Categorical combinators [12, 21, 43] and more recently oe-calculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the -calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oe-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oe-calculus, named the Env-calculus in [23], called here the confluent oe-calculus.

We present an extensive, annotated bibliography of the abstract machines designed for each of the main programming paradigms (imperative, object oriented, functional, logic and concurrent). We conclude that whilst a large number of efficient abstract machines have been designed for particular language implementations, relatively little work has been done to design abstract machines in a systematic fashion.

We show the soundness of a -calculus B where de Bruijn indices are used, substitution is explicit, and reduction is step-wise. This is done by interpreting B in the classical calculus where the explicit substitution becomes implicit and de Bruijn indices become named variables. This is the first flat semantics of explicit substitution and step-wise reduction and the first clear account of exactly when ff-reduction is needed. Keywords: Explicit Substitution, de Bruijn indices, Variable names, Soundness. 1. Introduction Variables play a very demanding role in the reduction and substitution of the -calculus. This has lead in many cases to using explicit rather than implicit substitution. Implementations of the -calculus provide their own explicit substitution procedures as in Nuprl 9 and Automath 23 . Furthermore, research on theories of explicit substitution has been striving lately 5;12;13;22;4;18 . In this paper, we extend the calculus of [13] (which is influenced by Automath...