Modular Construction of Fixed Point Combinators and Clocked Boehm Trees

Fixed point combinators (and their generalization: looping combinators) are classic notions belonging to the heart of λ-calculus and logic. We start with an exploration of the structure of fixed point combinators (fpc’s), vastly generalizing the well-known fact that if Y is an fpc, Y (SI) is again an fpc, generating the Böhm sequence of fpc’s. Using the infinitary λ-calculus we devise infinitely many other generation schemes for fpc’s. In this way we find schemes and building blocks to construct new fpc’s in a modular way. Having created a plethora of new fixed point combinators, the task is to prove that they are indeed new. That is, we have to prove their βinconvertibility. Known techniques via Böhm Trees do not apply, because all fpc’s have the same Böhm Tree (BT). Therefore, we employ ‘clocked BT’s’, with annotations that convey information of the tempo in which the data in the BT are produced. BT’s are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for λ-terms. The corresponding equality is strictly intermediate between =β and =BT, the equality in the classical models of λ-calculus. An analogous approach pertains to Lévy–Longo and Berarducci trees. Finally, we increase the discrimination power by a precision of the clock notion that we call ‘atomic clock’. The theory of sage birds (technically called fixed point combinators) is a fascinating and basic part of combinatory logic; we have only scratched the surface. R. Smullyan [17].