Defining Functions

Programs in Haskell

A Haskell program is a set of declarations. Declarations:

Further declarations:

The order of declarations is not important.

Modules

The definitions are organized into modules, a module is a unit of compilation.

Structure of a Haskell module:

Every Haskell module is a plain text file with an .hs or .lhs extension.

Let us create an empty text file with the name X.hs.

Load this module to the interpreter.

:l X.hs

Reload the currently loaded module.

:r

Definitions have to be added to one or more text files, all of them are called modules. Names for modules consist of letters and numerals, and they should start with capital letters, e.g. Char, List, Ratio.

When clicking on files with .hs extension on Windows systems, the GHCi is launched automatically and it loads the module with the corresponding name.

After the module is loaded, the prompt in GHCi changes to Main*>. Here, the * symbol signals that all of definitions from the module are now visible, together with the contents of modules that were imported from it. (That is, together with Prelude, which is always automatically available.)

Comments

This is a line comment.

 -- Comment til the end of the line

And this is a multi-line comment.

{- comment -}
{- comments
 {- embedded into -}
  each other -}

Line comments start with the -- symbol and they end at the end of the line.

Multi-line comments are delimited by {- and -} symbols. Multi-line comments may be embedded into each other.

Function Definitions

A one-parameter function, without a type:

f x = x + 1

Ask the compiler what type was inferred for this function.

Test>

A two-parameter function, with a type:

g :: Integer -> Integer -> Integer
g a b = a * b + 1

Usage:

Test>
3 :: Integer

Function and variable names consist of letters and numerals, but they have to start with a lowercase letter. They may contain the “underscore” (_) and “prime” (') characters, e.g. f_1, f', and f''.

Exercise: Mountain [*]

Generate the following list: [1, 2, …, n-1, n, n-1, …, 2, 1]

mountain :: Integer -> [Integer]

Test>
[1, 2, 3, 2, 1] :: [Integer]
Test>
[1] :: [Integer]
Test>
[] :: [Integer]
Test>
[] :: [Integer]

Exercise: Are Triangle Sides

Given sides of a triangle as a, b, c, decide whether the given triangle can be constructed.

areTriangleSides :: Real a => a -> a -> a -> Bool

Test>
True :: Bool
Test>
False :: Bool

Exercise: Prelude.even [*]

Redefine the even function from the Prelude module, which determines a given integer is even.

even :: Integer -> Bool

Test>
False :: Bool
Test>
True :: Bool

Because even is already defined in Prelude, we can only redefine this function if we hide its definition from Prelude.

import declarations can be only added to the beginning of the module.

More definitions can be also hidden by using hiding.

Exercise: Prelude.odd

Redefine the odd function from the Prelude module, which determines a given integer is odd.

odd :: Integer -> Bool

Test>
True :: Bool
Test>
[True, False, True, False, True] :: [Bool]

Exercise: Divisibility [*]

Define the divisibility.

divides :: Integer -> Integer -> Bool

Test>
True :: Bool
Test>
False :: Bool

Exercise: Leap Years

Determine a given year is a leap year.

isLeapYear :: Integer -> Bool

Test>
[1952, 1956, 1960] :: [Integer]
Test>
False :: Bool
Test>
True :: Bool

Exercise: Sum of Squares

Calculate the sum of squares from 1 to n.

sumSquaresTo :: Integer -> Integer

Test>
0 :: Integer
Test>
30 :: Integer
Test>
0 :: Integer

Exercise: Divisors [*]

Define a function that generates a list of divisors for a given number.

divisors :: Integer -> [Integer]

Test>
[1, 2, 3, 4, 6, 12] :: [Integer]

Exercise: Proper Divisors

Define a function that generates a list of proper divisors for a given number.

properDivisors :: Integer -> [Integer]

Test>
[2, 3, 4, 6] :: [Integer]

Exercise: Regular Polygons

Define regular polygons if the radius of the circumcircle and the number of sides are given.

regularPolygon :: Double -> Int -> Diagram

Test>

Hint: Use the polygon function, and there may be some conversion or rounding needed. Points of the polygon can be given in the (r * cos φ, r * sin φ) form.

Exercise: Stars

Define stars by connecting every mth node of an n-side regular polygon if the radius of the circumcircle, n, m are given.

star :: Double -> Int -> Int -> Diagram

Test>
Test>

Defining Operators

(.*.) :: Double -> Double -> Double
a .*. b = a * b + 1

Precedence and associativity for an operator can be set by the infix, infixl, and infixr keywords.

infixl 7 .*.

Query the precedence and associativity of an already defined operator.

Main*> :i (+)
infixl 6 +

Operator names may consist of the following ASCII symbols: !?.#$%@&*+-~^/|\<=>:. However, the following keywords are reserved (so they cannot be used as operators): =, .., |, <-, ->, =>, ::, \, @, ~.

(These must be familiar, except the last three.)

Operators are used in infix notation, that is, the name of the operator is put between the operands.

With the infix n o declaration, the o operator will have no associativity and precedence of n. The precedence may vary between 0 and 9. infixl means “left associative”, and infixr means “right associative”.