530 research outputs found
Minimal bounds and members of effectively closed sets
We show that there exists a non-empty 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
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
An important theorem of geometric measure theory (first proved by Besicovitch
and Davies for Euclidean space) says that every analytic set of non-zero
-dimensional Hausdorff measure contains a closed subset of
non-zero (and indeed finite) -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)
set of reals in Cantor space, there is always a
subset on non-zero -measure definable from
Kleene's . On the other hand, there are 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
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
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -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., , , and the
domain ) this theory is -equivalent to first order arithmetic
Reverse mathematics of compact countable second-countable spaces
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 of
second-order arithmetic, but we find that many unexpected pathologies can occur
in weaker subsystems. In particular, we show that does not
prove that every compact discrete countable second-countable space is finite
and that 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
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