Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 35, Number 4 (1994), 573-594.
Normal Forms in Combinatory Logic
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, $R$.
Notre Dame J. Formal Logic, Volume 35, Number 4 (1994), 573-594.
First available in Project Euclid: 20 December 2002
Permanent link to this document
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. https://projecteuclid.org/euclid.ndjfl/1040408614