Confluence and preservation of strong normalisation in an explicit substitutions calculus

Abstract

Explicit substitutions calculi are formal systems that implement /spl beta/-reduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The /spl lambda//sub /spl sigma//-calculus of explicit substitutions, proposed by M. Abadi et al. (1991), is a first-order rewriting system that implements substitution and renaming mechanism of /spl lambda/-calculus. However; /spl lambda//sub /spl sigma// does not preserve strong normalisation of /spl lambda/-calculus and it is not a confluent system. Typed variants of /spl lambda//sub /spl sigma// without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the /spl lambda//sub /spl zeta//-calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation.