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International Conference on Theory and Applications of Models of Computation

TAMC 2019: Theory and Applications of Models of Computation pp 448–466Cite as

Combinatorial Properties of Fibonacci Arrays

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11436))

Abstract

The non-trivial extension of Fibonacci words to Fibonacci arrays was proposed by Apostolico and Brimkov in order to study repetitions in arrays. In this paper we investigate several combinatorial as well as formal language theoretic properties of Fibonacci arrays. In particular, we show that the set of all Fibonacci arrays is a 2D primitive language (under certain conditions), count the number of borders in Fibonacci arrays, and show that the set of all Fibonacci arrays is a non-recognizable language. We also show that the set of all square Fibonacci arrays is a two dimensional code.

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References

  1. Alfred, S.P., Ingmar, L.: The Fabulous Fibonacci Numbers. Prometheus Books (2007)

    Google Scholar 

  2. Ali, D.: Twelve simple algorithms to compute Fibonacci numbers. CoRR. http://arxiv.org/abs/1803.07199 (2018)

  3. Amir, A., Benson, G.: Two-dimensional periodicity in rectangular arrays. SIAM J. Comput. 27(1), 90–106 (1998)

    Article  MathSciNet  Google Scholar 

  4. Anselmo, M., Giammarresi, D., Madonia, M.: Two dimensional prefix codes of pictures. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 46–57. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38771-5_6

    Chapter  MATH  Google Scholar 

  5. Anselmo, M., Giammarresi, D., Madonia, M.: Prefix picture codes: a decidable class of two-dimensional codes. Int. J. Found. Comput. Sci. 25(8), 1017–1031 (2014)

    Article  MathSciNet  Google Scholar 

  6. Anselmo, M., Madonia, M.: Two-dimensional comma-free and cylindric codes. Theor. Comput. Sci. 658, 4–17 (2017)

    Article  MathSciNet  Google Scholar 

  7. Apostolico, A., Brimkov, V.E.: Fibonacci arrays and their two-dimensional repetitions. Theor. Comput. Sci. 237(1–2), 263–273 (2000)

    Article  MathSciNet  Google Scholar 

  8. Berthé, V., Vuillon, L.: Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discret. Math. 223(1–3), 27–53 (2000)

    Article  MathSciNet  Google Scholar 

  9. Berthé, V., Vuillon, L.: Palindromes and two-dimensional Sturmian sequences. J. Autom. Lang. Comb. 6(2), 121–138 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Bozapalidis, S., Grammatikopoulou, A.: Picture codes. RAIRO-Theor. Inform. Appl. 40(4), 537–550 (2006)

    Article  MathSciNet  Google Scholar 

  11. Carlitz, L.: Fibonacci representations II. Fibonacci Q. 8, 113–134 (1970)

    MathSciNet  MATH  Google Scholar 

  12. Chang, C., Wang, H.: Comparison of two-dimensional string matching algorithms. In: 2012 International Conference on Computer Science and Electronics Engineering, vol. 3, pp. 608–611. IEEE (2012)

    Google Scholar 

  13. Dallapiccola, R., Gopinath, A., Stellacci, F., Dal, N.L.: Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles. Opt. Express 16(8), 5544–5555 (2008)

    Article  Google Scholar 

  14. Fu, K.S.: Syntactic Methods in Pattern Recognition, vol. 112. Elsevier, Amsterdam (1974)

    MATH  Google Scholar 

  15. Gamard, G., Richomme, G., Shallit, J., Smith, T.J.: Periodicity in rectangular arrays. Inf. Process. Lett. 118, 58–63 (2017)

    Article  MathSciNet  Google Scholar 

  16. Geizhals, S., Sokol, D.: Finding maximal 2-dimensional palindromes. In: The Proceedings of the 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, vol. 54, no. 19, pp. 1–12. Dagstuhl (2016)

    Google Scholar 

  17. Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 215–267. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59126-6_4

    Chapter  Google Scholar 

  18. Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation, vol. 24. Pearson (2006)

    Google Scholar 

  19. Kulkarni, M.S., Mahalingam, K.: Two-dimensional palindromes and their properties. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 155–167. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53733-7_11

    Chapter  MATH  Google Scholar 

  20. Lothaire, M.: Algebraic Combinatorics on Words, vol. 90. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  21. Luca, A.: A combinatorial property of the Fibonacci words. Inf. Process. Lett. 12(4), 193–195 (1981)

    Article  MathSciNet  Google Scholar 

  22. Margenstern, M.: An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl. Math. Comput. 218(16), 8005–8018 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Margenstern, M.: Fibonacci words, hyperbolic tilings and grossone. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 3–11 (2015)

    Article  MathSciNet  Google Scholar 

  24. Mignosi, F., Restivo, A., Silva, P.V.: On Fine and Wilf’s theorem for bidimensional words. Theor. Comput. Sci. 292(1), 245–262 (2003)

    Article  MathSciNet  Google Scholar 

  25. Minsky, M., Papert, S.: Perceptrons. The MIT Press, Cambridge (1969)

    MATH  Google Scholar 

  26. Pal, D., Masami, I.: Primitive words and palindromes. In: Context-Free Languages and Primitive Words, pp. 423–435. World Scientific (2014)

    Google Scholar 

  27. Yu, S.S., Zhao, Y.K.: Properties of Fibonacci languages. Discret. Math. 224(1–3), 215–223 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    Manasi S. Kulkarni, Kalpana Mahalingam & Sivasankar Mohankumar

Authors
  1. Manasi S. Kulkarni
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  2. Kalpana Mahalingam
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  3. Sivasankar Mohankumar
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Corresponding author

Correspondence to Kalpana Mahalingam .

Editor information

Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Waseda University, Kitakyushu, Japan

    Junzo Watada

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Kulkarni, M.S., Mahalingam, K., Mohankumar, S. (2019). Combinatorial Properties of Fibonacci Arrays. In: Gopal, T., Watada, J. (eds) Theory and Applications of Models of Computation. TAMC 2019. Lecture Notes in Computer Science(), vol 11436. Springer, Cham. https://doi.org/10.1007/978-3-030-14812-6_28

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  • DOI: https://doi.org/10.1007/978-3-030-14812-6_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-14811-9

  • Online ISBN: 978-3-030-14812-6

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