Sharing vs. non-sharing fixed-point combinator

Question

This is the usual definition of the fixed-point combinator in Haskell:

fix :: (a -> a) -> a
fix f = let x = f x in x

On https://wiki.

Answer 1

I think you already received excellent answers, from a GHC/Haskell perspective. I just wanted to chime in and add a few historical/theoretical notes.

The correspondence between unfolding and cyclic views of recursion is rigorously studied in Hasegawa's PhD thesis: https://www.springer.com/us/book/9781447112211

(Here's a shorter paper that you can read without paying Springer: https://link.springer.com/content/pdf/10.1007%2F3-540-62688-3_37.pdf)

Hasegawa assumes a traced monoidal category, a requirement that is much less stringent than the usual PCPO assumption of domain theory, which forms the basis of how we think about Haskell in general. What Hasegawa showed was that one can define these "sharing" fixed point operators in such a setting, and established that they correspond to the usual unfolding view of fixed points from Church's lambda-calculus. That is, there is no way to tell them apart by making them produce different answers.

Hasegawa's correspondence holds for what's known as central arrows; i.e., when there are no "effects" involved. Later on, Benton and Hyland extended this work and showed that the correspondence holds for certain cases when the underlying arrow can perform "mild" monadic effects as well: https://pdfs.semanticscholar.org/7b5c/8ed42a65dbd37355088df9dde122efc9653d.pdf

Unfortunately, Benton and Hyland only allow effects that are quite "mild": Effects like the state and environment monads fit the bill, but not general effects like exceptions, lists, or IO. (The fixed point operators for these effectful computations are known as mfix in Haskell, with the type signature (a -> m a) -> m a, and they form the basis of the recursive-do notation.)

It's still an open question how to extend this work to cover arbitrary monadic effects. Though it doesn't seem to be receiving much attention these days. (Would make a great PhD topic for those interested in the correspondence between lambda-calculus, monadic effects, and graph-based computations.)