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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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Cited by 14 (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.
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 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 2 (0 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.
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
unknown title
"... Is there any cartesianclosed category of continuous domains that would be closed under Jones and Plotkin’s probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higherorder languages. We relax the question, and look for quasico ..."
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Cited by 1 (0 self)
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Is there any cartesianclosed category of continuous domains that would be closed under Jones and Plotkin’s probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higherorder languages. We relax the question, and look for quasicontinuous dcpos instead. We introduce a natural class of such quasicontinuous dcpos, the
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 (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.
in a highertype setting
"... 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 prog ..."
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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. 1
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
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.