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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 ..."
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Cited by 248 (10 self)
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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:
A Theory of Recursive Domains with Applications to Concurrency
 In Proc. of LICS ’98
, 1997
"... Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains. ..."
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Cited by 24 (14 self)
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Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains.
Constructive algebraic integration theory without choice”, in Mathematics, Algorithms and Proofs
 Dagstuhl Seminar Proceedings, 05021, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2005
"... Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpret ..."
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Cited by 8 (5 self)
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Abstract. We present a constructive algebraic integration theory. The theory is constructive in the sense of Bishop, however we avoid the axiom of countable, or dependent, choice. Thus our results can be interpreted in any topos. Since we avoid impredicative methods the results may also be interpreted in MartinL type theory or in a predicative topos in the sense of Moerdijk and Palmgren. We outline how to develop most of Bishop’s theorems on integration theory that do not mention points explicitly. Coquand’s constructive version of the Stone representation theorem is an important tool in this process. It is also used to give a new proof of Bishop’s spectral theorem.
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
Quantitative Concept Analysis
"... Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy ..."
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Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.
Towards Measurable Types for Dynamical Process Modeling Languages
, 2010
"... Process modeling languages such as “Dynamical Grammars ” are highly expressive in the processes they model using stochastic and deterministic dynamical systems, and can be given formal semantics in terms of an operator algebra. However such process languages may be more limited in the types of objec ..."
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Process modeling languages such as “Dynamical Grammars ” are highly expressive in the processes they model using stochastic and deterministic dynamical systems, and can be given formal semantics in terms of an operator algebra. However such process languages may be more limited in the types of objects whose dynamics is easily expressible. For many applications in biology, the dynamics of spatial objects in particular (including combinations of discrete and continuous spatial structures) should be formalizable at a high level of abstraction. We suggest that this may be achieved by formalizating such objects within a type system endowed with type constructors suitable for complex dynamical objects. To this end we review and illustrate the operator algebraic formulation of heterogeneous process modeling and semantics, extending it to encompass partial differential equations and intrinsic graph grammar dynamics. We show that in the operator approach to heterogeneous dynamics, types require integration measures. From this starting point, “measurable ” object types can be enriched with generalized metrics under which approximation can be defined. The resulting measurable and “metricated ” types can be built up systematically by type constructors such as vectors, products, and labelled graphs. We find conditions under which functions and quotients can be added as constructors of measurable and metricated types. 1 Measureable Types V25.nb 1
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"... Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra ∗ ..."
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Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra ∗
The connected Vietoris powerlocale
, 2008
"... The Vietoris powerlocale V X is a pointfree analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V c X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the cate ..."
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The Vietoris powerlocale V X is a pointfree analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V c X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the category of locales. The strength gives rise to a product map × : V c X × V c Y → V c (X × Y), showing that the product of two of these connected sublocales is again connected. If X is locally connected then V c X is overt. In the case where X is the localic completion Y of a generalized metric space Y, the points of V c Y are characterized as certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric. The results are applied to the particular case of the pointfree real line R, giving a choicefree constructive version of the Intermediate Value Theorem and Rolle’s Theorem. The work is constructive in the sense of toposvalidity with natural numbers object. Its geometric aspects (preserved under inverse image functors) are stressed, and exploited to give a pointwise development of the pointfree locale theory. The connected Vietoris powerlocale itself is a geometric construction. 1