In this manuscript we study properties of multidimensional shifts. More precisely, we study the necessary and sufficient conditions for a shift to be sofic, i.e. the boundary between sofic shifts and effective ones. To this end, we use different versions of algorithmic complexity (a.k.a. Kolmogorov complexity). In the first part of the work we suggest new necessary conditions of soficness for multidimensional shift. These conditions are expressed in terms of Kolmogorov complexity with bounded ressources. We discuss several applications of this technique. In particular, we construct an example of a two-dimensional effective and non sofic shift that has a very low combinatorial complexity : the number of global admissible N x N patterns grows only polynomially in N. We also show that the technique developed by S.Kass and K.Madden is equivalent to a special case of our method. In the second part, we discuss properties of subshifts defined in terms of density of letters. More specifically, we study two-dimensional subshifts $S(\rho)$ in the binary alphabet (white and black cells) where a configuration is admissible if every pattern of size N x N contains at most $N^\rho$ black cells. We show that $S(^\rho)$ is sofic for every $\rho<1$. Moreover, all effectif subshifts of these shifts are also sofic. The proof of this result is principally based on the construction of a self-simulating point-fixed tile set, with several new ingredients: an ad hoc model of computation based on a non deterministic cellular automaton (which allows to implement efficiently massively parallel calculations) and some properties of flows in a specific type of planar graphs.

翻译：本文研究了多维移位的性质。具体而言，我们探讨了移位成为sofic（即sofic移位与有效移位之间的边界）的充要条件。为此，我们使用了不同版本的算法复杂度（即柯尔莫哥洛夫复杂度）。在第一部分工作中，我们提出了多维移位sofic性的新必要条件，这些条件以有界资源的柯尔莫哥洛夫复杂度表示。我们讨论了该技术的若干应用，尤其是构造了一个具有极低组合复杂度的二维有效非sofic移位示例：其全局可容许的N×N模式数量仅呈多项式增长。此外，我们证明了S.Kass和K.Madden发展的技术等价于我们方法的特例。在第二部分中，我们讨论了由字母密度定义的子移位的性质。具体而言，研究了二进制字母表（黑白单元格）中的二维子移位$S(\rho)$：若每个N×N大小的模式中黑色单元格数量不超过$N^\rho$，则称该构型为可容许的。我们证明了对于所有$\rho<1$，$S(\rho)$是sofic的，且这些移位的所有有效子移位也是sofic的。该结果的证明主要基于自模拟定点图块集的构造，其中包含若干新要素：一个基于非确定性元胞自动机的特设计算模型（可高效实现大规模并行计算）以及特定平面图中流的一些性质。