) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de Paris-Sud, 91405 Orsay Cedex, France, fkesner,puelg@lri.fr. Abstract. We study preservation of fi-strong normalization by d and dn , two confluent -calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of fi-strong normalization of one calculus to that of another one, and apply said technique to reduce preservation of fi-strong normalization of d and dn to that of f , another calculus having no composition operator. Then, preservation of fi-strong normalization of f is shown using the same technique as in [2]. As a consequence, d and dn become the fir...

. We introduce a simply typed -calculus " which has both contexts and environments as first-class values. In ", holes in contexts are represented by ordinary variables of appropriate types and hole filling is represented by the functional application together with a new abstraction mechanism which takes care of packing and unpacking of the term which is used to fill in the holes of the context. " is a conservative extension of the simply typed fi-calculus, enjoys subject reduction property, is confluent and strongly normalizing. The traditional method of defining substitution does not work for our calculus. So, we also introduce a new method of defining substitution. Although we introduce the new definition of substitution out of necessity, the new definition turns out to be conceptually simpler than the traditional definition of substitution. 1 Introduction Informally speaking, a context (in -calculus) is a -term with some holes in it. For example, writing [ ] for a hole, y: [ ] is a...

We refine the simulation technique introduced in [10] to show strong normalization of -calculi with explicit substitutions via termination of cut elimination in proof nets [12]. We first propose a notion of equivalence relation for proof nets that extends the one in [9], and we show that cut elimination modulo this equivalence relation is terminating. We then show strong normalization of the typed version of the l - calculus with de Bruijn indices (a calculus with full composition defined in [8]) using a translation from typed l to proof nets. Finally, we propose a version of typed l with named variables which helps to better understand the complex mechanism of the explicit weakening notation introduced in the l -calculus with de Bruijn indices [8]. 1

. We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simply-typed call-by-name ~-calculus enriched with operators of explicit unary substitutions. The ~-calculus, dened by Curien & Herbelin, is a variant of -calculus with a let operator that exhibits symmetries such as terms/contexts and call-byname /call-by-value reduction. The ~-calculus embeds various standard -calculi (and Gentzen's style sequent calculi too) and as an application we derive the strong normalization of Parigot's simply-typed -calculus with explicit substitution. Introduction Explicit substitution in -calculus The traditional theory of -calculus relies on -reduction, that is the capture by a function of its argument followed by the process of substituting this argument to the places where it is used. The ...

a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, who also believed in me from the very beginning, encouraged me to work in this area, showed me the beauty of logic, and, above all, honoured me with her friendship. This thesis would not exist if it were not for their constant support. Thanks to my old friends, Cecilia C. Crespo, Santiago M. Peric'as, and, especially, Mat'ias Giovannini, for being always a wonderful critic of my work. Many thanks to Mathias Kegelmann for showing me the thrill of theorem proving; and to my former supervisor, Achim Jung, for introducing me to semantics.

Abstract. We characterise the strongly normalising terms of a composition-free calculus of explicit substitutions (with or without garbage collection) by means of an intersection type assignment system. The main novelty is a cut-rule which allows to forget the context of the minor premise when the context of the main premise does not have an assumption for the cut variable.

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.

This paper proposes an implementation of objects and functions via a calculus with explicit substitutions which is confluent and preserves strong normalization. The source calculus corresponds to the combination of the \sigma-calculus of Abadi and Cardelli [AC96] and the \lambda-calculus, and the target calculus corresponds to an extension of the former calculus with explicit substitutions. The interesting feature of our calculus is that substitutions are separated -- and treated accordingly -- in two different kinds: those used to encode ordinary substitutions and those encoding invoke substitutions. When working with explicit substitutions, this differentiation is essential to encode \lambda-calculus into \sigma-calculus in a conservative way, following the style proposed in [AC96].

In the framework of explicit substitutions there is two termination properties: preservation of strong normalization (PSN), and strong normalization (SN). Since there are not easily proved, only one of them is usually established (and sometimes none). We propose here a connection between them which helps to get SN when one already has PSN. For this purpose, we formalize a general proof technique of SN which consists in expanding substitutions into “pure ” λ-terms and to inherit SN of the whole calculus by SN of the “pure ” calculus and by PSN. We apply it successfully to a large set of calculi with explicit substitutions, allowing us to establish SN, or, at least, to trace back the failure of SN to that of PSN. Contents