0 PROGRAMMING IN HASKELL Based on lecture notes by Graham Hutton The book “Learn You a Haskell for Great Good” (and a few other sources) Modules

1 Type Declarations In Haskell, a new name for an existing type can be defined using a type declaration. type String = [Char] String is a synonym for the type [Char].

2 Type declarations can be used to make other types easier to read. For example, given origin :: Pos origin = (0,0) left :: Pos  Pos left (x,y) = (x-1,y) type Pos = (Int,Int) we can define:

3 Like function definitions, type declarations can also have parameters. For example, given type Pair a = (a,a) we can define: mult :: Pair Int  Int mult (m,n) = m*n copy :: a  Pair a copy x = (x,x)

4 Type declarations can be nested: type Pos = (Int,Int) type Trans = Pos  Pos However, they cannot be recursive: type Tree = (Int,[Tree])

5 Data Declarations A completely new type can be defined by specifying its values using a data declaration. data Bool = False | True Bool is a new type, with two new values False and True.

6 Note: zThe two values False and True are called the constructors for the type Bool. zType and constructor names must begin with an upper-case letter. zData declarations are similar to context free grammars. The former specifies the values of a type, the latter the sentences of a language.

7 answers :: [Answer] answers = [Yes,No,Unknown] flip :: Answer  Answer flip Yes = No flip No = Yes flip Unknown = Unknown data Answer = Yes | No | Unknown we can define: Values of new types can be used in the same ways as those of built in types. For example, given

8 The constructors in a data declaration can also have parameters. For example, given data Shape = Circle Float | Rect Float Float square :: Float  Shape square n = Rect n n area :: Shape  Float area (Circle r) = pi * r^2 area (Rect x y) = x * y we can define:

9 Note: zShape has values of the form Circle r where r is a float, and Rect x y where x and y are floats. zCircle and Rect can be viewed as functions that construct values of type Shape: Circle :: Float  Shape Rect :: Float  Float  Shape

10 Recursive Types In Haskell, new types can be declared in terms of themselves. That is, types can be recursive. data Nat = Zero | Succ Nat Nat is a new type, with constructors Zero :: Nat and Succ :: Nat  Nat.

11 Note: zA value of type Nat is either Zero, or of the form Succ n where n :: Nat. That is, Nat contains the following infinite sequence of values: Zero Succ Zero Succ (Succ Zero)   

12 zWe can think of values of type Nat as natural numbers, where Zero represents 0, and Succ represents the successor function 1+. zFor example, the value Succ (Succ (Succ Zero)) represents the natural number 1 + (1 + (1 + 0))3 =

13 Using recursion, it is easy to define functions that convert between values of type Nat and Int: nat2int :: Nat  Int nat2int Zero = 0 nat2int (Succ n) = 1 + nat2int n int2nat :: Int  Nat int2nat 0 = Zero int2nat (n+1) = Succ (int2nat n)

14 Two naturals can be added by converting them to integers, adding, and then converting back: However, using recursion the function add can be defined without the need for conversions: add :: Nat  Nat  Nat add m n = int2nat (nat2int m + nat2int n) add Zero n = n add (Succ m) n = Succ (add m n)

15 For example: add (Succ (Succ Zero)) (Succ Zero) Succ (add (Succ Zero) (Succ Zero)) = Succ (Succ (add Zero (Succ Zero)) = Succ (Succ (Succ Zero)) = Note: zThe recursive definition for add corresponds to the laws 0+n = n and (1+m)+n = 1+(m+n).

16 Arithmetic Expressions Consider a simple form of expressions built up from integers using addition and multiplication. 1 +  32

17 Using recursion, a suitable new type to represent such expressions can be declared by: For example, the expression on the previous slide would be represented as follows: data Expr = Val Int | Add Expr Expr | Mul Expr Expr Add (Val 1) (Mul (Val 2) (Val 3))

18 Using recursion, it is now easy to define functions that process expressions. For example: size :: Expr  Int size (Val n) = 1 size (Add x y) = size x + size y size (Mul x y) = size x + size y eval :: Expr  Int eval (Val n) = n eval (Add x y) = eval x + eval y eval (Mul x y) = eval x * eval y

19 Note: zThe three constructors have types: Val :: Int  Expr Add :: Expr  Expr  Expr Mul :: Expr  Expr  Expr zMany functions on expressions can be defined by replacing the constructors by other functions using a suitable fold function. For example: eval = fold id (+) (*)

20 Binary Trees In computing, it is often useful to store data in a two-way branching structure or binary tree

21 Using recursion, a suitable new type to represent such binary trees can be declared by: For example, the tree on the previous slide would be represented as follows: data Tree = Leaf Int | Node Tree Int Tree Node (Node (Leaf 1) 3 (Leaf 4)) 5 (Node (Leaf 6) 7 (Leaf 9))

22 We can now define a function that decides if a given integer occurs in a binary tree: occurs :: Int  Tree  Bool occurs m (Leaf n) = m==n occurs m (Node l n r) = m==n || occurs m l || occurs m r But… in the worst case, when the integer does not occur, this function traverses the entire tree.

23 Now consider the function flatten that returns the list of all the integers contained in a tree: flatten :: Tree  [Int] flatten (Leaf n) = [n] flatten (Node l n r) = flatten l ++ [n] ++ flatten r A tree is a search tree if it flattens to a list that is ordered. Our example tree is a search tree, as it flattens to the ordered list [1,3,4,5,6,7,9].

24 Search trees have the important property that when trying to find a value in a tree we can always decide which of the two sub-trees it may occur in: This new definition is more efficient, because it only traverses one path down the tree. occurs m (Leaf n) = m==n occurs m (Node l n r) | m==n = True | m<n = occurs m l | m>n = occurs m r

25 Exercise A binary tree is complete if the two sub-trees of every node are of equal size. Define a function that decides if a binary tree is complete. data Tree = Leaf Int | Node Tree Int Tree Node (Node (Leaf 1) 3 (Leaf 4)) 5 (Node (Leaf 6) 7 (Leaf 9)) occurs :: Int  Tree  Bool occurs m (Leaf n) = m==n occurs m (Node l n r) = m==n || occurs m l || occurs m r

26 Modules So far, we’ve been using built-in functions provided in the Haskell prelude. This is a subset of a larger library that is provided with any installation of Haskell. (Google for Hoogle to see a handy search engine for these.) Examples of other modules: - lists - concurrent programming - complex numbers - char - sets - …

27 Example: Data.List To load a module, we need to import it: numUniques :: (Eq a) => [a] -> Int numUniques = length. nub import Data.List All the functions in this module are immediately available: This is a function in Data.List that removes duplicates from a list. function concatenation

28 You can also load modules from the command prompt: ghci> :m + Data.List Or several at once: import Data.List (nub, sort) import Data.List hiding (nub) ghci> :m + Data.List Data.Map Data.Set Or import only some, or all but some:

29 If duplication of names is an issue, can extend the namespace: import qualified Data.Map When the Data.Map gets a bit long, we can provide an alias: import qualified Data.Map as M And now we can just type M.filter, and the normal list filter will just be filter. This imports the functions, but we have to use Data.Map to use them – like Data.Map.filter.

30 Data.List has a lot more functionality than we’ve seen. A few examples: ghci> intersperse '.' "MONKEY" "M.O.N.K.E.Y" ghci> intersperse 0 [1,2,3,4,5,6] [1,0,2,0,3,0,4,0,5,0,6] ghci> intercalate " " ["hey","there","guys"] "hey there guys" ghci> intercalate [0,0,0] [[1,2,3],[4,5,6], [7,8,9]] [1,2,3,0,0,0,4,5,6,0,0,0,7,8,9]

31 And even more: ghci> transpose [[1,2,3],[4,5,6], [7,8,9]] [[1,4,7],[2,5,8],[3,6,9]] ghci> transpose ["hey","there","guys"] [" htg","ehu","yey","rs","e"] ghci> concat ["foo","bar","car"] "foobarcar" ghci> concat [[3,4,5],[2,3,4],[2,1,1]] [3,4,5,2,3,4,2,1,1]

32 And even more: ghci> and $ map (>4) [5,6,7,8] True ghci> and $ map (==4) [4,4,4,3,4] False ghci> any (==4) [2,3,5,6,1,4] True ghci> all (>4) [6,9,10] True

33 A nice example: adding functions Functions are often represented as vectors: 8x^3 + 5x^2 + x - 1 is [8,5,1,-1]. So we can easily use List functions to add these vectors: ghci> map sum $ transpose [[0,3,5,9], [10,0,0,9],[8,5,1,-1]] [18,8,6,17]

34 There are a ton of these functions, so I could spend all semester covering just lists. More examples: group, sort, dropWhile, takeWhile, partition, isPrefixOf, find, findIndex, delete, words, insert,… Instead, I’ll make sure to post a link to a good overview of lists on the webpage, in case you need them. In essence, if it’s a useful thing to do to a list, Haskell probably supports it!

35 The Data.Char module: includes a lot of useful functions that will look similar to python, actually. Examples: isAlpha, isLower, isSpace, isDigit, isPunctuation,… ghci> all isAlphaNum "bobby283" True ghci> all isAlphaNum "eddy the fish!"Fal se ghci> groupBy ((==) `on` isSpace) "hey guys its me" ["hey"," ","guys"," ","its"," ","me"]

36 The Data.Char module has a datatype that is a set of comparisons on characters. There is a function called generalCategory that returns the information. (This is a bit like the Ordering type for numbers, which returns LT, EQ, or GT.) ghci> generalCategory ' ' Space ghci> generalCategory 'A' UppercaseLetter ghci> generalCategory 'a' LowercaseLetter ghci> generalCategory '.' OtherPunctuation ghci> generalCategory '9' DecimalNumber ghci> map generalCategory " ¥ t¥nA9?|" [Space,Control,Control,UppercaseLetter,DecimalNumber,OtherPunctuation,M athSymbol] ]

37 There are also functions that can convert between Ints and Chars: ghci> map digitToInt "FF85AB" [15,15,8,5,10,11] ghci> intToDigit 15 'f' ghci> intToDigit 5 '5' ghci> chr 97 'a' ghci> map ord "abcdefgh" [97,98,99,100,101,102,103,104]

38 Neat application: Ceasar ciphers A primitive encryption cipher which encodes messages by shifted them a fixed amount in the alphabet. Example: hello with shift of 3 encode :: Int -> String -> String encode shift msg = let ords = map ord msg shifted = map (+ shift) ords in map chr shifted

39 Now to use it: ghci> encode 3 "Heeeeey" "Khhhhh|" ghci> encode 4 "Heeeeey" "Liiiii}" ghci> encode 1 "abcd" "bcde" ghci> encode 5 "Marry Christmas! Ho ho ho!” "Rfww~%Hmwnxyrfx&%Mt%mt%mt&"

40 Decoding just reverses the encoding: decode :: Int -> String -> String decode shift msg = encode (negate shift) msg ghci> encode 3 "Im a little teapot" "Lp#d#olwwoh#whdsrw" ghci> decode 3 "Lp#d#olwwoh#whdsrw" "Im a little teapot" ghci> decode 5. encode 5 $ "This is a sentence " "This is a sentence"

41 Making our own modules We specify our own modules at the beginning of a file. For example, if we had a set of geometry functions: module Geometry ( sphereVolume, sphereArea, cubeVolume, cubeArea, cuboidArea, cuboidVolume ) where

42 Then, we put the functions that the module uses: sphereVolume :: Float -> Float sphereVolume radius = (4.0 / 3.0) * pi * (radius ^ 3) sphereArea :: Float -> Float sphereArea radius = 4 * pi * (radius ^ 2) cubeVolume :: Float -> Float cubeVolume side = cuboidVolume side side side …

43 Note that we can have “private” helper functions, also: cuboidVolume :: Float -> Float -> Float -> Float cuboidVolume a b c = rectangleArea a b * c cuboidArea :: Float -> Float -> Float -> Float cuboidArea a b c = rectangleArea a b * 2 + rectangl eArea a c * 2 + rectangleArea c b * 2 rectangleArea :: Float -> Float -> Float rectangleArea a b = a * b

44 Can also nest these. Make a folder called Geometry, with 3 files inside it: Sphere.hs Cubiod.hs Cube.hs Each will hold a separate group of functions. To load: import Geometry.Sphere Or (if functions have same names): import qualified Geometry.Sphere as Sphere

45 The modules: module Geometry.Sphere ( volume, area ) where volume :: Float -> Float volume radius = (4.0 / 3.0) * pi * (radius ^ 3) area :: Float -> Float area radius = 4 * pi * (radius ^ 2)

46 module Geometry.Cuboid ( volume, area ) where volume :: Float -> Float -> Float -> Float volume a b c = rectangleArea a b * c …