Simplified Reducibility Proofs of Church-Rosser for β- and βη-reduction

  • Authors:

  • Fairouz Kamareddine;Vincent Rahli

  • Affiliations:

  • ULTRA Group, MACS, Heriot-Watt University, Edinburgh, Scotland, UK;ULTRA Group, MACS, Heriot-Watt University, Edinburgh, Scotland, UK

  • Venue:
  • Electronic Notes in Theoretical Computer Science (ENTCS)
  • Year:
  • 2009

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Abstract

The simplest proofs of the Church Rosser Property are usually done by the syntactic method of parallel reduction. On the other hand, reducibility is a semantic method which has been used to prove a number of properties in the @l-calculus and is well known to offer on one hand very general proofs which can be applied to a number of instantiations, and on the other hand, to be quite mysterious and inflexible. In this paper, we concentrate on simplifying a semantic method based on reducibility for proving Church-Rosser for both @b- and @b@h-reduction. Interestingly, this simplification results in a syntactic method (so the semantic aspect disappears) which is nonetheless projectable into a semantic method. Our contributions are as follows:*We give a simplification of a semantic proof of CR for @b-reduction which unlike some common proofs in the literature, avoids any type machinery and is solely carried out in a completely untyped setting. *We give a new proof of CR for @b@h-reduction which is a generalisation of our simple proof for @b-reduction. *Our simplification of the semantic proof results into a syntactic proof which is projectable into a semantic method and can hence be used as a bridge between syntactic and semantic methods.