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Stone duality for FirstOrder Logic: a nominal approach
 In Howard Barringer Festschrift
, 2011
"... 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 firstorder logic as a nominal logical theor ..."
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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 firstorder 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 predicateinterpretedasaset, 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 firstorder logic? The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces. Nominal algebra lets us extend Boolean algebras to ‘FOLalgebras’, and nominal sets let us
Stone duality for nominal Boolean algebras with NEW
 In Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011), volume 6859 of Lecture Notes in Computer Science
, 2011
"... Abstract. We define Boolean algebras over nominal sets with a functionsymbol Nmirroring the N‘fresh name ’ quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality. 1 ..."
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Abstract. We define Boolean algebras over nominal sets with a functionsymbol Nmirroring the N‘fresh name ’ quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality. 1
Resource combinatory algebras
"... Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down fou ..."
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Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λcalculus. We also show that the ideal completion of a resource combinatory (resp. lambda, lambdaabstraction) algebra induces a “classical ” combinatory (resp. lambda, lambdaabstraction) algebra, and that any model of the classical λcalculus raising from a resource lambdaalgebra determines a λtheory which equates all terms having the same Böhm tree. 1
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"... Representation and duality of the untyped λcalculus in nominal lattice and topological semantics, with a proof of topological completeness ..."
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Representation and duality of the untyped λcalculus in nominal lattice and topological semantics, with a proof of topological completeness
Under consideration for publication in Math. Struct. in Comp. Science Imaginary groups: lazy monoids and reversible computation
, 2012
"... By constructions in monoid and group theory we exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms, and the category of partially ordered groups and group homomorphisms, such that the unit of the adjunction is injective. We also prove a similar resul ..."
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By constructions in monoid and group theory we exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms, and the category of partially ordered groups and group homomorphisms, such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of lazy homomorphism for a function f between partiallyordered monoids such that f (m ◦m′) ≤ f (m) ◦ f (m′). Every monoid can be endowed with the discrete partial ordering (m ≤ m ′ if and only if m = m′) so our constructions provide a way of embedding monoids into groups. A simple counterexample (the twoelement monoid with a nontrivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of homomorphism to this new notion of lazy homomorphism. The computational significance of this is that a monoid is an abstract model of computation—or at least of ‘operations’—and similarly a group models reversible
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"... Universal algebra over lambdaterms and nominal terms: the connection in logic between nominal techniques and higherorder variables ..."
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Universal algebra over lambdaterms and nominal terms: the connection in logic between nominal techniques and higherorder variables
ORDERED MODELS OF THE LAMBDA CALCULUS ∗
, 2013
"... Vol. 9(4:21)2013, pp. 1–29 www.lmcsonline.org ..."
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