A Concrete Framework for Environment Machines

We materialize the common understanding that calculi with explicit substitutions provide an intermediate step between an abstract specification of substitution in the lambda-calculus and its concrete implementations. To this end, we go back to Curien's original calculus of closures (an early calculus with explicit substitutions), we extend it minimally so that it can also express one-step reduction strategies, and we methodically derive a series of environment machines from the specification of two one-step reduction strategies for the lambda-calculus: normal order and applicative order. The derivation extends Danvy and Nielsen's refocusing-based construction of abstract machines with two new steps: one for coalescing two successive transitions into one, and the other for unfolding a closure into a term and an environment in the resulting abstract machine. The resulting environment machines include both the Krivine machine and the original version of Krivine's machine, Felleisen et al.'s CEK machine, and Leroy's Zinc abstract machine.

Functional runtime systems within the lambda-sigma calculus

We define a weak λ-calculus, λσw, as a subsystem of the full λ-calculus with explicit substitutions λσ[uArr ]. We claim that λσw could be the archetypal output language of functional compilers, just as the λ-calculus is their universal input language. Furthermore, λσ[uArr ] could be the adequate theory to establish the correctness of functional compilers. Here we illustrate these claims by proving the correctness of four simplified compilers and runtime systems modelled as abstract machines. The four machines we prove are the Krivine machine, the SECD, the FAM and the CAM. Thus, we give the first formal proofs of Cardelli's FAM and of its compiler.