated by Lambek grammars with additives.…”

Section: Linguistic Introductionmentioning

confidence: 99%

Infinitary Action Logic with Exponentiation

Kuznetsov¹,

Speranski²

2020

Preprint

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We introduce infinitary action logic with exponentiation-that is, the multiplicativeadditive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is Π 1 1 -complete. Second, we show that the closure ordinal of its derivability operator is ω CK 1 .(N * \ N ) / N . The Kleene star, as shown in the next session, is axiomatized by means of an ωrule, which raises algorithmic complexity of the system to very high levels. Morrill and Valentín, however, in order to avoid undecidability, formulate an incomplete set of rules for Kleene star. Historically, Kleene star first appeared in the study of events, or actions within a transition system: in the original paper [Kleene 1956] it was used when describing events in neural networks. If A denotes a class of actions, then A * means actions of class A repeated several (possibly zero) times; [Pratt 1991] proposes action algebras, an extension of Kleene algebras with residuals. Though Pratt's work was independent from Lambek, these residuals actually coincide with Lambek divisions. In the presence of residuals, usage of infinitary systems for axiomatizing Kleene star becomes inevitable, due to complexity reasons (see below). 1 The second family of connectives we use to extend the Lambek calculus is the family of subexponential modalities, or subexponentials for short. Their linguistic motivation, going back to Morrill, is as follows. The Lambek calculus itself has a limited capability of treating relativization, or dependent clauses. For example, "that" in "book that John read" gets syntactic type (CN \ CN ) /(S / N ) (CN stands for "common noun," a noun without article), because the dependent clause "John read" lacks a noun phrase ("John read the book") to become a complete sentence (S). The place where the lacking noun phrase should be placed is called gap: "John read []." In more complicated situations, however, this does not work: in the phrase "book that John read yesterday" the dependent clause "John read yesterday" has a gap in the middle: "John read [] yesterday," and is neither of type S / N , nor N \ S. This syntactic phenomenon is called medial extraction and can be handled by adding a special modality, denoted by !, which allows permutation rules. Now "that" receives syntactic type (CN \ CN ) /(S / !N ), and "John read [] yesterday" is indeed of type S / !N , since by permutation !N reaches its place to fill the gap.There is a more sophisticated phenomenon called parasitic extraction: in the example "paper that John signed without reading" the dependent clause includes two gaps: "John signed [] without reading []" which should both be filled with the same "paper." ...

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Conjunctive Categorial Grammars

Cited by 9 publications

References 33 publications

Subexponentials in non-commutative linear logic

Subexponentials in non-commutative linear logic

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

Infinitary Action Logic with Exponentiation

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