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Extending a λ-calculus with explicit substitution which preserves strong normalisation into a confluent calculus on open terms

  • FAIROUZ KAMAREDDINE (a1) and ALEJANDRO RÍOS (a1)
    • Published online: 01 July 1997
Abstract

The last 15 years have seen an explosion in work on explicit substitution, most of which is done in the style of the λσ-calculus. In Kamareddine and Ríos (1995a), we extended the λ-calculus with explicit substitutions by turning de Bruijn's meta-operators into object-operators offering a style of explicit substitution that differs from that of λσ. The resulting calculus, λs, remains as close as possible to the λ-calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine and Ríos, 1995a), is extended in this paper to a confluent calculus on open terms: the λse-caculus. Since the establishment of these results, another calculus, λζ, came into being in Muñoz Hurtado (1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that λse still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical β-reduction, whereas λζ is not. To prove confluence we introduce a generalisation of the interpretation method (cf. Hardin, 1989; Curien et al., 1992) to a technique which uses weak normal forms (instead of strong ones). We consider that this extended method is a useful tool to obtain confluence when strong normalisation of the subcalculus of substitutions is not available. In our case, strong normalisation of the corresponding subcalculus of substitutions se, is still a challenging open problem to the rewrite community, but its weak normalisation is established here via an effective strategy.

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We are grateful to the anonymous referees for their comments and suggestions. We are also grateful to Hans Zantema for his interest in the strong normalisation property of our calculus and for his proof of termination of our σ-σ-transition rule. Kamareddine is grateful to Boston University and, in particular, to Assaf Kfoury and Joe Wells for their hospitality during work on this article. This work was carried out under EPSRC grant GR/K25014.
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Journal of Functional Programming
  • ISSN: 0956-7968
  • EISSN: 1469-7653
  • URL: /core/journals/journal-of-functional-programming
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Extending a λ-calculus with explicit substitution which preserves strong normalisation into a confluent calculus on open terms

  • FAIROUZ KAMAREDDINE (a1) and ALEJANDRO RÍOS (a1)
    • Published online: 01 July 1997
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