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Fixed point equations inside the algebra of normal forms

Böhm C.

Fundamenta Informaticae

1999

Vol. 37, Nr 4

329-342

Solutions of the equations X=ZX and X=XZ are found and discussed for Z, X normal terms of the lambda-calculus. Obviously fixed point combinators are of no help. Solutions will be independent from any kind of gödelization or coding of data structures, they will be provided by typeless self-application. Different approaches will be shown: algebraic properties, one side invertibility and idempotency. Certain subsets of proper combinators and Church alge-bras between them will be proved to be domains consisting only of fixed points of combinators.