Descriptive Complexity of Sensitivity of Cellular Automata

We study the computational complexity of determining whether a cellular automaton is sensitive to initial conditions. We show that this problem is ${\mathrm{\Pi }}_2^0$-complete in dimension 1 and ${\mathrm{\Sigma }}_3^0$-complete in dimension 2 and higher. This solves a question posed by Sablik and Theyssier.