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Topological and Limitspace subcategories of Countablybased Equilogical Spaces
, 2001
"... this paper we show that the two approaches are equivalent for a ..."
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Cited by 22 (4 self)
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this paper we show that the two approaches are equivalent for a
Exhaustible sets in highertype computation
 Logical Methods in Computer Science
"... Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no examp ..."
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Cited by 13 (12 self)
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Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications. 1.
A proof of strong normalisation using domain theory
 In LICS’06
, 2006
"... U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show tha ..."
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U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and MartinLöf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for MartinLöf dependent type theory extended with bar recursion, and with some form of proofirrelevance. 1
Term rewriting for normalization by evaluation
 19–42, International Workshop on Implicit Computational Complexity (ICC’99
"... We extend normalization by evaluation ( rst presented in [5]) from the pure typedcalculus to general higher type term rewriting systems. We distinguish between computational rules and proper rewrite rules, and de ne a domain theoretic model intended to explain why normalization by evaluation for th ..."
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Cited by 12 (3 self)
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We extend normalization by evaluation ( rst presented in [5]) from the pure typedcalculus to general higher type term rewriting systems. We distinguish between computational rules and proper rewrite rules, and de ne a domain theoretic model intended to explain why normalization by evaluation for the former is much more e cient. Normalization by evaluation is proved to be correct w.r.t. this model. 1
Full Abstraction, Totality and PCF
 Math. Structures Comput. Sci
, 1997
"... ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The ..."
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Cited by 8 (1 self)
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ion, Totality and PCF Gordon Plotkin Abstract Inspired by a question of Riecke, we consider the interaction of totality and full abstraction, asking whether full abstraction holds for Scott's model of cpos and continuous functions if one restricts to total programs and total observations. The answer is negative, as there are distinct operational and denotational notions of totality. However, when two terms are each total in both senses then they are totally equivalent operationally iff they are totally equivalent in the Scott model. Analysing further, we consider sequential and parallel versions of PCF and several models: Scott's model of continuous functions, Milner's fully abstract model of PCF and their effective submodels. We investigate how totality differs between these models. Some apparently rather difficult open problems arise, essentially concerning whether the sequential and parallel versions of PCF have the same expressive power, in the sense of total equivale...
Normalization By Evaluation
 Prospects for Hardware Foundations, LNCS 1546
, 1998
"... . We extend normalization by evaluation (first presented in [4]) from the pure typed calculus to general higher type term rewrite systems. This work also gives a theoretical explanation of the normalization algorithm implemented in the Minlog system. 1 Introduction In interactive proof systems it ..."
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Cited by 8 (2 self)
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. We extend normalization by evaluation (first presented in [4]) from the pure typed calculus to general higher type term rewrite systems. This work also gives a theoretical explanation of the normalization algorithm implemented in the Minlog system. 1 Introduction In interactive proof systems it is crucial to have a term rewriting machinery available, in order to ease the burden of equational reasoning. Quite often term rewriting can be reduced to normalization and therefore it is essential to implement normalization of terms efficiently. By the same token, one then can also effectively normalize whole proofs (which can be written as derivation terms, using the CurryHoward correspondence). Normalization is used when extracting terms from formal proofs. For an application concerning circuits cf. [12]. It is well known that implementing normalization of terms in the usual recursive fashion is quite inefficient. However, it is possible to compute the long normal form of a term by ev...
Categories of Domains With Totality
 Preprint Series, Inst. Math. Univ. Oslo
, 2000
"... We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of ..."
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We investigate domains with totality where density in general does not hold. We define three categories of domains X with totality X satisfying certain structural properties. We then define the ordered set of evaluation structures. These will induce domains with totality. We show that the set of evaluation structures in a natural way is closed under dependent sums and products and under direct limits.
The Largest Topological Subcategory of Countablybased Equilogical Spaces
, 1998
"... There are two main approaches to obtaining "topological" cartesianclosed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed  for example, the category of sequential spaces. Under the other, one generalises the notion of s ..."
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Cited by 4 (1 self)
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There are two main approaches to obtaining "topological" cartesianclosed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed  for example, the category of sequential spaces. Under the other, one generalises the notion of space  for example, to Scott's notion of equilogical space. In this paper we show that the two approaches are equivalent for a large class of objects. We first observe that the category of countablybased equilogical spaces has, in a precisely defined sense, a largest full subcategory that can be simultaneously viewed as a full subcategory of topological spaces. This category consists of certain "!projecting" topological quotients of countablybased topological spaces, and contains, in particular, all countablybased spaces. We show that this category is cartesian closed with its structure inherited, on the one hand, from the category of sequential spaces, and, on the other, from the cate...
Limit Spaces and Transfinite Types
, 1998
"... We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types. ..."
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We give a characterisation of an extension of the KleeneKreisel continuous functionals to objects of transfinite types using limit spaces of transfinite types.
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 808
"... Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no examp ..."
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Abstract. We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela–Ascoli type characterization of compact subsets of function spaces. We also show that, in the nonempty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications. 1.