Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 68, Issue 1 (2003), 132-152.
The Church-Rosser property in dual combinatory logic
Dual combinators emerge from the aim of assigning formulas containing $\leftarrow$ as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.
J. Symbolic Logic, Volume 68, Issue 1 (2003), 132-152.
First available in Project Euclid: 21 February 2003
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Bimbó, Katalin. The Church-Rosser property in dual combinatory logic. J. Symbolic Logic 68 (2003), no. 1, 132--152. doi:10.2178/jsl/1045861508. https://projecteuclid.org/euclid.jsl/1045861508