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Combinatorial Complexity of Regular Languages

  • Arseny M. Shur
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)

Abstract

We study combinatorial complexity (or counting function) of regular languages, describing these functions in three ways. First, we classify all possible asymptotically tight upper bounds of these functions up to a multiplicative constant, relating each particular bound to certain parameters of recognizing automata. Second, we show that combinatorial complexity equals, up to an exponentially small term, to a function constructed from a finite number of polynomials and exponentials. Third, we describe oscillations of combinatorial complexity for factorial, prefix-closed, and arbitrary regular languages. Finally, we construct a fast algorithm for calculating the growth rate of complexity for regular languages, and apply this algorithm to approximate growth rates of complexity of power-free languages, improving all known upper bounds for growth rates of such languages.

Keywords

Complexity Function Combinatorial Complexity Regular Language Counting Function Factorial Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D’Alessandro, F., Intrigila, B., Varricchio, S.: On the structure of counting function of sparse context-free languages. Theor. Comp. Sci. 356, 104–117 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Balogh, J., Bollobas, B.: Hereditary properties of words. RAIRO – Inf. Theor. Appl. 39, 49–65 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Brandenburg, F.-J.: Uniformly growing k-th power free homomorphisms. Theor. Comput. Sci. 23, 69–82 (1983)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Carpi, A.: On the repetition threshold for large alphabets. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 226–237. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Cassaigne, J.: Special factors of sequences with linear subword complexity. In: Dassow, J., Rozenberg, G., Salomaa, A. (eds.) Developments in Language Theory, II, pp. 25–34. World Scientific, Singapore (1996)Google Scholar
  6. 6.
    Choffrut, C., Karhumäki, J.: Combinatorics of words, ch. 6. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of formal languages, vol. 1, pp. 329–438. Springer, Berlin (1997)Google Scholar
  7. 7.
    Chomsky, N., Miller, G.A.: Finite state languages. Inf. and Control 1(2), 91–112 (1958)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Crochemore, M., Mignosi, F., Restivo, A.: Automata and forbidden words. Inform. Processing Letters 67(3), 111–117 (1998)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. Theory and applications, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  10. 10.
    Dejean, F.: Sur un Theoreme de Thue. J. Comb. Theory, Ser. A 13(1), 90–99 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Edlin, A.: The number of binary cube-free words of length up to 47 and their numerical analysis. J. Diff. Eq. and Appl. 5, 153–154 (1999)MathSciNetGoogle Scholar
  12. 12.
    Ehrenfeucht, A., Rozenberg, G.: On subword complexities of homomorphic images of languages. RAIRO Inform. Theor. 16, 303–316 (1982)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Franklin, J.N.: Matrix theory. Prentice-Hall Inc., Englewood Cliffs NJ (1968)zbMATHGoogle Scholar
  14. 14.
    Gantmacher, F.R.: Application of the theory of matrices. Interscience, New York (1959)Google Scholar
  15. 15.
    Govorov, V.E.: Graded algebras. Mat. Zametki 12, 197–204 (1972) (Russian)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ibarra, O., Ravikumar, B.: On sparseness, ambiguity and other decision problems for acceptors and transducers. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 1986. LNCS, vol. 210, pp. 171–179. Springer, Heidelberg (1986)Google Scholar
  17. 17.
    Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105, 335–347 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Lothaire, M.: Combinatorics on words. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  19. 19.
    Milnor, J.: Growth of finitely generated solvable groups. J. Diff. Geom. 2, 447–450 (1968)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Morse, M., Hedlund, G.A.: Symbolic dynamics. Amer. J. Math. 60, 815–866 (1938)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ochem, P., Reix, T.: Upper bound on the number of ternary square-free words. In: Electronic proceedings of Workshop on words and automata (WOWA 2006), S.-Petersburg, p. 8 (2006)Google Scholar
  22. 22.
    Richard, C., Grimm, U.: On the entropy and letter frequencies of ternary square-free words. Electronic J. Combinatorics 11(1), p. 14 (2004)Google Scholar
  23. 23.
    Shur, A.M.: The structure of the set of cube-free Z-words in a two-letter alphabet. Izv. Ross. Akad. Nauk Ser. Mat. 64, 201–224 (2000) (Russian); English translation in Izv. Math. 64, 847–871 (2000)Google Scholar
  24. 24.
    Shur, A.M.: Combinatorial complexity of rational languages. Discr. Anal. and Oper. Research, Ser. 1, 12(2), 78–99 (2005)MathSciNetGoogle Scholar
  25. 25.
    Shur, A.M.: Rational approximations of polynomial factorial languages. Int. J. Found. Comput. Sci. 18, 655–665 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Thue, A.: Über unendliche Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 7, 1–22 (1906)Google Scholar
  27. 27.
    Trofimov, V.I.: Growth functions of some classes of languages. Cybernetics 6, 9–12 (1981) (Russian)MathSciNetGoogle Scholar
  28. 28.
    Trofimov, V.I.: Growth functions of finitely generated semigroups. Semigroup Forum 21, 351–360 (1980)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Ufnarovsky, V.A.: On the growth of algebras. Proceedings of Moscow University 1(4), 59–65 (1978) (Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Arseny M. Shur
    • 1
  1. 1.Ural State University 

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