Deterministic Two-Dimensional Languages over One-Letter Alphabet

  • Marcella Anselmo
  • Maria Madonia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4728)


We study the family DREC(1) of deterministic tiling recognizable two-dimensional languages in the case of a one-letter alphabet. The family coincides with both the class of languages accepted by deterministic on-line tessellation acceptors (\(\mathcal{L}\)(DOTA)(1)) and the one of languages recognized by 2-way alternating finite automata (\(\mathcal{L}\)(2AFA)(1)). We show that DREC(1) is complex enough to contain languages that cannot be realized by classical operations, while other languages constructed using classical operations cannot be deterministically recognized. Furthermore we prove that there are unambiguously recognizable languages that cannot be deterministically recognized even in the case of one-letter alphabet. In particular \(\mathcal{L}\)(DOTA)(1) is different from \(\mathcal{L}\)(OTA)(1) (its non-deterministic counterpart).


Automata and Formal Languages Determinism Two-dimensional languages 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anselmo, M., Giammarresi, D., Madonia, M.: From determinism to non-determinism in recognizable two-dimensional languages. In: Procs. DLT 2007. LNCS, vol. 4588, pp. 36–47. Springer, Heidelberg (2007)Google Scholar
  2. 2.
    Anselmo, M., Giammarresi, D., Madonia, M.: New Operators and Regular Expressions for two-dimensional languages over one-letter alphabet. Theoretical Computer Science 340(2), 408–431 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Anselmo, M., Giammarresi, D., Madonia, M., Restivo, A.: Unambiguous Recognizable two-dimensional languages. RAIRO: Theoretical Informatics and Applications 40(2), 227–294 (2006)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertoni, A., Goldwurm, M., Lonati, V.: On the complexity of unary tiling-recognizable picture languages. In: STACS 2007. LNCS, Springer, Heidelberg (2007)Google Scholar
  5. 5.
    Blum, M., Hewitt, C.: Automata on a two-dimensional tape. In: IEEE Symposium on Switching and Automata Theory, pp. 155–160 (1967)Google Scholar
  6. 6.
    Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, London (1974)zbMATHGoogle Scholar
  7. 7.
    Giammarresi, D.: Two-dimensional languages and recognizable functions. In: Rozenberg, G., Salomaa, A. (eds.) Procs. in Dev. on Language Theory 1993, pp. 290–301. World Scientific Publishing Co., Singapore (1994)Google Scholar
  8. 8.
    Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. Journal Pattern Recognition and Artificial Intelligence 6(2&3), 241–256 (1992)CrossRefGoogle Scholar
  9. 9.
    Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., et al. (eds.) Handbook of Formal Languages, vol. III, pp. 215–268. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Ginsburg, S., Spanier, E.: Semigroups, Presburger formulas, and languages. Pacific Journal of Mathematics 16, 285–296 (1966)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Inoue, K., Nakamura, A.: Some properties of two-dimensional on-line tessellation acceptors. Information Sciences 13, 95–121 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Inoue, K., Nakamura, A.: Two-dimensional finite automata and unacceptable functions. Intern. J. Comput. Math. A7, 207–213 (1979)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Ito, A., Inoue, K., Takanami, I.: Deterministic two-dimensional On-line tesselation Acceptors are equivalent to two-way two-dimensional alternating finite automata through 180°-rotation. Theor. Comp. Sc. 66, 273–287 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Inoue, K., Takanami, I.: A characterization of recognizable picture languages. In: Nakamura, A., Saoudi, A., Inoue, K., Wang, P.S.P., Nivat, M. (eds.) ICPIA 1992. LNCS, vol. 654, Springer, Heidelberg (1992)Google Scholar
  15. 15.
    Inoue, K., Takanami, I., Taniguchi, H.: Two-dimensional alternating Turing machines. Theor. Comp. Sc. 27, 61–83 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lindgren, K., Moore, C., Nordahl, M.: Complexity of two-dimensional patterns. Journal of Statistical Physics 91(5-6), 909–951 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kari, J., Moore, C.: Rectangles and squares recognized by two-dimensional automata. In: Karhumäki, J., Maurer, H., Păun, G., Rozenberg, G. (eds.) Theory Is Forever. LNCS, vol. 3113, pp. 134–144. Springer, Heidelberg (2004), Google Scholar
  18. 18.
    Kinber, E.B.: Three-way Automata on Rectangular Tapes over a One-Letter Alphabet. Information Sciences 35, 61–77 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Matz, O.: Regular expressions and Context-free Grammars for picture languages. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 283–294. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  20. 20.
    Matz, O.: Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures. Theoretical Computer Science 270(1-2), 1–70 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Potthoff, A., Seibert, S., Thomas, W.: Nondeterminism versus determinism of finite automata over directed acyclic graphs. Bull. Belgian Math. Soc. 1, 285–298 (1994)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Marcella Anselmo
    • 1
  • Maria Madonia
    • 2
  1. 1.Dipartimento di Informatica ed Applicazioni, Università di Salerno I-84084 Fisciano (SA)Italy
  2. 2.Dip. Matematica e Informatica, Università di Catania, Viale Andrea Doria 6/a, 95125 CataniaItaly

Personalised recommendations