Bar recursive encodings of tree ordinals

We ask the attention for a de nition schema from higher order subrecursion the ory called bar recursion Bar recursion originates with Spector where bar recursion of all nite types is shown to characterize the class of provably total re cursive functions of analysis This class has also been characterized by Girard as those functions which are de nable in the second order typed lambda calculus or the polymorphic lambda calculus Analysis is a so called impredicative theory since it allows de ning a predicate by quanti cation over all predicates including the predicate that is de ned itself In terms may depend on types including the type of the term itself This phenomenon is also called impredica tivity Impredicativity is an informal notion which has many faces The question arises is there something impredicative about bar recursion The question is important for a better mathematical understanding of impredicativity especially of its computational impact We answer the question a rmatively by identifying and exploring a special feature of bar recursion which we consider to be another appearance of impredicativity the dependency of terms on order types Observe that in the case of higher order primitive recursion the order type of the recursion is xed In the case of bar recursion however there is a parameter which determines the order type of the recursion Lambda abstraction from this parameter is perfectly allowed thus yielding a polymorphic or better a poly order type recursor We explore the computational aspects of the dependency of terms on order types to some extent by constructing bar recursive encodings of trans nite recursors up to and beyond Trans nite recursors are of some interest for computer science they provide a strongly normalizing formalism for recursive functional programming Thus program termination is guaranteed for all evaluation strategies this is not the case when using xed point combinators