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A lambda calculus for real analysis
, 2005
"... Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoni ..."
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Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f: R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers ” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by highertype operators � and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to
Towards a convenient category of topological domains
 Kyoto University
, 2003
"... We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recu ..."
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We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countablybased topological spaces. Its convenience is a consequence of a connection with realizability models.
On the Relationship between Filter Spaces and Equilogical Spaces
, 1998
"... It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic la ..."
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It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of filter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we first clarify the notion of filter spacethere are at least three versions FIL a ' FIL b ' FIL c in the literature. We establish adjunctions between FIL a and ASSM and between FIL c and ASSM, and show that FIL b and FIL c are equivalent to reflective full subcategories of ASSM. The corresponding categories FIL b 0 and FIL c 0 of T 0 spaces are equivalent to full subcategories of EQU. Keywords: Categorical models and logics, domain theory and applications Author's address: Reinhold Heckmann, FB 14  Informatik, Universitat des Saarlandes, Postfach 151150, D66041 Saarbrucken, Germany Phone: +49 681 302 2454 Fax: +49 681 302 3065 email: heckmann@cs.un...
What do Types Mean?  From Intrinsic to Extrinsic Semantics
, 2001
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their ..."
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic" if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings.
Realizability as the connection between computable and constructive mathematics
 Proceedings of CCA 2005
, 2005
"... These are lecture notes for a tutorial seminar which I gave at a satellite seminar of “Computability and Complexity in Analysis 2004 ” in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted ..."
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These are lecture notes for a tutorial seminar which I gave at a satellite seminar of “Computability and Complexity in Analysis 2004 ” in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted at an audience which is familiar with computable mathematics but
RZ: A tool for bringing constructive and computable mathematics closer to programming practice
 CiE 2007: Computation and Logic in the Real World, volume 4497 of LNCS
, 2007
"... Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Obje ..."
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Abstract. Realizability theory can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between constructive mathematics and programming by translating specifications in constructive logic into annotated interface code in Objective Caml. The system supports a rich input language allowing descriptions of complex mathematical structures. RZ does not extract code from proofs, but allows any implementation method, from handwritten code to code extracted from proofs by other tools. 1
An Elementary Theory of Various Categories of Spaces in Topology
, 2005
"... In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclid ..."
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In Abstract Stone Duality the topology on a space X is treated, not as an infinitary lattice, but as an exponential space ΣX. This has an associated lambda calculus, in which monadicity of the selfadjunction Σ − a Σ − makes all spaces sober and gives subspaces the subspace topology, and the Euclidean principle Fσ ∧ σ = F> ∧ σ makes Σ the classifier for open subspaces. Computably based locally compact locales provide the leading model for these axioms, although the methods are also applicable to CCDop (constructively completely distributive lattices). In this paper we recover the textbook theories, using the additional axiom that the subcategory of overt discrete objects have a coreflection, the “underlying set ” functor. This subcategory is then a topos, and the whole category is characterised in the minimal situation as that of locally compact locales over that topos. However, by adding further axioms regarding the existence of equalisers and injectivity of Σ, we find the category of sober spaces or of locales over the topos as a reflective subcategory, whilst the whole category is cartesian closed and has all finite limits and colimits.
A Constructive Model of Uniform Continuity
"... Abstract. We construct a continuous model of Gödel’s system T and its logic HA ω in which all functions from the Cantor space 2 N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in partic ..."
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Abstract. We construct a continuous model of Gödel’s system T and its logic HA ω in which all functions from the Cantor space 2 N to the natural numbers are uniformly continuous. Our development is constructive, and has been carried out in intensional type theory in Agda notation, so that, in particular, we can compute moduli of uniform continuity of Tdefinable functions 2 N → N. Moreover, the model has a continuous Fan functional of type (2 N → N) → N that calculates moduli of uniform continuity. We work with sheaves, and with a full subcategory of concrete sheaves that can be presented as sets with structure, which can be regarded as spaces, and whose natural transformations can be regarded as continuous maps.
Computability in Computational Geometry
"... Abstract. We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective lim ..."
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Abstract. We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative measure for the approximation. The domaintheoretic model of computational geometry provides such an object directed theory, which supports two such quantitative measures, one based on the Hausdorff metric and one on the Lebesgue measure. With respect to a new data type for the Euclidean space, given by its nonempty compact and convex subsets, we show that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable.
Sheaf Toposes for Realizability
, 2001
"... We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part o ..."
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We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1