nLab looping combinator

Idea

In combinatory logic, in the ∞-calculus?, or more generally in type theory, a looping combinator is closely akin to a fixed-point combinator, but rather than producing a true fixed point, it produces a sequence of points each of which is the image of the next.

Definition

Definition

A term $Y$ is a looping combinator if for any function term $f$ to which $Y$ can be applied, we have a beta reduction

$$Y f \to_\beta f (Y' f)$$

where $Y'$ is a looping combinator.

This is a coinductive definition. Unraveled explicitly, it means that a looping combinator $Y = Y_0$ comes with a sequence of combinators $Y_n$ for $n\in\mathbb{N}$ and reductions

$$Y_n f \to_\beta f(Y_{n+1} f).$$

Implementing general recursion

A looping combinator is essentially just as good as a fixed-point combinator for implementing general recursion. See the discussion there for details.

Existence

Per Martin-Löf‘s original dependent type theory, which had the additional rule $\vdash Type:Type$, was shown to be inconsistent by Girard's paradox. In the 1980’s, Meyer, Reinhold, and Howe (see references) showed that this paradox could be modified to construct a looping combinator.

References

In the short paper

• Albert Meyer and Mark Reinhold, “‘Type’ is not a type”, POPL 1986

it was claimed that from Girard’s paradox one could actually construct a fixed-point combinator. The proof turned out to be flawed, but it was sufficient to produce a looping combinator. Details can be found in

• Mark Reinhold, “Typechecking is Undecidable When ‘Type’ is a Type”, 1989, citeseer

• Douglas Howe, “The Computational Behaviour of Girard’s Paradox”, Cornell University Technical Report, 1987, link.

Further analysis is in

• Herman Geuvers, Joep Verkoelen, On Fixed point and Looping Combinators in Type Theory (2009) [pdf, citeseer]