On the Complexity of Unary Tiling-Recognizable Picture Languages

  • Alberto Bertoni
  • Massimiliano Goldwurm
  • Violetta Lonati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

We give a characterization, in terms of computational complexity, of the family Rec1 of the unary picture languages that are tiling recognizable. We introduce quasi-unary strings to represent unary pictures and we prove that any unary picture language L is in Rec1 if and only if the set of all quasi-unary strings encoding the elements of L is recognizable by a one-tape nondeterministic Turing machine that is space and head-reversal linearly bounded. In particular, the result implies that the family of binary string languages corresponding to tiling-recognizable square languages lies between NTime(2n) and NTime(4n). This also implies the existence of a nontiling-recognizable unary square language that corresponds to a binary string language recognizable in nondeterministic time O(4n logn).

Classification: automata and formal languages, computational complexity.

Keywords

unary picture languages tiling systems Turing machine head reversal. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160(2), 781–793 (2004)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Anselmo, M., Giammarresi, D., Madonia, M.: Regular expressions for two-dimensional languages over one-letter alphabet. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds.) DLT 2004. LNCS, vol. 3340, pp. 63–75. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. J. Pattern Recognition and Artificial Intelligence, Special Issue on Parallel Image Processing, 31–42 (1992)Google Scholar
  4. 4.
    Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 215–268. Springer, Heidelberg (1997)Google Scholar
  5. 5.
    Giammarresi, D., et al.: Monadic second order logic over rectangular pictures and recognizability by tiling system. Information and Computation 125(1), 32–45 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Inoue, K., Takanami, I.: A survey of two-dimensional automata theory. In: Dassow, J., Kelemen, J. (eds.) IMYCS 1988. LNCS, vol. 381, pp. 72–91. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  7. 7.
    Kari, J., Moore, C.: New results on alternating and non-deterministic two-dimensional finite state automata. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 396–406. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proc. 13th Annual IEEE Symp. on Switching and Automata Theory, pp. 125–129. IEEE Computer Society Press, Los Alamitos (1972)Google Scholar
  10. 10.
    Meyer, A.R., Stockmeyer, L.J.: Words problems requiring exponential time. In: Proc. 5th ACM Symp. on Theory of Computing, pp. 1–9. ACM Press, New York (1973)Google Scholar
  11. 11.
    Seiferas, J.I., Fischer, M.J., Meyer, A.R.: Separating nondeterministic time complexity classes. Journal of ACM 25(1), 146–167 (1978)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Siromoney, R.: Advances in array languages. In: Ehrig, H., et al. (eds.) Graph Grammars 1986. LNCS, vol. 291, pp. 549–563. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Alberto Bertoni
    • 1
  • Massimiliano Goldwurm
    • 1
  • Violetta Lonati
    • 1
  1. 1.Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, Via Comelico 39/41, 20135 Milano –Italy

Personalised recommendations