Resource-Bounded Kolmogorov Complexity Provides an Obstacle to Soficness of Multidimensional Shifts

TL;DR

This paper introduces resource-bounded Kolmogorov complexity as a tool to analyze soficness in multidimensional shifts, providing new examples of shifts with low complexity and linking classical proofs to computational complexity concepts.

Contribution

It formulates necessary conditions for soficness using resource-bounded Kolmogorov complexity and constructs low-complexity non-sofic shifts, connecting complexity theory with symbolic dynamics.

Findings

Constructed effective, non-sofic shifts with polynomial growth of patterns.

Established necessary conditions for soficness based on resource-bounded Kolmogorov complexity.

Linked classical non-soficness proofs to unbounded computational resources in Kolmogorov complexity.

Abstract

We suggest necessary conditions of soficness of multidimensional shifts formulated in termsof resource-bounded Kolmogorov complexity. Using this technique we provide examples ofeffective and non-sofic shifts on $\mathbb{Z}^2$ with very low block complexity: the number of globallyadmissible patterns of size $n\times n$ grows only as a polynomial in $n$. We also show that moreconventional proofs of non-soficness for multi-dimensional effective shifts can be expressed interms of Kolmogorov complexity with unbounded computational resources.