Characterizing complexity classes by higher type

Primitive recursive definitions, part II
  • Andreas Goerdt
  • Helmut Seidl
Part III Communications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 464)


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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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Andreas Goerdt
    • 1
  • Helmut Seidl
    • 2
  1. 1.Fachbereich Mathematik Fachgebiet Praktische InformatikUniversität -GH- DuisburgDuisburg 1West-Germany
  2. 2.Fachbereich InformatikUniversität des SaarlandesSaarbrücken 11West-Germany

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