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Maximal Pairs of Computably Enumerable Sets in the Computably Lipschitz Degrees

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Abstract

A set A is computably Lipschitz or cl-reducible, for short, to a set B if A is Turing reducible to B by an oracle Turing machine with use function ϕ such that ϕ is bounded by the identity function up to an additive constant, i.e., ϕ(n)≤n+O(1). In this paper we study maximal pairs of computably enumerable (c.e.) cl-degrees or maximal pairs, for short, i.e., pairs of c.e. cl-degrees such that there is no c.e. cl-degree that is above both cl-degrees in this pair. Our main results are as follows. (1) A c.e. Turing degree contains a c.e. cl-degree that is half of a maximal pair if and only if this Turing degree contains a maximal pair if and only if this Turing degree is array noncomputable. (2) The cl-degrees of all weak truth-table complete sets are halves of maximal pairs while there is a Turing complete set A such that the cl-degree of A is not half of any maximal pair. In fact, any high c.e. Turing degree contains a c.e. cl-degree that is not half of a maximal pair. (3) Above any c.e. cl-degree there is a maximal pair. (4) There is a maximal pair which at the same time is a minimal pair. (5) There is a pair of c.e. cl-degrees that is not maximal and does not possess a least upper bound.

Moreover, we make some observations on the structure of the c.e. cl-degrees in general. For instance, we give a very simple proof of the fact that there are no maximal c.e. cl-degrees.

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Acknowledgements

We are grateful to the anonymous referees of Theory of Computing Systems for their helpful comments and corrections.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Klaus Ambos-Spies & Wolfgang Merkle

  2. Institute of Mathematical Science, Nanjing University, Nanjing, China

    Decheng Ding

  3. Department of Mathematics, Southeast University, Nanjing, China

    Yun Fan

Authors
  1. Klaus Ambos-Spies
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  2. Decheng Ding
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  3. Yun Fan
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  4. Wolfgang Merkle
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Corresponding author

Correspondence to Klaus Ambos-Spies.

Additional information

The main results of this paper where obtained when the second and third authors visited the University of Heidelberg in the spring of 2008, partially supported by the Sino-German binational grant “Algorithmic Foundation of Numerical Computations” (DFG 446 CHV 113/240/0-1 and NSCF 10711130658). The fourth author has been partially supported by the DFG grant “Computable Randomness and Dimension” (ME 1806/3-1).

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Ambos-Spies, K., Ding, D., Fan, Y. et al. Maximal Pairs of Computably Enumerable Sets in the Computably Lipschitz Degrees. Theory Comput Syst 52, 2–27 (2013). https://doi.org/10.1007/s00224-012-9424-1

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