530 research outputs found

    Minimal bounds and members of effectively closed sets

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    We show that there exists a non-empty Π10\Pi^0_1 class, with no recursive element, in which no member is a minimal cover for any Turing degree.Comment: 15 pages, 4 figures, 1 acknowledgemen

    Computability Theory

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    Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

    Finding subsets of positive measure

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    An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero ss-dimensional Hausdorff measure Hs\mathcal H^s contains a closed subset of non-zero (and indeed finite) Hs\mathcal H^s-measure. We investigate the question how hard it is to find such a set, in terms of the index set complexity, and in terms of the complexity of the parameter needed to define such a closed set. Among other results, we show that given a (lightface) Σ11\Sigma^1_1 set of reals in Cantor space, there is always a Π10(O)\Pi^0_1(\mathcal{O}) subset on non-zero Hs\mathcal H^s-measure definable from Kleene's O\mathcal O. On the other hand, there are Π20\Pi^0_2 sets of reals where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The Strength of the Besicovitch--Davies Theorem". The final publication of that paper is available at Springer via http://dx.doi.org/10.1007/978-3-642-13962-8_2

    Diagonalizations over polynomial time computable sets

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    AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP

    First Order Theories of Some Lattices of Open Sets

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    We show that the first order theory of the lattice of open sets in some natural topological spaces is mm-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., Rn\mathbb{R}^n, n≥1n\geq1, and the domain PωP\omega) this theory is mm-equivalent to first order arithmetic

    Reverse mathematics of compact countable second-countable spaces

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    We study the reverse mathematics of the theory of countable second-countable topological spaces, with a focus on compactness. We show that the general theory of such spaces works as expected in the subsystem ACA0\mathsf{ACA}_0 of second-order arithmetic, but we find that many unexpected pathologies can occur in weaker subsystems. In particular, we show that RCA0\mathsf{RCA}_0 does not prove that every compact discrete countable second-countable space is finite and that RCA0\mathsf{RCA}_0 does not prove that the product of two compact countable second-countable spaces is compact. To circumvent these pathologies, we introduce strengthened forms of compactness, discreteness, and Hausdorffness which are better behaved in subsystems of second-order arithmetic weaker than ACA0\mathsf{ACA}_0
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