Pierre-Louis Curien
Abstract
Categorical combinators [Curien 1986/1993; Hardin 1989; Yokouchi 1989] and more recently λσ-calculus [Abadi 1991; Hardin and Le´vy 1989], have been introduced to provide an explicit treatment of substitutions in the λ-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], Hardin [1989], Abadi [1991], and Hardin and Le´vy [1989] are the following:
(1) We present a confluent weak calculus of substitutions, where no variable clashes can be feared; (2) We solve a conjecture raised in Abadi [1991]: λσ-calculus is not confluent (it is confluent on ground terms only).
This unfortunate result is “repaired” by presenting a confluent version of λσ-calculus, named the λEnv-caldulus in Hardin and Le´vy [1989], called here the confluent λσ-calculus.
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Index Terms
- Confluence properties of weak and strong calculi of explicit substitutions
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Reviews
Gabriel M. Ciobanu
The lambda-sigma-calculus was introduced to provide explicit treatment of substitutions in the lambda-calculus. In this paper, the authors describe various calculi of explicit substitutions, emphasizing their confluence properties. They first define a “conditional weak” lambda-sigma-calculus, which is confluent. Then a “weak” calculus of explicit substitutions is defined, and critical pairs are solved by adding a new composition operation to substitutions. This weak lambda-sigma-calculus is confluent also. Using the De Bruijn notation to deal with name clashes, the authors add a new operator to handle substitutions under lambda. The resulting calculus is the lambda-sigma-calculus described in previous works. This lambda-sigma-calculus is ground-confluent, but not even locally confluent. A locally confluent theory called lambda-sigma_SP is obtained by adding a “surjective pairing” rule; lambda-sigma_SP is not confluent. Finally, a confluent lambda-sigma-calculus called lambdaEnv is obtained by introducing a new operator for substitutions that is able to cross the lambda abstractions. Moreover, a rather simple proof of termination is given for this confluent lambda-sigma-calculus. Mainly these calculi of explicit substitutions are presented as sigma-algebras with two sorts (lambda terms and substitutions), and then the techniques and results (particularly the Knuth-Bendix algorithm) for rewriting systems can be applied. Some confluence results are proven using the method described by Hardin for the categorical combinators.
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