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Modelling Alternatives

New data types are useful to model values with several alternatives. Example: Recording phone calls.

type History = [(Event, Time)]

type Time = Int

data Event = Call String

| Hangup

The number

called.

E.g. Call ”031-7721001”,

Hangup, etc.

CS776

Extracting a List of Calls

We can pattern match on values with components as usual.

Example: Extract a list of completed calls from a list of events.

calls :: History -> [(String, Time, Time)]

calls ((Call number, start) : (Hangup, end) : history)

= (number, start, end) : calls history

calls [(Call number, start)]

= [] -- a call is going on now

calls [] = []

CS776

Defining Recursive Data Types

data Tree a = Node a (Tree a) (Tree a)

| Leaf

deriving Show

Types of the

components.

Enables us to define polymorphic functions which work on a tree with any type of labels.

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Tree Insertion

Pattern

matching

works as

for lists.

Additional

requirement

insertTree :: Ord a => a -> Tree a -> Tree a

insertTree x Leaf = Node x Leaf Leaf

insertTree x (Node y l r)

| x < y = Node y (insertTree x l) r

| x > y = Node y l (insertTree x r)

| x==y = Node y l r

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Modelling Expressions

Let’s design a datatype to model arithmetic expressions -- not their values, but their structure.

- An expression can be:
- a number n
- a variable x
- an addition a+b
- a multiplication a*b

data Expr =

Num Int

|Var String

| Add Expr Expr

| Mul Expr Expr

A recursive data type !!

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Symbolic Differentiation

Differentiating an expression produces a new expression.

derive :: Expr -> String -> Expr

derive (Num n) x = Num 0

derive (Var y) x | x==y = Num 1

| x/=y = Num 0

derive (Add a b) x =

Add (derive a x) (derive b x)

derive (Mul a b) x = Add (Mul a (derive b x))

(Mul b (derive a x))

Variable to

differentiate w.r.t.

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Example

d (2*x) = 2

dx

derive (Mul (Num 2) (Var ”x”)) ”x”

Add (Mul (Num 2) (derive (Var ”x”) ”x”))

(Mul (Var ”x”) (derive (Num 2) ”x”))

Add (Mul (Num 2) (Num 1))

(Mul (Var ”x”) (Num 0))

2*1 + x*0

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Formatting Expressions

Expressions will be more readable if we convert them to strings.

formatExpr (Mul (Num 1) (Add (Num 2) (Num 3)))

”1*2+3”

formatExpr :: Expr -> String

formatExpr (Num n) = show n

formatExpr (Var x) = x

formatExpr (Add a b) =

formatExpr a ++ ”+” ++ formatExpr b

formatExpr (Mul a b) =

formatExpr a ++ ”*” ++ formatExpr b

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Quiz

NO!

Which brackets are necessary? 1+(2+3)

1+(2*3)

1*(2+3)

What kind of expression may need to be bracketed?

When does it need to be bracketed?

NO!

YES!

Additions

Inside multiplications.

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Idea

Give formatExpr an extra parameter, to tell it what context its argument appears in.

data Context = Multiply | AnyOther

formatExpr (Add a b) Multiply =

”(” ++

formatExpr (Add a b) AnyOther

++ ”)”

formatExpr (Mul a b) _ =

formatExpr a Multiply ++

”*” ++

formatExpr b Multiply

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ADT and Modules

CS776

module construct in Haskell

- Enables grouping a collection of related definitions
- Enables controlling visibility of names
- export public names to other modules
- import names from other modules
- disambiguation using fully qualified names
- Enables defining Abstract Data Types

CS776

module MTree ( Tree(Leaf,Branch), fringe )

wheredata Tree a = Leaf a | Branch (Tree a) (Tree a) fringe :: Tree a -> [a]fringe (Leaf x) = [x]fringe (Branch left right) =

fringe left ++ fringe right

- This definition exports all the names defined in the module including Tree-constructors.

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module Main (main) where

import

MTree ( Tree(Leaf,Branch), fringe )

main =

do print (fringe

(Branch (Leaf 1) (Leaf 2))

)

- Main explicitly imports all the names exported by the module MTree.

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module Fringe(fringe) where

import Tree(Tree(..))

fringe :: Tree a -> [a]

-- A different definition of fringe

fringe (Leaf x) = [x]

fringe (Branch x y) = fringe x

module QMain where

import Tree ( Tree(Leaf,Branch), fringe )

import qualified Fringe ( fringe )

qmain =

doprint (fringe (Branch (Leaf 1) (Leaf 2))) print(Fringe.fringe(Branch (Leaf 1) (Leaf 2)))

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Abstract Data Types

module TreeADT (Tree, leaf, branch, cell, left, right, isLeaf) where

data Tree a =

Leaf a | Branch (Tree a) (Tree a)

leaf = Leaf

branch = Branch

cell (Leaf a) = a

left (Branch l r) = l

right (Branch l r) = r

isLeaf (Leaf _) = True

isLeaf _ = False

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Other features

- Selective hiding

import Prelude hiding length

- Eliminating functions inherited on the basis of the representation.

module Queue( …operation names...) where

newtype Queue a = MkQ ([a],[a])

…operation implementation…

- Use of MkQ-constructor prevents equality testing, printing, etc of queue values.

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Kinds of functions

- Monomorphic (defined over one type)

capitalize : Char -> Char

- Polymorphic (defined similarly over all types)

length : [a] -> Int

- Overloaded (defined differently and over many types)

(==) : Char -> Char -> Bool

(==) : [(Int,Bool]] ->

[(Int,Bool]] -> Bool

CS776

Overloading problem in SML

fun add x y = x + y

- SML-90 treats this definition as ambiguous:

int -> int -> int

real -> real -> real

- SML-97 defaults it to:

int -> int -> int

- Ideally, add defined whenever + is defined on a type.

add :: (hasPlus a) => a -> a -> a

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Parametric vs ad hoc polymorphism

- Polymorphic functions use the same definition at each type.
- Overloaded functions may have a different definition at each type.

Class name.

class Eq a where

(==) :: a -> a -> Bool

(/=) :: a -> a -> Bool

x/=y = not (x==y)

Read:

“a is a type in class Eq, if it has the following methods”.

Class

methods

and types.

Default definition.

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Class Hierarchy and Instance Declarations

class Eq a => Ord a where

(<),(<=),(>=),(>) ::

a -> a -> Bool

max, min :: a -> a -> a

Read:

“Type a in class Eq is also in class Ord, if it provides the following methods…”

instance Eq Integer where

x==y = …primitive…

instance Eq a => Eq [a] where

[] == [] = True

x:xs == y:ys =

x == y && xs == ys

If a is in class Eq, then [a] is in class Eq, with the method definition given.

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Types of Overloaded Functions

a may be any type

in classOrd.

insert :: Ord a => a -> [a] -> [a]

insert x [] = []

insert x (y:xs)

| x<=y = x:y:xs

| x>y = y:insert x xs

f :: (Eq a) => a -> [a] -> Int

f x y = if x==y then 1 else 2

Because insert

uses a method

from class Ord.

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Show and Read

class Show a where

show :: a -> String

class Read a where

read :: String -> a

read . show = id(usually)

These are definitions are simplifications: there are more methods in reality.

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Derived Instances

data Tree a = Node a (Tree a) (Tree a)

| Leaf

deriving (Eq, Show)

Constructs a “default

instance” of class Show.

Works for standard classes.

Main> show (Node 1 Leaf (Node 2 Leaf Leaf))

"Node 1 Leaf (Node 2 Leaf Leaf)"

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Multi-Parameter Classes

Define relations between classes.

class Collection c a where

empty :: c

add :: a -> c -> c

member :: a -> c -> Bool

c is a collection with elements of type a.

instance Eq a =>

Collection [a] a where

empty = []

add = (:)

member = elem

instance Ord a =>

Collection (Tree a) a where

empty = Leaf

add = insertTree

member = elemTree

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Multiple Inheritance

class (Ord a, Show a) => a where

…

SortAndPrint function

…

Advanced Features:

Module, …

ADT, …

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Functional Dependencies

A functional dependency

class Collection c a | c -> a where

empty :: c

add :: a -> c -> c

member :: a -> c -> Bool

- Declares that c determines a: there can be only one instance for each type c.
- Helps the type-checker resolve ambiguities (tremendously).

add x (add y empty) -- x and y must be the same type.

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class MyFunctor f where

tmap :: (a -> b) -> f a -> f b

data Tree a = Branch (Tree a) (Tree a)

| Leaf a

deriving Show

instance MyFunctor Tree where

tmap f (Leaf x) = Leaf (f x)

tmap f (Branch t1 t2) =

Branch (tmap f t1) (tmap f t2)

tmap (*10) (Branch (Leaf 1) (Leaf 2))

CS776

Higher-Order Functions

- Functions are values in Haskell.
- “Program skeletons” take functions as parameters.

takeWhile :: (a -> Bool) -> [a] -> [a]

takeWhile p [] = []

takeWhile p (x:xs)

| p x = x:takeWhile p xs

| otherwise = []

Takes a prefix of a list, satisfying a predicate.

CS776

More Ways to Denote Functions

- below a b = b < a
- takeWhile (below 10) [1,5,9,15,20]
- takeWhile (\b -> b < 10) [1,5,9,15,20]
- takeWhile (<10) [1,5,9,15,20]

“Lambda” expression.

Function definition

in place.

Partial operator

application -- argument

replaces missing operand.

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Lazy Evaluation

fib = 1 : 1 :

[ a+b | (a,b)<- zip fib (tail fib) ]

- Expressions are evaluated only when their value is really needed!
- Function arguments, data structure components, are held unevaluated until their value is used.

nats = 0 : map (+1) nats

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Non-strict / Lazy Functional Language

- Parameter passing mechanism
- Call by name
- Call by need
- ( but not Call by value )
- Advantages
- Does not evaluate arguments not required to determine the final value of the function.
- “Most likely to terminate” evaluation order.

fun const x = 0; const (1/0) = 0;

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Practical Benefits

- Frees programmer from worrying about control issues:
- Best order for evaluation …
- To compute or not to compute a subexpression …
- Facilitates programming with potentially infinite value or partial value.
- Costs
- Overheads of building thunks to represent delayed argument.

CS776

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