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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
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
Interpolation and FEP for Logics of Residuated Algebras
"... A residuated algebra (RA) is a generalization of a residuated groupoid; instead of one basic binary operation · with residual operations \, /, it admits finitely many basic operations, and each n−ary basic operation is associated with n residual operations. A basic logical system for RAs was studied ..."
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A residuated algebra (RA) is a generalization of a residuated groupoid; instead of one basic binary operation · with residual operations \, /, it admits finitely many basic operations, and each n−ary basic operation is associated with n residual operations. A basic logical system for RAs was studied in e.g. [6, 8, 16, 15] under the name: Generalized Lambek Calculus GL. In this paper we study GL and its extensions in the form of sequent systems. We prove an interpolation property which allows to replace a substructure of the antecedent structure by a single formula in a provable sequent. Together with model constructions, based on nuclei [13], interpolation leads to proofs of Finite Embeddability Property of different classes of RAs, as e.g. all RAs, distributive latticeordered RAs, boolean RAs, Heyting RAs and double RAs. 1
Full Nonassociative Lambek Calculus with Distribution: Models and Grammars
"... We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Full Nonassociative Lambek Calculus with Distribution DFNL). We prove that formal grammars based on DFNL, also with assumptions, generate contextfree languages. The proof uses prooftheoretic tools (interp ..."
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We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Full Nonassociative Lambek Calculus with Distribution DFNL). We prove that formal grammars based on DFNL, also with assumptions, generate contextfree languages. The proof uses prooftheoretic tools (interpolation) and a construction of a finite model, employed in [13] in the proof of Strong Finite Model Property of DFNL. 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 whose underlying lattice is a Heyting algebra. Our proof also yields Finite Embeddability Property for boolean and Heyting algebras, supplied with an additional residuation structure. 1
Edited by Wojciech Buszkowski and Anne Preller
"... Abstract. We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cutelimination theorem and a normalization theorem for an extended system of this logic, its PTIME decidabilit ..."
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Abstract. We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cutelimination theorem and a normalization theorem for an extended system of this logic, its PTIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid.
Some Syntactic Interpretations in Different Systems of Full Lambek Calculus
"... [8] defines an interpretation of FL without 1 in its version without empty antecedents of sequents (employed in type grammars) and applies this interpretation to prove some general results on the complexity of substructural logics and the generative capacity of type grammars. Here this interpretati ..."
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[8] defines an interpretation of FL without 1 in its version without empty antecedents of sequents (employed in type grammars) and applies this interpretation to prove some general results on the complexity of substructural logics and the generative capacity of type grammars. Here this interpretation is extended for nonassociative logics (also with structural rules), logics with 1, logics with distributive laws for ∧,∨, logics with unary modalities, and multiplicative fragments. 1 Introduction and
MultiSorted Residuation
"... Abstract. Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and the ..."
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Abstract. Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and their residual operations [7]. In this paper we study a further generalization of GL which admits operations whose arguments and values can be of different sorts. This logic is called MultiSorted Lambek Calculus mL. We also consider its variants with lattice and boolean operations. We discuss some basic properties of these logics (completeness, decidability, complexity and others) and the corresponding algebras. 1
W. Buszkowski Type Logics in Grammar
"... Type logics are logics whose formulas are interpreted as types. For instance, A → B is a type of functions (procedures) which send inputs of type A to outputs of type B, and A⊗B is a type of pairs (f, g) such that f is of type A and g is of type B. The scope of possible realizations is huge: from co ..."
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Type logics are logics whose formulas are interpreted as types. For instance, A → B is a type of functions (procedures) which send inputs of type A to outputs of type B, and A⊗B is a type of pairs (f, g) such that f is of type A and g is of type B. The scope of possible realizations is huge: from con
COMPUTATIONAL COMPLEXITY OF NL1 WITH ASSUMPTIONS
"... We take into consideration Nonassociative Lambek Calculus with identity (NL1) enriched with a finite set of arbitrary assumptions and some of extensions of this system such as NL1 with permutation and Generalized Lambek Calculus (i.e. the system with nary operations) with identities. De Groote and ..."
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We take into consideration Nonassociative Lambek Calculus with identity (NL1) enriched with a finite set of arbitrary assumptions and some of extensions of this system such as NL1 with permutation and Generalized Lambek Calculus (i.e. the system with nary operations) with identities. De Groote and Lamarche in [2] established the polynomial time decidability for Classical Nonassociative Lambek Calculus. Buszkowski in [1] showed that systems of Nonassociative Lambek Calculus with assumptions are also decidable in polynomial time and generate contextfree languages. The same holds for systems with unary modalities, studied in Moortgat [5], nary operations, and the rule of permutation, studied in Jäger [3]. In order to obtain the PTIME decision procedure for NL1 with the finite set of nonlogical axioms we adapt the method used by Buszkowski [1]. This method does not rely on cut elimination which is not available for systems with additional assumptions. Then, using the results for NL1 we prove that considered extensions are decidable
Categorial Grammars and Substructural Logics
"... Substructural logics are formal logics whose Gentzenstyle sequent systems abandon some/all structural rules (Weakening, Contraction, Exchange, Associativity). They have extensively been studied in current literature on nonclassical logics from different points of view: as sequent axiomatizations of ..."
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Substructural logics are formal logics whose Gentzenstyle sequent systems abandon some/all structural rules (Weakening, Contraction, Exchange, Associativity). They have extensively been studied in current literature on nonclassical logics from different points of view: as sequent axiomatizations of relevant,