Snippet: Power Set

By Alex Beal
September 23, 2015

Below is an explanation of the intuition behind Olaf Klinke’s power set function posted to the haskell-cafe. See his explanation here.

One way of viewing the monad instance for list is as a container. We can gather elements into the container and apply functions to these elements that themselves produce containers of elements.

Another interpretation is to view the list monad as a way of encoding ambiguity or nondeterminism. A function of type a -> b must produce a single b for each a. A function of type a -> m b can produce a b lifted into an effect m. If that effect is ambiguity, then the function can produce multiple bs for each a–it doesn’t need to choose. The result is a sort of superposition.

> type Amb a = [a]
> -- An ambiguous Boolean that is both True and False.
> ambiguousBool :: Amb Bool
> ambiguousBool = [True, False]

Like all monadic effects, if a computation depends on ambiguity, it too becomes ambiguous. filterM, for example, allows filtering to happen ambiguously, where the result is an ambiguous list. If ambiguousBool is used to filter elements, the result is an ambiguous list where each elements has been both removed and kept, in other words, the power set.1

> import Control.Monad
> -- filterM :: Monad m => [a] -> (a -> m Bool) -> m [a]
> filterM (\_ -> ambiguousBool) [1,2,3] :: Amb [a]
> -- Result: [[1,2,3],[1,2],[1,3],[1],[2,3],[2],[3],[]]
> 
> powerSet :: [a] -> [[a]]
> powerSet = filterM (\_ -> ambiguousBool)

  1. Assuming the input list forms a set.