Abstract not found

) 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...

) Delia Kesner CNRS and LRI, B at 490, Universit e Paris-Sud - 91405 Orsay Cedex, France. e-mail:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and non-extensional #-calculi with explicit substitutions, where extensionality is interpreted by #-expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many well-known calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and non-extensional versions. 1 Introduction The #-calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #-reduction. The...

Explicit substitutions calculi are formal systems that implement fi-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 oe -calculus of explicit substitutions, proposed by Abadi, Cardelli, Curien andL evy, is a first-order rewriting system that implements substitution and renaming mechanism of -calculus. However, oe does not preserve strong normalisation of -calculus and it is not a confluent system. Typed variants of oe 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 i -calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation. 1. Explicit substitutions The -calculus is a higher-order theor...

The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oe-calculus. In (Kamareddine & R'ios, 1995a), we extended the -calculus with explicit substitutions by turning de Bruijn's meta-operators into objectoperators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the -calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine & R'ios, 1995a), is extended in this paper to a confluent calculus on open terms: the se-caculus. Since the establishment of these results, another calculus, i, came into being in (Mu~noz Hurtado, 1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that se still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi-reduction, whereas i is not. To ...

. We present the ! and !e calculi, the two-sorted (term and substitution) versions of the s (cf. [KR95a]) and se (cf. [KR96a]) calculi, respectively. We establish an isomorphism between the s-calculus and the term restriction of the !-calculus, which extends to an isomorphism between se and the term restriction of !e . Since the ! and !e calculi are given in the style of the oe-calculus (cf. [ACCL91]) they bridge calculi between s and oe and between se and oe and thus we are able to better understand one calculus in terms of the other. We improve our knowledge on the open problem of strong normalisation (SN) of the associated calculus of substitutions se by showing SN for two subcalculi (we use the isomorphism with !e for the proof of SN of one of them). Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !-calculus is a calculus in the oe-style that simulates one step fi-reduction, is confluent ...