Haskell Set Comprehensions

Question

Is there any possibility, such as a language extension, to use the syntax of list comprehensions syntax for Data.Set sets?

Example:

f :: Set a -> Set b -> Set (a,b)
f xs ys = [(x,y) | x <- xs , y <- ys] -- this is the set comprehension

Since sets are the mathematical structure that inspired list comprehensions it would be kind of strange not to have any possibility of using them on sets.

And yes, I am aware of MonadComprehensions for using the list-comp syntax with any Monad / MonadPlus type but AFAIK sets can't even be monads because of the Ord constraint in most functions.

Answer 1

In Theory, No

There are no language extensions which allow for "set comprehensions."

The differences between a Set and a List are:

1. The Set is unordered while the List is ordered
2. The elements of the Set are unique while the List may have duplicate elements
3. The type of the Set is an instance of Ord while the List has no type restrictions.

You can see that all possible Set s are a strict subset of all possible List s. This means that we can achieve "set comprehension" simply be using a list comprehension and converting it to a Set . Lazy evaluation will often make the "set generation" efficient deriving from most finite list comprehensions. However, list comprehensions resulting in infinite lists are unlikely to result in efficient "set comprehension."

Example:

import Data.Set

set :: Ord a => Set a
set = fromList [x * y | x <- [1..10], y <- [1..10]]

In Practice, Yes

Using the set-monad package, you can define a Set as an instance of Monad and using the language extension MonadComprehensions you can achieve a "set comprehension".

Example:

{-# LANGUAGE MonadComprehensions #-}
import Data.Set.Monad

set1 :: Set (Int,Int)
set1 = do 
         a <- fromList [1 .. 4]
         b <- fromList [1 .. 4]
         return (a,b)

-- Look a "set comprehension"
set2 :: Set (Int,Int)
set2 = [ (a,b) | (a,b) <- set1, even a, even b ]

Use whichever method makes the most sense for your project. Profile both before making a decision!