1. Functions and patterns
Haskell II
Functions and patterns
5-May-19

2. 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

3. 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

4. It’s all about types!
Getting the types right is critical to programming in Haskell
In GHCi, :type x or :t x will tell you the type of x
GHCi> :t "abc" "abc" :: [Char]
The :: can be read as “has the type”
[Char] is the representation for “list of char”
Is "abc" really a list of characters?
GHCi> :t ['a', 'b', 'c'] ['a', 'b', 'c'] :: [Char]
GHCi> ['a', 'b', 'c'] == "abc" True

5. More type definitions
GHCi> :t ["ab", "cd"] ["ab", "cd"] :: [[Char]]
[[Char]] is a list of lists of characters = a list of strings
GHCi> :t [] [] :: [a]
Here, a is a “type variable”—[] could be a list of any type
GHCi> :t [[]] [[]] :: [[a]]
[[a]] is a list of lists, all of which have the same type
GHCi> :t head head :: [a] -> a
head takes a list of any type a and returns a single value of type a

6. Type restrictions
GHCi> :t even even :: Integral a => a -> Bool
Integral a is a type restriction—it says a must be some Integral type
The -> indicates a function; in this case, a function from some type a (restricted to be integral) to a Boolean.
GHCi> :t (5, 'a', "abc") (5, 'a', "abc") :: Num t => (t, Char, [Char])
(5, 'a', "abc") is a tuple consisting of a Numeric type t, a Character, and a string (list of Char)
Unlike lists, tuples can (and often do) contain values of different types

7. Types of functions
GHCi> :t mod mod :: Integral a => a -> a -> a
GHCi> mod
GHCi> 20 `mod` 6 2
Functions in Haskell take a single argument
a -> a -> a associates as (a -> a) -> a
GHCi> (mod 20) 6 2
GHCi> :t (mod 20) (mod 20) :: Integral a => a -> a

8. Types of higher-order functions
GHCi> :t map map :: (a -> b) -> [a] -> [b]
GHCi> :t flip flip :: (a -> b -> c) -> b -> a -> c
GHCi> flip map [1, 2, 3, 4] even [False,True,False,True]
GHCi> :t flip map flip map :: [a] -> (a -> b) -> [b] compare this with map :: (a -> b) -> [a] -> [b]

9. Assorted Syntax
Comments are -- to end of line or {- 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]

10. Layout Grouping is done by indentation, not by braces or parentheses
Actually, braces can be used, but that’s unusual
The first nonblank character following where, let, or of determines the starting column
let x = a + b y = a * b in y / x
Every expression in the same group must begin in the same column
Make sure your text editor replaces tabs with spaces

11. Infinite Lists
[1..5] == [1, 2, 3, 4, 5]
[1..] == all positive integers
[5, ] == [5, 10, 15, 20, 25, 30]
[5, 10..] == positive multiples of 5
[x*x | x <- [1..]] == squares of positive integers
[x*x | x <- [1..], even x] == squares of positive even integers
[(x, y) | x <- [1..10], y <- [1..10], x < y]

13. 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!

14. 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

15. 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

16. 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)

17. Point free style
Functions that take parameters (“points”) can be written as functions with implicit parameters
GHCi> let square_all xs = map (\x -> x^2) xs
GHCi> square_all [1, 2, 3, 4, 5]
[1,4,9,16,25]
GHCi> let square_all = map (\x -> x^2)
GHCi> square_all [1, 2, 3, 4, 5] [1,4,9,16,25]
GHCi> let square_all = map (^2)
Point free style can result in easier to read, less cluttered code
It can also result in obscure code, so use with care

18. 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]

19. 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]

20. 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

21. 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 =

22. 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

23. 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'
Prelude> (flip elem) "aeiou" 'o'

24. 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 . double doubleFirst = (* 2) . head
Main> quadruple 3 12 Main> doubleFirst [3..10] 6

25. 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")

26. 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!")

27. Function Definition I Functions are defined with =
fact :: Int -> Int -- explicit type fact n = if n == 0 then 1 else n * fact (n - 1)
There is no requirement that you explicitly state the type of each function you define
Haskell is superb at inferring the types of functions
Using Hadley-Milner type inference
However: All experienced Haskell programmers do explicitly state the type of each function
Doing so catches almost all programming errors!
Haskell programming is all about getting the types right

28. Function Definition II
Functions are usually defined by cases
fact :: Int -> Int fact n | n == = 1 | otherwise = n * fact (n - 1)
fact :: Int -> Int fact n = case n of > 1 n -> n * fact (n - 1)
These are equivalent definitions

29. Function Definition III
You can separate the cases with “patterns”
fact :: Int -> Int fact 0 = 1 fact n = n * fact (n - 1)
How does this work?

30. 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 = 1 fact n = n * fact (n - 1)
fact 5 won’t match the first, but will match the second

31. 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

32. 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

33. Pattern Types III “As-patterns” have the form w@pattern
When the pattern matches, the w matches the whole of the thing matched
firstThree = take 3 all
Main> firstThree [1..10]
[1,2,3]

34. 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

35. 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

36. Disadvantages of Haskell
Unfamiliar
Slow
because compromises are less in favor of the machine

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

38. The End