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34
Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
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Pregroup Grammars and Contextfree Grammars
"... Pregroup grammars were introduced by Lambek [20] as a new formalism of typelogical grammars. They are weakly equivalent to contextfree grammars ..."
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Pregroup grammars were introduced by Lambek [20] as a new formalism of typelogical grammars. They are weakly equivalent to contextfree grammars
Lambek grammars, tree adjoining grammars and hyperedge replacement grammars
 In Proceedings of the TAG+ Conference. HAL  CCSD
, 2008
"... Two recent extension of the nonassociative Lambek calculus, the LambekGrishin calculus and the multimodal Lambek calculus, are shown to generate class of languages as tree adjoining grammars, using (tree generating) hyperedge replacement grammars as an intermediate step. As a consequence both ext ..."
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Two recent extension of the nonassociative Lambek calculus, the LambekGrishin calculus and the multimodal Lambek calculus, are shown to generate class of languages as tree adjoining grammars, using (tree generating) hyperedge replacement grammars as an intermediate step. As a consequence both extensions are mildly contextsensitive formalisms and benefit from polynomial parsing algorithms. 1
Nonassociative Lambek Calculus with Additives and ContextFree Languages, Francez Festschrift, O. Grunberg et al
 Eds.), LNCS
, 2009
"... Abstract. We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Distributive Full Nonassociative Lambek Calculus DFNL). We prove that categorial grammars based on DFNL, also enriched with assumptions, generate contextfree languages. The proof uses prooftheor ..."
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Abstract. We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Distributive Full Nonassociative Lambek Calculus DFNL). We prove that categorial grammars based on DFNL, also enriched with assumptions, generate contextfree languages. The proof uses prooftheoretic tools (interpolation) and a construction of a finite model, earlier employed in [11] in the proof of Finite Embeddability Property (FEP) of DFNL; our paper is selfcontained, since we provide a simplified version of the latter proof. We obtain analogous results for different variants of DFNL, e.g. BFNL, which admits negation ¬ such that ∧,∨, ¬ satisfy the laws of boolean algebra, and HFNL, corresponding to Heyting algebras with an additional residuation structure. Our proof also yields Finite Embeddability Property of booleanordered and Heytingordered residuated groupoids. The paper joins prooftheoretic and modeltheoretic techniques of modern logic with standard tools of mathematical linguistics. 1
Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic
 COMPUTER SCIENCE LOGIC, CZECH REPUBLIC
, 2010
"... We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to th ..."
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We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms.
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"... that this will be a book about categorial grammar. In the preface, however, several strains of categorial grammar are discussed, including combinatory categorial grammar (Steedman 2000), pregroup grammar (Lambek 1999), and abstract categorial grammar (de Groote 2001), and it is clarified that the bo ..."
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that this will be a book about categorial grammar. In the preface, however, several strains of categorial grammar are discussed, including combinatory categorial grammar (Steedman 2000), pregroup grammar (Lambek 1999), and abstract categorial grammar (de Groote 2001), and it is clarified that the book investigates only the tradition of typelogical grammar, or rather, those grammars based on Lambek categorial grammar (Lambek 1958). That being said, this book contains a good introduction to typelogical grammar and its first part would make a good textbook in an advanced course on the theory of typelogical grammar. In particular, exercises are sprinkled throughout the book that will be illuminating to the uninitiated reader. The book is neatly divided into three parts that are likely to be of varying levels of interest depending on the specific audience. Part I, titled “Lambek Categorial Grammar, ” gives a concise introduction to a number of aspects of Lambek categorial grammar, which is suitable for an audience interested in the basic intuitions and mechanics of that grammar. Part II, titled “Logical Categorial Grammar, ” introduces a number of extensions of Lambek’s grammar, each of which are motivated by linguistic
ProductFree Lambek Calculus Is NPComplete
 In Logical Foundations of Computer Science. Proceedings of the 2009 International Symposium on Logical Foundations of Computer Science
, 2009
"... In this paper we prove that the derivability problems for productfree Lambek calculus and productfree Lambek calculus allowing empty premises are NPcomplete. Also we introduce a new derivability characterization for these calculi. ..."
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In this paper we prove that the derivability problems for productfree Lambek calculus and productfree Lambek calculus allowing empty premises are NPcomplete. Also we introduce a new derivability characterization for these calculi.
Bounded and Ordered Satisfiability: Connecting Recognition with Lambekstyle Calculi to Classical Satisfiability Testing
"... this paper, when we mention the Lambek Calculus (LC) or Lambek Grammars (LG), we are referring to the productfree fragment ..."
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this paper, when we mention the Lambek Calculus (LC) or Lambek Grammars (LG), we are referring to the productfree fragment
The derivability problem for lambek calculus with one division
 Artificial Intelligence Preprint Series
, 2006
"... In this paper we prove that the derivability problem for Lambek calculus with one division is decidable in polynomial time and present an algorithm for it. ..."
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In this paper we prove that the derivability problem for Lambek calculus with one division is decidable in polynomial time and present an algorithm for it.
Untyping Typed Algebraic Structures
"... Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider “typed rings”. We prove some “untyping ” theorems: in some ..."
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Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider “typed rings”. We prove some “untyping ” theorems: in some algebras (semirings, Kleene algebras, residuated monoids), types can be reconstructed from valid untyped equalities. As a consequence, the corresponding untyped decision procedures can be extended to the typed setting.