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This volume results from two programs that took place at the
Institute for Mathematical Sciences at the National University of
Singapore: Aspects of Computation — in Celebration of the
Research Work of Professor Rod Downey (21 August to 15 September
2017) and Automata Theory and Applications: Games, Learning and
Structures (20-24 September 2021).The first program was dedicated
to the research work of Rodney G. Downey, in celebration of his
60th birthday. The second program covered automata theory whereby
researchers investigate the other end of computation, namely the
computation with finite automata, and the intermediate level of
languages in the Chomsky hierarchy (like context-free and
context-sensitive languages).This volume contains 17 contributions
reflecting the current state-of-art in the fields of the two
programs.
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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