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.

In this paper we introduce and study a new lambda-calculus with explicit substitution, lambda-xgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rules for explicit garbage collection; this simplifies several proofs. We show that lambda-xgc is a conservative extension which preserves strong normalisation (PSN) of the untyped lambda-calculus. The result is obtained in a modular way by first proving it for garbage-free reduction and then extending to `reductions in garbage'. This provides insight into the counterexample to PSN for lambda-sigma of Melliès (1995); we exploit the abstract nature of lambda-xgc to show how PSN is in conflict with any reasonable substitution composition rule (except for trivial composition rules of which we mention one). Key words: lambda calculus, explicit substitution, strong normalisation, garbage collection.

In [6] we have proposed a general higher-order unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higher-order patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an efficient implementation of the abstract algorithm and its generalization to constraint simplification, which has yielded good experimental results at the core of a higher-order constraint logic programming language.

. In this paper, we present several improvements of an algorithm for a higher-order unification based on the calculus of explicit substitutions. The main difference between our algorithm and the already known version is, that we try to postpone normalisation of oe-terms as long as possible, i.e. until some information of these oe-terms is necessary for the next step of the unification algorithm. 1 Introduction In this paper, we describe an improved version of a higher-order unification algorithm, which was presented in [DHK95]. The main idea of this algorithm is based on a calculus of explicit substitutions in a simply typed -theory (for definitions and details, see [ACCL90]), which integrates substitutions in the framework of the first-order formalism. In this calculus, substitutions are treated as the firstorder objects, i.e. all basic operations over substitutions, like an application, a composition and a concatenation are defined in the first-order theory (their semantic is descri...

: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the Curry-Howard isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambda-terms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambda-calculus with new operators. First, we consider typed meta-variables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing meta-variables. Unfortunately, the theory of explicit substitution calculi with typed meta-variables is more complex than that of lambda-calculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...

: Explicit substitutions calculi are formal systems that implement fi-reduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, for instance, to represent incomplete proofs in type based proof systems. The oe -calculus of explicit substitutions proposed by Abadi, Cardelli, Curien and L'evy gives an elegant way to deal with management of variable names and substitutions 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 an we present the full proofs of its main properties. This is, as far as we know, the...

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...

Extending the λ-calculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the - calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fi-reduction of g and the classical -calculus. Furthermore, we study the simply typed versions of s and gs, and show that well-typed terms are strongly normalizing and that other properties,...

. We show how to reduce the unification problem modulo fij- conversion and a first-order equational theory E, into a first-order unification problem in a union of two non-disjoint equational theories including E and a calculus of explicit substitutions. A rule-based unification procedure in this combined theory is described and may be viewed as an extension of the one initially designed by G. Dowek, T. Hardin and C. Kirchner for performing unification of simply typed -terms in a first-order setting via the oe-calculus of explicit substitutions. Additional rules are used to deal with the interaction between E and oe. 1 Introduction Unification modulo an equational theory plays an important role in automated deduction and in logic programming systems. For example, Prolog[NM88] is based on higher-order unification, ie. unification modulo the fij-conversion. In order to design more expressive higher-order logic programming systems enhanced with a first-order equational theory E,...

In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the Curry-Howard Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called places-holders. We present a theoretical approach to place-holders in type theory. In this approach place-holders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed -calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities.