Lowness for Difference Tests

Abstract

We show that being low for difference tests is the same as being computable and therefore lowness for difference tests is not the same as lowness for difference randomness. This is the first known example of a randomness notion where lowness for the randomness notion and lowness for the test notion do not coincide. Additionally, we show that for every incomputable set $A$, there is a difference test ${\mathcal{T}}^A$ relative to $A$ which cannot even be covered by finitely many unrelativized difference tests.