A Sufficient Condition for Non-soficness of Higher-dimensional Subshifts

A shift space is said to be sofic if it is the factor of a shift of finite type. In one dimension, there are complete characterizations of soficness. There are no characterizations in higher dimensions, and there are few examples of non-sofic Z shifts for d > 1. In this work we give a condition that implies non-soficness in higher-dimensional shift spaces, and we apply it to a variety of examples. 1. Background Definitions Let A = {1, . . . , n} be a finite alphabet, and let D ⊆ Z, d ≥ 1. For y ∈ A, ~v ∈ D and S ⊆ D, we will denote the symbol appearing in y at ~v as y~v and the configuration of symbols appearing in y in the locations in S as yS. When S is a one-dimensional subset of D of the form S = {~a + k~ ei | 0 ≤ k < m} for some 1 ≤ i ≤ d, we will call configuration yS a word (of length m). For d ≥ 1, the d-dimensional, full n-shift is the set AZd together with the Z-action defined by