Higher order data structures- cartesian closure versus λ-calculus

Abstract

We discuss connections of typed λ-calculus and cartesian closure and prove equivalence of the theories ‘up to abstraction’. This is a working out of ideas of Scott and Lambeck but in an abstract data type environment. The results serve as a basis for the discussion of higher order specifications. We demonstrate that higher order equations based on λ-calculus are more appropriate if the equivalence of λ-calculus and cartesian closure is to be preserved. We construct higher order theories for higher order specifications. For higher order models we discuss existence of initial models and completeness of higher order theories.