Relational semantics for full linear logic

Coumans, Dion; Gehrke, Mai; Rooijen, Lorijn van

doi:10.1016/j.jal.2013.07.005

Cited by 17 publications

(20 citation statements)

References 11 publications

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“…The ensuing translation stage computed in [15] takes (the notational counterpart of) the pure inequality i ⊥⊥ ≤ i as input. The quoted claim above is not explicitly justified, but is a consequence of the following observation, which appears early on in the text [15]: Implication sends joins in the first coordinate to meets, hence (·) ⊥ sends joins to meets. As (·) ⊥ is a bijection, it follows that it is a (bijective) lattice homomorphism L → L ∂ , where L ∂ is the lattice obtained by reversing the order in L.…”

Section: Lemma 25mentioning

confidence: 99%

“…Over the years, many extensions, variations and analogues of this result have appeared, including alternative proofs in e.g. [42], generalizations to arbitrary modal signatures [17], variations of the correspondence language [37,47], Sahlqvist-type results for hybrid logics [44], various substructural logics [31,43,15], mu calculus [48], enlargements of the Sahlqvist class to e.g. the inductive formulas of [27], to mention but a few.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Algorithmic correspondence and canonicity for non-distributive logics

Conradie

Palmigiano

2019

Annals of Pure and Applied Logic

139

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We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic. theoretical framework which would provide a mathematically grounded way to compare (hierarchies of) Sahlqvisttype classes belonging to different logical settings. 1 Along with this lack of uniformity, each Sahlqvist-type results is tied to a particular choice of relational semantics for the relevant logic. Such a choice could be motivated by the fact that a logic has a uniquely established set-based semantics, but for many logics, like substructural logics, this is not the case. Hence, it is desirable to have a modular Sahlqvist theory that would distinguish core characteristics from incidental details relating to a particular choice of relational semantics.A theory which subsumes the previous results and which satisfies the desiderata of uniformity and modularity is currently emerging, and has been dubbed unified correspondence [7]. It is built on duality-theoretic insights [11] and uniformly exports the state-of-the-art in Sahlqvist theory from normal modal logic to a wide range of logics which include, among others, intuitionistic and distributive lattice-based (normal modal) logics [10], non-normal (regular) modal logics of arbitrary modal signature [39], hybrid logics [14], and mu-calculus [5,6].The breadth of this work has also stimulated many and varied applications. Some are closely related to the core concerns of the theory itself, such as the understanding of the relationship between different methodologies for obtaining canonicity results [38,9], or of the phenomenon of pseudo-correspondence [12]. Other, possibly surprising applications include the dual characterizations of classes of finite lattices [20], and the identification of the syntactic shape of axioms which can be translated into analytic structural rules of a proper display calculus [28]. Finally, the insights of unified correspondence theory have made it possible ...

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Section: Lemma 25mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algorithmic correspondence and canonicity for non-distributive logics

Conradie

Palmigiano

2019

Annals of Pure and Applied Logic

139

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“…Residuated lattices. To describe some of the structures appearing in [8,11,3,6], let L * = L ∪ {T } with T a ternary relation symbol. An L * -structure has the form P = (X, Y, R, T ).…”

Section: Definable Operations Over Polaritiesmentioning

confidence: 99%

“…In the same sort of way that Kripke frames have been used to model Boolean modal logics, polarities have been used to provide a relational semantics for various non-distributive substructural logics. These include the implication-fusion fragments of relevant logic, BCK logic and others [8,11]; the full Lambek-Grishin calculus [3] and linear logic [6]; and logics with unary modalities [4,5]. Algebraically these systems are modelled by (typically non-distributive) lattice expansions, i.e.…”

Section: Introductionmentioning

confidence: 99%

Definable Operators on Stable Set Lattices

Goldblatt

2020

Stud Logica

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Structures based on polarities provide relational semantics for nondistributive logics. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. This lifts a fundamental result from modal model theory to the non-distributive level. The proof makes use of a relationship between canonical extensions and MacNeille completions.

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“…There is not yet a theory of operations on the stable set lattices of polarities that is as general as the construction by Jónsson and Tarski [74] of n-ary operations on Boolean set algebras from n+1-ary relations. But there has been extensive work on the expansion of a polarity P by ternary relations, as subsets of X × X × Y and X × Y × Y , that are used to define residuated binary operations on P + which model various connectives in substructural logics [25,28,12,16]. Also in [13,14] there are expansions of P by binary relations, on X and on Y and from X to Y and Y to X, that are used to model various unary modalities.…”

Section: Generating Canonical Varietiesmentioning

confidence: 99%

Canonical extensions and ultraproducts of polarities

Goldblatt

2018

Algebra Univers.

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Jónsson and Tarski's notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of latticebased algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.

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Relational semantics for full linear logic

Cited by 17 publications

References 11 publications

Algorithmic correspondence and canonicity for non-distributive logics

Algorithmic correspondence and canonicity for non-distributive logics

Definable Operators on Stable Set Lattices

Canonical extensions and ultraproducts of polarities

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