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June 2005 The Church-Rosser property in symmetric combinatory logic
Katalin Bimbó
J. Symbolic Logic 70(2): 536-556 (June 2005). DOI: 10.2178/jsl/1120224727


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.


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Katalin Bimbó. "The Church-Rosser property in symmetric combinatory logic." J. Symbolic Logic 70 (2) 536 - 556, June 2005.


Published: June 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1090.03003
MathSciNet: MR2140045
Digital Object Identifier: 10.2178/jsl/1120224727

Primary: 03B40
Secondary: 03B47 , 03B70

Keywords: Church-Rosser property , combinatory logic , dual combinatory logic , structurally free logic , substructural logics , symmetric combinatory logic , symmetric λ-calculus , type theory

Rights: Copyright © 2005 Association for Symbolic Logic


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Vol.70 • No. 2 • June 2005
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