This paper extends the recent work [CMT00] on the operational semantics and type system for a core language, called MiniML ref , which exploits the notion of closed type (see also [MTBS99]) to safely combine imperative and multi-stage programming. The main novelties are the identification of a larger set of closed types and the addition of a binder for useless variables. The resulting language is a conservative extension of MiniML ref , a simple imperative subset of SML. 1

We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply-typed terms, step by step simulation of β-reduction and full composition.

Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with meta-level substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambda-calculus which enjoys fundamental properties such as simulation of one-step beta-reduction, confluence on metaterms, preservation of beta-strong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proof-nets.

Abstract. In a previous paper [2] which appeared in the volume celebrating Klop’s 60th anniversary, we presented a labeled lambda-calculus to characterize the dag implementation of the weak lambda-calculus as described in Wadsworth’s dissertation [11]. In this paper, we simplify this calculus and present a simpler proof of the sharing property which allows the dag implementation. In order to avoid duplication of presentations, we mainly show here the modifications brought to the weak labeled lambda-calculus in [2]. The reader is therefore recommended to read first the companion article and later read our present paper. We are happy that this note can therefore be considered as establishing a new bridge between two friends and now senior colleagues, Jan Willem Klop and Henk