Dual combinators emerge from the aim of assigning formulas containing ← 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.
[1]
J. Roger Hindley,et al.
Introduction to combinators and λ-calculus
,
1986,
Acta Applicandae Mathematicae.
[2]
A. Church.
The calculi of lambda-conversion
,
1941
.
[3]
Katalin Bimbó.
Investigation into Combinatory Systems with Dual Combinators
,
2000,
Stud Logica.
[4]
S. Kleene.
Proof by Cases in Formal Logic
,
1934
.
[5]
J. Rosser.
A Mathematical Logic Without Variables. I
,
1935
.
[6]
Robert K. Meyer,et al.
Combinators and Structurally Free Logic
,
1997,
Log. J. IGPL.
[7]
John Staples.
Church-Rosser theorems for replacement systems
,
1975
.
[8]
Katalin Bimbó,et al.
Two Extensions of the Structurally Free Logic LC*
,
1998,
Log. J. IGPL.