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Infinite sets that admit fast exhaustive search
 In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science
, 2007
"... Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive sea ..."
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Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Highertype computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.
EXHAUSTIBLE SETS IN HIGHERTYPE COMPUTATION
, 2008
"... 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 C ..."
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Cited by 13 (12 self)
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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.
A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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Cited by 13 (3 self)
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
On Noetherian Spaces
"... A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of wellquasi order, in the sense that an Alexandroffdiscrete space is Noetherian iff its specialization quasiordering is well. For more general spaces, this opens the way to verify ..."
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Cited by 10 (5 self)
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A topological space is Noetherian iff every open is compact. Our starting point is that this notion generalizes that of wellquasi order, in the sense that an Alexandroffdiscrete space is Noetherian iff its specialization quasiordering is well. For more general spaces, this opens the way to verifying infinite transition systems based on nonwell quasi ordered sets, but where the preimage operator satisfies an additional continuity assumption. The technical development rests heavily on techniques arising from topology and domain theory, including sobriety and the de Groot dual of a stably compact space. We show that the category Nthr of Noetherian spaces is finitely complete and finitely cocomplete. Finally, we note that if X is a Noetherian space, then the set of all (even infinite) subsets of X is again Noetherian, a result that fails for wellquasi orders. 1.
On the ubiquity of certain total type structures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2007
"... It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel co ..."
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Cited by 4 (2 self)
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It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of KleeneKreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often nontrivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.
Computability of continuous solutions of highertype equations
, 2009
"... Given a continuous functional f: X → Y and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene–Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion ..."
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Cited by 1 (1 self)
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Given a continuous functional f: X → Y and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene–Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion functional ∀X: 2 X → 2. We also establish a version of the above for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include functionals defined on compact spaces X of analytic functions. Our development includes a discussion of the generality of our constructions, bringing QCB spaces into the picture, in addition to general topological considerations. Keywords and phrases. Highertype computability, Kleene–Kreisel spaces of continuous functionals, exhaustible set, searchable set, QCB space, admissible representation, topology in the theory of computation with infinite objects. 1
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.
Two Probabilistic Powerdomains in Topological Domain Theory
"... We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given b ..."
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We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ωcontinuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domaintheoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countablybased spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.
Domain representations of spaces of compact subsets
, 2010
"... We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domai ..."
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We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.
Algorithmic solution of highertype equations
, 2011
"... In recent work we developed the notion of exhaustible set as a highertype computational counterpart of the topological notion of compact set. In this paper we give applications to the computation of solutions of highertype equations. Given a continuous functional f: X → Y and y ∈ Y, we wish to co ..."
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In recent work we developed the notion of exhaustible set as a highertype computational counterpart of the topological notion of compact set. In this paper we give applications to the computation of solutions of highertype equations. Given a continuous functional f: X → Y and y ∈ Y, we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene– Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustibility condition. We also establish a version of this for computational metric spaces X and Y, where is X computationally complete and has an exhaustible set of Kleene–Kreisel representatives. Examples of interest include evaluation functionals defined on compact spaces X of bounded sequences of Taylor coefficients with values on spaces Y of real analytic functions defined on a compact set. A corollary is that it is semidecidable whether a function defined on such a compact set fails to be analytic, and that the Taylor coefficients of an analytic function can be computed extensionally from the function. Keywords and phrases. Highertype computability, Kleene–Kreisel spaces of continuous functionals, exhaustible set, searchable set, computationally compact set, QCB space, admissible representation, topology in the theory of computation. 1