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Fall 1994 Normal Forms in Combinatory Logic
Patricia Johann
Notre Dame J. Formal Logic 35(4): 573-594 (Fall 1994). DOI: 10.1305/ndjfl/1040408614


Let $R$ be a convergent term rewriting system, and let $CR$-equality on (simply typed) combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long $\beta$-normal forms in the lambda calculus. As a consequence, the translates of long $\beta R$-normal forms are easily seen to serve as canonical representatives for $CR$-equivalence classes of combinatory logic terms for non-empty, as well as for empty, $R$.


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Patricia Johann. "Normal Forms in Combinatory Logic." Notre Dame J. Formal Logic 35 (4) 573 - 594, Fall 1994.


Published: Fall 1994
First available in Project Euclid: 20 December 2002

zbMATH: 0830.03005
MathSciNet: MR1334292
Digital Object Identifier: 10.1305/ndjfl/1040408614

Primary: 03B40
Secondary: 68Q42

Rights: Copyright © 1994 University of Notre Dame


Vol.35 • No. 4 • Fall 1994
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