Functional Programming in

Haskell

Motivation through Concrete Examples

Adapted from Lectures by

Simon Thompson

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Functional Programming

Given the functionsabove invertColour flipH

sideBySide superimpose flipV

and the horse picture, how do you get …

(expression and evaluation)

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Definitions in Haskell

name :: Type

name = expression

blackHorse :: Picture

blackHorse = invertColour horse

rotate :: Picture -> Picture

rotate pic = flipH (flipV pic)

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Higher-level

Evaluation is about expressions and values, not storage locations.

• No need to allocate/deallocate storage: garbage collection.

• Values don't change over program execution: contrast x=x+1 etc. of Java, C, …

•… instead we describe relations between values by means of (fixed) functions.

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Declarative … proofs possible

Programs describe themselves: square n = n*n double n = 2*n

'The square of n is n*n, for every integer n.'

Programs are equations.So we can write proofs using the definitions.

square (double n)

= square (2*n)

= (2*n)*(2*n) = 2*2*n*n

= double (double (square n))

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Evaluation freedom

Evaluation can occur in any order ...

(4-3)+(2-1) (4-3)+(2-1) (4-3)+(2-1)

(4-3)+1 1+(2-1) 1+1

1+1 1+1 2

2 2

… and can choose to evaluate only what is needed, when it is needed: lazy evaluation (more later).

Can also evaluate in parallel … efficiently?

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History

First 'functional' language, LISP, defined c. 1960 … popular in AI in 70s/80s.

• Now represented best by Scheme.

• Weakly typed; allows side-effects and eval.

Next generation: ML (1980…), Miranda (1985…) and Haskell (1990…).

• Strongly-typed; ML allows references and thus side-effects.

• Miranda and Haskell: pure and lazy.

• FP (1982): heroic experiment by Backus (FORTRAN, ALGOL).

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Haskell and Hugs

Named after Haskell Brooks Curry: mathematician and logician; inventor of the -calculus.

Haskell 98 is the recent 'standard' version of Haskell.

Various implementations: Hugs (interpreter for Windows, Mac, Unix) and GHC, NHC, HBC (compilers).

http://www.haskell.org/

Basics: guards and base types

How many of three integers are equal … ?

howManyEqual :: Int -> Int -> Int -> Int

howManyEqual n m k

| n==m && m==k = 3

| n==m || m==k || k==n = 2

| otherwise = 1

If we reach herethey're not all equal …

… and if we reach herethey're all different.

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Regular and literate scripts

In a regular script there are definitions and comments:

-- FirstScript.hs

-- 5 October 2000

-- Double an integer.

double :: Int -> Int

double n = 2*n

Everything is program, except comments beginning --.

In a literate script there are comments and definitions:

FirstLit.lhs

5 October 2000

Double an integer.

> double :: Int -> Int

> double n = 2*n

Everything is comment, except program beginning > .

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How many pieces with n cuts?

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How many pieces with n cuts?

No cuts: 1 piece.

With the nth cut, you get n more pieces:

cuts :: Int -> Int

cuts n

| n==0 = 1

| n>0 = cuts (n-1) + n

| otherwise = 0

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The Pictures case study.

Using a powerful library of functions over lists.

• Pattern matching

• Recursion

• Generic functions

• Higher-order functions

• …

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Using Hugs

expr Evaluate expr

:type expr Give the type of expr

:l Blah Load the file Blah.hs

:r Reload the last file

:? Help: list commands

:e Edit the current file

:q Quit

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Functions over pictures

A function to flip a picture in a vertical mirror:

input outputflipV

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Functions over pictures

A function to invert the colours in a picture:

invertColour

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Functions over pictures

A function to superimpose two pictures:

superimpose

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Functions over pictures

A function to put one picture above another:

above

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Functions over pictures

A function to put two pictures side by side:

sideBySide

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A naïve implementation

type Picture = [String]

type String = [Char]

A Picture is a list of Strings.A String is a list of Char (acters).

.......##...

.....##..#..

...##.....#.

..#.......#.

..#...#...#.

..#...###.#.

.#....#..##.

..#...#.....

...#...#....

....#..#....

.....#.#....

......##....

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How are they implemented?

flipH Reverse the list of strings.

flipV Reverse each string.

rotate flipH then flipV (or v.versa).

above Join the two lists of strings.

sideBySide Join corresponding lines.

invertColour Change each Char … and each

line.

superimpose Join each Char … join each line.

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How are they implemented?

flipH reverse

flipV map reverse

rotate flipV . flipH

above ++

sideBySide zipWith (++)

invertColour map (map invertChar)

superimpose zipWith (zipWith combine)

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Lists and types

Haskell is strongly typed: detect all type errors before evaluation.

For each type t there is a type [t], 'list of t'.

reverse [] = []

reverse (x:xs) = reverse xs ++ [x]

reverse :: [a] -> [a]

a is a type variable: reverse works over any list type, returning a list of the same type.

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Flipping in a vertical mirror

flipV :: Picture -> Picture

flipV [] = []

flipV (x:xs) = reverse x : flipV xs

Run along the list, applying reverse to each element

Run along the list, applying … to every element.

General pattern of computation.

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Implementing the mapping pattern

map f [] = []

map f (x:xs) = f x : map f xs

map :: (a -> b) -> [a] -> [b]

Examples over pictures:

flipV pic = map reverse pic

invertColour pic = map invertLine pic

invertLine line = map invertChar line

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Functions as data

Haskell allows you to pass functions as arguments and return functions as results, put them into lists, etc. In contrast, in Pascal and C, you can only pass named functions, not functions you build dynamically.

map isEven = ??

map isEven :: [Int] -> [Bool]

It is a partial application, which gives a function:give it a [Int] and it will give you back a [Bool]

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Partial application in Pictures

flipV = map reverse

invertColour = map (map invertChar)

A function[Char]->[Char]

A function[[Char]]->[[Char]]

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Another pattern: zipping together

sideBySide [l1,l2,l3] [r1,r2,r3]

= [ l1++r1, l2++r2, l3++r3 ]

zipWith :: (a->b->c) -> [a] -> [b] -> [c]

zipWith f (x:xs) (y:ys)

= f x y : zipWith f xs ys

zipWith f xs ys = []

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In the case study …

sideBySide = zipWith (++)

Superimposing two pictures: need to combine individual elements:

combine :: Char -> Char -> Char

combine top btm

= if (top=='.' && btm=='.') then '.' else '#'

superimpose = zipWith (zipWith combine)

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Parsing

"((2+3)-4)"

is a sequence of symbols, but underlying it is a structure ...

-

4

2 3

+

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Arithmetical expressions

An expression is either

• a literal, such as 234 or a composite expression:

• the sum of two expressions (e1+e2)

• the difference of two expressions (e1-e2)

• the product of two expressions (e1*e2)

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How to represent these structures?

data Expr = Lit Int |

Sum Expr Expr |

Minus Expr Expr |

Times Expr Expr

Elements of this algebraic data type include

Lit 34 34

Sum (Lit 45) (Lit 3) (45+3)

Minus (Sum (Lit 2) (Lit 3)) (Lit 4) ((2+3)-4)

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Counting operators

data Expr = Lit Int | Sum Expr Expr | Minus ...

How many operators in an expression?

Definition using pattern matching

cOps (Lit n) = 0

cOps (Sum e1 e2) = cOps e1 + cOps e2 + 1

cOps (Minus e1 e2) = cOps e1 + cOps e2 + 1

cOps (Times e1 e2) = cOps e1 + cOps e2 + 1

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Evaluating expressions

data Expr = Lit Int | Sum Expr Expr | Minus ...

Literals are themselves …

eval (Lit n) = n

… in other cases, evaluate the two arguments and then combine the results …

eval (Sum e1 e2) = eval e1 + eval e2

eval (Minus e1 e2) = eval e1 - eval e2

eval (Times e1 e2) = eval e1 * eval e2

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List comprehensions

Example list x = [4,3,2,5][ n+2 | n<-x, isEven n]

run through the n in x …4 3 2 5

select those which are even …4 2

and add 2 to each of them6 4

giving the result[6,4]

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List comprehensions

Example lists x = [4,3,2] y = [12,17][ n+m | n<-x, m<-y]

run through the n in x …4 3 2

and for each, run through the m in y …12 17 12 17 12 17

add corresponding pairs16 21 15 20 14 19

giving the result[16,21,15,20,14,19]

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Quicksort

qsort [] = []

qsort (x:xs) =

qsort elts_lt_x

++ [x]

++ qsort elts_greq_x

where

elts_lt_x = [y | y <- xs, y < x]

elts_greq_x = [y | y <- xs, y >= x]

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MergeSort

mergeSort [] = []

mergeSort [x] = [x]

mergeSort xs | size >= 1 =

merge (mergeSort front) (mergeSort back)

where size = length xs `div` 2

front = take size xs

back = drop size xs

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Merging

x

y

x <= y?

merge [1, 3] [2, 4] 1 : merge [3] [2, 4]

1 : 2 : merge [3] [4]

1 : 2 : 3 : merge [] [4]

1 : 2 : 3 : [4] [1,2,3,4]

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Defining Merge

merge (x : xs) (y : ys)

| x <= y = x : merge xs (y : ys)

| x > y = y : merge (x : xs) ys

merge [] ys = ys

merge xs [] = xs

One list getssmaller.

Two possiblebase cases.

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Lazy evaluation

Only evaluate what is needed … infinite lists

nums :: Int -> [Int]

nums n = n : nums (n+1)

sft (x:y:zs) = x+y

sft (nums 3)

= sft (3: nums 4)

= sft (3: 4: nums 5)

= 7

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The list of prime numbers

primes = sieve (nums 2)

sieve (x:xs)

= x : sieve [ z | z<-xs, z `mod` x /= 0]

To sieve (x:xs) return x, together with the result of sieveing xs with all multiples of x removed.