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Haskell. Data Types/ADT/Modules Type/Class Hierarchy Lazy Functional Language. 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

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haskell

Haskell

Data Types/ADT/Modules

Type/Class Hierarchy

Lazy Functional Language

CS776

modelling alternatives
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
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
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.

CS776

tree insertion
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

CS776

modelling expressions
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 !!

CS776

symbolic differentiation
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.

CS776

example
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

CS776

formatting expressions
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

CS776

slide10
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.

CS776

slide11
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

CS776

module construct in haskell
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

slide14
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.

CS776

slide15
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.

CS776

slide16
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)))

CS776

abstract data types
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

CS776

other features
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.

CS776

kinds of functions
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
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

CS776

parametric vs ad hoc polymorphism
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.

CS776

class hierarchy and instance declarations
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.

CS776

types of overloaded functions
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.

CS776

show and read
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.

CS776

derived instances
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)"

CS776

multi parameter classes
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

CS776

multiple inheritance
Multiple Inheritance

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

SortAndPrint function

Advanced Features:

Module, …

ADT, …

CS776

functional dependencies
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.

CS776

slide30
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
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
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.

CS776

lazy evaluation
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

CS776

non strict lazy functional language
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;

CS776

slide35
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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