How to write a function whose fixed point is the Y combinator?

Question

I must find a function G which has a fixed point, and this fixed point must be the Y combinator.

Recall that YF = F(YF) for all F . I want a function G such that GY = Y .

How to write such a function in in Haskell?

Answer 1

Let

g w = \f -> f (w f)

Then, if

y f = f (y f)

we have

g y
-- definition of g
= \f -> f (y f)
-- property of y
= \f -> y f
-- eta-conversion
= y

Hence, y is a fixed point of g .

More than that: the set of fixed points of g is precisely the set of fixed point combinators. This is because above we did not exploit the definition of y , but merely the fact that it is a fixed point combinator.

Note that y is also a fixed point of id (anything is a fixed point of id ) and of const y (as @DanielWagner writes in his answer).

Answer 2

对于所有值v ，函数const v具有（唯一的，因此也是最小的）定点v 。

Answer 3

Daniel's answer is very good, but I'd like to add the other extreme. Everything is a fixed point of the identity function. Thus, take G to be

\x -> x

indeed:

GY = (\x -> x)Y = Y