Analogues of Chaitinʼs Omega in the computably enumerable sets☆
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.
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Cited by (10)
Kobayashi compressibility
2017, Theoretical Computer ScienceCitation Excerpt :This is an elegant characterization of the compressibility of c.e. sets. Other results on this topic, such as the work in [8,24,7,5], tend to focus on the Kolmogorov complexity of the initial segments of c.e. sets. In fact, Theorem 1.3 remains true if we add a third clause saying ‘every c.e. set X is computable by Ω with oracle-use bounded above by f’ where Ω is Chaitin's halting probability.
Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega
2016, Journal of Computer and System SciencesUniversal computably enumerable sets and initial segment prefix-free complexity
2013, Information and ComputationKolmogorov complexity and computably enumerable sets
2013, Annals of Pure and Applied LogicChaitin's ω as a Continuous Function
2020, Journal of Symbolic LogicAlgorithmic Randomness and Measures of Complexity
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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.