Finitely Stratified Polymorphism

We consider predicative type-abstraction disciplines based on type quantification with finitely stratified levels. The main technical result is that the functions representable in the finitely stratified polymorphic A-calculus are precisely the super-elementary functions, i.e. the class £4 in Grzegorczyk's subrecursive hierarchy. This implies that there is no super-elementary bound on the length of optimal normalization sequences, and that the equality problem for finitely stratified polymorphic A-expressions is not superelementary. We also observe that finitely stratified polymorphism augmented with type recursion admits functional algorithms that are not typable in the full second order A-calculus. Research partiaUy supported by DARPA (DOD) and monitored by the Avionics Laboratory, Air Force Wright Aeronautical Laboratories, Aeronautical Systems Division (AFSC), Wright-Patterson AFB, Ohio 45433-6543, under Contract F33615-87-C1499, ARPA Order Number 4976, Amendment 20; and by ONR grant N00014-84-K-0415. The view and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of ONR, DARPA, or the U.S. Government.