On (weak) fpc generators
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Corrado Böhm once observed that if Y is any fixed point combinator (fpc), then Y(λ yx.x(yx)) is again fpc. He thus discovered the first "fpc generating scheme" -- a generic way to build new fpcs from old. Continuing this idea, define an fpc generator to be any sequence of terms G_1,...,G_n such that Y is fpc YG_1... G_n is fpc In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several classes of such generators, and examine elementary properties like injectivity and constancy. We provide sufficient conditions for existence of fixed points of a given generator (G_1,..,G_n): an fpc Y such that Y = YG_1... G_n. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This generalizes Statman's conjecture on the non-existence of "double fpcs": fixed points of the generator (G) = (λ yx.x(yx)) discovered by Böhm.
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