On Higher Order Recursive Program Schemes

We define Higher Order Recursive Program Schemes (HRPSs) by allowing metasubstitutions (as in the λ-calculus) in right-hand sides of function and quantifier definitions. A study of several kinds of similarity of redexes makes it possible to lift properties of (first order) Recursive Program Schemes to the higher order case. The main result is the decidability of weak normalization in HRPSs, which immediately implies that HRPSs do not have full computational power. We analyze the structural properties of HRPSs and introduce several kinds of persistent expression reduction systems (PERSs) that enjoy similar properties. Finally, we design an optimal evaluation procedure for PERSs.

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