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
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)
Balogh, J., Bollobas, B.: Hereditary properties of words. RAIRO – Inf. Theor. Appl. 39, 49–65 (2005)
Brandenburg, F.-J.: Uniformly growing k-th power free homomorphisms. Theor. Comput. Sci. 23, 69–82 (1983)
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)
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)
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)
Chomsky, N., Miller, G.A.: Finite state languages. Inf. and Control 1(2), 91–112 (1958)
Crochemore, M., Mignosi, F., Restivo, A.: Automata and forbidden words. Inform. Processing Letters 67(3), 111–117 (1998)
Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. Theory and applications, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)
Dejean, F.: Sur un Theoreme de Thue. J. Comb. Theory, Ser. A 13(1), 90–99 (1972)
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)
Ehrenfeucht, A., Rozenberg, G.: On subword complexities of homomorphic images of languages. RAIRO Inform. Theor. 16, 303–316 (1982)
Franklin, J.N.: Matrix theory. Prentice-Hall Inc., Englewood Cliffs NJ (1968)
Gantmacher, F.R.: Application of the theory of matrices. Interscience, New York (1959)
Govorov, V.E.: Graded algebras. Mat. Zametki 12, 197–204 (1972) (Russian)
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)
Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Combin. Theory. Ser. A 105, 335–347 (2004)
Lothaire, M.: Combinatorics on words. Addison-Wesley, Reading (1983)
Milnor, J.: Growth of finitely generated solvable groups. J. Diff. Geom. 2, 447–450 (1968)
Morse, M., Hedlund, G.A.: Symbolic dynamics. Amer. J. Math. 60, 815–866 (1938)
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)
Richard, C., Grimm, U.: On the entropy and letter frequencies of ternary square-free words. Electronic J. Combinatorics 11(1), p. 14 (2004)
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)
Shur, A.M.: Combinatorial complexity of rational languages. Discr. Anal. and Oper. Research, Ser. 1, 12(2), 78–99 (2005)
Shur, A.M.: Rational approximations of polynomial factorial languages. Int. J. Found. Comput. Sci. 18, 655–665 (2007)
Thue, A.: Über unendliche Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl., Christiana 7, 1–22 (1906)
Trofimov, V.I.: Growth functions of some classes of languages. Cybernetics 6, 9–12 (1981) (Russian)
Trofimov, V.I.: Growth functions of finitely generated semigroups. Semigroup Forum 21, 351–360 (1980)
Ufnarovsky, V.A.: On the growth of algebras. Proceedings of Moscow University 1(4), 59–65 (1978) (Russian)
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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
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