Degree-Theoretic Aspects of Computably Enumerable Reals
نویسندگان
چکیده
A real is computable if its left cut, L( ); is computable. If (qi)i is a computable sequence of rationals computably converging to ; then fqig; the corresponding set, is always computable. A computably enumerable (c.e.) real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to : For example, every representation A of is Turing reducible to L( ): Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below degT L( ) necessarily contains a representation of :
منابع مشابه
Universal computably enumerable sets and initial segment prefix-free complexity
We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that s...
متن کاملComputably Enumerable Reals and Uniformly Presentable Ideals
We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that α = ∑ σ∈A 2 −|σ|. Note that ∑ σ∈A 2 −|σ| is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate ∑ σ∈A 2 −|σ| by μ(A). It is know...
متن کاملPresentations of computably enumerable reals
We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.
متن کاملStrong Jump-traceability I : the Computably Enumerable Case
Recent investigations in algorithmic randomness have lead to the discovery and analysis of the fundamental class K of reals called the K-trivial reals, defined as those whose initial segment complexity is identical with that of the sequence of all 1’s. There remain many important open questions concerning this class, such as whether there is a combinatorial characterization of the class and whe...
متن کاملThe Computably Enumerable Sets: Recent Results and Future Directions
We survey some of the recent results on the structure of the computably enumerable (c.e.) sets under inclusion. Our main interest is on collections of c.e. sets which are closed under automorphic images, such as the orbit of a c.e. set, and their (Turing) degree theoretic and dynamic properties. We take an algebraic viewpoint rather than the traditional dynamic viewpoint.
متن کامل