Fixed point combinators as fixed points of higher-order fixed point generators

TL;DR

This paper explores the structure of fixed point combinator generators, identifying classes with specific properties, providing conditions for their fixed points, and proposing conjectures about their algebraic structure.

Contribution

It introduces the concept of fpc generators, analyzes their properties, and formulates new conjectures on their fixed points and algebraic structure.

Findings

Identified robust classes of fpc generators based on properties like injectivity and constancy.

Provided sufficient conditions for the existence of fixed points of generator sequences.

Formulated conjectures relating to the structure of the monoid of fpc generators.

Abstract

Corrado B\"ohm once observed that if $Y$ is any fixed point combinator (fpc), then $Y(\lambda yx.x(yx))$ 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 $\textit{fpc generator}$ to be any sequence of terms $G_1,\dots,G_n$ such that \[ Y \in FPC \Rightarrow Y G_1 \cdots G_n \in FPC \] In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several robust classes of such generators, by examining their elementary properties like injectivity and (weak) constancy. We provide sufficient conditions for existence of fixed points of a given generator $(G_1,\cdots,G_n)$: an fpc $Y$ such that $Y = Y G_1 \cdots G_n$. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This statement generalizes Statman's conjecture on non-existence of "double fpcs": fixed points of the generator $(G) = (\lambda yx.x(yx))$ discovered by B\"ohm. Finally, we define and make a few observations about the monoid of (weak) fpc generators. This enables us to formulate new a conjecture about their structure.