An aperiodic set of 11 Wang tiles

A new aperiodic tile set containing 11 Wang tiles on 4 colors is presented. This tile set is minimal in the sense that no Wang set with less than 11 tiles is aperiodic, and no Wang set with less than 4 colors is aperiodic.

[1]  P. Paufler,et al.  Quasicrystals and Geometry , 1997 .

[2]  Robert E. Tarjan,et al.  Three Partition Refinement Algorithms , 1987, SIAM J. Comput..

[3]  Nicolas Ollinger,et al.  Substitutions and Strongly Deterministic Tilesets , 2012, CiE.

[4]  A S Kahr,et al.  ENTSCHEIDUNGSPROBLEM REDUCED TO THE AEA CASE. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Jarkko Kari The Nilpotency Problem of One-Dimensional Cellular Automata , 1992, SIAM J. Comput..

[6]  Antti Valmari Simple Bisimilarity Minimization in O(m log n) Time , 2010, Fundam. Informaticae.

[7]  Nicolas Ollinger Two-by-Two Substitution Systems and the Undecidability of the Domino Problem , 2008, CiE.

[8]  Gérard Cécé Foundation for a series of efficient simulation algorithms , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[9]  Zhi-Xiong Wen,et al.  Some Properties of the Singular Words of the Fibonacci Word , 1994, Eur. J. Comb..

[10]  Alexander Shen,et al.  Fixed Point and Aperiodic Tilings , 2008, Developments in Language Theory.

[11]  Lucian Ilie,et al.  Reducing NFAs by invariant equivalences , 2003 .

[12]  Jarkko Kari,et al.  Deterministic Aperiodic Tile Sets , 1999 .

[13]  Harry R. Lewis,et al.  Unsolvable classes of quantificational formulas , 1979 .

[14]  Leonid A. Levin Aperiodic Tilings: Breaking Translational Symmetry , 2005, Comput. J..

[15]  Victor Poupet Yet Another Aperiodic Tile Set , 2010, JAC.

[16]  Scott A. Smolka,et al.  CCS expressions, finite state processes, and three problems of equivalence , 1983, PODC '83.

[17]  A. Julien,et al.  Combinatorics and topology of the Robinson tiling , 2012, 1203.1387.

[18]  G. C. Shephard,et al.  Tilings and patterns. W. H. Freeman and Co. Ltd., Oxford 1987. IX + 700 p., 1395 figs., price £ 54.95, ISBN 0–7167–1193–1 , 1991 .

[19]  Hao Wang Proving theorems by pattern recognition — II , 1961 .

[20]  R. Robinson Undecidability and nonperiodicity for tilings of the plane , 1971 .

[21]  Robert L. Berger The undecidability of the domino problem , 1966 .

[22]  C. Goodman-Strauss MATCHING RULES AND SUBSTITUTION TILINGS , 1998 .

[23]  Hao Wang Notes on a class of tiling problems , 1975 .

[24]  Aimee S. A. Johnson,et al.  Putting The Pieces Together: Understanding Robinson’s Nonperiodic Tilings , 1997 .

[25]  Karel Culík,et al.  An aperiodic set of 13 Wang tiles , 1996, Discret. Math..

[26]  S. Mozes Tilings, substitution systems and dynamical systems generated by them , 1989 .

[27]  Bruno Poizat Une théorie finiement axiomatisable et superstable , 1982 .

[28]  Song-Sun Lin,et al.  Nonemptiness problems of Wang tiles with three colors , 2014, Theor. Comput. Sci..

[29]  Olivier Salon,et al.  Quelles tuiles ! (Pavages apériodiques du plan et automates bidimensionnels) , 1989 .

[30]  Ville Lukkarila The 4-way deterministic tiling problem is undecidable , 2009, Theor. Comput. Sci..

[31]  H. Hermes,et al.  A Simplified Proof for the Unsolvability of the Decision Problem in the Case , 1971 .

[32]  Hao Wang,et al.  Proving theorems by pattern recognition I , 1960, Commun. ACM.

[33]  Stål Aanderaa,et al.  Linear Sampling and the forall exists forall Case of the Decision Problem , 1974, J. Symb. Log..

[34]  H. Hermes Entscheidungsproblem und Dominospiele , 1970 .

[35]  John E. Hopcroft,et al.  An n log n algorithm for minimizing states in a finite automaton , 1971 .

[36]  Jarkko Kari,et al.  A small aperiodic set of Wang tiles , 1996, Discret. Math..

[37]  Peter Kulchyski and , 2015 .

[38]  Emmanuel Jeandel,et al.  Fixed Parameter Undecidability for Wang Tilesets , 2012, AUTOMATA & JAC.

[39]  L. Levin,et al.  Local rules and global order, or aperiodic tilings , 2005 .

[40]  Nicolas Ollinger,et al.  Combinatorial Substitutions and Sofic Tilings , 2010, JAC.