CHARACTERIZING COMPLEXITY CLASSES BY HIGHER TYPE PRIMITIVE RECURSIVE DEFINITIONS .2.

Higher type primitive recursive definitions (also known as Godel's system T) defining first-order functions (i.e. functions of type ind-->...-->ind, ind for individuals, higher types occur in between) can be classified into an infinite syntactic hierarchy: A definition is in the n'th stage of this hierarchy, a so called rank-n-definition, iff n is an upper bound on the levels of the types occurring in it. We interpret these definitions over finite structures and show for n-greater-than-or-equal-to- 1: Rank-(2n+2)-definitions characterize (in the sense of [Gu83], say) the complexity class DTIME(exp(n)(poly)) whereas rank-(2n+3)-definitions characterize DSPACE(exp(n)(poly)) (here exp0(x) = x, exp(n+1)(x) = 2exp(n)(x)). This extends the results that rank-1-definitions characterize LOGSPACE [Gu83], rank-2-definitions characterize PTIME, rank-3-definitions characterize PSPACE, rank-4-definitions characterize EXPTIME [Go89a].