Characterizing complexity classes by higher type
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Higher type primitive recursive definitions (also known as Gödel's system T) defining first-order functions (i.e. functions of type ind→...→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≥1: Rank-(2n+2)-definitions characterize (in the sense of [Gu83], say) the complexity class DTIME(expn(poly)) whereas rank-(2n+3)-definitions characterize DSPACE(expn(poly)) (here exp0(x) = x, expn+1(x)=2expnx). 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].
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