Bounded distributive lattice expansions

Gehrke, Mai; Jónsson, Bjàrni

doi:10.7146/math.scand.a-14428

Cited by 126 publications

(273 citation statements)

References 15 publications

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“…We now sketch the theory of canonical extensions for Distributive Lattice Expansions (DLE) -for the details, see [GJ94,GJ04]. Each map f ∶ UA n → UA can be extended to a map (UA σ ) n → UA σ in two canonical ways:…”

Section: Canonical Extension Of Distributive Lattice Expansionsmentioning

confidence: 99%

Coalgebraic completeness-via-canonicity for distributive substructural logics

Dahlqvist

Pym

2017

Journal of Logical and Algebraic Methods in Programming

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We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.

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Section: Canonical Extension Of Distributive Lattice Expansionsmentioning

confidence: 99%

Coalgebraic completeness-via-canonicity for distributive substructural logics

Dahlqvist

Pym

2017

Journal of Logical and Algebraic Methods in Programming

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“…The canonical extension encodes the topological dual frame in algebraic terms. A thorough explanation of canonical extensions may be found in [7]. One may obtain an incarnation of the canonical extension D σ of a bounded distributive lattice D as the lattice of all downsets of the frame (P D , τ, ≤) which is denoted by D(P D ).…”

Section: Duality For Distributive Latticesmentioning

confidence: 99%

“…The extension is effected by using the fact that the canonical extension of D is generated from the image of D by means of arbitrary meets and joins, see Theorem 3.25 in [7].…”

Section: Duality For Distributive Latticesmentioning

confidence: 99%

Distributive Lattice-Structured Ontologies

Bruun

Coumans

Gehrke

2009

Algebra and Coalgebra in Computer Science

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Abstract. In this paper we describe a language and method for deriving ontologies and ordering databases. The ontological structures arrived at are distributive lattices with attribution operations that preserve ∨, ∧ and ⊥. The preservation of ∧ allows the attributes to model the natural join operation in databases. We start by introducing ontological frameworks and knowledge bases and define the notion of a solution of a knowledge base. The import of this definition is that it specifies under what condition all information relevant to the domain of interest is present and it allows us to prove that a knowledge base always has a smallest, or terminal, solution. Though universal or initial solutions almost always are infinite in this setting with attributes, the terminal solution is finite in many cases. We describe a method for computing terminal solutions and give some conditions for termination and non-termination. The approach is predominantly coalgebraic, using Priestley duality, and calculations are made in the terminal coalgebra for the category of bounded distributive lattices with attribution operations.

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“…Canonical extensions were introduced in the 1950s by Jónsson and Tarski exactly for BAOs [JT51,JT52]. Thereafter their ideas have been developed further, which has led to a smooth theory of canonical extensions applicable in a broad setting [GH01,GJ94]. In [DGP05] canonical extensions of partially ordered algebras are defined to obtain relational semantics for the implication-fusion fragment of various substructural logics.…”

Section: Introductionmentioning

confidence: 99%

Relational semantics for full linear logic

Coumans¹,

Gehrke²,

Rooijen³

2014

Journal of Applied Logic

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Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic. In [DGP05] it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [Geh06] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form.In this paper we study full linear logic from an algebraic point of view. The main additional hurdle is the exponential. We analyse this operation algebraically and use canonical extensions to obtain relational semantics. Thus, we extend the work in [DGP05, Geh06] and use their approach to obtain relational semantics for full linear logic. Hereby we illustrate the strength of using canonical extension to retrieve relational semantics: it allows a modular and uniform treatment of additional operations and axioms.Traditionally, so-called phase semantics are used as models for (provability in) linear logic [Gir87]. These have the drawback that, contrary to our approach, they do not allow a modular treatment of additional axioms. However, the two approaches are related, as we will explain.

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Bounded distributive lattice expansions

Cited by 126 publications

References 15 publications

Coalgebraic completeness-via-canonicity for distributive substructural logics

Coalgebraic completeness-via-canonicity for distributive substructural logics

Distributive Lattice-Structured Ontologies

Relational semantics for full linear logic

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