Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts in dimensions $d > 1$ for subshifts of finite type and sofic shifts and in dimensions $d\geq 1$ for effective shifts. In particular, we prove that the factorization and embedding problems are $\Sigma^0_3$ -complete and $\Sigma^0_1$- complete respectively for SFTs, sofic and effective subshifts. Conjugacy on the other side is $\Sigma^0_1$-complete for SFTs and $\Sigma^0_3$-complete for sofic and effective shifts.