Final Semantics for untyped lambda-calculus

Proof principles for reasoning about various semantics of untyped \u3bb-calculus are discussed. The semantics are determined operationally by fixing a particular reduction strategy on \u3bb-terms and a suitable set of values, and by taking the corresponding observational equivalence on terms. These principles arise naturally as co-induction principles, when the observational equivalences are shown to be induced by the unique mapping into a final F-coalgebra, for a suitable functor F. This is achieved either by induction on computation steps or exploiting the properties of some, computationally adequate, inverse limit denotational model. The final F-coalgebras cannot be given, in general, the structure of a \u201cdenotational\u201d \u3bb-model. Nevertheless the \u201cfinal semantics\u201d can count as compositional in that it induces a congruence. We utilize the intuitive categorical setting of hypersets and functions. The importance of the principles introduced in this paper lies in the fact that they often allow to factorize the complexity of proofs (of observational equivalence) by \u201cstraight\u201d induction on computation steps, which are usually lengthy and error-prone