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 Z 2 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 .

Extended version: 18 pages, 5 figures