Ramified Structural Recursion and Corecursion

Abstract

We investigate feasible computation over a fairly general notion of data and codata. Specifically, we present a direct Bellantoni-Cookstyle normal/safe typed programming formalism, RS1 , that expresses feasible structural recursions and corecursions over data and codata specified by polynomial functors. (Lists, streams, finite trees, infinite trees, etc. are all directly definable.) A novel aspect of RS1 is that it embraces structure-sharing as in standard functional-programming implementations. As our data representations use sharing, our implementation of structural recursions are memoized to avoid the possibly exponentiallymany repeated subcomputations a naïve implementation might perform. We introduce notions of size for representations of data (accounting for sharing) and codata (using ideas from type-2 computational complexity) and establish that type-level 1 RS1 -functions have polynomial-bounded runtimes and satisfy a polynomial-time completeness condition. Also, restricting RS1 terms to particular types produces characterizations of some standard complexity classes (e.g., ω-regular languages, linear-space functions) and some less-standard classes (e.g., log-space streams).