A concrete framework for environment machines

Abstract

We materialize the common belief that calculi with explicit substitutions provide an intermediate step between an abstract specification of substitution in the λ-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 λ-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 idealized and the original versions of Krivine's machine, Felleisen et al.'s CEK machine, and Leroy's Zinc abstract machine.

A Rational Deconstruction of Landin's J Operator

“…Programmatically, the J operator has been superseded by control operators that capture the current continuation (i.e., both C and D) instead of the continuation of the caller (i.e., D), even though it is simple to simulate escape and call/cc in terms of J. Yet as we have shown here, both the SECD machine and the J operator fit in the functional correspondence [3,4,6,7,13,16,30,31] as well as in the syntactic correspondence [12,14,15,29,31,40], which made it possible for us to mechanically characterize them in new and precise ways. 7 In turn it was Landin who, through the J operator, invented what we know today as first-class continuations [49].…”

“…Symmetrically to the functional correspondence between evaluation functions and abstract machines that was sparked by the first rational deconstruction of the SECD machine [3,4,6,7,13,16,30,31], a syntactic correspondence exists between calculi and abstract machines, as investigated by Biernacka, Danvy, and Nielsen [12,14,15,29,31,40]. This syntactic correspondence is also derivational, and hinges not on defunctionalization but on a 'refocusing' transformation that mechanically connects an evaluation function defined as the iteration of one-step reduction, and an abstract machine.…”

“…As abundantly illustrated elsewhere [14,15], deforesting the intermediate terms yields a refocused evaluation function in the form of an abstract machine. Simplifying this machine (again as abundantly illustrated elsewhere [14,15]) precisely yields the caller-save, stackless SECD abstract machine of Section 6.1.…”

“…Simplifying this machine (again as abundantly illustrated elsewhere [14,15]) precisely yields the caller-save, stackless SECD abstract machine of Section 6.1. The following proposition, whose proof is routine [14,15], captures the raison d'être of this reduction semantics: Proposition 9 (syntactic correspondence) For any program t in the λ ρJ-calculus,…”

“…Biernacka and Danvy then showed that the refocusing technique could be applied to the very first calculus of explicit substitutions, Curien's simple calculus of closures [27], and that depending on the reduction order, it gave rise to a collection of both known and new environment-based abstract machines such as Felleisen et al's CEK machine (for left-to-right applicative order), the Krivine machine (for normal order), Krivine's machine (for normal order with generalized reduction), and Leroy's ZINC machine (for right-to-left applicative order with generalized reduction) [15]. They then turned to context-sensitive contraction functions, as first proposed by Felleisen [44], and showed that refocusing mechanically gives rise to an even larger collection of both known and new environment-based abstract machines for languages with computational effects such as Krivine's machine with call/cc, the λµ-calculus, static and dynamic delimited continuations, i/o, stack inspection, proper tail-recursion, and lazy evaluation [14].…”