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25
Geometric and higher order logic in terms of abstract Stone duality
 THEORY AND APPLICATIONS OF CATEGORIES
, 2000
"... The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this ..."
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The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.
Lifting as a KZdoctrine
 Proceedings of the 6 th International Conference, CTCS'95, volume 953 of Lecture Notes in Computer Science
, 1995
"... this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in ..."
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this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of nonordertheoretic axioms making the specialisation preorder !complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passagetothelimit operator appropriate for fixedpoint computations; stronger axioms make this operator be given by lubs of !chains
Relational parametricity for control considered as a computational effect
 Electr. Notes Theor. Comput. Sci
"... Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be ..."
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Replace this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be
An Elementary Theory of the Category of Locally Compact Locales
, 2003
"... The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a selfadjoint exponential # , and Scott continuity, F# = ##. ..."
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The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a selfadjoint exponential # , and Scott continuity, F# = ##.
Fundamentals of Object Oriented Database Modelling
, 1996
"... Solid theoretical foundations of object oriented databases (OODBs) are still missing. The work reported in this paper contains results on a formally founded object oriented datamodel (OODM) and is intended to contribute to the development of a uniform mathematical theory of OODBs. A clear distinctio ..."
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Solid theoretical foundations of object oriented databases (OODBs) are still missing. The work reported in this paper contains results on a formally founded object oriented datamodel (OODM) and is intended to contribute to the development of a uniform mathematical theory of OODBs. A clear distinction between objects and values turns out to be essential in the OODM. Types and classes are used to structure values and objects repectively. This can be founded on top of any underlying type system. We outline different approaches to type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topostheoretically defined semantics. On this basis the known solutions to the problems of unique object identification and genericity can be generalized. It turns out that extents of classes must be completely representable by values. Such classes are called valuerepresentable. As a consequence object identifiers degenerate to a pure...
The Type Concept in OODB Modelling and its Logical Implications
, 2000
"... Conceptual modelling requires a solid mathematical theory of concepts concerning the collection of concepts used in a specific, but broad enough field. The field considered in this paper is database modelling. Here object orientation in the widest sense has been identified as a unifying conceptual u ..."
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Conceptual modelling requires a solid mathematical theory of concepts concerning the collection of concepts used in a specific, but broad enough field. The field considered in this paper is database modelling. Here object orientation in the widest sense has been identified as a unifying conceptual umbrella that encompasses all relevant datamodels. The theory of object oriented databases has brought to light the fundamental distinction between the concepts of objects and values and correspondingly types and classes. This can be founded on top of any underlying type system. Thus, expressiveness of a datamodel basically depends on the type concept, from which the other concepts can be derived. In order to achieve a uniform mathematical theory we outline different type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topostheoretically defined semantics. On this basis the known solutions to the problems of unique ...
Inductive Construction of Repletion
 Appl. Categ. Structures
, 1997
"... Introduction In [2] Martin Hyland has proposed the notion of "Sreplete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains an ..."
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Introduction In [2] Martin Hyland has proposed the notion of "Sreplete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains and thus about meanings of functional programs where all functions between predomains are continuous and all endofunctions on domains have least fixpoints. Furthermore, SDT allows one to consider predomains as a full reflective subcategory of the ambient category of constructive sets. For this reason the ambient category is assumed to be a model of some sufficiently strong impredicative constructive type theory which will be used as the internal language for the ambient category of constructive sets in order to replace complicated external arguments by simpler proofs in the
Computability structures, simulations and realizability
, 2011
"... We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of si ..."
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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a very wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We also discuss a general notion of simulation between such computability structures, and show that such simulations correspond precisely to certain functors between the realizability models. Furthermore, we show that our class of computability structures has good closure properties — in particular, it is ‘cartesian closed ’ in a slightly relaxed sense. We also investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2category of computability structures and simulations may offer a framework for a general investigation of questions of computational power, abstraction and simulability for a wide range of computation models from across computer science.
A Note on SDT in Filter Spaces
, 1993
"... Introduction This work was motivated by the desire to integrate Evaluation Logic EL T (see [Mog93]) and Synthetic Domain Theory SDT (see [Hyl91]). A rst step in this direction was to consider several computational monads over the category Cpo of predomains, and check whether they preserve regular m ..."
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Introduction This work was motivated by the desire to integrate Evaluation Logic EL T (see [Mog93]) and Synthetic Domain Theory SDT (see [Hyl91]). A rst step in this direction was to consider several computational monads over the category Cpo of predomains, and check whether they preserve regular monos { in Cpo regular monos correspond to inductive subsets. The dominion of regular monos (in a left exact category) is the simplest candidate for a LCFlike logic, where assertions are Horn sequents with equalities among the atomic formulas. Although, most computational monads over Cpo are unproblematic, there are two important exceptions: continuations ( X ) , where is the cpo classifying open subsets Plotkin's powerdomain P p (X? ), where P p (X) is
Program Verification in Synthetic Domain Theory
, 1995
"... Synthetic Domain Theory provides a setting to consider domains as sets with certain closure properties for computing suprema of ascending chains. As a consequence the notion of domain can be internalized which allows one to construct and reason about solutions of recursive domain equations. Moreover ..."
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Synthetic Domain Theory provides a setting to consider domains as sets with certain closure properties for computing suprema of ascending chains. As a consequence the notion of domain can be internalized which allows one to construct and reason about solutions of recursive domain equations. Moreover, one can derive that all functions are continuous. In this thesis such a synthetic theory of domains (#domains) is developed based on a few axioms formulated in an adequate intuitionistic higherorder logic. This leads to an elegant theory of domains. It integrates the positive features of several approaches in the literature. In contrast to those, however, it is model independent and can therefore be formalized. A complete formalization of the whole theory of #domains has been coded into a proofchecker (Lego) for impredicative type theory. There one can exploit dependent types in order to express program modules and modular specifications. As an application of this theory an entirely fo...