Characterizing complexity classes by general recursive definitions in higher types

Abstract

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 of first-order functions, i.e., functions of type ind × … × ind → ind, ind for individuals; higher types only occur as detours in between. Interpreting these definitions over finite structures we show that rank ( n + 1) definitions characterize the complexity class of global functions ⋃ p(x) a poly DTIME ( exp n (p(x))) , where exp 0 ( x ) = x , exp n + 1 ( x ) = 2 exp n ( x ) . This generalizes the result of Y. Gurevich ( in “Proceedings, 24th Symposium on Foundations of Computer Science,” pp. 210–214, IEEE Comput. Soc. Press, New York, 1989) and V. Sazonov ( Elektron. Informationsverarb. Kybernet. 16 , 319–323, 1980) that rank 1 recursive definitions over finite structures characterize PTIME.