Explicit Substitutions and Intersection Types

Calculi of explicit substitutions have been introduced to give an account to the substitution process in lambda calculus. The idea is to introduce a notation for substitutions explicitely in the calculus. In other words one makes substitutions first class citizens whereas the classical lambda calculus leaves them in the meta-theory. Originally, the expression “explicit substitution” and the concept itself were introduced by Abadi, Cardelli, Curien and Levy [1]. The idea of the authors was to replace categorical combinators by a syntax that extend lambda calculus. Actually the idea of internalizing substitutions into the lambda calculus is older. A first credit should be given to Curien with his λρ calculus [4] which is the origin of the calculus of [1]. But Curry himself expressed the same wish much earlier. For him combinatory logic was a mean to analyze substitutions. In the introduction of his famous book [5] he noted the advantage of including a calculus of substitutions into the lambda calculus. Perhaps the idea of explicit substitution is not “explicit” there, but at least the program of building a simple system, which analyzes substitutions and which departs less of our intuition than combinatory logic, appears clearly. For Curry this system should not be far from lambda calculus. Nowadays, we would call such a system a “calculus of explicit substitution”. Another ancestor of calculi of explicit substitutions is de Bruijn λCξφ [7]. As its complicated name might indicate, λCξφ is not an easy calculus as it mixes up the concrete and the abstract syntax in the same framework. In [2], this calculus has been revisited in modern notations. At the noticeable exception of λχ [10] which uses de Bruijn levels, most of these calculi use de Bruijn indices [6] but more natural approaches have been proposed where names for variables are made explicit like in classical lambda-calculus. The most popular presentation is due to Roel Bloo and Kristoffer Rose [3], but Lins [11] proposed a calculus with the same features in 1985. The work presented here is the result of a cooperation with Daniel Dougherty and Stéphane Lengrand [9, 8].