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Two of the central concepts for the study of degree structures in
computability theory are computably enumerable degrees and minimal
degrees. For strong notions of reducibility, such as
$m$-deducibility or truth table reducibility, it is possible for
computably enumerable degrees to be minimal. For weaker notions of
reducibility, such as weak truth table reducibility or Turing
reducibility, it is not possible to combine these properties in a
single degree. This book considers how minimal weak truth table
degrees interact with computably enumerable Turing degrees and
obtain three main results. First, there are sets with minimal weak
truth table degree which bound noncomputable computably enumerable
sets under Turing reducibility. Second, no set with computable
enumerable Turing degree can have minimal weak truth table degree.
Third, no $\Delta^0_2$ set which Turing bounds a promptly simple
set can have minimal weak truth table degree.
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