## Growth rates of complexity of power-free languages.(English)Zbl 1196.68121

Summary: We present a new fast algorithm for calculating the growth rate of complexity for regular languages. Using this algorithm we develop a space and time efficient method to approximate growth rates of complexity of arbitrary power-free languages over finite alphabets. Through extensive computer-assisted studies we sufficiently improve all known upper bounds for growth rates of such languages, obtain a lot of new bounds and discover some general regularities.

### MSC:

 68Q45 Formal languages and automata
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### References:

 [1] Berstel, J.; Karhumäki, J., Combinatorics on words: a tutorial, Bull. eur. assoc. theoret. comput. sci., 79, 178-228, (2003) · Zbl 1169.68560 [2] Brandenburg, F.-J., Uniformly growing $$k$$-th power free homomorphisms, Theoret. comput. sci., 23, 69-82, (1983) · Zbl 0508.68051 [3] Carpi, A., On dejean’s conjecture over large alphabets, Theoret. comput. sci., 385, 137-151, (2007) · Zbl 1124.68087 [4] Crochemore, M.; Mignosi, F.; Restivo, A., Automata and forbidden words, Inform. process. lett., 67, 3, 111-117, (1998) · Zbl 1339.68145 [5] Cvetković, D.M.; Doob, M.; Sachs, H., Spectra of graphs. theory and applications, (1995), Johann Ambrosius Barth Heidelberg · Zbl 0824.05046 [6] Dejean, F., Sur un theoreme de thue, J. combin. theory ser. A, 13, 1, 90-99, (1972) · Zbl 0245.20052 [7] Edlin, A., The number of binary cube-free words of length up to 47 and their numerical analysis, J. difference equ. appl., 5, 153-154, (1999) · Zbl 0939.05007 [8] Franklin, J.N., Matrix theory, (1968), Prentice-Hall Inc. Englewood Cliffs, NJ · Zbl 0174.31501 [9] Gantmacher, F.R., Application of the theory of matrices, (1959), Interscience New York · Zbl 0085.01001 [10] Godsil, C.D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075 [11] Karhumäki, J.; Shallit, J., Polynomial versus exponential growth in repetition-free binary words, J. combin. theory. ser. A, 105, 335-347, (2004) · Zbl 1065.68080 [12] Kobayashi, Y., Repetition-free words, Theoret. comput. sci., 44, 175-197, (1986) · Zbl 0596.20058 [13] Kobayashi, Y., Enumeration of irreducible binary words, Discrete. appl. math., 20, 221-232, (1988) · Zbl 0673.68046 [14] R. Kolpakov, On the number of repetition-free words, in: Electronic Proceedings of Workshop on Words and Automata, WOWA’06, S.-Petersburg, 2006, #6. · Zbl 1249.68149 [15] Lothaire, M., Combinatorics on words, (1983), Addison-Wesley · Zbl 0514.20045 [16] P. Ochem, T. Reix, Upper bound on the number of ternary square-free words, in: Electronic Proceedings of Workshop on Words and Automata, WOWA’06, S.-Petersburg, 2006, #8. [17] Restivo, A.; Salemi, S., (), 196-206 [18] Richard, C.; Grimm, U., On the entropy and letter frequencies of ternary square-free words, Electron. J. combin., 11, 1, (2004), # R14 · Zbl 1104.68090 [19] 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 (2000), 847-871 · Zbl 0972.68131 [20] Shur, A.M., Combinatorial complexity of rational languages, Discr. anal. oper. research, ser. 1, 12, 2, 78-99, (2005), (Russian) · Zbl 1249.68107 [21] Shur, A.M., Comparing complexity functions of a language and its extendable part, RAIRO theor. inf. appl., 42, 647-655, (2008) · Zbl 1149.68055 [22] Shur, A.M., Calculating parameters and behavior types of combinatorial complexity for regular languages, Proc. inst. math. mech. UB RAS, 16, 2, 270-287, (2010) [23] Shur, A.M., (), 289-301 [24] Thue, A., Über unendliche zeichenreihen, Kra. vidensk. selsk. skrifter. I. mat.-nat. kl., christiana, 7, 1-22, (1906) · JFM 39.0283.01 [25] J.D. Currie, N. Rampersad, A proof of Dejean’s conjecture, http://arxiv.org/PScache/arxiv/pdf/0905/0905.1129v3.pdf. · Zbl 1215.68192 [26] M. Rao, Last Cases of Dejean’s conjecture, in: Proceedings of the 7th International Conference on Words, Salerno, Italy,-2009. #115. · Zbl 1230.68163
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