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 βηRequality 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 βR-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 β-normal forms in the lambda calculus. As a consequence, the translates of long βR-normal forms are easily seen to serve as canonical representatives for CR-equivalence classes of combinatory logic terms for nonempty, as well as for empty, R.