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INTRODUCTION TOFUNCTIONAL PROGRAMMING

Graham HuttonUniversity of Nottingham

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What is Functional Programming?

Functional programming is 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.

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

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.

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Example

Summing the integers 1 to 10 in Haskell:

sum [1..10]

The computation method is function application.

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Why is it Useful?

Again, there are many possible answers to this question, but generally speaking:

The abstract nature of functional programming leads to considerably simpler programs;

It also supports a number of powerful new ways to structure and reason about programs.

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This Course

A series of mini-lectures (with exercises) reviewing a number of basic concepts, using Haskell:The Hugs system;Types and classes I/II;Defining functions;List comprehensions;Recursive functions;Higher-order functions;Functional parsers;Defining types.

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These concepts will be tied together at the end by two extended programming examples, concerning a simple game and a simple compiler.

Note:

The material in this course is based upon my forthcoming book, Programming in Haskell;

Please ask questions during the lectures!

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LECTURE 1THE HUGS SYSTEM

Graham HuttonUniversity of Nottingham

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What is Hugs?An interpreter for Haskell, and the most

widely used implementation of the language;

An interactive system, which is well-suited for teaching and prototyping purposes;

Hugs is freely available from:www.haskell.org/hugs

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The Standard Prelude

When Hugs is started it first loads the library file Prelude.hs, and then repeatedly prompts the user for an expression to be evaluated.

For example:

> 2+3*414

> (2+3)*420

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> length [1,2,3,4]4

> product [1,2,3,4]24

> take 3 [1,2,3,4,5][1,2,3]

The standard prelude also provides many useful functions that operate on lists. For example:

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Function Application

In mathematics, function application is denoted using parentheses, and multiplication is often denoted using juxtaposition or space.

f(a,b) + c d

Apply the function f to a and b, and add the result to the product of c

and d.

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In Haskell, function application is denoted using space, and multiplication is denoted using *.

f a b + c*d

As previously, but in Haskell syntax.

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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).

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Examples

Mathematics Haskell

f(x)

f(x,y)

f(g(x))

f(x,g(y))

f(x)g(y)

f x

f x y

f (g x)

f x (g y)

f x * g y

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My First Script

double x = x + x

quadruple x = double (double x)

When developing a Haskell script, it is useful to keep two windows open, one running an editor for the script, and the other running Hugs.

Start an editor, type in the following two function definitions, and save the script as test.hs:

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% hugs test.hs

Leaving the editor open, in another window start up Hugs with the new script:

> quadruple 1040

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

Now both Prelude.hs and test.hs are loaded, and functions from both scripts can be used:

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factorial n = product [1..n]

average ns = sum ns `div` length ns

Leaving Hugs open, return to the editor, add the following two definitions, and resave:

div is enclosed in back quotes, not forward;

x `f` y is just syntactic sugar for f x y.

Note:

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> :reloadReading file "test.hs"

> factorial 103628800

> average [1..5]3

Hugs does not automatically reload scripts when they are changed, so a reload command must be executed before the new definitions can be used:

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ExercisesTry out some of the other functions from the standard prelude using Hugs.

Work through "My First Script" using Hugs.

Show how the functions last and init from the standard prelude could be re-defined using other functions from the prelude.

Note: there are many possible answers!

(1)

(2)

(3)

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LECTURE 2TYPES AND CLASSES (I)

Graham HuttonUniversity of Nottingham

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What is a Type?

A type is a collection of related values.

Bool

Bool Bool

The logical valuesFalse and True.

All functions that map a logical

value to a logical value.

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Types in Haskell

False :: Bool

not :: Bool Bool

not False :: Bool

False && True :: Bool

We use the notation e :: T to mean that evaluating the expression e will produce a value of type T.

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Every expression must have a valid type, which is calculated prior to evaluating the expression by a process called type inference;

Haskell programs are type safe, because type errors can never occur during evaluation;

Type inference detects a very large class of programming errors, and is one of the most powerful and useful features of Haskell.

Note:

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Basic TypesHaskell has a number of basic types, including:

Char

String

Integer

Int

Bool - Logical values

- Single characters

- Strings of characters

- Fixed-precision integers

- Arbitrary-precision integers

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List Types

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

[’a’,’b’,’c’,’d’] :: [Char]

In general:

A list is sequence of values of the same type:

[T] is the type of lists with elements of type T.

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The type of a list says nothing about its length:

[False,True] :: [Bool]

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

[[’a’],[’b’,’c’]] :: [[Char]]

Note:

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

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Tuple TypesA 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.

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The type of a tuple encodes its arity:

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

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

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

(True,[’a’,’b’]) :: (Bool,[Char])

Note:

The type of the components is unrestricted:

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Function Types

not :: Bool Bool

isDigit :: Char Bool

In general:

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

T1 T2 is the type of functions that map arguments of type T1 to results of type T2.

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The argument and result types are unrestricted. For example, functions with multiple arguments or results are possible using lists or tuples:

Note:

add :: (Int,Int) Intadd (x,y) = x+y

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

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Exercises

[’a’,’b’,’c’]

(’a’,’b’,’c’)

[(False,’0’),(True,’1’)]

[isDigit,isLower,isUpper]

What are the types of the following values?(1)

Check your answers using Hugs.(2)

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LECTURE 3TYPES AND CLASSES (II)

Graham HuttonUniversity of Nottingham

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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. In turn, this function takes an integer y and

returns the result x+y.

Curried Functions

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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:

Note:

Functions that take their arguments one at a time are called curried functions.

add :: (Int,Int) Int

add’ :: Int (Int Int)

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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, which in turn takes an integer y and returns a function,

which finally takes an integer z and returns the result x*y*z.

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Curry Conventions

The arrow associates to the right.

Int Int Int Int

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

Means Int (Int (Int Int)).

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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.

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Polymorphic TypesThe function length calculates the length of any list, irrespective of the type of its elements.

> length [1,3,5,7]4

> length ["Yes","No"]2

> length [isDigit,isLower,isUpper]3

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This idea is made precise in the type for length by the inclusion of a type variable:

length :: [a] Int

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

integer.

A type with variables is called polymorphic.

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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)]

Note:

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Overloaded TypesThe arithmetic operator + calculates the sum of any two numbers of the same numeric type.

For example:

> 1+23

> 1.1 + 2.23.3

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This idea is made precise in the type for + by the inclusion of a class constraint:

(+) :: Num a a a a

For any type a in the class Num of numeric types, + takes two values of type a

and returns another.

A type with constraints is called overloaded.

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Classes in HaskellA class is a collection of types that support certain operations, called the methods of the class.

EqTypes whose values can be compared for

equality and difference using

(==) :: a a Bool(/=) :: a a Bool

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Eq - Equality types

Ord - Ordered types

Show - Showable types

Haskell has a number of basic classes, including:

Read - Readable types

Num - Numeric types

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(==) :: Eq a a a Bool

(<) :: Ord a a a Bool

show :: Show a a String

read :: Read a String a

() :: Num a a a a

Example methods:

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Exercises

second xs = head (tail xs)swap (x,y) = (y,x)pair x y = (x,y)double x = x*2palindrome xs = reverse xs == xstwice f x = f (f x)

What are the types of the following functions?(1)

Check your answers using Hugs.(2)

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LECTURE 4DEFINING FUNCTIONS

Graham HuttonUniversity of Nottingham

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Conditional ExpressionsAs in most programming languages, functions can be defined using conditional expressions.

abs :: Int Intabs n = if n 0 then n else -n

abs takes an integer n and returns n if it is non-negative and

-n otherwise.

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Conditional expressions can be nested:

signum :: Int Intsignum n = if n < 0 then -1 else if n == 0 then 0 else 1

In Haskell, conditional expressions must always have an else branch, which avoids any possible ambiguity problems with nested conditionals.

Note:

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Guarded EquationsAs an alternative to conditionals, functions can also be defined using guarded equations.

abs n | n 0 = n | otherwise = -n

As previously, but using guarded equations.

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Guarded equations can be used to make definitions involving multiple conditions easier to read:

The catch all condition otherwise is defined in the prelude by otherwise = True.

Note:

signum n | n < 0 = -1 | n == 0 = 0 | otherwise = 1

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Pattern MatchingMany functions have a particularly clear definition using pattern matching on their arguments.

not :: Bool Boolnot False = Truenot True = False

not maps False to True, and True to False.

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Functions can often be defined in many different ways using pattern matching. For example

(&&) :: Bool Bool BoolTrue && True = TrueTrue && False = FalseFalse && True = False False && False = False

True && True = True_ && _ = False

can be defined more compactly by

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False && _ = FalseTrue && b = b

However, the following definition is more efficient, as it avoids evaluating the second argument if the first argument is False:

The underscore symbol _ is the wildcard pattern that matches any argument value.

Note:

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List PatternsIn Haskell, every non-empty list is constructed by repeated use of an operator : called “cons” that adds a new element to the start of a list.

[1,2,3]

Means 1:(2:(3:[])).

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The cons operator can also be used in patterns, in which case it destructs a non-empty list.

head :: [a] ahead (x:_) = x

tail :: [a] [a]tail (_:xs) = xs

head and tail map any non-empty list to its first and remaining

elements.

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Lambda Expressions

A function can be constructed without giving it a name by using a lambda expression.

x x+1

The nameless function that takes a number x and returns the

result x+1.

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Why Are Lambda's Useful?Lambda expressions can be used to give a formal meaning to functions defined using currying.

For example:

add x y = x+y

add = x (y x+y)

means

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compose f g x = f (g x)

is more naturally defined by

compose f g = x f (g x)

Lambda expressions are also useful when defining functions that return functions as results.

For example,

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Exercises

Assuming that else branches were optional in conditional expressions, give an example of a nested conditional with ambiguous meaning.

(1)

Give three possible definitions for the logical or operator || using pattern matching.

(2)

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Consider a function safetail that behaves in the same way as tail, except that safetail maps the empty list to the empty list, whereas tail gives an error in this case. Define safetail using:

(i) a conditional expression; (ii) guarded equations; (iii) pattern matching.

Hint:

The prelude function null :: [a] Bool can be used to test if a list is empty.

(3)