Note: this is the completed version of lecture MonoidFoldable.

In class exercise: Semigroup, Monoid and Foldable

> module MonoidFoldable where
> -- https://www.seas.upenn.edu/~cis5520/current/lectures/stub/04-classes/MonoidFoldable.html

> import Prelude hiding (all, any, and, or)
> import qualified Data.List as List
> import Test.HUnit

Monoids

First, make sure that you have read the 'Semigroups and Monoids' section of HW 02's SortedList module.

Note that this section defines the following function that tailors a fold operation to a specific instance of the Monoid class.

> foldList :: Monoid b => [b] -> b
> foldList = List.foldr (<>) mempty

For example, because the String type is an instance of this class (using ++ for mappend) we can foldList a list of Strings to a single string.

> -- >>> foldList ["C", "I", "S", "5", "5", "2" ]
> -- "CIS552"

Having instances of the Monoid type class available makes the foldList function more powerful because we can use it with more types. The result type b above must be an instance of this class. For the String type, that is good news! There is really only way to make the String type an instance of the Monoid type class. As a result, Haskell includes this type class instance in the Prelude---it is available in any module and foldList above just works with Strings.

However, for some types, there are multiple but equally reasonable instances of the Monoid class.

The homework assignment shows you that numbers can instantiate this class in multiple ways. We have 0 as an identity element for the associative operator + and 1 as an identity element for the associative operator *. The assignment resolves this tension by introducing two new types.

Like numbers, Booleans can also be made an instance of the Monoid class in two different ways. So for practice, we will do the same thing here.

First, we define two newtype -- i.e. wrappers for the Bool type.

> newtype And = MkAnd { getAnd :: Bool } deriving (Eq,Show)
> newtype Or = MkOr { getOr :: Bool } deriving (Eq,Show)

Above, newtype is like data, but is restricted to a single variant. It is typically used to create a new name for an existing type. This new name allows us to have multiple instances for the same type (as below) or to provide type abstraction (like SortedList in the HW).

Make sure that you understand these definitions. We are defining a type And with single data constructor (also called And). The argument of this data constructor is a record with a single field, called getAnd. What this means is that And and getAnd allow us to convert Bools to And and back.

> -- >>>  :t MkAnd
> -- MkAnd :: Bool -> And

> -- >>> :t getAnd
> -- getAnd :: And -> Bool

Your job is to complete these instances that can tell us whether any of the booleans in a list are true, or whether all of the booleans in a list are true. (See two test cases below for an example of the behavior.)

> -- >>> getOr (foldList (fmap MkOr [True, False, True])) 
> -- True

> -- >>> getAnd (foldList (fmap MkAnd [True, False, True]))
> -- False

> instance Semigroup And where
>   (<>) :: And -> And -> And
>   (<>) = \x y -> MkAnd (getAnd x && getAnd y)

> instance Monoid And where
>   mempty :: And
>   mempty  = MkAnd True

> instance Semigroup Or where
>   (<>) :: Or -> Or -> Or
>   (<>) = \x y -> MkOr (getOr x || getOr y)

> instance Monoid Or where
>   mempty :: Or
>   mempty  = MkOr False

Because And and Or wrap boolean values, we can be sure that our instances have the right properties by testing the truth tables. (There are more concise to write these tests, but we haven't covered them yet.)

> monoidAnd :: Test
> monoidAnd = TestList [      
>   MkAnd False <> (MkAnd False <> MkAnd False) ~?= (MkAnd False <> MkAnd False) <> MkAnd False,
>   MkAnd False <> (MkAnd False <> MkAnd True)  ~?= (MkAnd False <> MkAnd False) <> MkAnd True,
>   MkAnd False <> (MkAnd True  <> MkAnd False) ~?= (MkAnd False <> MkAnd True)  <> MkAnd False,
>   MkAnd False <> (MkAnd True  <> MkAnd True)  ~?= (MkAnd False <> MkAnd True)  <> MkAnd True,
>   MkAnd True  <> (MkAnd False <> MkAnd False) ~?= (MkAnd True  <> MkAnd False) <> MkAnd False,
>   MkAnd True  <> (MkAnd False <> MkAnd True)  ~?= (MkAnd True  <> MkAnd False) <> MkAnd True,
>   MkAnd True  <> (MkAnd True  <> MkAnd False) ~?= (MkAnd True  <> MkAnd True)  <> MkAnd False,
>   MkAnd True  <> (MkAnd True  <> MkAnd True)  ~?= (MkAnd True  <> MkAnd True)  <> MkAnd True, 
>   MkAnd True  <> mempty    ~?= MkAnd True,
>   MkAnd False <> mempty    ~?= MkAnd False,
>   mempty    <> MkAnd True  ~?= MkAnd True,
>   mempty    <> MkAnd False ~?= MkAnd False ]

> monoidOr :: Test
> monoidOr = TestList [
>   MkOr False <> (MkOr False <> MkOr False) ~?= (MkOr False <> MkOr False) <> MkOr False,
>   MkOr False <> (MkOr False <> MkOr True)  ~?= (MkOr False <> MkOr False) <> MkOr True,
>   MkOr False <> (MkOr True  <> MkOr False) ~?= (MkOr False <> MkOr True)  <> MkOr False,
>   MkOr False <> (MkOr True  <> MkOr True)  ~?= (MkOr False <> MkOr True)  <> MkOr True,
>   MkOr True  <> (MkOr False <> MkOr False) ~?= (MkOr True  <> MkOr False) <> MkOr False,
>   MkOr True  <> (MkOr False <> MkOr True)  ~?= (MkOr True  <> MkOr False) <> MkOr True,
>   MkOr True  <> (MkOr True  <> MkOr False) ~?= (MkOr True  <> MkOr True)  <> MkOr False,
>   MkOr True  <> (MkOr True  <> MkOr True)  ~?= (MkOr True  <> MkOr True)  <> MkOr True, 
>   MkOr True  <> mempty    ~?= MkOr True,
>   MkOr False <> mempty    ~?= MkOr False,
>   mempty    <> MkOr True  ~?= MkOr True,
>   mempty    <> MkOr False ~?= MkOr False ]

Foldable

Now, make sure that you have read the section marked The Foldable Typeclass in the MergeSort module.

We can use your Monoid instances for Or and And to generalize operations to any data structure that is in the Foldable type class.

The Foldable type class requires an implementation of the foldMap operation. This operation takes a function that can convert any element in the data structure to a Monoid type, and then uses the Monoid operations to combine all of the elements together.

> -- >>> :t foldMap
> -- foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m

For example, the definition of foldMap for the list type looks something like this.

foldMap :: (Monoid m) => (a -> m) -> [a] -> m
foldMap f l = case l of
                 [] -> mempty  -- use identity element 
                 x:xs -> f x <> foldMap f xs  -- use combining function

Using foldMap we can compute whether all of the booleans stored in a data structure are True

> -- >>> and [True,True,False] 
> -- False

> and :: Foldable t => t Bool -> Bool
> and x = getAnd (foldMap MkAnd x)

Your job is to define these three related operations

> or :: Foldable t => t Bool -> Bool
> or =  getOr . foldMap MkOr

> all :: Foldable t => (a -> Bool) -> t a -> Bool
> all f = getAnd . foldMap (MkAnd . f)

> any :: Foldable t => (a -> Bool) -> t a -> Bool
> any f = getOr . foldMap (MkOr . f)

so that the following tests pass

> -- >>> or [True, False, False, True] 
> -- True

> -- >>> all (>0) [1::Int,2,4,18] 
> -- True

> -- >>> all (>0) [1::Int,-2,4,18] 
> -- False

> -- >>> any (>0) [1::Int,2,4,18]
> -- True

> -- >>> any (>0) [-1::Int,-2,-4,-18]
> -- False

Trees are Foldable

Recall our familiar Tree type. Haskell can derive the Functor instance for this type so we ask it to do so.

> data Tree a = Empty | Branch a (Tree a) (Tree a) deriving (Eq)

And here is an example Tree.

> t1 :: Tree String
> t1 = Branch "d" (Branch "b" (l "a" ) (l "c")) (Branch "f" (l "e") (l "g")) where
>        l x = Branch x Empty Empty

We could make this type an instance of Foldable using the definition of foldrTree from the TreeFolds module.

But, for practice, complete the instance using foldMap.

> instance Foldable Tree where
>   foldMap :: (Monoid m) => (a -> m) -> Tree a -> m
>   
>   foldMap _f Empty = mempty
>   foldMap f (Branch k l r) = foldMap f l <> f k <> foldMap f r
>

With this instance, we can for example, verify that all of the sample strings above have length 1.

> -- >>> all (\x -> length x == 1) t1
> -- True

Finally, look at the documentation for the Foldable class and find some other tree operations that we get automatically for free.

> -- >>> :info Foldable

Oblig-main

> main :: IO ()
> main = do
>   _ <- runTestTT $ TestList [monoidAnd, monoidOr]
>   return ()

CIS 5520: Advanced Programming

Fall 2024

In class exercise: Semigroup, Monoid and Foldable

Monoids

Foldable

Trees are Foldable

Oblig-main