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Combinatorial Complexity of Regular Languages

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Computer Science – Theory and Applications (CSR 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5010))

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balogh, J., Bollobas, B.: Hereditary properties of words. RAIRO – Inf. Theor. Appl. 39, 49–65 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brandenburg, F.-J.: Uniformly growing k-th power free homomorphisms. Theor. Comput. Sci. 23, 69–82 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. Chomsky, N., Miller, G.A.: Finite state languages. Inf. and Control 1(2), 91–112 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  8. Crochemore, M., Mignosi, F., Restivo, A.: Automata and forbidden words. Inform. Processing Letters 67(3), 111–117 (1998)

    Article  MathSciNet  Google Scholar 

  9. Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. Theory and applications, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)

    MATH  Google Scholar 

  10. Dejean, F.: Sur un Theoreme de Thue. J. Comb. Theory, Ser. A 13(1), 90–99 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  12. Ehrenfeucht, A., Rozenberg, G.: On subword complexities of homomorphic images of languages. RAIRO Inform. Theor. 16, 303–316 (1982)

    MathSciNet  MATH  Google Scholar 

  13. Franklin, J.N.: Matrix theory. Prentice-Hall Inc., Englewood Cliffs NJ (1968)

    MATH  Google Scholar 

  14. Gantmacher, F.R.: Application of the theory of matrices. Interscience, New York (1959)

    Google Scholar 

  15. Govorov, V.E.: Graded algebras. Mat. Zametki 12, 197–204 (1972) (Russian)

    MathSciNet  MATH  Google Scholar 

  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. Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105, 335–347 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lothaire, M.: Combinatorics on words. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  19. Milnor, J.: Growth of finitely generated solvable groups. J. Diff. Geom. 2, 447–450 (1968)

    MathSciNet  MATH  Google Scholar 

  20. Morse, M., Hedlund, G.A.: Symbolic dynamics. Amer. J. Math. 60, 815–866 (1938)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. Shur, A.M.: Combinatorial complexity of rational languages. Discr. Anal. and Oper. Research, Ser. 1, 12(2), 78–99 (2005)

    MathSciNet  Google Scholar 

  25. Shur, A.M.: Rational approximations of polynomial factorial languages. Int. J. Found. Comput. Sci. 18, 655–665 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thue, A.: Über unendliche Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 7, 1–22 (1906)

    Google Scholar 

  27. Trofimov, V.I.: Growth functions of some classes of languages. Cybernetics 6, 9–12 (1981) (Russian)

    MathSciNet  Google Scholar 

  28. Trofimov, V.I.: Growth functions of finitely generated semigroups. Semigroup Forum 21, 351–360 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ufnarovsky, V.A.: On the growth of algebras. Proceedings of Moscow University 1(4), 59–65 (1978) (Russian)

    Google Scholar 

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Authors and Affiliations

  1. Ural State University,  

    Arseny M. Shur

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  1. Arseny M. Shur
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Editor information

Edward A. Hirsch Alexander A. Razborov Alexei Semenov Anatol Slissenko

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Shur, A.M. (2008). Combinatorial Complexity of Regular Languages. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_30

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  • DOI: https://doi.org/10.1007/978-3-540-79709-8_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79708-1

  • Online ISBN: 978-3-540-79709-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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