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User-defined datatypes and Trees

> module Datatypes where

> import Prelude hiding (Maybe,Just,Nothing,Either,Left,Right)
> import Test.HUnit

> -- https://www.seas.upenn.edu/~cis5520/current/lectures/stub/02-trees/Datatypes.html

User-defined datatypes

So far, we've mostly talked about how to use the types that appear in the Haskell standard library. We also discussed a few type synonyms, like

type XY = (Double, Double)

from the last lecture, but we haven't described any ways to define really new types.

Days

As a motivating example, suppose you are writing an application that deals with calendars, and you need to represent the days of the week. You might be tempted to String or Int, but both of these choices have downsides. If you use

type Day = String

there will be lots of Days that don't actually represent real days. Also, you will need to devise and adhere to some sort of standardization - is Monday represented by "Monday", "monday", "MONDAY", or "Mon"? Should you handle more than one of these?

The choice

type Day = Int

has similar problems. There are lots of integers that won't represent valid days. And you'll have to remember whether you pick Sunday or Monday to be the first day of the week, and whether it is represented by 0 or 1.

Haskell has a better solution: user-defined datatypes.

> data Day = Monday
>          | Tuesday
>          | Wednesday
>          | Thursday
>          | Friday
>          | Saturday
>          | Sunday
>     deriving (Show, Eq)

The new values (Monday, Tuesday, etc.) are called "constructors" or "data constructors". This is a very simple example of a datatype (basically just an enumeration), but we'll see more examples soon.

The deriving line enables printing and equality for this datatype. We'll see more about this next week, but it is common to do this for every datatype. (If we leave this line off, we won't be able to use the == operator or show function with these values. This is particularly important for working with ghci (which uses show to print out the result of evaluation) and HUnit which uses == to test equality.)

The great thing about datatype is that we can define functions by pattern matching! For example:

> nextWeekday :: Day -> Day
> nextWeekday Monday    = Tuesday
> nextWeekday Tuesday   = Wednesday
> nextWeekday Wednesday = Thursday
> nextWeekday Thursday  = Friday
> nextWeekday Friday    = Monday
> nextWeekday Saturday  = Monday
> nextWeekday Sunday    = Monday

This is great. Now we don't have to worry about the difference between "Monday" and "monday" or which integer corresponds to which day. If we make a typo (for example, write Frday instead of Friday), the compiler will tell us at compile time. If we forget to handle one of the days in some function, the compiler will warn us about that too (via the incomplete patterns warning, which we have enabled at the top of this file).

Let's write one more function on Days, to compute when a package will arrive by "two day shipping":

> twoBusinessDays :: Day -> Day
> twoBusinessDays d = nextWeekday (nextWeekday d)

Shapes

Datatypes can carry data values, too. For example, here is a datatype for representing shapes:

> data Shape
>     = Circle    Double Double Double
>     | Rectangle Double Double Double Double
>     deriving (Eq, Show)

Here, Circle and Rectangle are the constructors - every Shape value must be one or the other. Each constructor takes some arguments:

• A Circle is specified by three Doubles. These represent the x and y coordinates of the center and the radius.

• A Rectangle is specifed by four Doubles. The first two are the coordinates of the lower left corner, and the second two are the coordinates of the upper right corner.

We can pattern match on shapes. For example, here is a function that computes the area of any Shape:

> area :: Shape -> Double
> area (Circle x y r) = pi * r * r
> area (Rectangle llx lly urx ury) = width * height where
>     width  = urx - llx
>     height = ury - lly

Note that constructors are first-class Haskell values, and -- like any value -- they have types.

For example the types of Monday and Tuesday shouldn't surprise you:

Monday  :: Day
Tuesday :: Day

The constructors that take arguments have function types. For example, you must apply Circle to three Doubles to get a Shape:

Circle    :: Double -> Double -> Double -> Shape

Rectangle :: Double -> Double -> Double -> Double -> Shape

However, note that although data constructors have types that look like functions, they are not functions. From a syntactic point of view, the names of data constructors start with capital letters so that they cannot be confused with normal variables, which might be the names of functions.

Like functions, they can be applied to arguments. However, they have a special power that normal functions do not have: they can be used in pattern matching. If we have a Shape we can see what data constructor we used to create that Shape.

Datatype values also look different from function values. If we ask Haskell to evaluate an expression of type Day, we are going to get one of the data constructors. (i.e. Monday, Tuesday, etc.) On the other hand, function values are lambda expressions (i.e. \x -> x + 3) or primitive operators.

Datatypes vs. Objects

At this point, you might think a little about the comparison between representing shapes via datatypes in Haskell or via objects in a language like Java. Maybe your introductory Java class uses subtyping to represent the same relationship between Circle's and Rectangles. One way it could have done so is with this code.

interface Shape {
   public double area();
}
class Circle implements Shape {
   // fields
   private final double x;
   private final double y;
   private final double r;

   // constructor definition omitted

   public double area() {
     return Math.pi * r * r;
   }
} 
class Rectangle implements Shape {
   // fields
   private final double llx;
   private final double lly;
   private final double urx;
   private final double ury;

   // constructor definition omitted  

   public double area() {
      double width  = urx - llx;
      double height = ury - lly;
      return width * height;
   }
}

In these class definitions, we have two object types Circle and Rectangle that carry their associated data. The constructors for these objects act like the data constructors of a datatype. The area method works for both circles and rectangles, using dynamic dispatch instead of pattern matching to select the appropriate code.

One difference between these two implementations is that Haskell datatypes are easy to extend with new operations but harder to extend with new variants. If we want a new operation on Shapes, we need only define a new function by pattern matching. However, to add a new kind of Shape we need to add a new variant to the datatype definition and update each of the function definitions with a new case in the pattern matching. (The compiler will tell us if we miss any.)

In Java, it is the reverse situation. We can easily add a new kind of Shape by adding a new object that implements the interface. However, to add a new operation, we need to edit the interface to include the new method and then update all of the existing classes with the new method definition. (The compiler will tell us if we miss any.)

This contrast between these two ways of defining data types is commonly called the expression problem.

Records

One nice feature of the Java implementation is that it gives names to the individual fields.

Suppose we would like to represent a point in space, using its x and y coordinates. One way we might do so is with the following datatype definition:

    data Point = NewPoint Double Double

This line defines the Point type as the application of the data constructor NewPoint to two Doubles, the x and y coordinates of the point. However, when we construct a point using this data constructor, we need to remember that the first double corresponds to the x value and the second corresponds to the y value.

> point :: Point
> point = NewPoint 2.0 1.0  -- Be careful, the order matters

We can do better in Haskell by giving names to the individual arguments of data constructors, like this:

> data Point = NewPoint { x :: Double, y :: Double }
>     deriving (Show, Eq)

This form of datatype definition is called a record.

The names x and y, are called field names and provide more documentation about the type. They let us know that we intend the first argument to be the x coordinate and the second to be the y coordinate. We construct a Point we can use these names for better readability.

> point1 :: Point
> point1 = NewPoint { x = 2.0, y = 1.0 }  -- explicit field names

We can also use these names for more flexibility during construction. For example, we can swap the order of the arguments.

> point2 :: Point
> point2 = NewPoint { y = 1.0, x = 2.0 }  -- Same as `point1`

Furthermore, each field name also defines a selector for that component of the data structure. The selector is a function with the same name as the field that can be used to directly extract that component from the type.

> x1 :: Double
> x1 = x point1

> -- Access the x component of point2
> -- >>> x point2

When taking arguments that use records, we can either use the record selectors to access their components, or use pattern-matching.

> distFromOrigin :: Point -> Double
> distFromOrigin NewPoint {x = px, y = py} = sqrt (px * px + py * py)

Now, rewrite this function using selectors x and y:

> distFromOrigin' :: Point -> Double
> distFromOrigin' p = sqrt (x p * x p + y p * y p)

Which version is easier to read? Opinions differ.

Things to watch out for with records in Haskell:

• Records must be defined as part of a datatype definition.

• The selectors are first-class functions. This is really great for higher-order programming, i.e. we can easily access all of the x components from a list of points with

     map x points

• Record selectors must be unique within a module. If Point has an x field name, then no other datatype in that module can use x as a field name.

• Record selectors are just normal variable names. So if Point has an x component, there cannot be another top-level definition in the module called x. (Recall that all toplevel definitions in a module must be unique, but local definitions may shadow toplevel definitions. )

• It's idiomatic to "pun" when pattern matching records. For example, we name the variables using the same names as the selectors. (Note that this can also be confusing, and some Haskellers advise against this practice.)

> distFromOrigin'' :: Point -> Double
> distFromOrigin'' NewPoint {x = x, y = y} = sqrt (x * x + y * y)

• Records are immutable in Haskell. There is no way to modify a component when it is created. However, there is an easy way to construct new values that share components with existing structures.

> point3 :: Point
> point3 = point1 { x = 2.0 }

This syntax is called record update. It is a purely functional way to create a new record value by reusing the components of an existing record. Above, point3 is a Point withxcomponent equal to 2.0, and all other components (which is onlyyhere) the same aspoint1`. When records have many components, this can be a very useful feature.

• Haskell's record system is far from perfect. It's strange that records must be defined as part of data constructors. The fact that different record types in the same module cannot share fields names can be awkward. It can be tedious to work with purely functional record update when you have nested records. Some of these issues are addressed by language extensions (such as DuplicateRecordFields) and by sophisticated libraries. However, these are advanced topics so we won't be covering them here.

Recursive Datatypes

Datatypes can be defined recursively. That is, their constructors can take other elements of the same type as arguments.

For example, we could define a type representing nonempty lists of integers:

> data IntListNE = ISingle Int
>                | ICons Int IntListNE

So that the list [1,2,3] is represented as:

> oneTwoThree :: IntListNE
> oneTwoThree = ICons 1 (ICons 2 (ISingle 3))

For comparison with Haskell's built-in lists, it might help to think of this as:

> oneTwoThree' :: IntListNE
> oneTwoThree' = 1 `ICons` (2 `ICons` ISingle 3)  -- backticks for infix

Nonempty lists are great in that we can write a total head function (i.e. this version of head is not partial like the one for regular lists.)

> -- >>> safeHead oneTwoThree
> -- 1
> safeHead :: IntListNE -> Int
> 
> safeHead (ISingle i) = i
> safeHead (ICons i _) = i

We can define functions by recursion on IntListNEs too, of course. Write a function to calculate the sum of a non-empty list of integers.

> -- >>> sumOfIntListNE oneTwoThree
> -- 6
> sumOfIntListNE :: IntListNE -> Int
> 
> sumOfIntListNE (ISingle i)    = i
> sumOfIntListNE (ICons i rest) = i + sumOfIntListNE rest

Polymorphic Datatypes

It would sure be annoying to have a seperate kind of list for each type of data! Luckily, we know Haskell's list type is polymorphic: you can have a list of type [a] for any a.

We can define new polymorphic datatypes too. For example, we can easily make the non-empty lists above polymorphic.

As another example, here is the definition of the Maybe type (from the Prelude) that we've used in past lectures:

> data Maybe a = Nothing | Just a

Notice that the type Maybe itself takes an argument: the type variable a. We're also allowed to use that type variable in the constructors. So Just is a constructor that can be applied to values of any type and will create a Maybe with the same type:

Just :: a -> Maybe a

Thus, Just and Nothing work at any type:

> noInt :: Maybe Int
> noInt = Nothing

> justTrue :: Maybe Bool
> justTrue = Just True

> justThree :: Maybe Int
> justThree = Just 3

A number of other polymorphic datatypes appear in the standard library. For example, here's a standard datatype to carry around values that could have either of two types:

> data Either a b = Left a | Right b

Either is often useful for error handling. Sometimes, returning a Maybe a isn't quite good enough because you'd like to give a helpful error message. Either String a works a little better: instead of Nothing, use Left msg in the case of an error, and instead of Just v, use Right v in case things are... all right.

For example, here's a safer integer division function:

> safeDiv :: Int -> Int -> Either String Int
> safeDiv _ 0 = Left "You can't divide by zero, silly."
> safeDiv x y = Right $ x `div` y

Of course, Either is more useful when things can go wrong in more than one way.

Trees

Now let's play a bit with a bigger example: trees. Here's one way to define binary trees that have data at the internal nodes in Haskell:

> data Tree a = Empty    -- No data
>             | Branch a (Tree a) (Tree a)  -- data of type a, left and right subtrees
>                deriving (Eq, Show)

For example, we can represent the following tree

          5
        /   \
       2     9
      / \     \
     1   4     7

like this:

> exTree :: Tree Int
> exTree = Branch 5 (Branch 2 (Branch 1 Empty Empty) (Branch 4 Empty Empty))
>                   (Branch 9 Empty (Branch 7 Empty Empty))

We can write simple functions on trees by recursion:

> -- | increment all integers in the tree
> -- >>> treePlus (Branch 2 Empty Empty) 3
> -- Branch 5 Empty Empty
> treePlus :: Tree Int -> Int -> Tree Int
> 
> treePlus (Branch x l r) k = Branch (x+k) (treePlus l k) (treePlus r k)
> treePlus Empty          k = Empty

We can accumulate all of the elements in a tree into a list:

> -- >>> infixOrder exTree
> -- [1,2,4,5,9,7]
> infixOrder :: Tree a -> [a]
> infixOrder Empty = []
> infixOrder (Branch x l r) = infixOrder l ++ [x] ++ infixOrder r

... visiting the nodes in different orders ....

> -- >>> prefixOrder exTree
> -- [5,2,1,4,9,7]
> 
> prefixOrder :: Tree a -> [a]
> 
> prefixOrder Empty = []
> prefixOrder (Branch x l r) = [x] ++ prefixOrder l ++ prefixOrder r

(NOTE: This is a simple way of defining a tree walk in Haskell, but it is not the best way. In particular, the infixOrder function is not linear in the number of nodes in the tree. Why? Can you think of a way to rewrite infixOrder so that it runs in linear time? One solution is to use a DList.)

But, of course, what we should really do is reimplement our higher-order patterns for trees!

> treeMap :: (a -> b) -> Tree a -> Tree b
> treeMap f Empty = Empty
> treeMap f (Branch x l r) = Branch (f x) (treeMap f l) (treeMap f r)

So that, for example, to increment each node in a Tree Int by one, we could write this:

> -- >>> treeIncr (Branch 1 (Branch 2 Empty Empty) Empty)
> -- Branch 2 (Branch 3 Empty Empty) Empty
> treeIncr :: Tree Int -> Tree Int
> treeIncr = treeMap (+1)

The type of foldr will have to change a little bit. Now the function that tells us how to combine the current element with the recursive result will have to take two results - one for each branch!

> foldTree :: b -> (a -> b -> b -> b) -> Tree a -> b
> foldTree b f Empty = b
> foldTree b f (Branch x l r) = f x (foldTree b f l) (foldTree b f r)

Now, let's reimplement infixOrder:

> -- 
> infixOrder' :: Tree a -> [a]
> infixOrder' = foldTree [] (\x ll rl -> ll ++ [x] ++ rl)

> testInfixOrder' = "infixOrder'" ~: infixOrder' exTree ~?= [1,2,4,5,9,7]

And we can get the other orderings just as quickly!

> prefixOrder' :: Tree a -> [a]
> prefixOrder' = foldTree [] (\x ll rl -> [x] ++ ll ++ rl)

> postfixOrder' :: Tree a -> [a]
> postfixOrder' = foldTree [] (\x ll rl -> ll ++ rl ++ [x])

> testPrefixOrder'  = "prefixOrder'" ~: prefixOrder' exTree ~?= [5,2,1,4,9,7]
> testPostfixOrder' = "postfixOrder'" ~: postfixOrder' exTree ~?= [1,4,2,7,9,5]

> main :: IO ()
> main = do
>   runTestTT $ TestList [
>     "safeHead" ~: safeHead oneTwoThree ~?= 1,
>     "sumOfIntListNE" ~: sumOfIntListNE oneTwoThree ~?= 6,
>     "treeIncr" ~: treeIncr (Branch 1 (Branch 2 Empty Empty) Empty) ~?=
>                                        Branch 2 (Branch 3 Empty Empty) Empty,
>     "treePlus" ~: treePlus (Branch 2 Empty Empty) 3 ~?= Branch 5 Empty Empty,
>     "infixOrder" ~: infixOrder exTree ~?= [1,2,4,5,9,7],
>     "prefixOrder" ~: prefixOrder exTree ~?= [5,2,1,4,9,7]
>      ]
>   return ()

Part of this lecture is taken from "Learn You a Haskell for Great Good".