Advertisement

Complexity of Regex Crosswords

  • Stephen FennerEmail author
  • Daniel PadéEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

Abstract

In a regular expression crossword puzzle, one is given two non-empty lists \(\langle {} \langle {} R_1,\ldots , R_m \rangle {}\) and \(\langle {} C_1, \ldots , C_n \rangle {} \rangle {}\) over some alphabet, and the challenge is to fill in an \(m\times n\) grid of characters such that the string formed by the \(i^\text {th}\) row is in \(L(R_i)\) and the string in the \(j^\text {th}\) column is in \(L(C_j)\). We consider a restriction of this puzzle where all the \(R_i\) are equal to one another and similarly the \(C_j\). We consider a 2-player version of this puzzle, showing it to be Open image in new window -complete. Using a reduction from Open image in new window , we also give a new, simple proof of the known result that the existence problem of a solution for the restricted (1-player) puzzle is Open image in new window -complete.

Keywords

Complexity Regular expressions Regex crossword Picture language NP-complete 

Notes

Acknowledgments

We would like to thank Thomas Thierauf for several interesting discussions on this topic and to Joshua Cooper for finding for us a particularly challenging and fun regex crossword puzzle. We are also grateful to Klaus-Jörn Lange for suggesting the connection between our work and the theory of picture languages.

References

  1. 1.
  2. 2.
  3. 3.
    MIT Mystery Hunt. http://www.mit.edu/~puzzle
  4. 4.
  5. 5.
  6. 6.
    Berger, R.: The undecidability of the domino problem. No. 66 in memoirs of the American Mathematical Society. American Mathematical Society, Providence, Rhode Island (1966). mR0216954MathSciNetCrossRefGoogle Scholar
  7. 7.
    Black, L.: Can you do the regular expression crossword? I programmer, February 2013. http://www.i-programmer.info/news/144-graphics-and-games/5450-can-you-do-the-regular-expression-crossword.html
  8. 8.
    Fenner, S.: The complexity of some regex crossword problems (2014)Google Scholar
  9. 9.
    Giammarresi, D., Restivo, A.: Recognizable picture languages. Int. J. Pattern Recogn. Artif. Intell. 31–46 (1992)Google Scholar
  10. 10.
    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).  https://doi.org/10.1007/978-3-642-59126-6_4. Chap. 96CrossRefGoogle Scholar
  11. 11.
    Latteux, M., Simplot, D.: Recognizable picture languages and domino tiling. Theor. Comput. Sci. 178(1–2), 275–283 (1997). NoteMathSciNetCrossRefGoogle Scholar
  12. 12.
    Rosenfeld, A., Rheinboldt, W.: Picture Languages: Formal Models for Picture Recognition. Computer Science and Applied Mathematics. Elsevier Inc., Academic Press Inc., New York (1979)zbMATHGoogle Scholar
  13. 13.
    Takahashi, G.: Are regex crosswords NP-hard? cS Stack Exchange question 30143, answered by FrankW, September 2014. http://cs.stackexchange.com/questions/30143/are-regex-crosswords-np-hard
  14. 14.
    Takahashi, G.: Are regex crosswords NP-hard? CS Stack Exchange question 30143, answered by FrankW (2014). http://cs.stackexchange.com/questions/30143/are-regex-crosswords-np-hard

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of South CarolinaColumbiaUSA

Personalised recommendations