AbstractCategorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently lambda-sigma-calculus [Abadi 1991; Hardin and Lévy 1989], have been introduced to provide an explicit treatment of substitutions in the lambda-calculus. we reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper with respect to Curien [1986/1993], Harding [1989], Abadi [1991], and Hardin and Lévy [1989] are the following:

This unfortunate result is ``repaired'' by presenting a confluent version of the lambda-sigma-calculus, named the lambda

- We present a confluent weak calculus of substitutions, where no variable clashes can be feared;
- We sove a conjecture raised in Abadi [1991]: lambda-sigma-calculus is not confluent (it is confluent on ground terms only).
Env-calculus in Hardin and Lévy [1989], called here the confluent lambda-sigma-calculus.The abstract is also available as a LaTeX file, a DVI file, or a PostScript file.

Categories and Subject Descriptors: F.3.2 [Logics and Meanings of Programs]: Semantics of Programming Languages --operational semantics; F.3.3 [Logics and Meanings of Programs]: Studies of Program Constructs --functional constructs; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic --lambda calenders and related systems

General Terms: Languages, Theory

Additional Key Words and Phrases: Confluency, explicit substitutions

Selected papers that cite this one

- P.-L. Curien, T. Hardin, and A. Ríos. Strong normalizations of substitutions. Journal of Logic and Computation, 6(6):799-817, December 1996.

- César Muñoz. Confluence and preservation of strong normalisation in an explicit substitutions calculus. In Proceedings, 11th Annual IEEE Symposium on Logic in Computer Science, pages 440-447, New Brunswick, New Jersey, 27-30 July 1996. IEEE Computer Society Press.

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