Hypergraph Lambek grammars

Pshenitsyn, Tikhon

doi:10.1016/j.jlamp.2022.100798

Cited by 3 publications

(18 citation statements)

References 10 publications

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“…Now let us define the hypergraph Lambek calculus. Details concerning its motivation can be found in [13,15]; we also provide a motivation relating the hypergraph Lambek calculus with linear logic in the next subsection.…”

Section: Hypergraph Lambek Calculus and Hypergraph Lambek Grammarsmentioning

confidence: 99%

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From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

Pshenitsyn

2023

Electron. Proc. Theor. Comput. Sci.

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We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL 0 ; then we prove that any DPO grammar can be converted into an equivalent HMEL 0grammar. We also define the conjunctive Kleene star, which behaves similarly to this exponential modality, and establish a similar result. If we add neither the exponential modality nor the conjunctive Kleene star to HL, then we can still use the same encoding and show that any DPO grammar with a linear restriction on the length of derivations can be converted into an equivalent HL-grammar.

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Section: Hypergraph Lambek Calculus and Hypergraph Lambek Grammarsmentioning

confidence: 99%

“…The hypergraph Lambek calculus HL and hypergraph Lambek grammars are novel approaches described in [13,15]. They are based on logical grounds: HL generalizes the Lambek calculus introduced in [8].…”

Section: Introductionmentioning

confidence: 99%

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

Pshenitsyn

2023

Electron. Proc. Theor. Comput. Sci.

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“…Our goal is to show that the hypergraph Lambek calculus HL studied in [10,11,12] generalizes the multimodal Lambek calculus NL♦. In order to reach it, firstly, we would like to discuss how HL works.…”

Section: Hypergraph Lambek Calculus As a General Theory Of Residuationmentioning

confidence: 99%

“…Another generalized logic of pure residuation is the hypergraph Lambek calculus HL defined in [11]. It is defined as a sequent calculus that operates on hypergraph formulas and sequents using the hyperedge replacement operation [1].…”

Section: Introductionmentioning

confidence: 99%

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Multimodality in the Hypergraph Lambek Calculus

Pshenitsyn

2023

Electron. Proc. Theor. Comput. Sci.

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The multimodal Lambek calculus NL♦ is an extension of the Lambek calculus that includes several product operations (some of them being commutative or/and associative), unary modalities, and corresponding residual implications. In this work, we relate this calculus to the hypergraph Lambek calculus HL. The latter is a general pure logic of residuation defined in a sequent form; antecedents of its sequents are hypergraphs, and the rules of HL involve hypergraph transformation. Our main result is the embedding of the multimodal Lambek calculus (with at most one associative product) in HL. It justifies that HL is a very general Lambek-style logic and also provides a novel syntactic interface for NL♦: antecedents of sequents of NL♦ are represented as tree-like hypergraphs in HL, and they are derived from each other by means of hyperedge replacement. The advantage of this embedding is that commutativity and associativity are incorporated in the sequent structure rather than added as separate rules. Besides, modalities of NL♦ are represented in HL using the product and the division of HL, which explicitizes their residual nature.

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Powerful and NP-Complete: Hypergraph Lambek Grammars

Pshenitsyn

2021

Graph Transformation

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Hypergraph Lambek grammars (HL-grammars) is a novel logical approach to generating graph languages based on the hypergraph Lambek calculus. In this paper, we establish a precise relation between HL-grammars and hypergraph grammars based on the double pushout (DPO) approach: we prove that HL-grammars generate the same class of languages as DPO grammars with the linear restriction on lengths of derivations. This can be viewed as a complete description of the expressive power of HL-grammars and also as an analogue of the Pentus theorem, which states that Lambek grammars generate the same class of languages as context-free grammars. As a corollary, we prove that HL-grammars subsume contextual hyperedge replacement grammars.

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Hypergraph Lambek grammars

Cited by 3 publications

References 10 publications

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

Multimodality in the Hypergraph Lambek Calculus

Powerful and NP-Complete: Hypergraph Lambek Grammars

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