Characterizing complexity classes by higher type primitive recursive definitions

Higher type primitive recursive definitions (also known as Godel´s system T) defining first-order functions can be classified into an infinite syntactic hierarchy. A definition is in the nth level of this hierarchy, a so-called rank-n definition, if and only if n is an upper bound on the levels of the types occurring in it. The author interprets these definitions over finite structures and shows that rank-2 definitions characterize PTIME, rank-3 definitions characterize PSPACE, and rank-4 definitions EXPTIME