Notre Dame Journal of Formal Logic

Normal Forms in Combinatory Logic

Patricia Johann


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, $R$.

Article information

Notre Dame J. Formal Logic, Volume 35, Number 4 (1994), 573-594.

First available in Project Euclid: 20 December 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B40: Combinatory logic and lambda-calculus [See also 68N18]
Secondary: 68Q42: Grammars and rewriting systems


Johann, Patricia. Normal Forms in Combinatory Logic. Notre Dame J. Formal Logic 35 (1994), no. 4, 573--594. doi:10.1305/ndjfl/1040408614.

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