# Two-Dimensional Palindromes and Their Properties

• Manasi S. Kulkarni
• Kalpana Mahalingam
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

## Keywords

Combinatorics on words Two-Dimensional words Two-Dimensional palindromes Recognizable languages Primitivity Symmetry

## References

1. 1.
Amir, A., Benson, G.: Two-dimensional periodicity in rectangular arrays. SIAM J. Comput. 27(1), 90–106 (1998)
2. 2.
Berthé, V., Vuillon, L.: Palindromes and two-dimensional Sturmian sequences. J. Automata Lang. Comb. 6(2), 121–138 (2001)
3. 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. 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. 5.
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 215–267. Springer, Heidelberg (1997)
6. 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:
7. 7.
Hopcroft, J.E., Ullman, J.D.: Formal Languages and Their Relation to Automata. Addison-Wesley Longman Inc., Boston (1969)
8. 8.
Kiryati, N., Gofman, Y.: Detecting symmetry in grey level images: the global optimization approach. Int. J. Comput. Vision 29(1), 29–45 (1998)
9. 9.
Kulkarni, M.S., Mahalingam, K.: Two-dimensional primitive words. Manuscript (in preparation)Google Scholar
10. 10.
Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge (1997)
11. 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:
12. 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)
13. 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:
14. 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. 15.
Pickover, C.A.: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton University Press, Princeton (2002)
16. 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. 17.
Yu, S.S.: Languages and Codes. Tsang Hai Book Publishing Co., Taichung (2005)Google Scholar
18. 18.
Yu, S.S.: Palindrome words and reverse closed languages. Int. J. Comput. Math. 75(4), 389–402 (2000)

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## Authors and Affiliations

1. 1.Department of Mathematics Indian Institute of Technology MadrasChennaiIndia