Functional Programming in Haskell Part 2 : Abstract dataypes and infinite structures
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1 Functional Programming in Haskell Part 2 : Abstract dataypes and infinite structures Madhavan Mukund Chennai Mathematical Institute 92 G N Chetty Rd, Chennai , India madhavan@cmi.ac.in madhavan Madras Christian College, 10 December 2003 p.1
2 Haskell review Program Collection of function definitions Madras Christian College, 10 December 2003 p.2
3 Haskell review Program Collection of function definitions Computation Rewriting using definitions Madras Christian College, 10 December 2003 p.2
4 Haskell review Program Collection of function definitions Computation Rewriting using definitions Functions are associated with input and output types Madras Christian College, 10 December 2003 p.2
5 Haskell review Program Collection of function definitions Computation Rewriting using definitions Functions are associated with input and output types isdigit :: Char -> Bool Madras Christian College, 10 December 2003 p.2
6 Haskell review Program Collection of function definitions Computation Rewriting using definitions Functions are associated with input and output types isdigit :: Char -> Bool isdigit 0 = True isdigit 1 = True... isdigit 9 = True isdigit c = False Madras Christian College, 10 December 2003 p.2
7 Haskell review Program Collection of function definitions Computation Rewriting using definitions Functions are associated with input and output types isdigit :: Char -> Bool isdigit 0 = True isdigit 1 = True... isdigit 9 = True isdigit c = False isdigit c (c >= 0 && c <= 9 ) = True otherwise = False Madras Christian College, 10 December 2003 p.2
8 Haskell review... Basic collective type is a list Madras Christian College, 10 December 2003 p.3
9 Haskell review... Basic collective type is a list Define list functions by induction on structure Madras Christian College, 10 December 2003 p.3
10 Haskell review... Basic collective type is a list Define list functions by induction on structure Example Adding up a list of integers Madras Christian College, 10 December 2003 p.3
11 Haskell review... Basic collective type is a list Define list functions by induction on structure Example Adding up a list of integers sum :: [Int] -> Int sum [] = 0 sum (x:l) = x + (sum l) Madras Christian College, 10 December 2003 p.3
12 Haskell review... Basic collective type is a list Define list functions by induction on structure Example Adding up a list of integers sum :: [Int] -> Int sum [] = 0 sum (x:l) = x + (sum l) (Conditional) polymorphism Madras Christian College, 10 December 2003 p.3
13 Haskell review... Basic collective type is a list Define list functions by induction on structure Example Adding up a list of integers sum :: [Int] -> Int sum [] = 0 sum (x:l) = x + (sum l) (Conditional) polymorphism Most general type of sum is sum :: (Num a) => [a] -> a where Num a is true for any type a that supports basic arithmetic operations +, -,... Madras Christian College, 10 December 2003 p.3
14 Today s agenda Adding new types Madras Christian College, 10 December 2003 p.4
15 Today s agenda Adding new types Defining abstract datatypes Provide an interface that hides the implementation Madras Christian College, 10 December 2003 p.4
16 Today s agenda Adding new types Defining abstract datatypes Provide an interface that hides the implementation Using infinite data structures Madras Christian College, 10 December 2003 p.4
17 User defined datatypes The data declaration adds new datatypes Madras Christian College, 10 December 2003 p.5
18 User defined datatypes The data declaration adds new datatypes Enumerated types data Signal = Red Yellow Green Madras Christian College, 10 December 2003 p.5
19 User defined datatypes The data declaration adds new datatypes Enumerated types data Signal = Red Yellow Green Can use this type in a function such as stopwhen :: Signal -> Bool stopwhen Red = True stopwhen c = False Madras Christian College, 10 December 2003 p.5
20 User defined datatypes The data declaration adds new datatypes Enumerated types data Signal = Red Yellow Green Can use this type in a function such as stopwhen :: Signal -> Bool stopwhen Red = True stopwhen c = False What if we write instead stopwhen2 :: Signal -> Bool stopwhen2 c (c == Red) = True otherwise = False Madras Christian College, 10 December 2003 p.5
21 User defined types and type classes stopwhen2 requires Eq Signal Madras Christian College, 10 December 2003 p.6
22 User defined types and type classes stopwhen2 requires Eq Signal How about nextlight :: Signal -> Signal nextlight Green = Yellow nextlight Yellow = Red nextlight Red = Green Madras Christian College, 10 December 2003 p.6
23 User defined types and type classes stopwhen2 requires Eq Signal How about nextlight :: Signal -> Signal nextlight Green = Yellow nextlight Yellow = Red nextlight Red = Green Displaying result of nextlight requires Show Signal Madras Christian College, 10 December 2003 p.6
24 User defined types and type classes stopwhen2 requires Eq Signal How about nextlight :: Signal -> Signal nextlight Green = Yellow nextlight Yellow = Red nextlight Red = Green Displaying result of nextlight requires Show Signal Show a is true of type a if there is a function show :: a -> String that allows values of a to be displayed Madras Christian College, 10 December 2003 p.6
25 Adding user defined types to type classes Simplest solution is data Signal = Red Yellow Green deriving (Eq, Show, Ord) Madras Christian College, 10 December 2003 p.7
26 Adding user defined types to type classes Simplest solution is data Signal = Red Yellow Green deriving (Eq, Show, Ord) Fixes default values Madras Christian College, 10 December 2003 p.7
27 Adding user defined types to type classes Simplest solution is data Signal = Red Yellow Green deriving (Eq, Show, Ord) Fixes default values Red == Red, Red /= Yellow,... Madras Christian College, 10 December 2003 p.7
28 Adding user defined types to type classes Simplest solution is data Signal = Red Yellow Green deriving (Eq, Show, Ord) Fixes default values Red == Red, Red /= Yellow,... show Red = "Red", show Yellow = "Yellow",... Madras Christian College, 10 December 2003 p.7
29 Adding user defined types to type classes Simplest solution is data Signal = Red Yellow Green deriving (Eq, Show, Ord) Fixes default values Red == Red, Red /= Yellow,... show Red = "Red", show Yellow = "Yellow",... Red < Yellow < Green Madras Christian College, 10 December 2003 p.7
30 Adding user defined types to type classes... Or, provide your own functions Madras Christian College, 10 December 2003 p.8
31 Adding user defined types to type classes... Or, provide your own functions data Signal = Red Yellow Green deriving (Eq) Madras Christian College, 10 December 2003 p.8
32 Adding user defined types to type classes... Or, provide your own functions data Signal = Red Yellow Green deriving (Eq) instance Show Signal where show Yellow = "Yellow" show c = "Black" Madras Christian College, 10 December 2003 p.8
33 Adding user defined types to type classes... Or, provide your own functions data Signal = Red Yellow Green deriving (Eq) instance Show Signal where show Yellow = "Yellow" show c = "Black" instance Ord Signal where Green <= Yellow = True Yellow <= Red = True Red <= Green = True x <= y = False Madras Christian College, 10 December 2003 p.8
34 Adding user defined types to type classes... Or, provide your own functions data Signal = Red Yellow Green deriving (Eq) instance Show Signal where show Yellow = "Yellow" show c = "Black" instance Ord Signal where Green <= Yellow = True Yellow <= Red = True Red <= Green = True x <= y = False In the class Ord, >, >=,... are defined in terms of <= Madras Christian College, 10 December 2003 p.8
35 Adding user defined types to type classes... Or, provide your own functions data Signal = Red Yellow Green deriving (Eq) instance Show Signal where show Yellow = "Yellow" show c = "Black" instance Ord Signal where Green <= Yellow = True Yellow <= Red = True Red <= Green = True x <= y = False In the class Ord, >, >=,... are defined in terms of <= Note: <= need not even be a consistent ordering! Madras Christian College, 10 December 2003 p.8
36 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Madras Christian College, 10 December 2003 p.9
37 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors Madras Christian College, 10 December 2003 p.9
38 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors The constructor Nil takes zero arguments Madras Christian College, 10 December 2003 p.9
39 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors The constructor Nil takes zero arguments A constant, like Red, Green,... Madras Christian College, 10 December 2003 p.9
40 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors The constructor Nil takes zero arguments A constant, like Red, Green,... The constructor Node has three arguments Madras Christian College, 10 December 2003 p.9
41 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors The constructor Nil takes zero arguments A constant, like Red, Green,... The constructor Node has three arguments First component Int: value stored at the node Madras Christian College, 10 December 2003 p.9
42 Recursive datatypes A binary tree to store integers at each node data Btreeint = Nil Node Int Btreeint Btreeint Nil and Node are constructors The constructor Nil takes zero arguments A constant, like Red, Green,... The constructor Node has three arguments First component Int: value stored at the node Other two components Btreeint: left and right subtrees Madras Christian College, 10 December 2003 p.9
43 Recursive datatypes... Example The tree 6 would be written as 4 8 Node 6 (Node 4 Nil Nil) (Node 8 (Node 7 Nil Nil) Nil) 7 Madras Christian College, 10 December 2003 p.10
44 Functions on recursive datatypes Define by induction on the structure of datatype Madras Christian College, 10 December 2003 p.11
45 Functions on recursive datatypes Define by induction on the structure of datatype How many values are there in the tree? Madras Christian College, 10 December 2003 p.11
46 Functions on recursive datatypes Define by induction on the structure of datatype How many values are there in the tree? size :: Btreeint -> Int size Nil = 0 size (Node n t1 t2) = 1 + (size t1) + (size t2) Madras Christian College, 10 December 2003 p.11
47 Functions on recursive datatypes Define by induction on the structure of datatype How many values are there in the tree? size :: Btreeint -> Int size Nil = 0 size (Node n t1 t2) = 1 + (size t1) + (size t2) List out all values in the tree Madras Christian College, 10 December 2003 p.11
48 Functions on recursive datatypes Define by induction on the structure of datatype How many values are there in the tree? size :: Btreeint -> Int size Nil = 0 size (Node n t1 t2) = 1 + (size t1) + (size t2) List out all values in the tree listout :: Btreeint -> [Int] listout Nil = [] listout (Node n t1 t2) = [n] ++ listout t1 ++ listout t2 Madras Christian College, 10 December 2003 p.11
49 Polymorphic recursive datatypes A binary tree to store arbitrary values at each node? data Btree a = Nil Node a (Btree a) (Btree a) Madras Christian College, 10 December 2003 p.12
50 Polymorphic recursive datatypes A binary tree to store arbitrary values at each node? data Btree a = Nil Node a (Btree a) (Btree a) What if we want to use Btree a as a search tree Madras Christian College, 10 December 2003 p.12
51 Polymorphic recursive datatypes A binary tree to store arbitrary values at each node? data Btree a = Nil Node a (Btree a) (Btree a) What if we want to use Btree a as a search tree Values in the tree must have a natural ordering Madras Christian College, 10 December 2003 p.12
52 Polymorphic recursive datatypes A binary tree to store arbitrary values at each node? data Btree a = Nil Node a (Btree a) (Btree a) What if we want to use Btree a as a search tree Values in the tree must have a natural ordering Conditional polymorphism! (Ord a) => data Btree a = Nil Node a (Btree a) (Btree a) Madras Christian College, 10 December 2003 p.12
53 Polymorphic recursive datatypes... Built in list type is a polymorphic recursive datatype Madras Christian College, 10 December 2003 p.13
54 Polymorphic recursive datatypes... Built in list type is a polymorphic recursive datatype data Mylist a = Emptylist Append a (Mylist a) Madras Christian College, 10 December 2003 p.13
55 Polymorphic recursive datatypes... Built in list type is a polymorphic recursive datatype data Mylist a = Emptylist Append a (Mylist a) Since lists are built in, they can use special symbols [] and : for constructors Emptylist and Append Madras Christian College, 10 December 2003 p.13
56 Adding recursive datatypes to type classes Can inherit type classes from underlying type Madras Christian College, 10 December 2003 p.14
57 Adding recursive datatypes to type classes Can inherit type classes from underlying type data Btree a = Nil Node a (Btree a) (Btree a) deriving (Eq, Show) Madras Christian College, 10 December 2003 p.14
58 Adding recursive datatypes to type classes Can inherit type classes from underlying type data Btree a = Nil Node a (Btree a) (Btree a) deriving (Eq, Show) Note: Not Eq (Btree a) but Eq a => Eq (Btree a) Madras Christian College, 10 December 2003 p.14
59 Adding recursive datatypes to type classes Can inherit type classes from underlying type data Btree a = Nil Node a (Btree a) (Btree a) deriving (Eq, Show) Note: Not Eq (Btree a) but Eq a => Eq (Btree a) Derived == checks that trees have same structure Madras Christian College, 10 December 2003 p.14
60 Adding recursive datatypes to type classes Can inherit type classes from underlying type data Btree a = Nil Node a (Btree a) (Btree a) deriving (Eq, Show) Note: Not Eq (Btree a) but Eq a => Eq (Btree a) Derived == checks that trees have same structure 6 8 /= 8 6 Madras Christian College, 10 December 2003 p.14
61 Adding recursive datatypes to type classes... Or we can define our own functions Madras Christian College, 10 December 2003 p.15
62 Adding recursive datatypes to type classes... Or we can define our own functions instance (Eq a) => Eq (Btree a) where t1 == t2 = (listout t1 == listout t2) Madras Christian College, 10 December 2003 p.15
63 Adding recursive datatypes to type classes... Or we can define our own functions instance (Eq a) => Eq (Btree a) where t1 == t2 = (listout t1 == listout t2) 6 8 == 8 6 because [6,8] == [6,8] Madras Christian College, 10 December 2003 p.15
64 Adding recursive datatypes to type classes... Or we can define our own functions instance (Eq a) => Eq (Btree a) where t1 == t2 = (listout t1 == listout t2) 6 8 == 8 6 because [6,8] == [6,8] Madras Christian College, 10 December 2003 p.15
65 Adding recursive datatypes to type classes... Or we can define our own functions instance (Eq a) => Eq (Btree a) where t1 == t2 = (listout t1 == listout t2) 6 8 == 8 6 because [6,8] == [6,8] Madras Christian College, 10 December 2003 p.15
66 Declarative programming with abstract datatypes Rotate right transformation used to balance trees x y y x t 3 t 1 t 1 t 2 t 2 t 3 Madras Christian College, 10 December 2003 p.16
67 Declarative programming with abstract datatypes Rotate right transformation used to balance trees x y y x t 3 t 1 t 1 t 2 t 2 t 3 rotateright (Node x (Node y t1 t2) t3) = Node y t1 (Node x t2 t3) Madras Christian College, 10 December 2003 p.16
68 Abstract datatypes Example Queues Madras Christian College, 10 December 2003 p.17
69 Abstract datatypes Example Queues Stores sequence of values in FIFO fashion Madras Christian College, 10 December 2003 p.17
70 Abstract datatypes Example Queues Stores sequence of values in FIFO fashion Append items at the tail of queue Madras Christian College, 10 December 2003 p.17
71 Abstract datatypes Example Queues Stores sequence of values in FIFO fashion Append items at the tail of queue Want a datatype Queue a with functions addq :: (Queue a) -> a -> (Queue a) removeq :: (Queue a) -> (a,queue a) isemptyq :: (Queue a) -> Bool emptyqueue :: (Queue a) Madras Christian College, 10 December 2003 p.17
72 Abstract datatypes... Implement a queue as a list Madras Christian College, 10 December 2003 p.18
73 Abstract datatypes... Implement a queue as a list data Queue a = Qu [a] Madras Christian College, 10 December 2003 p.18
74 Abstract datatypes... Implement a queue as a list data Queue a = Qu [a] addq :: (Queue a) -> a -> (Queue a) addq (Qu l) d = Qu (d:l) removeq :: (Queue a) -> (a,queue a) removeq (Qu l)) = (last l,qu (init l)) isemptyq :: (Queue a) -> Bool isemptyq (Qu []) = True isemptyq q = False emptyqueue :: (Queue a) emptyqueue = Qu [] Madras Christian College, 10 December 2003 p.18
75 Modules Group together the definitions for Queue a in a separate reusable module Madras Christian College, 10 December 2003 p.19
76 Modules Group together the definitions for Queue a in a separate reusable module module Queue where data Queue a = Qu [a] addq :: (Queue a) -> a -> (Queue a)... emptyqueue :: (Queue a)... Madras Christian College, 10 December 2003 p.19
77 Modules Group together the definitions for Queue a in a separate reusable module module Queue where data Queue a = Qu [a] addq :: (Queue a) -> a -> (Queue a)... emptyqueue :: (Queue a)... Use these definitions in another file import Queue Madras Christian College, 10 December 2003 p.19
78 Modules Group together the definitions for Queue a in a separate reusable module module Queue where data Queue a = Qu [a] addq :: (Queue a) -> a -> (Queue a)... emptyqueue :: (Queue a)... Use these definitions in another file import Queue How do we prevent unauthorized access to a queue? remsec :: (Queue a) -> (a,queue a) remsec (Qu (x:y:l)) = (y, Qu (x:l)) Madras Christian College, 10 December 2003 p.19
79 Modules... Restrict visibility outside module Madras Christian College, 10 December 2003 p.20
80 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Madras Christian College, 10 December 2003 p.20
81 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Constructor Qu is not visible if you do import Queue Madras Christian College, 10 December 2003 p.20
82 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Constructor Qu is not visible if you do import Queue Can override imported function with local definition Madras Christian College, 10 December 2003 p.20
83 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Constructor Qu is not visible if you do import Queue Can override imported function with local definition All Haskell programs implicitly import Prelude Madras Christian College, 10 December 2003 p.20
84 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Constructor Qu is not visible if you do import Queue Can override imported function with local definition All Haskell programs implicitly import Prelude Redefine builtin functions using import Prelude hiding (max) Madras Christian College, 10 December 2003 p.20
85 Modules... Restrict visibility outside module module Queue(addq,removeq,isemptyq,emptyqueue) where... Constructor Qu is not visible if you do import Queue Can override imported function with local definition All Haskell programs implicitly import Prelude Redefine builtin functions using import Prelude hiding (max) Madras Christian College, 10 December 2003 p.20
86 Rewriting revisted More than one expression may qualify for rewriting Madras Christian College, 10 December 2003 p.21
87 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x Madras Christian College, 10 December 2003 p.21
88 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) Madras Christian College, 10 December 2003 p.21
89 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) sqr 7 7*7 49 Madras Christian College, 10 December 2003 p.21
90 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) sqr 7 7*7 49 (4+3)*(4+3) (4+3)*7 7*7 49 Madras Christian College, 10 December 2003 p.21
91 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) sqr 7 7*7 49 (4+3)*(4+3) (4+3)*7 7*7 49 If there are multiple expressions to rewrite, Haskell chooses outermost expression Madras Christian College, 10 December 2003 p.21
92 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) sqr 7 7*7 49 (4+3)*(4+3) (4+3)*7 7*7 49 If there are multiple expressions to rewrite, Haskell chooses outermost expression Outermost reduction Lazy rewriting Evaluate argument to a function only when needed. Madras Christian College, 10 December 2003 p.21
93 Rewriting revisted More than one expression may qualify for rewriting sqr x = x*x sqr (4+3) sqr 7 7*7 49 (4+3)*(4+3) (4+3)*7 7*7 49 If there are multiple expressions to rewrite, Haskell chooses outermost expression Outermost reduction Lazy rewriting Evaluate argument to a function only when needed. Eager rewriting evaluate arguments before evaluating function Madras Christian College, 10 December 2003 p.21
94 Lazy rewriting Haskell evaluates arguments only when needed Madras Christian College, 10 December 2003 p.22
95 Lazy rewriting Haskell evaluates arguments only when needed power :: Float -> Int -> Float power x n = if (n == 0) then 1.0 else x * (power x (n-1)) Madras Christian College, 10 December 2003 p.22
96 Lazy rewriting Haskell evaluates arguments only when needed power :: Float -> Int -> Float power x n = if (n == 0) then 1.0 else x * (power x (n-1)) power (8.0/0.0) Madras Christian College, 10 December 2003 p.22
97 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) Madras Christian College, 10 December 2003 p.23
98 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) from 2 Madras Christian College, 10 December 2003 p.23
99 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) from 2 2:(from 3) 2:(3:(from 4)) 2:(3:(4:(from 5)))... Madras Christian College, 10 December 2003 p.23
100 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) from 2 2:(from 3) 2:(3:(from 4)) 2:(3:(4:(from 5)))... Limit is the infinite list [2,3,4,5,...] Madras Christian College, 10 December 2003 p.23
101 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) from 2 2:(from 3) 2:(3:(from 4)) 2:(3:(4:(from 5)))... Limit is the infinite list [2,3,4,5,...] Haskell can (and will) generate it incrementally, till you stop it, or it runs out of memory Madras Christian College, 10 December 2003 p.23
102 Infinite lists! The following definition makes sense in Haskell from n = n : from (n+1) from 2 2:(from 3) 2:(3:(from 4)) 2:(3:(4:(from 5)))... Limit is the infinite list [2,3,4,5,...] Haskell can (and will) generate it incrementally, till you stop it, or it runs out of memory Can write [2..] to denote [2,3,4,...] Madras Christian College, 10 December 2003 p.23
103 Why infinite lists? Can sometimes simplify a problem Madras Christian College, 10 December 2003 p.24
104 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Madras Christian College, 10 December 2003 p.24
105 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry Madras Christian College, 10 December 2003 p.24
106 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry The Sieve of Eratosthenes Madras Christian College, 10 December 2003 p.24
107 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry The Sieve of Eratosthenes Start with [2,3,4,...] Madras Christian College, 10 December 2003 p.24
108 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry The Sieve of Eratosthenes Start with [2,3,4,...] Transfer smallest number into list of primes and delete all its multiples Madras Christian College, 10 December 2003 p.24
109 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry The Sieve of Eratosthenes Start with [2,3,4,...] Transfer smallest number into list of primes and delete all its multiples Repeat second step Madras Christian College, 10 December 2003 p.24
110 Why infinite lists? Can sometimes simplify a problem Consider the problem of computing the n th prime number Idea: generate all prime numbers and wait for the n th entry The Sieve of Eratosthenes Start with [2,3,4,...] Transfer smallest number into list of primes and delete all its multiples Repeat second step forever! Madras Christian College, 10 December 2003 p.24
111 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] Madras Christian College, 10 December 2003 p.25
112 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? Madras Christian College, 10 December 2003 p.25
113 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] Madras Christian College, 10 December 2003 p.25
114 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] 2:sieve [y y <- [3..], mod y 2 > 0] Madras Christian College, 10 December 2003 p.25
115 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] 2:sieve [y y <- [3..], mod y 2 > 0] 2:sieve (3:[y y <- [4..], mod y 2 > 0]) Madras Christian College, 10 December 2003 p.25
116 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] 2:sieve [y y <- [3..], mod y 2 > 0] 2:sieve (3:[y y <- [4..], mod y 2 > 0]) 2:3:sieve [z z <- [y <- [4..], mod y 2 > 0], mod z 3 > 0]... Madras Christian College, 10 December 2003 p.25
117 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] 2:sieve [y y <- [3..], mod y 2 > 0] 2:sieve (3:[y y <- [4..], mod y 2 > 0]) 2:3:sieve [z z <- [y <- [4..], mod y 2 > 0], mod z 3 > 0]... 2:3:sieve [z z <- [5,7,9...], mod z 3 > 0]... Madras Christian College, 10 December 2003 p.25
118 The Sieve of Eratosthenes primes = sieve [2..] where sieve (x:l) = x : sieve [y y <- l, mod y x > 0] How does this work? sieve [2..] 2:sieve [y y <- [3..], mod y 2 > 0] 2:sieve (3:[y y <- [4..], mod y 2 > 0]) 2:3:sieve [z z <- [y <- [4..], mod y 2 > 0], mod z 3 > 0]... 2:3:sieve [z z <- [5,7,9...], mod z 3 > 0]... 2:3:sieve [5,7,11...]... Madras Christian College, 10 December 2003 p.25
119 The n th prime We now have an infinite list primes of primes Madras Christian College, 10 December 2003 p.26
120 The n th prime We now have an infinite list primes of primes nthprime n = head (drop (n-1) primes) Madras Christian College, 10 December 2003 p.26
121 The n th prime We now have an infinite list primes of primes nthprime n = head (drop (n-1) primes) Drop the first n 1 numbers from primes Madras Christian College, 10 December 2003 p.26
122 The n th prime We now have an infinite list primes of primes nthprime n = head (drop (n-1) primes) Drop the first n 1 numbers from primes To take the head of the rest, only need to compute one more entry in the list Madras Christian College, 10 December 2003 p.26
123 The n th prime We now have an infinite list primes of primes nthprime n = head (drop (n-1) primes) Drop the first n 1 numbers from primes To take the head of the rest, only need to compute one more entry in the list Once enough has been computed, the rest of primes is ignored! Madras Christian College, 10 December 2003 p.26
124 Concluding remarks Functional programming provides a framework for declarative programming Madras Christian College, 10 December 2003 p.27
125 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Madras Christian College, 10 December 2003 p.27
126 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Rapid prototyping Madras Christian College, 10 December 2003 p.27
127 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Rapid prototyping Haskell has a powerful typing mechanism Madras Christian College, 10 December 2003 p.27
128 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Rapid prototyping Haskell has a powerful typing mechanism Conditional polymorphism Madras Christian College, 10 December 2003 p.27
129 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Rapid prototyping Haskell has a powerful typing mechanism Conditional polymorphism Modules with hiding and overriding for abstract datatypes Madras Christian College, 10 December 2003 p.27
130 Concluding remarks Functional programming provides a framework for declarative programming Provably correct programs Rapid prototyping Haskell has a powerful typing mechanism Conditional polymorphism Modules with hiding and overriding for abstract datatypes Lazy evaluation permits infinite data structures Madras Christian College, 10 December 2003 p.27
131 For more information Software and other resources Quick tutorial A Gentle Introduction to Haskell by Paul Hudak et al Textbooks The Craft of Functional Programming by Simon Thompson Introduction to Functional Programming in Haskell by Richard Bird Madras Christian College, 10 December 2003 p.28
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