Results 1  10
of
22
Domain Theory in Logical Form
 Annals of Pure and Applied Logic
, 1991
"... The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and system ..."
Abstract

Cited by 253 (10 self)
 Add to MetaCart
(Show Context)
The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:
Constructive points of Powerlocales
 Math. Proc. Cambridge Philos. Soc
, 1995
"... Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as s ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
(Show Context)
Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. We then also describe the points of the Vietoris powerlocale. In each of two special cases, an exponential $ D ($ being the Sierpinsky locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular. 1
The regularlocallycompact coreflection of stably locally compact locale
 Journal of Pure and Applied Algebra
, 2001
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally comp ..."
Abstract

Cited by 20 (8 self)
 Add to MetaCart
(Show Context)
The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,
Topical Categories of Domains
, 1997
"... this paper are algebraic dcpos, and many of the points discussed here will be needed later in the special case. 2 They provide a simple example to illustrate the "Display categories" in Section 3.2 ..."
Abstract

Cited by 19 (18 self)
 Add to MetaCart
this paper are algebraic dcpos, and many of the points discussed here will be needed later in the special case. 2 They provide a simple example to illustrate the "Display categories" in Section 3.2
Entailment Relations and Distributive Lattices
, 1998
"... . To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
. To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures. 1 Introduction Most spaces associated to mathematical structures: spectrum of a ring, space of valuations of a field, space of bounded linear functionals, . . . can be represented as distributive lattices. The key to have a natural definition in these cases is to use the notion of entailment relation due to Dana Scott. This note explains the connection between entailment relations and distributive lattices. An entailment relation may be seen as a logical description of a distributive lattice. Furthermore, most operations on distributive lattices are simpler when formulated as operations on entailment relations. A special kind of distribu...
Locales Are Not Pointless
 Theory and Formal Methods 1994: Proceedings of the Second Imperial College Department of Computing Workshop on Theory and Formal Methods, Mller
, 1994
"... The KripkeJoyal semantics is used to interpret the fragment of intuitionistic logic containing ; ! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be pr ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
The KripkeJoyal semantics is used to interpret the fragment of intuitionistic logic containing ; ! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems. Openness and properness of maps between locales are characterized by descriptions using the logic, and it is proved that a locale is open iff its lower powerlocale has a greatest point. The entire account is constructive and holds for locales over any topos. 1
Strongly complete logics for coalgebras
, 2006
"... Coalgebras for a functor T on a category X model many different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary strongly complete specification languages for Setbased coalgebras. We show how to associate a finitary logic to any finitesets preservin ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
Coalgebras for a functor T on a category X model many different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary strongly complete specification languages for Setbased coalgebras. We show how to associate a finitary logic to any finitesets preserving functor T and prove the logic to be strongly complete under a mild condition on T. The proof is based on the following result. An endofunctor on a variety has a presentation by operations and equations iff it preserves sifted colimits. 1
Computably based locally compact spaces
, 2003
"... ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalen ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
(Show Context)
ASD (Abstract Stone Duality) is a reaxiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambdacalculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth’s effectively given domains and Jung’s Strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the waybelow relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott’s domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.
Inside every model of Abstract Stone Duality lies an Arithmetic Universe
 Electronic Notes in Theoretical Computer Science 122
, 2005
"... The first paper published on Abstract Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an Nindexed least fixed poin ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
The first paper published on Abstract Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an Nindexed least fixed point axiom, here we show that this full subcategory is an arithmetic universe, having a free semilattice (“collection of Kuratowskifinite subsets”) and a free monoid (“collection of lists”) on any overt discrete object. Each finite subset is represented by its pair (, ♦) of modal operators, although a tight correspondence with these depends on a stronger Scottcontinuity axiom. Topologically, such subsets are both compact and open and also involve proper open maps. In applications of ASD this can eliminate lists in favour of a continuationpassing interpretation. Contents 6. Admissible implies modal 17 1. Introduction 1 7. K
Free modal algebras: a coalgebraic perspective
"... Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1