Polyomino tilings, cellular automata and codicity

Lecture Notes in Computer Science, 2007

We give a characterization, in terms of computational complexity, of the family Rec1 of the unary picture languages that are tiling recognizable. We introduce quasi-unary strings to represent unary pictures and we prove that any unary picture language L is in Rec1 if and only if the set of all quasi-unary strings encoding the elements of L is recognizable by a one-tape nondeterministic Turing machine that is space and head-reversal linearly bounded. In particular, the result implies that the family of binary string languages corresponding to tiling-recognizable square languages lies between NTime(2 n) and NTime(4 n). This also implies the existence of a nontiling-recognizable unary square language that corresponds to a binary string language recognizable in nondeterministic time O(4 n log n).

2005

Discrete & Computational Geometry, 2006

Let T be a regular tiling of R 2 which has the origin 0 as a vertex, and suppose that ϕ : R 2 → R 2 is a homeomorphism such that i) ϕ(0) = 0, ii) the image under ϕ of each tile of T is a union of tiles of T , and iii) the images under ϕ of any two tiles of T are equivalent by an orientation-preserving isometry which takes vertices to vertices. It is proved here that there is a subset Λ of the vertices of T such that Λ is a lattice and ϕ| Λ is a group homomorphism.

Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01, 2001

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity at least n. This adds a quantitative angle to classical results on non-recursivity of tilings -that we also develop in terms of Turing degrees of unsolvability.

Discrete Applied Mathematics, 2009

On square or hexagonal lattices tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. For pseudo-hexagon polyominoes not containing arbitrarily large square factors we also have a linear algorithm. The results are extended to more general tiles.

2006

Beauquier and Nivat introduced and gave a characterization of the class of pseudo-square polyominoes, i.e. those polyominoes that tile the plane by translation: a polyomino tiles the plane by translation if and only if its boundary word W may be factorized as W = XY X Y . In this paper we consider the subclass PSP of pseudo-square polyominoes which are also parallelogram. By using the Beauquier-Nivat characterization we provide by means of a rational language the enumeration of the subclass of psp-polyominoes with a fixed planar basis according to the semiperimeter. The case of pseudo-square convex polyominoes is also analyzed. * Lab. Combinatoire et d'Informatique Mathématique, Un. Québec Montréal, CP 8888 Succ. Centreville, Montréal (QC) Canada H3C 3P8

In this paper, we construct a weakly universal cellular automaton with two states only on the tiling {11,3}. The cellular automaton is rotation invariant and it is a true planar one.

2011 International Conference on High Performance Computing & Simulation, 2011

In this paper, we look at the possibility to implement the algorithm to construct a discrete line devised by the first author in cellular automata. It turns out that such an implementation is feasible.

Theoretical Computer Science, 2011

It has been proved that, among the polyominoes that tile the plane by translation, the so-called squares tile the plane in at most two distinct ways. In this paper, we focus on double squares, that is, the polyominoes that tile the plane in exactly two distinct ways. Our approach is based on solving equations on words, which allows to exhibit properties about their shape. Moreover, we describe two infinite families of double squares. The first one is directly linked to Christoffel words and may be interpreted as segments of thick straight lines. The second one stems from the Fibonacci sequence and reveals some fractal features.

In this paper, we show how the study of two-way automata on words may relevantly be extended to the study of two-way automata on one-dimensional overlapping tiles that generalize finite words. Indeed, over tiles, languages recognizable by finite two-way automata (resp. multi-pebble automata) coincide with languages definable by Kleene's (resp. Kleene's extended) regular expressions.