On the Mints Hierarchy in First-Order Intuitionistic Logic
  We stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy).   We study the decidability and complexity of these fragments. We prove that even the $\Delta_2$ level is undecidable and that $\Sigma_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $\Sigma_1$ is complete for co-Nexptime. 
Aleksy Schubert and Pawe{\l} Urzyczyn and Konrad Zdanowski
cs.LO