### Abstract

The typed λμ-calculus is known to be strongly normalizing and weakly Church-Rosser, and hence becomes confluent. In fact, Parigot formulated a parallel reduction to prove confluence of the typed λμ-calculus by "Tait-and-Martin-Löf" method. However, the diamond property does not hold for his parallel reduction. The confluence for type-free λμ-calculus cannot be derived from that of the typed λμ-calculus and is not confirmed yet as far as we know. We analyze granularity of the reduction rules, and then introduce a new parallel reduction such that both renaming reduction and consecutive structural reductions are considered as one step parallel reduction. It is shown that the new formulation of parallel reduction has the diamond property, which yields a correct proof of the confluence for type free λμ-calculus. The diamond property of the new parallel reduction is also applicable to a call-by-value version of the λμ-calculus containing the symmetric structural reduction rule.

Original language | English |
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Pages (from-to) | 52-66 |

Number of pages | 15 |

Journal | Electronic Notes in Theoretical Computer Science |

Volume | 42 |

DOIs | |

Publication status | Published - Jan 1 2001 |

Event | Computing: The Australasian Theory Symposium (CATS 2001) - Gold Coast, Australia Duration: Jan 29 2001 → Jan 30 2001 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Electronic Notes in Theoretical Computer Science*,

*42*, 52-66. https://doi.org/10.1016/S1571-0661(04)80878-8