Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ff-conversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being well-tied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the t-calculus: a -calculus `a la de Bruijn which can be translated into a -calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...

The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate eta-reduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of beta-contraction.

. An eective strategy for implementing higher order unication (HOU) based on the se - style of explicit substitution is proposed. The strategy is based on a se-unication method recently developed by the authors. A pre-cooking translation for applying the se -style of unication to HOU in the pure -calculus is presented. Correctness and completeness of the proposed strategy and of the pre-cooking translation are shown and their applicability to HOU in the pure -calculus is illustrated. 1 Introduction In [DHK00], a higher order unication (HOU) method was based on the -style of explicit substitution [ACCL91]. In [ARK00], HOU was studied in the s e -style of explicit substitution [KR97]. It is claimed in [ARK00] that s e -unication has the advantages of enabling quicker detection of redices and of having a clearer semantics. In this paper, we set out to provide an eective strategy for implementing s e -unication and a pre-cooking translation for applying it to HOU in the ...

This article is a brief review of the type free λ-calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λ-calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λ-calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λ-calculus is introduced and some of its advantages are outlined.

A unication method based on the se -style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure -calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the -style of explicit substitution. Correctness and completeness properties of the proposed se-unication method are shown and its advantages, inherited from the qualities of the se -calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the -calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the -calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the -style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambda-calculi. 1

. A pre-cooking translation for applying the se -style of uni- cation to higher order unication (HOU) in the pure -calculus is presented. The pre-cooking jointly with a back translation complement a se - unication method recently developed by the authors. Their correctness and completeness are shown and additionally we show why avoiding the use of substitution objects makes se-HOU more ecient than -HOU. Keywords: Higher order unication, explicit substitution, -calculus 1 Introduction In [DHK00], a higher order unication (HOU) method was based on the - style of explicit substitution [ACCL91]. In [ARK00b], HOU was studied in the s e -style of explicit substitution [KR97]. It is claimed in [ARK00b] that s e - unication has the advantages of enabling quicker detection of redices and of having a clearer semantics. In this paper, we set out to provide a pre-cooking translation for applying s e -unication to HOU in the -calculus and we focus on the advantages of us...

A higher order unification (HOU) method based on the ...-style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...-style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.

Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the -calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are unable to handle local substitutions, and b) have answered (positively or negatively) the question of the termination of the underlying calculus of substitutions. The exception to a) is the calculus of [9] where substitution is handled both locally and globally. However, the calculus of [9] does not satisfy properties like conuence and termination. The exception to b) is the s e -calculus [11] for which termination of the s e -calculus, the underlying calculus of substitutions, remains unsolved. This paper has two aims: (i) To provide a calculus a la de Bruijn which deals with local substitution and whose underlying calculus of substitutions is terminating and conuent.

The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substitutions are treated in this paper: the and the s e which have proved useful for solving practical problems like higher order uni cation, and the suspension calculus related to the implementation of the language Prolog. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. Additionally, we prove that and s e as well as and the suspension calculus are non comparable while s e is more adequate than the suspension calculus.