Cylindric algebras

Henkin, Leon; Tarski, Alfred

doi:10.1090/pspum/002/0124250

Cited by 94 publications

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“…Both RLA and PAL and a third algebra called Krasner algebra (cf. Hereth Correia & Pöschel, 2004) are very similar in nature to Henkin & Tarski's (1961) Cylindric Set Algebras (CSA). CSA, RLA, PAL and Krasner algebra all have the expressive power of first order logic (FOL) with equality (cf.…”

Section: Algebras As a Representation Of Logical Structuresmentioning

confidence: 96%

An Application of Relation Algebra to Lexical Databases

Priss

Old

2006

Conceptual Structures: Inspiration and Application

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Abstract. This paper presents an application of relation algebra to lexical databases. The semantics of knowledge representation formalisms and query languages can be provided either via a set-theoretic semantics or via an algebraic structure. With respect to formalisms based on n-ary relations (such as relational databases or power context families), a variety of algebras is applicable. In standard relational databases and in formal concept analysis (FCA) research, the algebra of choice is usually some form of Cylindric Set Algebra (CSA) or Peircean Algebraic Logic (PAL). A completely different choice of algebra is a binary Relation Algebra (RA). In this paper, it is shown how RA can be used for modelling FCA applications with respect to lexical databases.

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Section: Algebras As a Representation Of Logical Structuresmentioning

confidence: 96%

An Application of Relation Algebra to Lexical Databases

Priss

Old

2006

Conceptual Structures: Inspiration and Application

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“…The formulation of Lemma 1.1 is deliberately analogous to that of Henkin's result [HMT71,2.5.4] to emphasize the potential analogous applications. It is not difficult to see that Lemma 1.1 implies Theorem l(i), (ii), cf.…”

Section: Decidability Of Relation Algebras With Weakened Associativitmentioning

confidence: 99%

Decidability of relation algebras with weakened associativity

Németi¹

1987

Proc. Amer. Math. Soc.

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Tarski showed that mathematics can be built up in the equational theory EqRA of relation algebras (RA’s), hence EqRA is undecidable. He raised the problem "how much associativity of relation composition is needed for this result." Maddux defined the classes NA ⊃ WA ⊃ SA ⊃ RA \operatorname {NA} \supset \operatorname {WA} \supset \operatorname {SA} \supset \operatorname {RA} by gradually weakening the associativity of relation composition, and he proved that the equational theory of SA is still undecidable. We showed, elsewhere, that mathematics can be built up in SA, too. In the present paper we prove that the equational theories of WA and NA are already decidable. Hence mathematics cannot be built up in WA or NA. This solves a problem in the book by Tarski and Givant.

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“…(And (6) says the same about starting with a ≡ 1 -step, and then taking a ≡ 0 one.) Note that the axiomatisation given in [6] contains slightly complicated forms of (5) and (6). As it is shown by Venema [15], on the basis of (1), (2) and (4), the 'Henkin-axioms' are equivalent to (5) and (6).…”

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confidence: 95%

“…This view is implicit in the algebraisation of finite variable fragments using finite dimensional cylindric algebras [6], and is made explicit in [15,12].…”

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A Note on Axiomatisations of Two-Dimensional Modal Logics

Kurucz

2013

Logic and Its Applications

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Abstract. We analyse the role of the modal axiom corresponding to the first-order formula "∃y (x = y)" in axiomatisations of two-dimensional propositional modal logics.One of the several possible connections between propositional multi-modal logics and classical first-order logic is to consider finite variable fragments of the latter as 'multi-dimensional' modal formalisms: First-order variable-assignment tuples are regarded as possible worlds in Kripke frames, and each first-order quantification ∃v i and ∀v i as 'coordinate-wise' modal operators 3 i and 2 i in these frames. This view is implicit in the algebraisation of finite variable fragments using finite dimensional cylindric algebras [6], and is made explicit in [15,12].Here we look at axiomatisation questions for the two-dimensional case from this modal perspective. (For basic notions in modal logic and its Kripke semantics, consult e.g. [2,3].) We consider the propositional multi-modal language ML δ 2 having the usual Boolean operators, unary modalities 3 0 and 3 1 (and their duals 2 0 , 2 1 ), and a constant δ:Formulas of this language can be embedded into the two-variable fragment of first-order logic by mapping propositional variables to binary atoms P (v 0 , v 1 ) (with this fixed order of the two available variables), diamonds 3 i to quantification ∃v i , and the 'diagonal' constant δ to the equality atom v 0 = v 1 . Semantically, we look at first-order models as multimodal Kripke frames (fitting to the above language) of the formWe call frames of this kind square frames. The above embedding is validitypreserving in the sense that a modal ML δ 2 -formula ϕ is valid in all square frames iff its translation ϕ † is a first-order validity.

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Cylindric algebras

Cited by 94 publications

References 0 publications

An Application of Relation Algebra to Lexical Databases

An Application of Relation Algebra to Lexical Databases

Decidability of relation algebras with weakened associativity

A Note on Axiomatisations of Two-Dimensional Modal Logics

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