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An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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Cited by 17 (6 self)
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
General Synthetic Domain Theory  A Logical Approach
 Math. Struct. in Comp. Sci
, 1997
"... Synthetic Domain Theory (SDT) is a version of Domain Theory where "all functions are continuous". In [14, 12] there has been developed a logical and axiomatic version of SDT which is special in the sense that it captures the essence of Domain Theory `a la Scott but rules out other important noti ..."
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Cited by 10 (1 self)
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Synthetic Domain Theory (SDT) is a version of Domain Theory where "all functions are continuous". In [14, 12] there has been developed a logical and axiomatic version of SDT which is special in the sense that it captures the essence of Domain Theory `a la Scott but rules out other important notions of domain. In this article we will give a logical and axiomatic account of General Synthetic Domain Theory (GSDT) aiming to grasp the structure common to all notions of domain as advocated by various authors. As in [14, 12] the underlying logic is a sufficiently expressive version of constructive type theory. We start with a few basic axioms giving rise to a core theory on top of which we study various notions of predomains as wellcomplete and replete Sspaces [9], define the appropriate notion of domain and verify the usual induction principles. 1
Computational Adequacy in an Elementary Topos
 Proceedings CSL ’98, Springer LNCS 1584
, 1999
"... . We place simple axioms on an elementary topos which suffice for it to provide a denotational model of callbyvalue PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their settheoretic counterparts within the topos. The main result characterises whe ..."
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Cited by 9 (4 self)
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. We place simple axioms on an elementary topos which suffice for it to provide a denotational model of callbyvalue PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their settheoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1consistent (i.e. its internal logic validates only true \Sigma 0 1 sentences). 1 Introduction One axiomatic approach to domain theory is based on axiomatizing properties of the category of predomains (in which objects need not have a "least" element). Typically, such a category is assumed to be bicartesian closed (although it is not really necessary to require all exponentials) with natural numbers object, allowing the denotations of simple datatypes to be determined by universal properties. It is well known...
Axioms and (Counter)examples in Synthetic Domain Theory
 Annals of Pure and Applied Logic
, 1998
"... this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the p ..."
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Cited by 8 (7 self)
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this paper we adopt the most popular choice, the internal logic of an elementary topos (with nno), also chosen, e.g., in [23, 8, 26]. The principal benefits are that models of the logic (toposes) are ubiquitous, and the methods for constructing and analysing them are very wellestablished. For the purposes of the axiomatic part of this paper, we believe that it would also be
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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Cited by 7 (0 self)
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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.
Computer Theorem Proving in Math
"... We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. ..."
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Cited by 1 (0 self)
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We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context.
Domains in H
"... We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Int ..."
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We give various internal descriptions of the category !Cpo of !complete posets and !continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domainlike sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a settheoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...
Studying repleteness in the category of cpos
"... We consider the notion of replete object in the category of directed complete partial orders and Scottcontinuous functions, and show that, contrary to previous expectations, there are nonreplete objects. The same happens in the case of ωcomplete posets. Synthetic Domain Theory developed from an i ..."
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We consider the notion of replete object in the category of directed complete partial orders and Scottcontinuous functions, and show that, contrary to previous expectations, there are nonreplete objects. The same happens in the case of ωcomplete posets. Synthetic Domain Theory developed from an idea of Dana Scott: it is consistent with intuitionistic set theory that all functions between domains are continuous. He never wrote about this point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. [10,13,11,8,12], see also [14]. SDT can now be recognized as defining the “good properties ” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of sets. One of the problems addressed early in the theory was the identification of the sets to be considered as the Scott domains. As one would expect in a synthetic approach, the collection of these should be determined by the “good properties ” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from [6,15,5] and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that A is replete (wrt. Σ) if it is orthogonal to all f: X ✲ Y in 1
The King's Buildings Edinburgh EH9 3JZ, Scotland
"... We provide an internal characterization of the category!Cpo of!complete posets and!continuous functions within the model H of SDT recently introduced by the authors. It follows that!cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness. ..."
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We provide an internal characterization of the category!Cpo of!complete posets and!continuous functions within the model H of SDT recently introduced by the authors. It follows that!cpos lie between the two extreme synthetic notions of domain given by repleteness and wellcompleteness.