Evaluation under lambda abstraction

Abstract

In light of the usual definition of values [18] as terms in weak head normal form (WHNF), a λ-abstraction is regarded as a value, and therefore no expressions under λ-abstraction can get evaluated and the sharing of computation under λ has to be achieved through program transformations such as λ-lifting and supercombinators. In this paper we generalise the notion of head normal form (HNF) and introduce the definition of generalised head normal form (GHNF). We then define values as terms in GHNF with flexible heads, and study a call-by-value λ-calculus λ ^v _hd corresponding to this new notion of values. After establishing a version of the normalisation theorem in λ ^v _hd , we construct an evaluation function evalhd for λ ^v _hd which evaluates under λ-abstraction. We prove that a program can be evaluated in λ ^v _hd to a term in GHNF if and only if it can be evaluated in the usual λ-calculus to a term in HNF. We also present an operational semantics for λ ^v _hd via a SECD machine. We argue that lazy functional programming languages can be implemented through λ ^v _hd and an implementation of λ ^v _hd can significantly enhance the degree of sharing in evaluation. This establishes a solid foundation for implementing run-time code generation through λ ^v _hd