13 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
Comparing the degrees of enumerability and the closed Medvedev degrees
We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees
Comparing the strength of diagonally non-recursive functions in the absence of Sigma^0_2 induction
We prove that the statement “there is a k such that for every f there is a k-bounded diagonally nonrecursive function relative to f” does not imply weak König’s lemma over . This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of
Comparing the strength of diagonally non-recursive functions in the absence of Σ02 induction
We prove that the statement there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f does not imply weak K\ onig\u27s lemma over RCA0+BΣ02. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally non-recursive function may fail in the absence of IΣ02
Degrees of members of ∏01 classes
Abstract unable to be displayed accurately. Please see eThesis for full detail
Kolmogorov complexity and the Recursion Theorem
Several classes of DNR functions are characterized in terms of Kolmogorov
complexity. In particular, a set of natural numbers A can wtt-compute a DNR
function iff there is a nontrivial recursive lower bound on the Kolmogorov
complexity of the initial segments of A. Furthermore, A can Turing compute a
DNR function iff there is a nontrivial A-recursive lower bound on the
Kolmogorov complexity of the initial segements of A. A is PA-complete, that is,
A can compute a {0,1}-valued DNR function, iff A can compute a function F such
that F(n) is a string of length n and maximal C-complexity among the strings of
length n. A solves the halting problem iff A can compute a function F such that
F(n) is a string of length n and maximal H-complexity among the strings of
length n. Further characterizations for these classes are given. The existence
of a DNR function in a Turing degree is equivalent to the failure of the
Recursion Theorem for this degree; thus the provided results characterize those
Turing degrees in terms of Kolmogorov complexity which do no longer permit the
usage of the Recursion Theorem.Comment: Full version of paper presented at STACS 2006, Lecture Notes in
Computer Science 3884 (2006), 149--16