3,124 research outputs found
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
Enumeration 1-Genericity in the Local Enumeration Degrees.
We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a'=b', but also that, if a is nonzero, then we can find such b satisfying 0e<b<a. We conclude by proving the existence of both a nonzero low and a properly Σ02 nonsplittable enumeration 1-generic degree, hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees
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
Turing Degrees and Randomness for Continuous Measures
We study degree-theoretic properties of reals that are not random with
respect to any continuous probability measure (NCR). To this end, we introduce
a family of generalized Hausdorff measures based on the iterates of the
"dissipation" function of a continuous measure and study the effective nullsets
given by the corresponding Solovay tests. We introduce two constructions that
preserve non-randomness with respect to a given continuous measure. This
enables us to prove the existence of NCR reals in a number of Turing degrees.
In particular, we show that every -degree contains an NCR element.Comment: 22 page
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