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0 PROGRAMMING IN HASKELL Chapter 11 - Interactive Programs, Declaring Types and Classes

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Introduction 1 To date, we have seen how Haskell can be used to write batch programs that take all their inputs at the start and give all their outputs at the end. batch program inputsoutputs

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2 However, we would also like to use Haskell to write interactive programs that read from the keyboard and write to the screen, as they are running. interactive program inputsoutputs keyboard screen

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The Problem 3 Haskell programs are pure mathematical functions: However, reading from the keyboard and writing to the screen are side effects: zHaskell programs have no side effects. zInteractive programs have side effects.

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The Solution 4 Interactive programs can be written in Haskell by using types to distinguish pure expressions from impure actions that may involve side effects. IO a The type of actions that return a value of type a.

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5 For example: IO Char IO () The type of actions that return a character. The type of purely side effecting actions that return no result value. z() is the type of tuples with no components. Note:

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Basic Actions 6 The standard library provides a number of actions, including the following three primitives: getChar :: IO Char zThe action getChar reads a character from the keyboard, echoes it to the screen, and returns the character as its result value:

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7 zThe action putChar c writes the character c to the screen, and returns no result value: putChar :: Char IO () zThe action return v simply returns the value v, without performing any interaction: return :: a IO a

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Sequencing 8 A sequence of actions can be combined as a single composite action using the keyword do. For example: a :: IO (Char,Char) a = do x getChar getChar y getChar return (x,y)

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Derived Primitives 9 getLine :: IO String getLine = do x getChar if x == '\n' then return [] else do xs getLine return (x:xs) zReading a string from the keyboard:

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10 putStr :: String IO () putStr [] = return () putStr (x:xs) = do putChar x putStr xs zWriting a string to the screen: zWriting a string and moving to a new line: putStrLn :: String IO () putStrLn xs = do putStr xs putChar '\n'

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Example 11 We can now define an action that prompts for a string to be entered and displays its length: strlen :: IO () strlen = do putStr "Enter a string: " xs getLine putStr "The string has " putStr (show (length xs)) putStrLn " characters"

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12 For example: > strlen Enter a string: abcde The string has 5 characters zEvaluating an action executes its side effects, with the final result value being discarded. Note:

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Hangman 13 Consider the following version of hangman: zOne player secretly types in a word. zThe other player tries to deduce the word, by entering a sequence of guesses. zFor each guess, the computer indicates which letters in the secret word occur in the guess.

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14 zThe game ends when the guess is correct. hangman :: IO () hangman = do putStrLn "Think of a word: " word sgetLine putStrLn "Try to guess it:" guess word We adopt a top down approach to implementing hangman in Haskell, starting as follows:

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15 The action sgetLine reads a line of text from the keyboard, echoing each character as a dash: sgetLine :: IO String sgetLine = do x getCh if x == '\n' then do putChar x return [] else do putChar '-' xs sgetLine return (x:xs)

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16 primitive getCh :: IO Char Note: zThe action getCh reads a character from the keyboard, without echoing it to the screen. zThis useful action is not part of the standard library, but is a special Hugs primitive that can be imported into a script as follows:

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17 The function guess is the main loop, which requests and processes guesses until the game ends. guess :: String IO () guess word = do putStr "> " xs getLine if xs == word then putStrLn "You got it!" else do putStrLn (diff word xs) guess word

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18 The function diff indicates which characters in one string occur in a second string: For example: > diff "haskell" "pascal" "-as--ll" diff :: String String String diff xs ys = [if elem x ys then x else '-' | x xs]

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Exercise 19 Implement the game of nim in Haskell, where the rules of the game are as follows: zThe board comprises five rows of stars: 1: * * * * * 2: * * * * 3: * * * 4: * * 5: *

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20 zTwo players take it turn about to remove one or more stars from the end of a single row. zThe winner is the player who removes the last star or stars from the board. Hint: Represent the board as a list of five integers that give the number of stars remaining on each row. For example, the initial board is [5,4,3,2,1].

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Type Declarations 21 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].

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22 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:

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23 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)

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

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Data Declarations 25 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.

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26 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.

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27 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

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28 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:

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29 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

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30 Not surprisingly, data declarations themselves can also have parameters. For example, given data Maybe a = Nothing | Just a safediv :: Int Int Maybe Int safediv _ 0 = Nothing safediv m n = Just (m `div` n) safehead :: [a] Maybe a safehead [] = Nothing safehead xs = Just (head xs) we can define:

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Recursive Types 31 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.

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32 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)

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33 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 =

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34 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)

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35 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)

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36 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).

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Arithmetic Expressions 37 Consider a simple form of expressions built up from integers using addition and multiplication. 1 + 32

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38 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))

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39 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

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40 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 (+) (*)

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Binary Trees 41 In computing, it is often useful to store data in a two-way branching structure or binary tree. 5 7 96 3 41

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42 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))

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43 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.

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44 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].

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45 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

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Exercises 46 (1) Using recursion and the function add, define a function that multiplies two natural numbers. (2) Define a suitable function fold for expressions, and give a few examples of its use. (3) 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.

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