A classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅'. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.
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References
K. Ambos-Spies, C. G. Jockusch, Jr., R. A. Shore, and R. I. Soare, An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), pp. 109--128.
Mathematical Reviews (MathSciNet):
MR719661
R. Downey, N. Greenberg, J. Miller, and R. Weber, Prompt simplicity, array computability and cupping, Computational prospects of infinity (Chong, Feng, Slaman, Woodin, and Yang, editors), Lecture Notes Series, vol. 15, World Scientific, Institute for Mathematical Sciences, National University of Singapore, 2008, pp. 59--78.
R. Downey, N. Greenberg, and R. Weber, Totally $\omega$-computably enumerable degrees I: bounding critical triples, Journal of Mathematical Logic, vol. 7 (2007), pp. 145--171.
R. Downey and D. R. Hirschfeldt, Algorithmic randomness and complexity, Springer-Verlag, Berlin,to appear.
R. Downey, D. R. Hirschfeldt, A. Nies, and S. A. Terwijn, Calibrating randomness, Bulletin of Symbolic Logic, vol. 12 (2006), pp. 411--491.
C. Jockusch, Jr. and R. I. Soare, $\Pi^0_1$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
Mathematical Reviews (MathSciNet):
MR316227
A. Kučera and T. A. Slaman, Low upper bounds of ideals, to appear.
Jeanleah Mohrherr, A refinement of low $n$ and high $n$ for the r.e. degrees, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 5--12.
Mathematical Reviews (MathSciNet):
MR832857
André Nies, Reals which compute little, Proceedings of logic colloquium 2002, Lecture Notes in Logic, vol. 27, 2002, pp. 261--275.
--------, Lowness properties and randomness, Advances in Mathenatics, vol. 197 (2005), pp. 274--305.
--------, Computability and randomness, Clarendon Press, Oxford,to appear.
Robert I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Heidelberg, 1987.
Mathematical Reviews (MathSciNet):
MR882921
--------, Computability theory and applications, Springer-Verlag, Heidelberg,to appear.
