Lightweight invertible enumerations in Haskell

In a previous post I introduced a new Haskell library for enumerations (now on Hackage as simple-enumeration). The Data.Enumeration module defines a type Enumeration a, represented simply by a function Integer -> a which picks out the value of type a at a given index. This representation has a number of advantages, including the ability to quickly index into very large enumerations, and the convenience that comes from having Functor, Applicative, and Alternative instances for Enumeration.

I’ve just uploaded version 0.2 of the package, which adds a new Data.Enumeration.Invertible module with a new type, IEnumeration a, representing invertible enumerations. Whereas a normal enumeration is just a function from index to value, an invertible enumeration is a bijection between indices and values. In particular, alongside the Integer -> a function for picking out the value at an index, an invertible enumeration also stores an inverse function a -> Integer (called locate) for finding the index of a given value.

On the one hand, this comes at a cost: because the type parameter a now occurs both co- and contravariantly, IEnumeration i s no longer an instance of Functor, Applicative, or Alternative. There is a mapE combinator provided for mapping IEnumeration a to IEnumeration b, but in order to work it needs both an a -> b function and an inverse b -> a.

On the other hand, we also gain something: of course the ability to look up the index of a value is nifty, and beyond that we also get a combinator

functionOf :: IEnumeration a -> IEnumeration b -> IEnumeration (a -> b)

which works as long as the IEnumeration a is finite. This is not possible to implement with normal, non-invertible enumerations: we have to take an index and turn it into a function a -> b, but that function has to take an a as input and decide what to do with it. There’s nothing we can possibly do with a value of type a unless we have a way to connect it back to the IEnumeration a it came from.

Here’s a simple example of using the functionOf combinator to enumerate all Bool -> Bool functions, and then locating the index of not:

>>> bbs = functionOf (boundedEnum @Bool) (boundedEnum @Bool)
>>> card bbs
Finite 4
>>> locate bbs not
>>> map (select bbs 2) [False, True]

And here’s an example of enumerating recursive trees, which is parallel to an example given in my previous post. Note, however, how we can no longer use combinators like <$>, <*>, and <|>, but must explicitly use <+> (disjoint sum of enumerations) and >< (enumeration product) in combination with mapE. In return, though, we can find the index of any given tree in addition to selecting trees by index.

data Tree = L | B Tree Tree
  deriving Show

toTree :: Either () (Tree, Tree) -> Tree
toTree = either (const L) (uncurry B)

fromTree :: Tree -> Either () (Tree, Tree)
fromTree L       = Left ()
fromTree (B l r) = Right (l,r)

trees :: IEnumeration Tree
trees = infinite $ mapE toTree fromTree (unit <+> (trees >< trees))

>>> locate trees (B (B L (B L L)) (B (B L (B L L)) (B L (B L L))))
>>> select trees 123
B (B L (B L L)) (B (B L (B L L)) (B L (B L L)))

Of course, the original Data.Enumeration module remains available; there is clearly an inherent tradeoff to invertibility, and you are free to choose either style depending on your needs. Other than the tradeoffs outlined above and a couple other minor exceptions, the two modules export largely identical APIs.


About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
This entry was posted in combinatorics, haskell, projects and tagged , , , , , , . Bookmark the permalink.

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