Abstract
The tiling problem is the decision problem to determine if a given finite collection of Wang tiles admits a valid tiling of the plane. In this work we give a new proof of this fact based on tiling simulations of certain piecewise affine transformations. Similar proof is also shown to work in the hyperbolic plane, thus answering an open problem posed by R.M.Robinson 1971 [9].
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References
Berger, R.: Undecidability of the Domino Problem. Memoirs of the American Mathematical Society 66, 72 (1966)
Blondel, V., Bournez, O., Koiran, P., Papadimitriou, C., Tsitsiklis, J.: Deciding stability and mortality of piecewise affine dynamical systems. Theoretical Computer Science 255, 687–696 (2001)
Goodman-Strauss, C.: A strongly aperiodic set of tiles in the hyperbolic plane. Inventiones Mathematicae 159, 119–132 (2005)
Hooper, P.K.: The undecidability of the Turing machine immortality problem. The Journal of Symbolic Logic 31, 219–234 (1966)
Kari, J.: The Tiling Problem Revisited (extended abstract). In: Durand-Lose, J., Margenstern, M. (eds.) MCU 2007. LNCS, vol. 4664, pp. 72–79. Springer, Heidelberg (2007)
Kari, J.: A small aperiodic set of Wang tiles. Discrete Mathematics 160, 259–264 (1996)
Koiran, P., Cosnard, M., Garzon, M.: Computability with low-dimensional dynamical systems. Theoretical Computer Science 132, 113–128 (1994)
Margenstern, M.: About the domino problem in the hyperbolic plane, a new solution. Manuscript, 109 (2007) (and also see arXiv:cs/0701096, same title), http://www.lita.univ-metz.fr/~margens/
Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12, 177–209 (1971)
Robinson, R.M.: Undecidable tiling problems in the hyperbolic plane. Inventiones Mathematicae 44, 259–264 (1978)
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Kari, J. (2008). On the Undecidability of the Tiling Problem. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds) SOFSEM 2008: Theory and Practice of Computer Science. SOFSEM 2008. Lecture Notes in Computer Science, vol 4910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77566-9_7
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DOI: https://doi.org/10.1007/978-3-540-77566-9_7
Publisher Name: Springer, Berlin, Heidelberg
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