Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Relatively recursive reals and real functions
Theoretical Computer Science - Special issue on real numbers and computers
Computable analysis: an introduction
Computable analysis: an introduction
Divergence bounded computable real numbers
Theoretical Computer Science - Real numbers and computers
A Note On the Turing Degrees of Divergence Bounded Computable Reals
Electronic Notes in Theoretical Computer Science (ENTCS)
On the divergence bounded computable real numbers
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
A computability theory of real numbers
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
On the turing degrees of divergence bounded computable reals
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
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Each real number x can be assigned a degree of unsolvability by using, for example, the degree of unsolvability of its binary or decimal expansion, or of its Dedekind cut, or of some other representation of x. We show that the degree of unsolvability assigned to x in any such way is the same regardless of the representation used. This gives to each real number a unique degree of unsolvability. If x is of computably enumerable degree, there is a computable sequence of rationals which converges to x with a modulus of convergence having the same degree of unsolvability as x itself. In contrast, if x is computable relative to the halting set but is not of computably enumerable degree, this is not true. Specifically, if {rn} is any computable sequence of rationals converging to such a real number x, the modulus of convergence of {rn} must have degree of unsolvability strictly higher than that of x. Thus there is an inherent gap between the degree of unsolvability of such an x and the degree of unsolvability of the modulus of convergence of an approximating computable sequence of rationals; this gap is bridged (in the sense of the "join operator" of degree theory) by a set of natural numbers which measures the twists and turns of the computable sequence {rn}.