Advertisement

A decidability result about convex polyominoes

  • Danièle Beauquier
  • Michel Latteux
  • Karine Slowinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 583)

Abstract

A polyomino contour can be represented as a word over a four letter alphabet A. Each letter induces a unit line pointing one of the four directions (right, left, up and down). According to[b], checking whether a rational language R⊂A* contains a polyomino contour word is undecidable. We restrict the problem to convex polyominoes and we prove that, in this case, the problem turns out to be decidable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    D. Beauquier, A Undecidable Problem about Rational Sets and Contour Words of Polyominoes, Information Processing Letters 37 (1991), 257–263.Google Scholar
  2. [Be]
    R. Berger, The undecidability of the domino problem, Memoirs Amer. Math. Soc. 66 (1966), 72.Google Scholar
  3. [BLS]
    D. Beauquier, M. Latteux, K. Slowinski, A decidability result about convex polyominoes, Technical Report I.T. 214 (1991), University of Lille, France.Google Scholar
  4. [BN]
    D. Beauquier, M. Nivat, On translating one polyomino to tile the plane, Discrete and Computational Geometry, to appear.Google Scholar
  5. [Dal]
    J. Dassow, Convexity and Simplicity of Chain Code Picture Languages, Rostock. Math. Kolloq. 34 (1988), 53–60.Google Scholar
  6. [Da2]
    J. Dassow, Graph-theoretical Properties and Chain Code Picture Languages, J. Inf. Process. Cybern. EIK25 (1989), 423–433.Google Scholar
  7. [DH]
    J. Dassow, F. Hinz, Decision Problems and Regular Chain Code Picture Languages, to appear in Discrete Applied Mathematics.Google Scholar
  8. [DV]
    M.P. Delest and G. Viennot, Algebraic Languages and Polyominoes enumeration, Theoretical Computer Science 34 (1984), 169–206.Google Scholar
  9. [El]
    D. Ellard, Poyominoes and Enumeration, Math. Gazette 66 (1982), 130–314.Google Scholar
  10. [G1]
    S.W. Golomb, Polyominoes, Georges Allen and Unwin Ltd (London, 1966).Google Scholar
  11. [G2]
    S.W. Golomb, Tiling with sets of Polyominoes, S. Combinatorial Theory 9 (1970), 60–71.Google Scholar
  12. [GS]
    B. Grünbaum, G.C. Shephard, Tilings and Patterns, Freeman & Company (New York, 1986).Google Scholar
  13. [HU]
    J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley (Reading MA, 1979).Google Scholar
  14. [Ja]
    M. Jantzen, Confluent string rewriting, EATCS Monog. on T.C.S. 14 (Springer-Verlag, 1988).Google Scholar
  15. [Pi]
    J.E. Pin, Variétés de langages formels, Masson (1984).Google Scholar
  16. [Ro]
    R.M. Robinson, Undecidability and non Periodicity of Tilings of the Plane, Inventione Math. 12 (1971), 177–209.Google Scholar
  17. [Sh]
    H.D. Shapiro, Theorical limitations on the efficient user of parallel memories, I.E.E. Trans. Computing (1978).Google Scholar
  18. [S1]
    K. Slowinski, Systèmes de réécriture et langages de mots de figure, Ph. D. Thesis (1992), University of Lille, France.Google Scholar
  19. [Wa]
    H. Wang, Notes on a class of tiling problems, Fundam. Mathematics 82 (1975), 295–305.Google Scholar
  20. [WV]
    H.A.G. Wiljshoff, J. Van Leeuwen, Periodic versus arbitrary tessellations of the plane using polyominoes of a single type, Inf. Control 62 (1984), 1–25.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Danièle Beauquier
    • 1
  • Michel Latteux
    • 2
  • Karine Slowinski
    • 2
  1. 1.C.N.R.S.-U.A. 248-L.I.T.P., Institut Blaise PascalParis Cedex 05France
  2. 2.C.N.R.S.-U.A. 369-L.I.F.L., Université de Lille 1Villeneuve d'Ascq CedexFrance

Personalised recommendations