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Generalized de Bruijn Words and the State Complexity of Conjugate Sets

  • Daniel Gabric
  • Štěpán Holub
  • Jeffrey ShallitEmail author
Conference paper
  • 154 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11612)

Abstract

We consider a certain natural generalization of de Bruijn words, and use it to compute the exact maximum state complexity for the language consisting of the conjugates of a single word. In other words, we determine the state complexity of cyclic shift on languages consisting of a single word.

Notes

Acknowledgments

We thank the anonymous referees for helpful comments and suggestions.

References

  1. 1.
    van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Circuits and trees in oriented linear graphs. Simon Stevin 28, 203–217 (1951)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anisiu, M., Blázsik, Z., Kása, Z.: Maximal complexity of finite words. Pure Math. Appl. 13, 39–48 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    de Bruijn, N.G.: A combinatorial problem. Proc. Konin. Neder. Akad. Wet. 49, 758–764 (1946)MathSciNetzbMATHGoogle Scholar
  4. 4.
    de Bruijn, N.G.: Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of \(2^n\) zeros and ones that show each \(n\)-letter word exactly once. Technical report 75-WSK-06, Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands (1975)Google Scholar
  5. 5.
    Brzozowski, J.A.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010).  https://doi.org/10.25596/jalc-2010-071MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Etzion, T.: An algorithm for generating shift-register cycles. Theoret. Comput. Sci. 44, 209–224 (1986).  https://doi.org/10.1016/0304-3975(86)90118-0MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Flaxman, A., Harrow, A.W., Sorkin, G.B.: Strings with maximally many distinct subsequences and substrings. Electron. J. Combin. 11(1), 8 (2004). http://www.combinatorics.org/Volume11/Abstracts/v11i1r8.htmlMathSciNetzbMATHGoogle Scholar
  8. 8.
    Flye Sainte-Marie, C.: Question 48. L’Intermédiaire Math. 1, 107–110 (1894)Google Scholar
  9. 9.
    Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24, 195–221 (1982).  https://doi.org/10.1137/1024041MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Good, I.J.: Normal recurring decimals. J. London Math. Soc. 21, 167–169 (1946)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hemmati, F., Costello Jr., D.J.: An algebraic construction for q-ary shift register sequences. IEEE Trans. Comput. 27(12), 1192–1195 (1978).  https://doi.org/10.1109/TC.1978.1675025MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iványi, A.: On the \(d\)-complexity of words. Ann. Univ. Sci. Budapest. Sect. Comput. 8, 69–90 (1987)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Jirásek, J., Jirásková, G.: Cyclic shift on prefix-free languages. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 246–257. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38536-0_22CrossRefGoogle Scholar
  14. 14.
    Jirásková, G., Okhotin, A.: State complexity of cyclic shift. RAIRO Theor. Inform. Appl. 42(2), 335–360 (2008).  https://doi.org/10.1051/ita:2007038MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lempel, A.: \(m\)-ary closed sequences. J. Combin. Theory 10, 253–258 (1971)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Martin, M.H.: A problem in arrangements. Bull. Am. Math. Soc. 40, 859–864 (1934)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194(6), 1266–1268 (1970). In Russian. English translation in Soviet Math. Doklady 11(5), 1373–1375 (1970)MathSciNetGoogle Scholar
  18. 18.
    Ralston, A.: De Bruijn sequences — a model example of the interaction of discrete mathematics and computer science. Math. Mag. 55, 131–143 (1982).  https://doi.org/10.2307/2690079MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shallit, J.: On the maximum number of distinct factors of a binary string. Graphs Combin. 9(2–4), 197–200 (1993).  https://doi.org/10.1007/BF02988306MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sloane, N.J.A. et al.: The on-line encyclopedia of integer sequences (2019). https://oeis.org

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  • Daniel Gabric
    • 1
  • Štěpán Holub
    • 2
  • Jeffrey Shallit
    • 1
    Email author
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic

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