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.

We present a systematic construction of environment-based abstract machines from context-sensitive calculi of explicit substitutions, and we illustrate it with ten calculi and machines for applicative order with an abort operation, normal order with generalized reduction and call/cc, the lambda-mu-calculus, delimited continuations, stack inspection, proper tail-recursion, and lazy evaluation. Most of the machines already exist but have been obtained independently and are only indirectly related to the corresponding calculi. All of the calculi are new and they make it possible to directly reason about the execution of the corresponding machines. In connection with the functional correspondence between evaluation functions and abstract machines initiated by Reynolds, the present syntactic correspondence makes it possible to construct reduction-free normalization functions out of reductionbased ones, which was an open problem in the area of normalization by evaluation.

We present direct proofs of termination of evaluation for typed delimited-control operators shift and reset using a variant of Tait's method with context-based reducibility predicates. We address both call by value and call by name, and for each reduction strategy we consider a type-and-effect systemà la Danvy and Filinski as well as a system with a fixed answer type. The call-by-value type-andeffect system we present is a refinement of Danvy and Filinski's original type system, whereas the call-by-name type-and-effect system is new. From the normalization proofs, we extract call-by-value and call-by-name evaluators in continuation-passing style with two layers of continuations; by construction, these evaluators are instances of normalization by evaluation.

Abstract. We present an abstract machine and a reduction semantics for the lambdacalculus extended with control operators that give access to delimited continuations in the CPS hierarchy. The abstract machine is derived from an evaluator in continuationpassing style (CPS); the reduction semantics (i.e., a small-step operational semantics with an explicit representation of evaluation contexts) is constructed from the abstract machine; and the control operators are the shift and reset family.We also present new applications of delimited continuations in the CPS hierarchy: finding list prefixes and normalization by evaluation for a hierarchical language of units and products.

We present an abstract machine and a reduction semantics for the lambda-calculus extended with control operators that give access to delimited continuations in the CPS hierarchy. The abstract machine is derived from an evaluator in continuation-passing style (CPS); the reduction semantics (i.e., a small-step operational semantics with an explicit representation of evaluation contexts) is constructed from the abstract machine; and the control operators are the shift and reset family. At level n of the CPS hierarchy, programs can use the control operators shift_i and reset_i for 1 <= i <= n , the evaluator has n + 1 layers of continuations, the abstract machine has n + 1 layers of control stacks, and the reduction semantics has n + 1 layers of evaluation contexts.<br /> <br /> We also present new applications of delimited continuations in the CPS hierarchy: finding list prefixes and normalization by evaluation for a hierarchical language of units and products.