# [Haskell-cafe] Optimizing Fold Expressions

Conal Elliott conal at conal.net
Tue Apr 2 05:00:28 CEST 2013

```You can use a general fold and unfold, without any type-specific
programming if you re-express Expr as the least fixed point of its
underlying "base functor":

> data ExprF a = Add a a | Sub a a | Mul a a | Eq  a a | B Bool | I Int
>   deriving (Show,Functor)
>
> data Expr = Fix ExprF

Then use the standard definitions:

> newtype Fix f = Roll { unRoll :: f (Fix f) }
>
> fold :: Functor f => (f b -> b) -> (Fix f -> b)
> fold h = h . fmap (fold h) . unRoll
>
> unfold :: Functor f => (a -> f a) -> (a -> Fix f)
> unfold g = Roll . fmap (unfold g) . g

Also handy:

> hylo :: Functor f => (f b -> b) -> (a -> f a) -> (a -> b)
> hylo h g = fold h . unfold g

For details, see Jeremy Gibbons's paper "Calculating functional programs".
There are probably easier sources as well.

-- Conal

On Sat, Mar 30, 2013 at 11:45 AM, J. J. W. <bsc.j.j.w at gmail.com> wrote:

> Dear all,
>
> I was wondering whether it was possible to write fold expressions more
> elegantly. Suppose I have the following
> datastructure:
>
> data Expr = Add Expr Expr
>           | Sub Expr Expr
>           | Mul Expr Expr
>           | Eq  Expr Expr
>           | B Bool
>           | I Int
>           deriving Show
>
>  type ExprAlgebra r = (r -> r -> r, -- Add
>                       r -> r -> r, -- Sub
>                       r -> r -> r, -- Mul
>                       r -> r -> r, -- Eq
>                       Bool   -> r, -- Bool
>                       Int -> r     -- Int
>                       )
>
> foldAlgebra :: ExprAlgebra r -> Expr -> r
> foldAlgebra alg@(a, b, c ,d, e, f) (Add x y) = a (foldAlgebra alg x)
> (foldAlgebra alg y)
> foldAlgebra alg@(a, b, c ,d, e, f) (Sub x y) = b (foldAlgebra alg x)
> (foldAlgebra alg y)
> foldAlgebra alg@(a, b, c ,d, e, f) (Mul x y) = c (foldAlgebra alg x)
> (foldAlgebra alg y)
> foldAlgebra alg@(a, b, c ,d, e, f) (Eq  x y) = d (foldAlgebra alg x)
> (foldAlgebra alg y)
> foldAlgebra alg@(a, b, c ,d, e, f) (B b')    = e b'
> foldAlgebra alg@(a, b, c ,d, e, f) (I i)     = f i
>
> If I am correct, this works, however if we for example would like to
> replace all Int's by booleans (note: this is to illustrate my problem):
>
> replaceIntByBool = foldAlgebra (Add, Sub, Mul, Eq, B, \x -> if x == 0 then
> B False else B True)
>
> As you can see, a lot of "useless" identity code. Can I somehow optimize
> this? Can someone give me some pointers how I can write this more clearly
> (or with less code?) So I constantly don't have to write Add, Sub, Mul, for
> those things that I just want an "identity function"?
>
> Thanks in advance!
>
> Jun Jie
>
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