Decidable properties of intersection type systems

We give positive answers to the following decision problems for the intersection type system λΛ and its variations; (1) the type checking problem of normalizing terms for λΛ, (2) the inhabitation problem for the system λΛ without (ΛI)-rule, and (3) the same problem for a typed counterpart of λΛ (or the intersection type system à la Church). Our result (1) contrasts with the well-known negative answer to the type checking problem (of all terms) for λΛ, while (2) and (3) contrast with Urzyczyn's negative answer to the inhabitation problem for λΛ.