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The purpose of this article is to study the algorithmic complexity of theBesicovitch stability of noisy subshifts of finite type, a notion studied in aprevious article. First, we exhibit an unstable aperiodic tiling, and then seehow it can serve as a building block to implement several reductions fromclassical undecidable problems on Turing machines. It will follow that thequestion of stability of subshifts of finite type is undecidable, and thestrongest lower bound we obtain in the arithmetical hierarchy is$\Pi_2$-hardness. Lastly, we prove that this decision problem, which requiresto quantify over an uncountable set of probability measures, has a $\Pi_4$upper bound.

Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings