Picture Array Generation Using Pure 2D Context-Free Grammar Rules

  • K. G. SubramanianEmail author
  • M. Geethalakshmi
  • N. Gnanamalar David
  • Atulya K. Nagar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9448)


Pure two-dimensional context-free grammar (P2DCFG) is a simple but effective non-isometric 2D grammar model to generate picture arrays. This 2D grammar uses only one kind of symbol as in a pure string grammar and rewrites in parallel all the symbols in a column or row by a set of context-free type rules. P2DCFG and a variant called (l / u)P2DCFG, which was recently introduced motivated by the “leftmost” rewriting mode in string context-free grammars, have been investigated for different properties. In this paper we introduce another variant of P2DCFG that corresponds to “rightmost” rewriting in string context-free grammars. The resulting grammar is called (r / d)P2DCFG and rewrites in parallel all the symbols only in the rightmost column or the lowermost row of a picture array by a set of context-free type rules. Unlike the case of string context-free grammars, the picture language families of P2DCFG and the two variants (l / u)P2DCFG and (r / d)P2DCFG are mutually incomparable, although they are not disjoint. We also examine the effect of regulating the rewriting in a (r / d)P2DCFG by suitably adapting two well-known control mechanisms in string grammars, namely, control words and matrix control.



The authors are grateful to the referees for their very useful comments which helped improve the presentation of the paper.


  1. 1.
    Bersani, M.M., Frigeri, A., Cherubini, A.: On some classes of 2D languages and their relations. In: Aggarwal, J.K., Barneva, R.P., Brimkov, V.E., Koroutchev, K.N., Korutcheva, E.R. (eds.) IWCIA 2011. LNCS, vol. 6636, pp. 222–234. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  2. 2.
    Bersani, M.M., Frigeri, A., Cherubini, A.: Expressiveness and complexity of regular pure two-dimensional context-free languages. Int. J. Comput. Math. 90, 1708–1733 (2013)zbMATHCrossRefGoogle Scholar
  3. 3.
    Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  4. 4.
    Freund, R.: Control mechanisms on #-context-free array grammars. In: Păun, Gh. (ed.) Mathematical Aspects of Natural and Formal Languages, pp. 97–137. World Scientific Publishing, Singapore (1994)Google Scholar
  5. 5.
    Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 215–267. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  6. 6.
    Křivka, Z., Martín-Vide, C., Meduna, A., Subramanian, K.G.: A variant of pure two-dimensional context-free grammars generating picture languages. In: Barneva, R.P., Brimkov, V.E., Šlapal, J. (eds.) IWCIA 2014. LNCS, vol. 8466, pp. 123–133. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  7. 7.
    Maurer, H.A., Salomaa, A., Wood, D.: Pure grammars. Inform. Control 44, 47–72 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Mishra, J.: Classification of linear fractals through L-system. In: Proceedings of the International Conferences Emerging Trends in Engineering and Technology, pp. 1–5 (2008)Google Scholar
  9. 9.
    Nakamura, A.: Picture languages. In: Davis, L.S. (ed.) Foundations of Image Understanding. International Series in Engineering and Computer Science, pp. 127–155. Kluwer Academic Publishers, Norwell (2001)CrossRefGoogle Scholar
  10. 10.
    Rosenfeld, A.: Picture Languages. Academic Press, Reading (1979)zbMATHGoogle Scholar
  11. 11.
    Rosenfeld, A., Siromoney, R.: Picture languages - a survey. Lang. Des. 1, 229–245 (1993)Google Scholar
  12. 12.
    Rozenberg, G., Salomaa, A.: The Mathematical Theory of L-Systems. Academic Press, New York (1980)zbMATHGoogle Scholar
  13. 13.
    Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages (3 Volumes). Springer, Berlin (1997)Google Scholar
  14. 14.
    Salomaa, A.: Formal Languages. Academic Press, Reading (1973)zbMATHGoogle Scholar
  15. 15.
    Siromoney, R., Subramanian, K.G., Rangarajan, K.: Control on kolam arrays. Inform. Control 32, 272–275 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Subramanian, K.G., Ali, R.M., Geethalakshmi, M., Nagar, A.K.: Pure 2D picture grammars and languages. Discrete Appl. Math. 157(16), 3401–3411 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Subramanian, K.G., Nagar, A.K., Geethalakshmi, M.: Pure 2D picture grammars (P2DPG) and P2DPG with regular control. In: Brimkov, V.E., Barneva, R.P., Hauptman, H.A. (eds.) IWCIA 2008. LNCS, vol. 4958, pp. 330–341. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  18. 18.
    Subramanian, K.G., Rangarajan, K., Mukund, M. (eds.): Formal Models, Languages and Applications. Series in Machine Perception and Artificial Intelligence, vol. 66. World Scientific, Singapore (2006) zbMATHGoogle Scholar
  19. 19.
    Wang, P.S.-P. (ed.): Array Grammars, Patterns and Recognizers. Series in Computer Science, vol. 18. World Scientific, Singapore (1989) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • K. G. Subramanian
    • 1
    Email author
  • M. Geethalakshmi
    • 2
  • N. Gnanamalar David
    • 3
  • Atulya K. Nagar
    • 1
  1. 1.Department of Mathematics and Computer ScienceFaculty of Science, Liverpool Hope UniversityLiverpoolUK
  2. 2.Department of MathematicsQueen Mary’s CollegeChennaiIndia
  3. 3.Department of MathematicsMadras Christian CollegeChennaiIndia

Personalised recommendations