Theory of Computing Systems

, Volume 52, Issue 1, pp 28–47 | Cite as

On the Gap Between Trivial and Nontrivial Initial Segment Prefix-Free Complexity

  • Martijn Baartse
  • George BarmpaliasEmail author


An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 n ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a \(\varDelta^{0}_{3}\) unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be \(\varDelta^{0}_{2}\) hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound \(\varDelta^{0}_{4}\) that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499–1502, 2006).

Our second result is that if f is \(\varDelta^{0}_{2}\) then there exists a non-empty \(\varPi^{0}_{1}\) class of reals X with nontrivial prefix-free complexity which satisfy (⋆). This implies that in this case there uncountably many nontrivial reals X satisfying (⋆) in various well known classes from computability theory and algorithmic randomness; for example low for Ω, non-low for Ω and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452–463, 2011).


Kolmogorov complexity Initial segment prefix-free complexity K-triviality Low for Ω 



Barmpalias was supported by a research fund for international young scientists No. 611501-10168 and an International Young Scientist Fellowship number 2010-Y2GB03 from the Chinese Academy of Sciences. Partial support was also obtained by the Grand project: Network Algorithms and Digital Information of the Institute of Software, Chinese Academy of Sciences.


  1. 1.
    Barmpalias, G.: Compactness arguments with effectively closed sets for the study of relative randomness. J. Log. Comput. (2010). doi: 10.1093/logcom/exq036 Google Scholar
  2. 2.
    Barmpalias, G.: Elementary differences between the degrees of unsolvability and the degrees of compressibility. Ann. Pure Appl. Log. 161(7), 923–934 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barmpalias, G.: Relative randomness and cardinality. Notre Dame J. Form. Log. 51(2), 195–205 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Barmpalias, G., Lewis, A.E.M., Stephan, F.: Π 0 1 classes, LR degrees and Turing degrees. Ann. Pure Appl. Log. 156(1), 21–38 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bienvenu, L., Merkle, W., Nies, A.: Solovay functions and K-triviality. In: Schwentick, T., Dürr, C. (eds.) STACS. LIPIcs, vol. 9, pp. 452–463. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2011) Google Scholar
  6. 6.
    Barmpalias, G., Sterkenburg, T.F.: On the number of infinite sequences with trivial initial segment complexity. Theor. Comput. Sci. 412(52), 7133–7146 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Barmpalias, G., Vlek, C.S.: Kolmogorov complexity of initial segments of sequences and arithmetical definability. Theor. Comput. Sci. 412(41), 5656–5667 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chaitin, G.J.: Information-theoretical characterizations of recursive infinite strings. Theor. Comput. Sci. 2, 45–48 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Csima, B.F., Montalbán, A.: A minimal pair of K-degrees. Proc. Am. Math. Soc. 134(5), 1499–1502 (2006) (electronic) zbMATHCrossRefGoogle Scholar
  10. 10.
    Downey, R., Greenberg, N.: Turing degrees of reals of positive packing dimension. Inf. Process. Lett. 108, 198–203 (2008) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Berlin (2010) zbMATHCrossRefGoogle Scholar
  12. 12.
    Downey, R., Hirschfeldt, D.R., Miller, J.S., Nies, A.: Relativizing Chaitin’s halting probability. J. Math. Log. 5(2), 167–192 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hirschfeldt, D.R., Nies, A., Stephan, F.: Using random sets as oracles. J. Lond. Math. Soc. 75(3), 610–622 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kučera, A., Slaman, T.: Turing incomparability in Scott sets. Proc. Am. Math. Soc. 135(11), 3723–3731 (2007) zbMATHCrossRefGoogle Scholar
  15. 15.
    Miller, J.S.: The K-degrees, low for K degrees, and weakly low for K sets. Notre Dame J. Form. Log. 50(4), 381–391 (2010) CrossRefGoogle Scholar
  16. 16.
    Nies, A.: Computability and Randomness. Oxford University Press, London (2009) zbMATHCrossRefGoogle Scholar
  17. 17.
    Reimann, J., Slaman, T.: Measures and their random reals. Submitted (2010) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Computer Science InstituteTechnical University CottbusCottbusGermany
  2. 2.State Key Lab of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingPeople’s Republic of China

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