Analogues of Chaitinʼs Omega in the computably enumerable sets

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Abstract

We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefix-free version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete c.e. set has maximum initial segment Kolmogorov complexity. Moreover, every c.e. set with maximum initial segment prefix-free complexity is the disjoint union of two c.e. sets with the same property; and is also the disjoint union of two c.e. sets of lesser initial segment complexity.

Highlights

► There are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets. ► These complete sets are characterized in the case of one of the measures of initial segment complexity. ► Every c.e. set can be split into two c.e. disjoint parts of the same prefix-free complexity but this fails for plain complexity.

References (27)

  • Rod G. Downey et al.

    Randomness and reducibility

    J. Comput. System Sci.

    (2004)
  • Wolfgang Merkle et al.

    On C-degrees, H-degrees and T-degrees

  • Klaus Ambos-Spies

    Anti-mitotic recursively enumerable sets

    Math. Log. Q.

    (1985)
  • Janis M. Barzdin

    Complexity of programs to determine whether natural numbers not greater than n belong to recursively enumerable set

    Soviet Math. Dokl.

    (1968)
  • George Barmpalias

    Computably enumerable sets in the Solovay and the strong weak truth table degrees

  • George Barmpalias

    On strings with trivial Kolmogorov complexity

    Int. J. Softw. Inform.

    (2011)
  • George Barmpalias, Universal computably enumerable sets and initial segment prefix-free complexity, in...
  • George Barmpalias, Angsheng Li, Kolmogorov complexity and computably enumerable sets, submitted for...
  • Gregory J. Chaitin

    A theory of program size formally identical to information theory

    J. Assoc. Comput. Mach.

    (1975)
  • Cristian Calude et al.

    Recursively enumerable reals and Chaitin Ω numbers

    Theoret. Comput. Sci.

    (2001)
  • Rod Downey et al.

    Algorithmic Randomness and Complexity

    (2010)
  • Rod Downey et al.

    Randomness, computability and density

    SIAM J. Comput.

    (2002)
  • Rod Downey et al.

    Splitting theorems in recursion theory

    Ann. Pure Appl. Logic

    (1993)
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    This research was partially done during the Dagstuhl Seminar on Computability, Complexity and Randomness (12021) and the programme ‘Semantics and Syntax’ in the Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK. Barmpalias was supported by the Research Fund for International Young Scientists number 611501-10168 from the National Natural Science Foundation of China, and an International Young Scientist Fellowship number 2010-Y2GB03 from the Chinese Academy of Sciences; partial support was also received from the project Network Algorithms and Digital Information number ISCAS2010-01 from the Institute of Software, Chinese Academy of Sciences. Hölzl was supported by a Feodor Lynen postdoctoral research fellowship from the Alexander von Humboldt Foundation. Lewis was supported by a Royal Society University Research Fellowship. This work was partially supported by Bulgarian National Science Fund under contract D002-258/18.12.08. We are indebted to the referees, and in particular one of the referees who suggested Theorem 2.7 and simplified the proof of Theorem 2.4.

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