The finite model property for various fragments of intuitionistic linear logic

Okada, Mitsuhiro; Terui, Kazushige

doi:10.2307/2586501

Cited by 74 publications

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“…Results that we obtain in Section 5 include the cut-elimination property for several logical systems, the decidability of logics and of varieties of residuated structures, the finite model property, and the finite embeddability property. The homomorphism theorem generalizes and simplifies ideas found in a variety of papers [2][3][4]16,18,22,24,25] that address the above types of problems in otherwise seemingly unrelated ways. Thus the notion of residuated frame provides a unifying framework for the analysis of various logical and algebraic properties and for their proof in a general setting.…”

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confidence: 74%

Residuated frames with applications to decidability

Galatos

Jipsen

2012

Trans. Amer. Math. Soc.

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Abstract. Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several varieties of residuated lattice-ordered groupoids, including the variety of involutive FL-algebras.Substructural logics and their algebraic formulation as varieties of residuated lattices and FL-algebras provide a general framework for a wide range of logical and algebraic systems, such as Classical propositional logic ↔ Boolean algebras Intuitionistic logic ↔ Heyting algebras Lukasiewicz logic ↔ MV-algebras Abelian logic ↔ abelian lattice-ordered groups Basic fuzzy logic ↔ BL-algebras Monoidal t-norm logic ↔ MTL-algebras Intuitionistic linear logic ↔ ILL-algebras Full Lambek calculus ↔ FL-algebras as well as lattice-ordered groups, symmetric relation algebras and many other systems.In this paper we introduce residuated frames and show that they provide relational semantics for substructural logics and representations for residuated structures. Our approach is driven by the applications of the theory. As is the case with Kripke frames for modal logics, residuated frames provide a valuable tool for solving both algebraic and logical problems. Moreover we show that there is a direct link between Gentzen-style sequent calculi and our residuated frames, which gives insight into the connection between a cut-free proof system and the finite embeddability property for the corresponding variety of algebras.We begin with an overview of residuated structures and certain types of closure operators called nuclei. This leads to the definition of residuated frames (Section 3) and Gentzen frames (Section 4), illustrated by several examples. We then prove a general homomorphism theorem in the setting of Gentzen frames (Thm. 4.2) and Date: August 25, 2008.

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confidence: 74%

Residuated frames with applications to decidability

Galatos

Jipsen

2012

Trans. Amer. Math. Soc.

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“…The sequent proof systems we mentioned in this paper have the sub-formula property, so that provability is decidable in each case, using a simple proof search algorithm [26]. Surprisingly, the concept of types can be used to cut off useless branches.…”

Section: Applicationsmentioning

confidence: 99%

Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic

Pous¹

2010

Computer Science Logic

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International audienceWe 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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“…[5] defines phase spaces for CNL and employs them in proofs of some meta-logical theorems; phase spaces for InNL (denoted there by CNL − ) are defined, but not worked out. Phase spaces for associative substructural logics were studied in [9,1,15]. The present paper is mostly proof-theoretic (but we avoid proof nets).…”

Section: Introductionmentioning

confidence: 99%

Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity

Buszkowski

2017

B Sect Log

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In [5] we study Nonassociative Lambek Calculus (NL) augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus (CNL). Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.

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The finite model property for various fragments of intuitionistic linear logic

Cited by 74 publications

References 10 publications

Residuated frames with applications to decidability

Residuated frames with applications to decidability

Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic

Involutive Nonassociative Lambek Calculus: Sequent Systems and Complexity

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