## Journal of Symbolic Logic

### The Church-Rosser property in dual combinatory logic

Katalin Bimbó

#### Abstract

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.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 132-152.

Dates
First available in Project Euclid: 21 February 2003

https://projecteuclid.org/euclid.jsl/1045861508

Digital Object Identifier
doi:10.2178/jsl/1045861508

Mathematical Reviews number (MathSciNet)
MR1959314

Zentralblatt MATH identifier
1045.03017

#### Citation

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

#### References

• K. Bimbó Investigation into combinatory systems with dual combinators, Studia Logica, vol. 66 (2000), pp. 285--296.
• K. Bimbó and J. M. Dunn Two extensions of the structurally free logic $LC$, Logic Journal of IGPL, vol. 6 (1998), pp. 403--424.
• A. Church The calculi of lambda-conversion, 1st ed., Princeton University Press, Princeton,1941.
• H. B. Curry and R. Feys Combinatory logic, 1st ed., vol. I, North-Holland, Amsterdam,1958.
• H. B. Curry, J. R. Hindley, and J. P. Seldin Combinatory logic, vol. II, North-Holland, Amsterdam,1972.
• J. M. Dunn and R. K. Meyer Combinatory logic and structurally free logic, Logic Journal of IGPL, vol. 5 (1997), pp. 505--537.
• J. R. Hindley An abstract form of the Church-Rosser theorem, I, Journal of Symbolic Logic, vol. 34 (1969), pp. 545--560.
• J. R. Hindley and J. P. Seldin Introduction to combinators and $\lambda$-calculus, Cambridge University Press, Cambridge (UK),1986.
• S. C. Kleene Proof by cases in formal logic, Annals of Mathematics, vol. 35 (1934), pp. 529--544.
• J. Lambek From categorial grammar to bilinear logic, Substructural logics (K. Došen and Schroeder-Heister P., editors), Clarendon and Oxford University Press, Oxford (UK),1993, pp. 207--237.
• R. K. Meyer, K. Bimbó, and J. M. Dunn Dual combinators bite the dust, (abstract), Bulletin of Symbolic Logic, vol. 4 (1998), pp. 463--464.
• B. K. Rosen Tree-manipulating systems and Church-Rosser theorems, Journal of the Association for Computing Machinery, vol. 20 (1973), pp. 160--187.
• J. B. Rosser A mathematical logic without variables, Annals of Mathematics, vol. 2 (1936), pp. 127--150.
• J. Staples Church-Rosser theorems for replacement systems, Algebra and logic (J. N. Crossley, editor), Lecture Notes in Mathematics, vol. 450, Springer, Berlin,1975, pp. 291--307.