程序代做CS代考 Haskell PROGRAMMING IN HASKELL – cscodehelp代写

PROGRAMMING IN HASKELL
Chapter 8 – Declaring Types and Classes
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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 :: Pos
origin = (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::a → Paira copy x = (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 always 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 :: [Answer]
answers = [Yes,No,Unknown]
flip :: Answer → Answer flip Yes = No
flip No = Yes
flip Unknown = 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 :: Float → Shape square n = 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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Not surprisingly, data declarations themselves can also have parameters. For example, given
data Maybe a = Nothing | Just a
we can define:
safediv :: Int → Int → Maybe Int safediv _ 0 = Nothing
safediv m n = Just (m `div` n)
safehead :: [a] → Maybe a safehead [] = Nothing safehead xs = Just (head xs)
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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 :: Nat → Int
nat2int Zero = 0
nat2int (Succ n) = 1 + nat2int n
int2nat :: Int → Nat
int2nat 0 = Zero
int2nat n = Succ (int2nat (n-1))
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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:
addZero n=n
add (Succ m) n = Succ (add m n)
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For example:
add (Succ (Succ Zero)) (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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Succ (add (Succ Zero) (Succ Zero))
Succ (Succ (add Zero (Succ Zero))
Succ (Succ (Succ Zero))

Arithmetic Expressions
Consider a simple form of expressions built up from integers using addition and multiplication.
+ 1*
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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
| 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 :: Expr → Int
size (Val n) = 1
size (Add x y) = size x + size y size (Mul x y) = size x + size y
eval :: Expr → Int
eval (Val n) = 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 → Expr → Expr
❚ Many functions on expressions can be defined by replacing the constructors by other functions using a suitable fold function. For example:
eval = folde 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.
5
37 1469
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Using recursion, a suitable new type to represent such binary trees can be declared by:
data Tree a = Leaf a
| Node (Tree a) a (Tree a)
For example, the tree on the previous slide would be represented as follows:
t :: Tree Int
t = Node (Node (Leaf 1) 3 (Leaf 4)) 5
(Node (Leaf 6) 7 (Leaf 9))
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We can now define a function that decides if a given value occurs in a binary tree:
occurs::Orda ⇒ a → Treea → Bool occursx(Leafy) =x==y occurs x (Node l y r) = x == y
|| occurs x l
|| occurs x r
But… in the worst case, when the value 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 values contained in a tree:
flatten :: Tree a → [a]
flatten (Leaf x) = [x] flatten (Node l x r) = flatten l
++ [x]
++ 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 x (Leaf y) = x == y
occurs x (Node l y r) | x == y = True
|xy =occursxr
This new definition is more efficient, because it only traverses one path down the tree.
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Exercises
(1) Using recursion and the function add, define a function that multiplies two natural numbers.
(2) Define a suitable function folde for expressions, and give a few examples of its use.
(3) 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.
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