Transcript Functional Programming - SLU Mathematics and Computer Science

PROGRAMMING IN HASKELL
Modules
Based on lecture notes by Graham Hutton
The book “Learn You a Haskell for Great Good”
(and a few other sources)
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Type Declarations
In Haskell, a new name for an existing type can be
defined using a type declaration.
type String = [Char]
String is a synonym for the type [Char].
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Type declarations can be used to make other types
easier to read. For example, given
type Pos = (Int,Int)
we can define:
origin
origin
:: Pos
= (0,0)
left
:: Pos  Pos
left (x,y) = (x-1,y)
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Like function definitions, type declarations can also
have parameters. For example, given
type Pair a = (a,a)
we can define:
mult
:: Pair Int  Int
mult (m,n) = m*n
copy
copy x
:: a  Pair a
= (x,x)
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Type declarations can be nested:
type Pos
= (Int,Int)
type Trans = Pos  Pos
However, they cannot be recursive:
type Tree = (Int,[Tree])
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Data Declarations
A completely new type can be defined by specifying
its values using a data declaration.
data Bool = False | True
Bool is a new type, with two
new values False and True.
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Note:
The two values False and True are called the
constructors for the type Bool.
Type and constructor names must begin with
an upper-case letter.
Data declarations are similar to context free
grammars. The former specifies the values of a
type, the latter the sentences of a language.
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Values of new types can be used in the same ways
as those of built in types. For example, given
data Answer = Yes | No | Unknown
we can define:
answers
answers
:: [Answer]
= [Yes,No,Unknown]
flip
::
flip Yes
=
flip No
=
flip Unknown =
Answer  Answer
No
Yes
Unknown
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The constructors in a data declaration can also have
parameters. For example, given
data Shape = Circle Float
| Rect Float Float
we can define:
square
square n
:: Float  Shape
= Rect n n
area
:: Shape  Float
area (Circle r) = pi * r^2
area (Rect x y) = x * y
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Note:
Shape has values of the form Circle r where r is
a float, and Rect x y where x and y are floats.
Circle and Rect can be viewed as functions that
construct values of type Shape:
Circle :: Float  Shape
Rect
:: Float  Float  Shape
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Recursive Types
In Haskell, new types can be declared in terms of
themselves. That is, types can be recursive.
data Nat = Zero | Succ Nat
Nat is a new type, with constructors
Zero :: Nat and Succ :: Nat  Nat.
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Note:
A value of type Nat is either Zero, or of the form
Succ n where n :: Nat. That is, Nat contains the
following infinite sequence of values:
Zero
Succ Zero
Succ (Succ Zero)



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We can think of values of type Nat as natural
numbers, where Zero represents 0, and Succ
represents the successor function 1+.
For example, the value
Succ (Succ (Succ Zero))
represents the natural number
1 + (1 + (1 + 0))
= 3
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Using recursion, it is easy to define functions that
convert between values of type Nat and Int:
nat2int
nat2int Zero
:: Nat  Int
= 0
nat2int (Succ n) = 1 + nat2int n
int2nat
:: Int  Nat
int2nat 0
= Zero
int2nat (n+1)
= Succ (int2nat n)
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Two naturals can be added by converting them to
integers, adding, and then converting back:
add
:: Nat  Nat  Nat
add m n = int2nat (nat2int m + nat2int n)
However, using recursion the function add can be
defined without the need for conversions:
add Zero
n = n
add (Succ m) n = Succ (add m n)
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For example:
=
=
=
add (Succ (Succ Zero)) (Succ Zero)
Succ (add (Succ Zero) (Succ Zero))
Succ (Succ (add Zero (Succ Zero))
Succ (Succ (Succ Zero))
Note:
The recursive definition for add corresponds to
the laws 0+n = n and (1+m)+n = 1+(m+n).
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Arithmetic Expressions
Consider a simple form of expressions built up from
integers using addition and multiplication.
+

1
2
3
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Using recursion, a suitable new type to represent
such expressions can be declared by:
data Expr = Val Int
| Add Expr Expr
| Mul Expr Expr
For example, the expression on the previous slide
would be represented as follows:
Add (Val 1) (Mul (Val 2) (Val 3))
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Using recursion, it is now easy to define functions
that process expressions. For example:
size
size (Val n)
:: Expr  Int
= 1
size (Add x y) = size x + size y
size (Mul x y) = size x + size y
eval
eval (Val n)
:: Expr  Int
= n
eval (Add x y) = eval x + eval y
eval (Mul x y) = eval x * eval y
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Note:
The three constructors have types:
Val :: Int  Expr
Add :: Expr  Expr  Expr
Mul :: Expr  Expr  Expr
Many functions on expressions can be defined
by replacing the constructors by other functions
using a suitable fold function. For example:
eval = fold id (+) (*)
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Binary Trees
In computing, it is often useful to store data in a
two-way branching structure or binary tree.
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7
3
1
4
6
9
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Using recursion, a suitable new type to represent
such binary trees can be declared by:
data Tree = Leaf Int
| Node Tree Int Tree
For example, the tree on the previous slide would
be represented as follows:
Node (Node (Leaf 1) 3 (Leaf 4))
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(Node (Leaf 6) 7 (Leaf 9))
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We can now define a function that decides if a given
integer occurs in a binary tree:
occurs
:: Int  Tree  Bool
occurs m (Leaf n)
= m==n
occurs m (Node l n r) = m==n
|| occurs m l
|| occurs m r
But… in the worst case, when the integer does not
occur, this function traverses the entire tree.
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Now consider the function flatten that returns the
list of all the integers contained in a tree:
flatten
:: Tree  [Int]
flatten (Leaf n)
= [n]
flatten (Node l n r) = flatten l
++ [n]
++ flatten r
A tree is a search tree if it flattens to a list that is
ordered. Our example tree is a search tree, as it
flattens to the ordered list [1,3,4,5,6,7,9].
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Search trees have the important property that when
trying to find a value in a tree we can always decide
which of the two sub-trees it may occur in:
occurs m (Leaf n)
= m==n
occurs m (Node l n r) | m==n = True
| m<n
= occurs m l
| m>n
= occurs m r
This new definition is more efficient, because it only
traverses one path down the tree.
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Exercise
Node (Node (Leaf 1) 3 (Leaf 4))
5 (Node (Leaf 6) 7 (Leaf 9))
A binary tree is complete if the two sub-trees of
every node are of equal size. Define a function
that decides if a binary tree is complete.
data Tree = Leaf Int
| Node Tree Int Tree
occurs
:: Int  Tree  Bool
occurs m (Leaf n)
= m==n
occurs m (Node l n r) = m==n
|| occurs m l
|| occurs m r
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Modules
So far, we’ve been using built-in functions
provided in the Haskell prelude. This is a subset
of a larger library that is provided with any
installation of Haskell. (Google for Hoogle to see a
handy search engine for these.)
Examples of other modules:
- lists
- concurrent programming
- complex numbers
- char
- sets
-…
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Example: Data.List
To load a module, we need to import it:
import Data.List
All the functions in this module are immediately
available:
numUniques :: (Eq a) => [a] -> Int
numUniques = length . nub
function
concatenation
This is a function in Data.List
that removes duplicates from a
list.
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You can also load modules from the command
prompt:
ghci> :m + Data.List
Or several at once:
ghci> :m + Data.List Data.Map Data.Set
Or import only some, or all but some:
import Data.List (nub, sort)
import Data.List hiding (nub)
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If duplication of names is an issue, can extend the
namespace:
import qualified Data.Map
This imports the functions, but we have to use
Data.Map to use them – like Data.Map.filter.
When the Data.Map gets a bit long, we can
provide an alias:
import qualified Data.Map as M
And now we can just type M.filter, and the
normal list filter will just be filter.
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Data.List has a lot more functionality than we’ve
seen. A few examples:
ghci> intersperse '.' "MONKEY"
"M.O.N.K.E.Y"
ghci> intersperse 0 [1,2,3,4,5,6]
[1,0,2,0,3,0,4,0,5,0,6]
ghci> intercalate " " ["hey","there","guys"]
"hey there guys"
ghci> intercalate [0,0,0] [[1,2,3],[4,5,6],
[7,8,9]]
[1,2,3,0,0,0,4,5,6,0,0,0,7,8,9]
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And even more:
ghci> transpose [[1,2,3],[4,5,6],
[7,8,9]]
[[1,4,7],[2,5,8],[3,6,9]]
ghci> transpose ["hey","there","guys"] ["
htg","ehu","yey","rs","e"]
ghci> concat ["foo","bar","car"]
"foobarcar"
ghci> concat [[3,4,5],[2,3,4],[2,1,1]]
[3,4,5,2,3,4,2,1,1]
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And even more:
ghci> and $ map (>4) [5,6,7,8]
True
ghci> and $ map (==4) [4,4,4,3,4]
False
ghci> any (==4) [2,3,5,6,1,4]
True
ghci> all (>4) [6,9,10]
True
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A nice example: adding functions
Functions are often represented as vectors: 8x^3 +
5x^2 + x - 1 is [8,5,1,-1].
So we can easily use List functions to add these
vectors:
ghci> map sum $ transpose [[0,3,5,9],
[10,0,0,9],[8,5,1,-1]]
[18,8,6,17]
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There are a ton of these functions, so I could spend
all semester covering just lists.
More examples: group, sort, dropWhile, takeWhile,
partition, isPrefixOf, find, findIndex, delete, words,
insert,…
Instead, I’ll make sure to post a link to a good
overview of lists on the webpage, in case you need
them.
In essence, if it’s a useful thing to do to a list,
Haskell probably supports it!
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The Data.Char module: includes a lot of useful
functions that will look similar to python, actually.
Examples: isAlpha, isLower, isSpace, isDigit,
isPunctuation,…
ghci> all isAlphaNum "bobby283"
True
ghci> all isAlphaNum "eddy the fish!"Fal
se
ghci> groupBy ((==) `on` isSpace)
"hey guys its me"
["hey"," ","guys"," ","its"," ","me"]
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The Data.Char module has a datatype that is a set
of comparisons on characters. There is a function
called generalCategory that returns the information.
(This is a bit like the Ordering type for numbers,
which returns LT, EQ, or GT.)
ghci> generalCategory ' '
Space
ghci> generalCategory 'A'
UppercaseLetter
ghci> generalCategory 'a'
LowercaseLetter
ghci> generalCategory '.'
OtherPunctuation
ghci> generalCategory '9'
DecimalNumber
ghci> map generalCategory " ¥t¥nA9?|"
[Space,Control,Control,UppercaseLetter,DecimalNumber,OtherPunctuation,M
athSymbol] ]
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There are also functions that can convert between
Ints and Chars:
ghci> map digitToInt "FF85AB"
[15,15,8,5,10,11]
ghci> intToDigit 15
'f'
ghci> intToDigit 5
'5'
ghci> chr 97
'a'
ghci> map ord "abcdefgh"
[97,98,99,100,101,102,103,104]
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Neat application: Ceasar ciphers
A primitive encryption cipher which encodes
messages by shifted them a fixed amount in the
alphabet.
Example: hello with shift of 3
encode :: Int -> String -> String
encode shift msg =
let ords = map ord msg
shifted = map (+ shift) ords
in map chr shifted
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Now to use it:
ghci> encode 3 "Heeeeey"
"Khhhhh|"
ghci> encode 4 "Heeeeey"
"Liiiii}"
ghci> encode 1 "abcd"
"bcde"
ghci> encode 5 "Marry Christmas! Ho ho ho!”
"Rfww~%Hmwnxyrfx&%Mt%mt%mt&"
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Decoding just reverses the encoding:
decode :: Int -> String -> String
decode shift msg =
encode (negate shift) msg
ghci> encode 3 "Im a little teapot"
"Lp#d#olwwoh#whdsrw"
ghci> decode 3 "Lp#d#olwwoh#whdsrw"
"Im a little teapot"
ghci> decode 5 . encode 5 $ "This is a sentence
"
"This is a sentence"
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Making our own modules
We specify our own modules at the beginning of a
file. For example, if we had a set of geometry
functions:
module Geometry
( sphereVolume
, sphereArea
, cubeVolume
, cubeArea
, cuboidArea
, cuboidVolume
) where
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Then, we put the functions that the module uses:
sphereVolume :: Float -> Float
sphereVolume radius = (4.0 / 3.0) * pi *
(radius ^ 3)
sphereArea :: Float -> Float
sphereArea radius = 4 * pi * (radius ^ 2)
cubeVolume :: Float -> Float
cubeVolume side = cuboidVolume side side side
…
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Note that we can have “private” helper functions,
also:
cuboidVolume :: Float -> Float -> Float
-> Float
cuboidVolume a b c = rectangleArea a b * c
cuboidArea :: Float -> Float ->
Float -> Float
cuboidArea a b c = rectangleArea a b * 2 + rectangl
eArea a c * 2 + rectangleArea c b * 2
rectangleArea :: Float -> Float -> Float
rectangleArea a b = a * b
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Can also nest these. Make a folder called
Geometry, with 3 files inside it:
• Sphere.hs
• Cubiod.hs
• Cube.hs
Each will hold a separate group of functions.
To load:
import Geometry.Sphere
Or (if functions have same names):
import qualified Geometry.Sphere as Sphere
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The modules:
module Geometry.Sphere
( volume
, area
) where
volume :: Float -> Float
volume radius = (4.0 / 3.0) * pi * (radius ^
3)
area :: Float -> Float
area radius = 4 * pi * (radius ^ 2)
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module Geometry.Cuboid
( volume
, area
) where
volume :: Float -> Float -> Float -> Float
volume a b c = rectangleArea a b * c
…
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