Abstract not found

The λσ-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λσ-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations.

The aim of this paper is to present the s-calculus which is a very simple -calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of -terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oe-calculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the s-calculus is obtained by the "interpretation method" ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the AE-calculus (cf. [BBLRD95]). The relation between s and AE is also studied.

This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of fi-reduction including local and global fi- reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to...

This paper is concerned with the study of cyclic - graphs. The starting point is to treat a -graph as a system of recursion equations involving -terms, and to manipulate such systems in an unrestricted manner, using equational logic, just as is possible for firstorder term rewriting. Surprisingly, now the confluence property breaks down in an essential way. Confluence can be restored by introducing a restraining mechanism on the `copying' operation. This leads to a family of -graph calculi, which are inspired by the family of oe-calculi (-calculi with explicit substitution) . However, these concern acyclic expressions only. In this paper we are not concerned with optimality questions for acyclic -reduction. We also indicate how Wadsworth's interpreter can be simulated in the -graph rewrite rules that we propose. Introduction As shown in recent years, first-order orthogonal term rewriting [8, 19] has quite pleasant confluent extensions to the case where cycles are admitted (term grap...

We introduce the new framework of Labeled Terms Rewriting Systems (T l RS), a general framework to express sharing in Term Rewriting Systems (TRS). For Orthogonal T l RS, an important subclass of T l RS, we characterize optimal derivations. This result is applied to weak -calculi, showing the optimality of the lazy strategy, that is, the call-by-name with sharing strategy. The result is also valid in the presence of ffi -rules, as in PCF. Orthogonal T l RS is also useful as a calculus for proving syntactic properties of functional languages. 1 Compilation of the -calculus Most compilers for functional languages translate their source language into some enriched -calculus [17], and then, compile this intermediate language to a low-level language, such as mutually recursive supercombinators, as in LML [2, 10], or categorical combinators, as in CAML [4]. These low-level languages define different forms of weak fi-reduction. We now describe two of these low-level languages, superc...

) 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...

In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #x-calculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as # # [24], # s [19], # d [21], and # f [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.

The specification and derivation of substitution for the de Bruijn representation of - terms is used to illustrate programming with a function-sequence monad. The resulting program is improved by interactive program transformation methods into an efficient implementation that uses primitive machine arithmetic. These transformations illustrate new techniques that assist the discovery of the arithmetic structure of the solution. Introduction Substitution is one of many problems in computer science that, once understood in one context, is understood in all contexts. Why, then, must a different substitution function be written for every abstract syntax implemented? This paper shows how to define substitution once and use the monadic structure of the definition to instantiate it on different abstract syntax structures. It also shows how to interactively derive an efficient implementation of substitution from this very abstract definition. The authors are supported in part by a grant fr...