Comments on: Fixed Point Datatypes

By: Albert Y. C. Lai

Albert Y. C. Lai — Wed, 09 Jul 2008 19:52:44 +0000

Today on a math IRC channel I saw the question

“what is sqrt(6+sqrt(6+… ?”

It is an instance of a rather popular kind of exercises in calculus.

Letting f(x) = sqrt(6 + x), the question asks

“what is f(f(… ?”

The solution is to first ensure that the sequence s0 = some starting point, s1 = f(s0), s2 = f(s1), … converges, then find out what it converges to, by solving x = sqrt(6+x), i.e., x = f(x).

I think it illustrates a strong analogy between calculus fixed points and programming fixed points, including the self-application part “what is f(f(… ?”

By: web design

web design — Wed, 09 Jul 2008 14:01:59 +0000

Type algebra is the hot new thing, it seems. What with this and all the zipper / type calculus hackery of late.

By: bradenshep

bradenshep — Tue, 08 Jul 2008 04:25:24 +0000

Thanks for the comment.

I’m not sure how much of this is semantics and how much is content, but:

When I refer to self-application, I meant taking a type like Pair a b and using Pair a b again for b: Pair a (Pair a b). Perhaps self-application isn’t the right term for that. I think it mostly comes from when I was first learning about fixpoints, this “self-application” leap was what made it slip into focus for me. I couldn’t figure out how fix could take an arbitrary function and find a fixpoint of it while I was thinking about it from the f(x) = x viewpoint.

I do understand the point about a_n converging to 0, and thinking about the constant functor definitely improved my understanding.

By: Derek Elkins

Derek Elkins — Tue, 08 Jul 2008 03:31:55 +0000

The notion of fixpoint in programming is the same as in “calculus” (or better algebra). If F is a functor, then G is it’s fixed point if F(G) ~ G where ~ is being used for isomorphic to. Just as in algebra or calculus, one way (that need not always work) at arriving at a fixed point is via iteration. Let’s take f(x) = x^2 as an example. The sequence a_0 = 1/2 and a_n = f(a_{n-1}) converges to 0, one of the fixed points of x^2. There is nothing inherent about self-application in the programming notion of fixed point and in fact none of your examples have any self-application. There is a definite relationship between recursion and fixed points, though fixed points aren’t inextricably linked to recursion. For example the constant functor for any constant or the identity functor both have fixed points as well as the functor PA=AxA (namely P1) and other examples.