Characterizing complexity classes by general recursive definitions in higher types

  • Andreas Goerdt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 385)


General recursive definitions in higher (finite) types, a different notation of finitely typed λ-terms with if-then-else and fixpoints, 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 restrict attention to definitions defining first-order functions, i.e. functions of type ind x...x ind→ind, ind for individuals, higher types only occur as detour in between.

Interpreting these definitions over finite structures we show: Rank-(n+1)-definitions characterize the complexity class
$$\begin{gathered}\cup DTIME(exp_n (p(x)),(exp_o (x) = x,exp_{n + 1} (x) = 2^{exp_n (x)} ). \hfill \\p(x) a poly \hfill \\\end{gathered}$$
This generalizes the result of Gurevich, Sazonov [Gu 83, Sa 80], that “normal” recursive definitions over finite structures characterize PTIME.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Andreas Goerdt
    • 1
  1. 1.Fachbereich Mathematik Fachgebiet Praktische InformatikUniversität -GH- DuisburgDuisburg 1West-Germany

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