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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 17 (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.
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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Cited by 15 (8 self)
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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.
A rich hierarchy of functionals of finite types (manuscript
, 2008
"... Abstract. We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the socalled Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric ..."
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Cited by 5 (1 self)
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Abstract. We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the socalled Urysohn space constructed by P. Urysohn. One of the properties of the Urysohn space is that every other separable metric space can be isometrically embedded into it. We discuss why the Urysohn space may be considered as the universal model of possibly infinitary outputs of algorithms. The main result is that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over the Urysohn space. As a preparation for this, we prove an effective density theorem that is also of independent interest. 1.
An Effective TietzeUrysohn Theorem for QCBSpaces
 Journal of Universal Computer Science, Vol
, 2009
"... Abstract: The TietzeUrysohn Theorem states that every continuous realvalued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we in ..."
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Cited by 3 (0 self)
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Abstract: The TietzeUrysohn Theorem states that every continuous realvalued function defined on a closed subspace of a normal space can be extended to a continuous function on the whole space. We prove an effective version of this theorem in the Type Two Model of Effectivity (TTE). Moreover, we introduce for qcbspaces a slightly weaker notion of normality than the classical one and show that this property suffices to establish an Extension Theorem for continuous functions defined on functionally closed subspaces. Qcbspaces are known to form an important subcategory of the category Top of topological spaces. QCB is cartesian closed in contrast to Top.
Partial Combinatory Algebras of Functions
, 2009
"... We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative st ..."
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We employ the notions of ‘sequential function ’ and ‘interrogation ’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorderenriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is, that every realizability topos is a quotient of a realizability topos on a total combinatory algebra. AMS Subject Classification (2000): 03B40,68N18
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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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
Acknowledgments I am grateful for discussions with (in alphabetic order) Philipp Gerhardy,
, 2008
"... We are considering typed hierarchies of total, continuous functionals over base types that are complete, separable metric spaces. P. Urysohn [17, 18] constructed a complete, separable metric space U. One of the properties of U is that every other separable metric space can be isometrically embedded ..."
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We are considering typed hierarchies of total, continuous functionals over base types that are complete, separable metric spaces. P. Urysohn [17, 18] constructed a complete, separable metric space U. One of the properties of U is that every other separable metric space can be isometrically embedded into U. We discuss why U may be considered as the universal model of possibly infintary outputs of algorithms, and show that all our typed hierarchies may be topologically embedded, type by type, into the corresponding hierarchy over U. Restricting our base types to effective, separable Banach spaces, we also prove a density theorem and an effective embedding theorem. These are our main technical results.
A Note on Closed Subsets in Quasizerodimensional Qcbspaces
"... Abstract: We introduce the notion of quasizerodimensionality as a substitute for the notion of zerodimensionality, motivated by the fact that the latter behaves badly in the realm of qcbspaces. We prove that the category QZ of quasizerodimensional qcb0spaces is cartesian closed. Prominent exa ..."
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Abstract: We introduce the notion of quasizerodimensionality as a substitute for the notion of zerodimensionality, motivated by the fact that the latter behaves badly in the realm of qcbspaces. We prove that the category QZ of quasizerodimensional qcb0spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces of the KleeneKreisel continuous functionals equipped with the respective sequential topology. Moreover, we characterise some types of closed subsets of QZspaces in terms of their ability to allow extendability of continuous functions. These results are related to a problem in Computable Analysis.
A Note on Closed Subsets in Quasizerodimensional Qcbspaces (Extended Abstract)
"... Abstract. We introduce the notion of quasizerodimensionality as a substitute for the notion of zerodimensionality, motivated by the fact that the latter behaves badly in the realm of qcbspaces. We prove that the category QZ of quasizerodimensional qcb0spaces is cartesian closed. Prominent exa ..."
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Abstract. We introduce the notion of quasizerodimensionality as a substitute for the notion of zerodimensionality, motivated by the fact that the latter behaves badly in the realm of qcbspaces. We prove that the category QZ of quasizerodimensional qcb0spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces in the sequential hierarchy of the KleeneKreisel continuous functionals. Moreover, we characterise some types of closed subsets of QZspaces in terms of their ability to allow extendability of continuous functions. These results are related to an open problem in Computable Analysis.
Semidecidability of may, must . . . in a highertype setting
, 2009
"... We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic pro ..."
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We show that, in a fairly general setting including highertypes, may, must and probabilistic testing are semidecidable. The case of must testing is perhaps surprising, as its mathematical definition involves universal quantification over the infinity of possible outcomes of a nondeterministic program. The other two involve existential quantification and integration. We also perform first steps towards the semidecidability of similar tests under the simultaneous presence of nondeterministic and probabilistic choice. Keywords: Nondeterministic and probabilistic computation, highertype computability theory and exhaustible sets, may and must testing, operational and denotational semantics, powerdomains.