preprint · Logical Methods in Computer Science · 2020
Fixed point combinators as fixed points of higher-order fixed point generators
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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.
Cite this paper
Polonsky, A. (2020). Fixed point combinators as fixed points of higher-order fixed point generators. Logical Methods in Computer Science, Volume 16, Issue 3. https://doi.org/10.23638/lmcs-16(3:7)2020
Polonsky, Andrew. “Fixed point combinators as fixed points of higher-order fixed point generators.” Logical Methods in Computer Science, vol. Volume 16, Issue 3, 2020, https://doi.org/10.23638/lmcs-16(3:7)2020.
Polonsky, Andrew. 2020. “Fixed point combinators as fixed points of higher-order fixed point generators.” Logical Methods in Computer Science Volume 16, Issue 3. https://doi.org/10.23638/lmcs-16(3:7)2020.
Polonsky, A. (2020) ‘Fixed point combinators as fixed points of higher-order fixed point generators’, Logical Methods in Computer Science, Volume 16, Issue 3. Available at: https://doi.org/10.23638/lmcs-16(3:7)2020.
@misc{polonsky2020fixed,
title = {{Fixed point combinators as fixed points of higher-order fixed point generators}},
author = {Polonsky, Andrew},
journal = {Logical Methods in Computer Science},
volume = {Volume 16, Issue 3},
year = {2020},
doi = {10.23638/lmcs-16(3:7)2020}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.