The logic of distributive bilattices

Bou, Fèlix; Rivieccio, Umberto

doi:10.1093/jigpal/jzq041

Cited by 29 publications

(43 citation statements)

References 25 publications

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“…Its interest comes mainly from the fact that any interlaced pre-bilattice can be represented as a special product of two lattices. This result is well-known for bounded pre-bilattices, and it has been more recently generalized to the unbounded case [17,5]. The interlacing conditions may be strengthened through the following definition, due to Ginsberg [14].…”

Section: Algebraic Preliminariesmentioning

confidence: 95%

“…The interest in bilattices has thus different sources: among others, computer science and A.I. (see especially the works of Ginsberg, Arieli and Avron), logic programming (Fitting), lattice theory and algebra [16] and, more recently, algebraic logic [4,5,24]. An up-todate review of the applications of this formalism and also of the motivation behind its study can be found in the dissertation [24].…”

Section: Introductionmentioning

confidence: 99%

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Priestley Duality for Bilattices

Jung

Rivieccio

2012

Stud Logica

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Abstract.We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.Keywords: bilattices, Priestley duality theory, bilattices with conflation, bilattices with implication, Brouwerian bilattices. IntroductionBilattices are algebraic structures introduced in 1988 by Matthew Ginsberg [14] as a uniform framework for inference in Artificial Intelligence. Since then they have found a variety of applications, sometimes in quite different areas from the original one. The interest in bilattices has thus different sources: among others, computer science and A.I. (see especially the works of Ginsberg, Arieli and Avron), logic programming (Fitting), lattice theory and algebra [16] and, more recently, algebraic logic [4,5,24]. An up-todate review of the applications of this formalism and also of the motivation behind its study can be found in the dissertation [24].In the present work we develop a Priestley-style duality theory for bilattices and some related algebras that are obtained by adding new operations to the basic algebraic language of bilattices. The main idea guiding our work is that it is possible to view bounded bilattices and related algebras as bounded lattices having two extra constants that satisfy certain properties. This approach will enable us to apply known results on duality theory for different classes of lattices to the study of bilattices and other algebras having a bilattice reduct.A duality theory for bilattices has already been introduced by Mobasher et al. in [16]. However, while the point of view of [16] is abstract and category-theoretic, in the present paper instead we are interested in a concrete study of the topological spaces that correspond to bilattices and related algebras. We will briefly review the results of [16] and discuss the differences between their approach and ours in Sections 1.2 and 1.6.The rest of the paper is organized as follows. In Section 1 we introduce some definitions and algebraic results on bilattices and language expansions thereof (bilattices with a dual negation operation, bilattices with implication) that will be needed to develop our duality theory. There are no new results here but it is included for ease of reference. Section 1.6 is crucial: it is here where we introduce the fundamental point of view with which we approach the topic. It is concluded with an overview of the various algebras that are considered in this paper.In Section 2 we recall some known results on duality theories for De Morgan algebras and N4-lattices, on which we will base our treatment of various classes of bilattices in Section 3. We start in Section 3.1 with a duali...

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Section: Algebraic Preliminariesmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Priestley Duality for Bilattices

Jung

Rivieccio

2012

Stud Logica

Self Cite

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“…We then have ⊨ for all ∈ Γ and ∕ ⊨ , in other words, Γ ∕ ⊨ . 8 The asymmetry between the two definitions being exactly the "twist" in the twist-structure construction of Section V. Proof: Only a small adjustment is needed. By Theorem 9 we know that the algebraic countermodel is of the form ⟨B, 0 ⟩ where 0 is the least bifilter of B.…”

Section: Completenessmentioning

confidence: 98%

“…Given a modal bilattice B, one first defines an (equivalence) relation ≈ by letting ≈ if ∧ = ⊕ [8,Definition 3.7]. One then shows that the quotient B/≈ can be endowed with algebraic operations that turn it into a bimodal Boolean algebra: for…”

Section: Definition 11 a Bimodal Boolean Algebra Is A Structurementioning

confidence: 99%

Kripke Semantics for Modal Bilattice Logic

Jung

Rivieccio

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-We employ the well-developed and powerful techniques of algebraic semantics and Priestley duality to set up a Kripke semantics for a modal expansion of Arieli and Avron's bilattice logic, itself based on Belnap's four-valued logic. We obtain soundness and completeness of a Hilbert-style derivation system for this logic with respect to four-valued Kripke frames, the standard notion of model in this setting. The proof is via intermediary relational structures which are analysed through a topological reading of one of the axioms of the logic. Both local and global consequence on the models are covered.

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“…Apart from mathematical logic ( [2,4,8,15,16,22,26,28]) and algebra, De Morgan algebras (and De Morgan bisemilattices) have applications in multi-valued simulations of digital circuits too ( [10,11]). …”

Section: Now Let Us Define De Morgan Algebras (Lattices)mentioning

confidence: 99%

Super-De Morgan functions and free De Morgan quasilattices

Movsisyan

Aslanyan

2014

Open Mathematics

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A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices).In this paper we give a functional representation of the free -generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on free generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of variables. MSC:08B05, 03C85, 06B25, 06D30, 06E30, 06B20, 08B20 Keywords Introduction and preliminariesIt is well known that the free Boolean algebra on free generators is isomorphic to the Boolean algebra of Boolean functions of variables ([7, 13, 17, 55]). The free bounded distributive lattice on free generators is isomorphic to the lattice of monotone Boolean functions of variables ([7, 13, 17, 55] [44,49,50]). In the paper [46] we have proved the completeness theorem for De Morgan functions.In this paper we give a functional representation of the free -generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on free generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of variables. *

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The logic of distributive bilattices

Cited by 29 publications

References 25 publications

Priestley Duality for Bilattices

Priestley Duality for Bilattices

Kripke Semantics for Modal Bilattice Logic

Super-De Morgan functions and free De Morgan quasilattices

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