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Two-Dimensional Palindromes and Their Properties

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Language and Automata Theory and Applications (LATA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10168))

Abstract

A two-dimensional word (2D) is a rectangular finite array of letters from the alphabet \(\varSigma \). A 2D word is said to be a 2D palindrome if it is equal to its reverse image. In this paper, we study some combinatorial properties of 2D palindromes. In particular, we provide a sufficient condition under which a 2D word is said to be a 2D palindrome, discuss the necessary and sufficient condition under which a 2D word can be decomposed into 2D palindromes, and find the relation between the set of all 2D palindromes and the set of all 2D primitive words. We also show that the set of all 2D palindromes is not a recognizable language, and study a special class of 2D palindromes, namely 2D palindrome square words.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. De Natale, F.G.B., Giusto, D.D., Maccioni, F.: A symmetry-based approach to facial features extraction. In: Proceedings of 13th International Conference on Digital Signal Processing, vol. 2, pp. 521–525 (1997)

    Google Scholar 

  4. Geizhals, S., Sokol, D.: Finding maximal 2-dimensional palindromes. In: Grossi, R., Lewenstein, M. (eds.) 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016), vol. 54, pp. 19:1–19:12 (2016)

    Google Scholar 

  5. Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 215–267. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  6. Hooda, A., Bronstein, M.M., Bronstein, A.M., Horaud, R.P.: Shape palindromes: analysis of intrinsic symmetries in 2D articulated shapes. In: Bruckstein, A.M., Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 665–676. Springer, Heidelberg (2012). doi:10.1007/978-3-642-24785-9_56

    Chapter  Google Scholar 

  7. Hopcroft, J.E., Ullman, J.D.: Formal Languages and Their Relation to Automata. Addison-Wesley Longman Inc., Boston (1969)

    MATH  Google Scholar 

  8. Kiryati, N., Gofman, Y.: Detecting symmetry in grey level images: the global optimization approach. Int. J. Comput. Vision 29(1), 29–45 (1998)

    Article  Google Scholar 

  9. Kulkarni, M.S., Mahalingam, K.: Two-dimensional primitive words. Manuscript (in preparation)

    Google Scholar 

  10. Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  11. Loy, G., Eklundh, J.-O.: Detecting symmetry and symmetric constellations of features. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 508–521. Springer, Heidelberg (2006). doi:10.1007/11744047_39

    Chapter  Google Scholar 

  12. Lyndon, R.C., Schützenberger, M.P.: The equation \(a^m= b^n c^p\) in a free group. Mich. Math. J. 9(4), 289–298 (1962)

    Article  MATH  Google Scholar 

  13. Matz, O.: Recognizable vs. regular picture languages. In: Bozapalidis, S., Rahonis, G. (eds.) CAI 2007. LNCS, vol. 4728, pp. 112–121. Springer, Heidelberg (2007). doi:10.1007/978-3-540-75414-5_7

    Chapter  Google Scholar 

  14. O’Mara, D., Owens, R.: Measuring bilateral symmetry in digital images. In: Proceedings of 1996 IEEE TENCON Digital Signal Processing Applications, vol. 1, pp. 151–156 (1996)

    Google Scholar 

  15. Pickover, C.A.: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  16. Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Symmetries of non-rigid shapes. In: 2007 IEEE 11th International Conference on Computer Vision. pp. 1–7 (2007)

    Google Scholar 

  17. Yu, S.S.: Languages and Codes. Tsang Hai Book Publishing Co., Taichung (2005)

    Google Scholar 

  18. Yu, S.S.: Palindrome words and reverse closed languages. Int. J. Comput. Math. 75(4), 389–402 (2000)

    Article  MathSciNet  MATH  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

Authors
  1. Manasi S. Kulkarni
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  2. Kalpana Mahalingam
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Corresponding author

Correspondence to Manasi S. Kulkarni .

Editor information

Editors and Affiliations

  1. Umeå University, Umeå, Sweden

    Frank Drewes

  2. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  3. University of Giessen, Giessen, Germany

    Bianca Truthe

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Kulkarni, M.S., Mahalingam, K. (2017). Two-Dimensional Palindromes and Their Properties. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2017. Lecture Notes in Computer Science(), vol 10168. Springer, Cham. https://doi.org/10.1007/978-3-319-53733-7_11

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  • DOI: https://doi.org/10.1007/978-3-319-53733-7_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53732-0

  • Online ISBN: 978-3-319-53733-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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