1-nov-15 haskell ii functions and patterns. data types int + - * / ^ even odd float + - * / ^ sin...

Apr 20, 2023

Haskell II

Functions and patterns

Data Types

Int + - * / ^ even odd Float + - * / ^ sin cos pi truncate Char ord chr isSpace isUpper … Bool && || not Lists : ++ head tail last init take Tuples fst snd Polymorphic: < <= == /= => > show

User-Defined Data Types

User-defined data types data Color = Red | Blue toString Red = "red"

toString Blue = "blue" data Tree a =

Leaf a | Branch (Tree a) (Tree a) Can be tricky to use

Assorted Syntax

Comments are -- to end of lineor {- to -} (these may be nested)

Types are capitalized, variables are not Indentation may be used in place of braces Infix operators: + - `mod` `not` Prefix operators: (+) (-) mod not Types: take :: Int -> [a] -> [a]

Layout

The first nonblank character following where, let, or of determines the starting column let x = a + b

y = a * bin y / x

If you start too far to the left, that may end an enclosing clause

You can use { } instead, but this is not usually done

Infinite Lists

[1..5] == [1, 2, 3, 4, 5] [1..] == all positive integers [5, 10..32] == [5, 10, 15, 20, 25, 30] [5, 10..] == positive multiples of 5 [x*x | x <- [1..]] == squares of positive ints [x*x | x <- [1..], even x] == squares of positive

even ints [(x, y) | x <- [1..10], y <- [1..10], x < y]

Functions are also data

Functions are “first-class objects” Functions can be assigned Functions can be passed as parameters Functions can be stored in data structures There are operations on functions

But functions can’t be tested for equality Theoretically very hard!

Anonymous Functions

Form is \ parameters -> body Example: \x y -> (x + y) / 2

the \ is pronounced “lambda” the x and y are the formal parameters

inc x = x + 1 this is shorthand for inc = \x -> x + 1

add x y = x + y this is shorthand for add = \x y -> x + y

Currying

Technique named after Haskell Curry Functions only need one argument Currying absorbs an argument into a function f a b = (f a) b, where (f a) is a curried function (avg 6) 8

7.0

Slicing

Functions may be “partially applied” inc x = x + 1

can be defined instead as inc = (+ 1) add x y = x + y

can be defined instead as add = (+) negative = (< 0)

map

map :: (a -> b) -> [a] -> [b] applies the function to all elements of the list Prelude> map odd [1..5]

[True,False,True,False,True] Prelude> map (* 2) [1..5]

[2,4,6,8,10]

filter

filter :: (a -> Bool) -> [a] -> [a] Returns the elements that satisfy the test Prelude> filter even [1..10]

[2,4,6,8,10] Prelude> filter (\x -> x>3 && x<10) [1..20]

[4,5,6,7,8,9]

iterate

iterate :: (a -> a) -> a -> [a] f x returns the list [x, f x, f f x, f f f x, …] Prelude> take 8 (iterate (2 *) 1)

[1,2,4,8,16,32,64,128] Prelude> iterate tail [1..3]

[[1,2,3],[2,3],[3],[], *** Exception: Prelude.tail: empty list

foldl

foldl :: (a -> b -> a) -> a -> [b] -> a foldl f i x starts with i, repeatedly applies f to i and the

next element in the list x Prelude> foldl (-) 100 [1..3]

94 94 = 100 - 1 - 2 - 3

foldl1

foldl1 :: (a -> a -> a) -> [a] -> a Same as: foldl f (head x) (tail x) Prelude> foldl1 (-) [100, 1, 2, 3]

94 Prelude> foldl1 (+) [1..100]

5050

flip

flip :: (a -> b -> c) -> b ->a -> c Reverses first two arguments of a function Prelude> elem 'o' "aeiou"

True Prelude> flip elem "aeiou" 'o'

True Prelude> (flip elem) "aeiou" 'o'

True

Function composition with (.)

(.) :: (a -> b) -> (c -> a) -> (c -> b) (f . g) x is the same as f (g x) double x = x + x

quadruple = double . doubledoubleFirst = (* 2) . head

Main> quadruple 312Main> doubleFirst [3..10]6

span

span :: (a -> Bool) -> [a] -> ([a], [a]) Break the lists into two lists

those at the front that satisfy the condition the rest

Main> span (<= 5) [1..10] ([1,2,3,4,5],[6,7,8,9,10])

Main> span (< 'm') "abracadabra" ("ab","racadabra")

break

break :: (a -> Bool) -> [a] -> ([a], [a]) Break the lists into two lists

those at the front that fail the condition the rest

Main> break (== ' ') "Haskell is neat!" ("Haskell"," is neat!")

Function Definition I

Functions are defined with = fact n =

if n == 0 then 1 else n * fact (n - 1)

Function Definition II

Functions are usually defined by cases fact n

| n == 0 = 1 | otherwise = n * fact (n - 1)

fact n = case n of 0 -> 1 n -> n * fact (n - 1)

These are “the same”

Function Definition III

You can separate the cases with “patterns” fact :: Int -> Int -- not essential

fact 0 = 1fact n = n * fact (n - 1)

How does this work?

Pattern Matching

Functions cannot in general be overloaded But they can be broken into cases Each case must have the same signature fact :: Int -> Int -- explicit signature

fact 0 = 1fact n = n * fact (n - 1)

fact 5 won’t match the first, but will match the second

Pattern Types I

A variable will match anything A wildcard, _, will match anything, but you can’t use

the matched value A constant will match only that value Tuples will match tuples, if same length and

constituents match Lists will match lists, if same length and constituents

match However, the pattern may specify a list of arbitrary length

Pattern Types II

(h:t) will match a nonempty list whose head is h and whose tail is t

second (h:t) = head t Main> second [1..5]

2

Pattern Types III

“As-patterns” have the form w@pattern When the pattern matches, the w matches the whole of

the thing matched firstThree all@(h:t) = take 3 all Main> firstThree [1..10]

[1,2,3]

Pattern Types IV

(n+k) matches any value equal to or greater than k; n is k less than the value matched

silly (n+5) = n Main> silly 20

15 This is the only arithmetical pattern; it does not

generalize to any other pattern

Advantages of Haskell

Extremely concise Easy to understand

no, really! No core dumps Polymorphism improves chances of re-use Powerful abstractions Built-in memory management

Disadvantages of Haskell

Unfamiliar Slow

because compromises are less in favor of the machine

quicksort

quicksort [] = []

quicksort (x:xs) = quicksort [y | y <- xs, y < x] ++ [x] ++ quicksort [y | y <- xs, y >= x]

The End