Algorithmic complexity and soficness of shifts in dimension two

TL;DR

This paper explores the conditions under which multidimensional shifts are sofic, using algorithmic complexity, and introduces new techniques to characterize and construct such shifts, including examples with low complexity.

Contribution

It provides new necessary conditions for soficness in multidimensional shifts using Kolmogorov complexity and constructs examples with polynomial complexity.

Findings

Constructed a 2D effective, non-sofic shift with polynomial pattern growth.

Proved shifts with density constraints are sofic for certain parameters.

Developed a new self-simulating tile set with advanced computational features.

Abstract

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.