[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

(Un)decidability of the Emptiness Problem for Multi-dimensional Context-Free Grammars

  • Conference paper
  • First Online:
Implementation and Application of Automata (CIAA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9223))

Included in the following conference series:

  • 509 Accesses

Abstract

We study how dimensionality and form of context-free productions affect the power of multi-dimensional context-free grammars over unary alphabets. Attention is paid to the emptiness decision problem. It is an open question whether or not it is decidable for two-dimensional Kolam type context-free grammars of Siromoney. We show that the undecidability can be proved in the three-dimensional setting. For the two-dimensional variant, we present several results revealing that the process of generating is still much more complex than that one of the classical one-dimensional context-free grammar.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 35.99
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 44.99
Price includes VAT (United Kingdom)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Blum, M., Hewitt, C.: Automata on a 2-dimensional tape. In: Proceedings of the 8th Annual Symposium on Switching and Automata Theory (SWAT 1967), FOCS 1967, pp. 155–160. IEEE Computer Society, Washington, DC (1967)

    Google Scholar 

  2. Bremner, A.: Positively prodigious powers or how Dudeney done it? Math. Mag. 84(2), 120–125 (2011)

    Article  MATH  Google Scholar 

  3. Drewes, F., Ewert, S., Klempien-Hinrichs, R., Kreowski, H.: Computing raster images from grid picture grammars. J. Automata, Lang. Comb. 8(3), 499–519 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 215–267. Springer, New York (1997)

    Chapter  Google Scholar 

  5. Hopcroft, J., Ullman, J.: Formal languages and their relation to automata. Addison-Wesley, Reading (1969)

    Google Scholar 

  6. Ito, T., Sakamoto, M., Okabe, H., Furutani, H., Kono, M., Ikeda, S.: Marker versus inkdot over three-dimensional patterns. Artif. Life Robot. 13(1), 65–68 (2008)

    Article  Google Scholar 

  7. Matiyasevich, Y.: Hilbert’s tenth problem: Diophantine equations in the twentieth century. In: Bolibruch, A., Osipov, Y., Sinai, Y. (eds.) Mathematical Events of the Twentieth Century, pp. 185–213. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  9. Nakamura, A.: Three-dimensional connected pictures are not recognizable by finite-state acceptors. Inf. Sci. 66(3), 225–234 (1992)

    Article  MATH  Google Scholar 

  10. Nakamura, A.: Two-dimensional connected pictures are not recognizable by finite-state acceptors. Inf. Sci. 69(1–2), 55–64 (1993)

    Article  MATH  Google Scholar 

  11. Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. J. Comput. Syst. Sci. 65(2), 393–414 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pradella, M., Cherubini, A., Reghizzi, S.C.: A unifying approach to picture grammars. Inf. Comput. 209(9), 1246–1267 (2011)

    Article  Google Scholar 

  13. Průša, D.: Two-dimensional Languages. Ph.D. thesis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic (2004)

    Google Scholar 

  14. Průša, D.: Non-recursive trade-offs between two-dimensional automata and grammars. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 352–363. Springer, Heidelberg (2014)

    Google Scholar 

  15. Schlesinger, M.I.: Matematiceskie sredstva obrabotki izobrazenij (Mathematic tools for image processing). Naukova Dumka, Kiev (1989) (in Russian)

    Google Scholar 

  16. Schlesinger, M.I., Hlaváč, V.: Ten Lectures on Statistical and Structural Pattern Recognition (Computational Imaging and Vision). 1st edn. Springer, Heidelberg, May 2012

    Google Scholar 

  17. Siromoney, G., Siromoney, R., Krithivasan, K.: Abstract families of matrices and picture languages. Comput. Graph. Image Proces. 1(3), 284–307 (1972)

    Article  MathSciNet  Google Scholar 

  18. Siromoney, G., Siromoney, R., Krithivasan, K.: Picture languages with array rewriting rules. Inf. Control 22(5), 447–470 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Uchida, Y., Ito, T., Sakamoto, M., Uchida, K., Ide, T., Katamune, R., Furutani, H., Kono, M., Yoshinaga, T.: Cooperating systems of four-dimensional finite automata. Artif. Life Robot. 16(4), 555–558 (2012)

    Article  Google Scholar 

Download references

Acknowledgement

The author would like to thank Markus Holzer for his suggestions that became the basis for this paper. This work was supported by the Czech Science Foundation under grant no. 15-04960S.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Průša .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Průša, D. (2015). (Un)decidability of the Emptiness Problem for Multi-dimensional Context-Free Grammars. In: Drewes, F. (eds) Implementation and Application of Automata. CIAA 2015. Lecture Notes in Computer Science(), vol 9223. Springer, Cham. https://doi.org/10.1007/978-3-319-22360-5_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-22360-5_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22359-9

  • Online ISBN: 978-3-319-22360-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics