Resource-bounded Kolmogorov complexity provides an obstacle to soficness of multidimensional shifts

摘要

We suggest necessary conditions for soficness of multidimensional shifts formulated in terms of resource-bounded Kolmogorov complexity. Using this technique we provide examples of effective and non-sofic shifts on Z2 with very low block complexity: the number of globally admissible patterns of size n×n grows only as a polynomial in n. We also show that more conventional proofs of non-soficness for multi-dimensional effective shifts, including the techniques of Pavlov (2013) [15] and Kass and Madden (2013) [6], can be expressed in terms of Kolmogorov complexity with unbounded computational resources.