Haskell - part 1

Introduction, types and classesMaterial adapted from Erik Meijer’s Functional ProgrammingA Jupyter notebook version of this material is available online in Google Colab. Although Colab does not support Haskell, the notebook can be downloaded and run in Jupyter Notebook, if the IHaskell extension is installed.

Introduction

The software crisis 

• How can we cope with the size and complexity of modern computer programs?  
• How can we reduce the time and cost of program development?  
• How can we increase our confidence that the finished programs work correctly?

Programming languages 

One approach to the software crisis is to design new programming languages that:

• Allow programs to be written clearly, concisely, and at a high-level of abstraction
• Support reusable software components
• Encourage the use of formal verification 

Permit rapid prototyping

• Provide powerful problem-solving tools
• Functional languages provide a particularly elegant framework in which to address these goals

What is a functional language?

Opinions differ, and it is difficult to give a precise definition, but generally speaking:

• Functional programming is a style of programming in which the basic method of computation is the application of functions to arguments
• A functional language is one that supports and encourages the functional style

Example

Summing the integers 1 to 10 in Java:

             total = 0;
             for (i = 1; i ≤ 10; ++i)
                total = total+i;

The computation method is variable assignment

Example

Summing the integers 1 to 10 in Haskell:

sum [1..10]

The computation method is function application

Historical background

• 1930s: Alonzo Church develops the lambda calculus, a simple but powerful theory of functions
• 1950s: John McCarthy develops Lisp, the first functional language, with some influences from the lambda calculus, but retaining variable assignments
• 1960s: Peter Landin develops ISWIM, the first pure functional language, based strongly on the lambda calculus, with no assignments
• 1970s: John Backus develops FP, a functional language that emphasizes higher-order functions and reasoning about programs
• 1970s: Robin Milner and others develop ML, the first modern functional language, which introduced type inference and polymorphic types
• 1970s - 1980s: David Turner develops a number of lazy functional languages, culminating in the Miranda system
• 1987: An international committee of researchers initiates the development of Haskell, a standard lazy functional language
• 2003: The committee publishes the Haskell 98 report, defining a stable version of the language
• 2003 - date: Standard distribution, library support, new language features, development tools, use in industry, influence on other languages, etc.

A taste of Haskell

f []     = []
f (x:xs) = f ys ++ [x] ++ f zs
    where
        ys = [a | a <- xs, a <= x]
        zs = [b | b <- xs, b > x]

First steps

2+3*4

(2+3)*4

sqrt (3^2 + 4^2)

The standard prelude

Haskell comes with a large number of standard library functions. In addition to the familiar numeric functions such as + and *, the library also provides many useful functions on lists

-- select the first element of a list:
head [1,2,3,4,5]

-- remove the first element from a list:
tail [1,2,3,4,5]

-- select the nth element of a list:
[1,2,3,4,5] !! 2

-- select the first n elements of a list:
take 3 [1,2,3,4,5]

-- remove the first n elements from a list:
drop 3 [1,2,3,4,5]

-- calculate the length of a list:
length [1,2,3,4,5]

-- calculate the sum of a list of numbers:
sum [1,2,3,4,5]

-- calculate the product of a list of numbers:
product [1,2,3,4,5]

-- append two lists:
[1,2,3] ++ [4,5]

-- reverse a list:
reverse [1,2,3,4,5]

Function application

• In mathematics, function application is denoted using parentheses, and multiplication is often denoted using juxtaposition or space, \(f(a,b) + c\cdot d\)
• Apply the function f to a and b, and add the result to the product of c and d
• In Haskell, function application is denoted using space, and multiplication is denoted using *

  f a b + c*d
  

• As previously, but in Haskell syntax
• Moreover, function application is assumed to have higher priority than all other operators

  f a + b
  

• Means (f a) + b, rather than f (a + b)

Examples

Haskell scripts

• As well as the functions in the standard library, you can also define your own functions

• New functions are defined within a script, a text file comprising a sequence of definitions

• By convention, Haskell scripts usually have a .hs suffix on their filename. This is not mandatory, but is useful for identification purposes

Script example

double x    = x + x

quadruple x = double (double x)

quadruple 10

take (double 2) [1, 2, 3, 4, 5, 6]

factorial n = product [1..n]

average ns  = sum ns `div` length ns

Note:

• div is enclosed in back quotes, not forward
• x `f` y is just syntactic sugar for f x y

factorial 10

average [1,2,3,4,5]

Naming requirements

Function and argument names must begin with a lower-case letter. For example:

myFun
fun1
arg_2
x’

By convention, list arguments usually have an s suffix on their name. For example:

xs
ns
nss

The layout rule

In a sequence of definitions, each definition must begin in precisely the same column:

a = 10
b = 20
c = 30

a = 10
 b = 20
c = 30

 a = 10
b = 20
 c = 30

The layout rule avoids the need for explicit syntax to indicate the grouping of definitions

-- implicit grouping
a = b + c
    where
        b = 1
        c = 2
d = a * 2

means

-- explicit grouping
a = b + c
    where
        {b = 1;
        c = 2}
d = a * 2

Useful GHCi commands

Exercises

1. Try out the code covered so far using GHCi.
2. Fix the syntax errors in the program below, and test your solution using GHCi

N = a ’div’ length xs
    where
        a = 10
       xs = [1,2,3,4,5]

1. Show how the library function last that selects the last element of a list can be defined using the functions introduced in this lecture.
2. Can you think of another possible definition?
3. Similarly, show how the library function init that removes the last element from a list can be defined in two different ways.

Types and classes 

What is a type?

A type is a name for a collection of related values. For example, in Haskell the basic type Bool contains the two logical values: False and True

Type errors

Applying a function to one or more arguments of the wrong type is called a type error

1 + False
-- Error

1 is a number and False is a logical value, but + requires two numbers

Types in Haskell

If evaluating an expression e would produce a value of type t, then e has type t, written

e :: t

Every well formed expression has a type, which can be automatically calculated at compile time using a process called type inference

• All type errors are found at compile time, which makes programs safer and faster by removing the need for type checks at run time
• In GHCi, the :type command calculates the type of an expression, without evaluating it:

not False

:type not False

Basic types

Haskell has a number of basic types, including:

List types

A list is sequence of values of the same type:

[False,True,False] :: [Bool]

[’a’,’b’,’c’,’d’]  :: [Char]

In general: [t] is the type of lists with elements of type t

Note:

• The type of a list says nothing about its length:

[False,True]       :: [Bool]

[False,True,False] :: [Bool]

• The type of the elements is unrestricted. For example, we can have lists of lists:

[['a'],['b','c']] :: [[Char]]

Tuple types

A tuple is a sequence of values of different types:

(False,True)     :: (Bool,Bool)

(False,'a',True) :: (Bool,Char,Bool)

In general: (t1,t2,...,tn) is the type of n-tuples whose ith components have type ti for any i in 1...n.

Note:

• The type of a tuple encodes its size:

(False,True)       :: (Bool,Bool)

(False,True,False) :: (Bool,Bool,Bool)

• The type of the components is unrestricted:

(’a’,(False,’b’)) :: (Char,(Bool,Char))

(True,[’a’,’b’])  :: (Bool,[Char])

Function types

A function is a mapping from values of one type to values of another type:

not     :: Bool → Bool

isDigit :: Char → Bool

In general: t1 → t2 is the type of functions that map values of type t1 to values to type t2

Note:

• The arrow → is typed at the keyboard as ->
• The argument and result types are unrestricted. For example, functions with multiple arguments or results are possible using lists or tuples:

add :: (Int, Int) -> Int
add (x, y) = x + y

zeroto :: Int -> [Int]
zeroto n = [0..n]

Curried functions

Functions with multiple arguments are also possible by returning functions as results:

add' :: Int -> (Int -> Int)
add' x y = x + y

add' takes an integer x and returns a function add’ x. In turn, this function takes an integer y and returns the result x+y

Note:

• add and add' produce the same final result, but add takes its two arguments at the same time, whereas add' takes them one at a time:

add :: (Int, Int) → Int

add' :: Int → (Int → Int)

• Functions that take their arguments one at a time are called curried functions, celebrating the work of Haskell Curry on such functions

• Functions with more than two arguments can be curried by returning nested functions:

mult :: Int -> (Int -> (Int -> Int))
mult x y z = x*y*z

mult takes an integer x and returns a function mult x, which in turn takes an integer y and returns a function mult x y, which finally takes an integer z and returns the result x*y*z

Why is Currying useful?

Curried functions are more flexible than functions on tuples, because useful functions can often be made by partially applying a curried function

For example:

add' 1 :: Int → Int

take 5 :: [Int] → [Int]

drop 5 :: [Int] → [Int]

Currying conventions

To avoid excess parentheses when using curried functions, two simple conventions are adopted:

The arrow → associates to the right

Int → Int → Int → Int

Means Int → (Int → (Int → Int))

• As a consequence, it is then natural for function application to associate to the left

mult x y z

Means ((mult x) y) z

Unless tupling is explicitly required, ALL functions in Haskell are normally defined in curried form

Polymorphic functions

A function is called polymorphic (“of many forms”) if its type contains one or more type variables

length :: [a] → Int

for any type a, length takes a list of values of type a and returns an integer

Note:

• Type variables can be instantiated to different types in different circumstances:

length [False,True] -- a = Bool

length [1,2,3,4]    -- a = Int

• Type variables must begin with a lower-case letter, and are usually named a, b, c, etc

• Many of the functions defined in the standard prelude are polymorphic.

For example:

fst  :: (a,b) → a

head :: [a] → a

take :: Int → [a] → [a]

zip  :: [a] → [b] → [(a,b)]

id   :: a → a

Overloaded functions

A polymorphic function is called overloaded if its type contains one or more class constraints

sum :: Num a ⇒ [a] → a

for any numeric type a, sum takes a list of values of type a and returns a value of type a

Note:

• Constrained type variables can be instantiated to any types that satisfy the constraints:

sum [1,2,3]       -- a = Int

sum [1.1,2.2,3.3] -- a = Float

sum ['a','b','c'] -- numeric type

• Haskell has a number of type classes, including:

• For example:

(+)  :: Num a ⇒ a → a → a

(==) :: Eq a  ⇒ a → a → Bool

(<)  :: Ord a ⇒ a → a → Bool

Exercises

1. What are the types of the following values?

['a','b','c']

('a','b','c')

[(False,'0'),(True,'1')]

([False,True],['0','1'])

[tail,init,reverse]

1. What are the types of the following functions?

second xs     = head (tail xs)

swap (x,y)    = (y,x)

pair x y      = (x,y)

double x      = x*2

palindrome xs = reverse xs == xs

twice f x     = f (f x)

1. Check your answers using Jupyter notebook, or GHCi.

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