Quickie: Generating the powerset

View literate file on Github

> import Control.Monad

The traditional way to generate the powerset of a list of elements is the following

> powerset :: [a] -> [[a]]
> powerset (x:xs) = map (x:) (powerset xs) ++ powerset xs
> powerset [] = [[]]
ghci> powerset [1,2,3]

But, I found this little gem of a way hidden in the Haskell wiki.

> powerset2 :: [a] -> [[a]]
> powerset2 = filterM (const [True,False])

This version runs a filter using the List as the underlying monad. Let’s see why this works on [1,2,3].

First, here is the definition of filterM

filterM          :: (Monad m) => (a -> m Bool) -> [a] -> m [a]
filterM _ []     =  return []
filterM p (x:xs) =  do
   flg <- p x
   ys  <- filterM p xs
   return (if flg then x:ys else ys)

and it’s expansion for [1,2,3] in powerset2

   powerset2 [1,2,3]
== filterM (const [True,False]) [1,2,3]
== [True,False] >>= \flg ->
   filterM (const [True,False]) [2,3] >>= \ys ->
   return (if b then (1:ys) else ys)

There we have it. Another use of List for computations binding on multiple return values (Bool in this case).

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