Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
Jonathan Stephenson, Department of Pure Mathematics, University of Waterloo
“Bases for Randomness”
We often regard the computational power of a set A as being synonymous with the class of sets A is able to compute. However, we might instead ask which sets are able to compute A. The class of such sets is called the cone above A. When the cone above A consists of few sets, we may think of A as being difficult to compute. We wish to make the term ”few sets” precise.
We will see that if A is incomputable, the class of sets which can compute A is of measure zero, so regarding measure zero as a condition indicating a class contains ”few sets” does not allow us to make a significant distinction in terms of computational difficulty.
We will say that A is a base for MLrandomness if the cone above A contains a set which is MLrandom relative to A. Those sets which are not bases for MLrandomness may be regarded as difficult to compute. This provides a better notion than measure zero, because any set which is low for MLrandomness is a base for MLrandomness, but that there are many sets which are not bases for MLrandomness.
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Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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