Journal of Symbolic Logic 70 (2):536 - 556 (2005)
|Abstract||Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property.|
|Keywords||No keywords specified (fix it)|
|Categories||No categories specified (fix it)|
|Through your library||Configure|
Similar books and articles
Katalin Bimbó (2003). The Church-Rosser Property in Dual Combinatory Logic. Journal of Symbolic Logic 68 (1):132-152.
Katalin Bimbó (2000). Investigation Into Combinatory Systems with Dual Combinators. Studia Logica 66 (2):285-296.
Katalin Bimbó (2005). The Church-Rosser Property in Symmetric Combinatory Logic. Journal of Symbolic Logic 70 (2):536-556.
Katalin Bimbó (2004). Semantics for Dual and Symmetric Combinatory Calculi. Journal of Philosophical Logic 33 (2):125-153.
Katalin Bimb� (2003). The Church-Rosser Property in Dual Combinatory Logic. Journal of Symbolic Logic 68 (1):132-152.
M. W. Bunder (1988). Arithmetic Based on the Church Numerals in Illative Combinatory Logic. Studia Logica 47 (2):129 - 143.
Kenneth Loewen (1968). The Church Rosser Theorem for Strong Reduction in Combinatory Logic. Notre Dame Journal of Formal Logic 9 (4):299-302.
J. Roger Hindley (1972). Introduction to Combinatory Logic. Cambridge [Eng.]University Press.
Sabine Broda & Luís Damas (1997). Compact Bracket Abstraction in Combinatory Logic. Journal of Symbolic Logic 62 (3):729-740.
Henk Barendregt, Martin Bunder & Wil Dekkers (1993). Systems of Illative Combinatory Logic Complete for First-Order Propositional and Predicate Calculus. Journal of Symbolic Logic 58 (3):769-788.
Haskell B. Curry (1958). Combinatory Logic. Amsterdam, North-Holland Pub. Co..
Katalin Bimbó (2005). Admissibility of Cut in LC with Fixed Point Combinator. Studia Logica 81 (3):399 - 423.
Barkley Rosser (1942). New Sets of Postulates for Combinatory Logics. Journal of Symbolic Logic 7 (1):18-27.
J. Roger Hindley (1986). Introduction to Combinators and [Lambda]-Calculus. Cambridge University Press.
Lou Goble (2007). Combinatory Logic and the Semantics of Substructural Logics. Studia Logica 85 (2):171 - 197.
Sorry, there are not enough data points to plot this chart.
Added to index2011-05-29
Total downloads1 ( #274,982 of 549,198 )
Recent downloads (6 months)0
How can I increase my downloads?