We formulate recursive characterizations of the class of elementary functions and the class of functions computable in polynomial space that do not require any explicit bounded scheme. More specifically, we use functions where the input variables can occur in different kinds of positions —normal and safe— in the vein of the Bellantoni and Cook’s characterization of the polytime functions.

1991 Mathematics Subject Classification: 03D15, 03D20, 68Q15.

ACM Classification: F.1.3.