Characterizing complexity classes by higher type primitive recursive definitions

Abstract

Higher type primitive recursive definitions (also known as Gödel's system T) defining first-order functions (i.e. functions of type ind x…x ind→ind, ind for individuals, note that the higher types are used as detour to define first-order functions) can be classificed into an infinite syntactic hierarchy: A definition belongs to the nth stage of this hirarchy (is of rank n) iff n is an upper bound on the levels of the types occurring in it. We interpret these definitions over finite structures and show: Rank-1-definitions characterize LOGSPACE (in the sense of Gurevich (1983); in fact, this result is from his paper), rank-2 definitions characterize PTIME, rank-3 PSPACE and rank-4 EXPTIME (=DTIME(2 poly)).