Stone Duality for Nominal Boolean Algebras with И

Gabbay, Murdoch J.; Litak, Tadeusz; Petrişan, Daniela

doi:10.1007/978-3-642-22944-2_14

Cited by 8 publications

(13 citation statements)

References 31 publications

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“…This property of N is one precise technical reason that the proofs of [GLP11] can be simpler than the proofs here.…”

Section: Maximal and Prime Filtersmentioning

confidence: 99%

Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

Gabbay

2017

Annals of Pure and Applied Logic

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Please cite this article in press as: M.J. Gabbay, M.J. Gabbay, Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness, Ann. Pure Appl. Logic (2016), http://dx.doi.org/10. 1016/j.apal.2016.10.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness AbstractWe give a semantics for the λ-calculus based on a topological duality theorem in nominal sets. A novel interpretation of λ is given in terms of adjoints, and λ-terms are interpreted absolutely as sets (no valuation is necessary).

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“…This property of N is one precise technical reason that the proofs of [GLP11] can be simpler than the proofs here.…”

Section: Maximal and Prime Filtersmentioning

confidence: 99%

Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

Gabbay

2017

Annals of Pure and Applied Logic

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“…We are aware of one other nominal representation theorem: a previous paper by the author with Litak and Petrişan [11]. This proved a Stone duality for Boolean algebras with (an axiomatisation of) the Gabbay-Pitts N-quantifier-N is defined in this paper in Definition 2.14.…”

Section: Discussionmentioning

confidence: 82%

“…The proofs of this paper follow the same general outline as those in [11], though they are significantly different because ∀ has more structure than N; notably we must account for substitution, which is not relevant in the case of N. This led us to notions such as sigma-and amgis-algebra, σ-topological space, ∀-Stone space, and so on. Still, at a sufficiently high level of abstraction the reader can think of this paper as " [11] plus σ and ∀".…”

Section: Discussionmentioning

confidence: 99%

“…Still, at a sufficiently high level of abstraction the reader can think of this paper as " [11] plus σ and ∀".…”

Section: Discussionmentioning

confidence: 99%

“…In [11] we treated the N-quantifier. In this paper we treated the universal quantification of first-order logic-if it was not surprising that the 'nominal' quantifier N has a 'nominal' representation, it is more striking that the 'traditional' quantifier ∀ has a similar 'nominal' representation.…”

Section: Discussionmentioning

confidence: 99%

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Stone duality for first-order logic: a nominal approach to logic and topology

Gabbay¹

EPiC Series in Computing

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What are variables, and what is universal quantification over a variable? Nominal sets are a notion of 'sets with names', and using equational axioms in nominal algebra these names can be given substitution and quantification actions. So we can axiomatise first-order logic as a nominal logical theory.We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement. Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quantification becomes an infinite sets intersection?Given answers to these questions, we can seek notions of topology. What is the general notion of topological space of which our sets representation of predicates makes predicates into 'open sets'; and what specific class of topological spaces corresponds to the image of nominal algebras for first-order logic?The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces. Nominal algebra lets us extend Boolean algebras to 'FOL-algebras', and nominal sets let us correspondingly extend Stone spaces to '∀-Stone spaces'. These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets.

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PNL to HOL: From the logic of nominal sets to the logic of higher-order functions

Dowek

Gabbay

2012

Theoretical Computer Science

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Stone Duality for Nominal Boolean Algebras with И

Cited by 8 publications

References 31 publications

Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness

Stone duality for first-order logic: a nominal approach to logic and topology

PNL to HOL: From the logic of nominal sets to the logic of higher-order functions

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