Theory of Computing Systems

, Volume 62, Issue 7, pp 1555–1572 | Cite as

Finite-State Independence

  • Verónica Becher
  • Olivier Carton
  • Pablo Ariel Heiber


In this work we introduce a notion of independence based on finite-state automata: two infinite words are independent if no one helps to compress the other using one-to-one finite-state transducers with auxiliary input. We prove that, as expected, the set of independent pairs of infinite words has Lebesgue measure 1. We show that the join of two independent normal words is normal. However, the independence of two normal words is not guaranteed if we just require that their join is normal. To prove this we construct a normal word x 1 x 2 x 3… where x 2n = x n for every n. This construction has its own interest.


Finite-state automata Infinite sequences Normal sequences Independence 



The authors acknowledge Alexander Shen for many fruitful discussions. The authors are members of the Laboratoire International Associé INFINIS, CONICET/Universidad de Buenos Aires–CNRS/Université Paris Diderot. Becher is supported by the University of Buenos Aires and CONICET.


  1. 1.
    Bauwens, B., Shen, A., Takahashi, H.: Conditional probabilities and van Lambalgen theorem revisited. Submitted (2016)Google Scholar
  2. 2.
    Becher, V., Carton, O.: Normal numbers and computer science. In: Berthé, V., Rigó, M. (eds.) Sequences, Groups, and Number Theory, Trends in Mathematics Series. Birkhauser/Springer (2017)Google Scholar
  3. 3.
    Becher, V., Carton, O., Heiber, P.A.: Normality and automata. J. Comput. Syst. Sci. 81(8), 1592–1613 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Becher, V., Heiber, P.A.: Normal numbers and finite automata. Theor. Comput. Sci. 477, 109–116 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo 27, 247–271 (1909)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bugeaud, Y.: Distribution Modulo One and Diophantine Approximation. Series: Cambridge Tracts in Mathematics 193. Cambridge University Press (2012)Google Scholar
  7. 7.
    Calude, C.S., Zimand, M.: Algorithmically independent sequences. Inf. Comput. 208(3), 292–308 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carton, O., Heiber, P.A.: Normality and two-way automata. Inf. Comput. 241, 264–276 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dai, J., Lathrop, J., Lutz, J., Mayordomo, E.: Finite-state dimension. Theor. Comput. Sci. 310, 1–33 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Downey, R.G., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Theory and Applications of Computability, p. xxvi 855. Springer, New York, NY (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press (2008)Google Scholar
  12. 12.
    Huffman, D.: A method for the construction of minimum-redundancy codes. In: Institute of Radio Engineers, vol. 40:9, pp. 1098–1101 (1952)Google Scholar
  13. 13.
    Hyde, K., Kjos-Hanssen, B.: Nondeterministic automatic complexity of almost square-free and strongly cube-free words. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds.) Computing and Combinatorics: 20Th International Conference, COCOON 2014, Atlanta, GA, USA, August 4-6, 2014. Proceedings, pp. 61–70. Springer International Publishing, Cham (2014)Google Scholar
  14. 14.
    Kautz, S.: Degrees of Random Sets. PhD Thesis, Cornell University (1991)Google Scholar
  15. 15.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley-Interscience, New York (1974)zbMATHGoogle Scholar
  16. 16.
    Li, M., Vitanyi, P.: An Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer Publishing Company, Incorporated (2008)Google Scholar
  17. 17.
    Nies, A.: Computability and Randomness. Clarendon Press (2008)Google Scholar
  18. 18.
    Perrin, D., Pin, J.-É.: Infinite Words. Elsevier (2004)Google Scholar
  19. 19.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press (2009)Google Scholar
  20. 20.
    Schnorr, C.P., Stimm, H.: Endliche automaten und zufallsfolgen. Acta Informatica 1, 345–359 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shallit, J., Wang, M.: Automatic complexity of strings. J. Autom. Lang. Comb. 6(4), 537–554 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shen, A., Uspensky, V., Vereshchagin, N.: Kolmogorov complexity and algorithmic randomness. Submitted (2016)Google Scholar
  23. 23.
    van Lambalgen, M.: Random Sequences. PhD Thesis, University of Amsterdam (1987)Google Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Departamento de Computación, Facultad de Ciencias Exactas y Naturales & ICCUniversidad de Buenos Aires & CONICETBuenos AiresArgentina
  2. 2.Institut de Recherche en Informatique FondamentaleUniversité Paris DiderotParisFrance
  3. 3.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos Aires & CONICETBuenos AiresArgentina

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