Introduction to Functional Languages

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Introduction to Functional Languages
1. Referential transparency, no side effects
“substitution of equals for equals”
2. Function deﬁnitions can be used
Suppose f is deﬁned to be the function (fn x=>exp), then f
(arg) can be replaced by exp[x := arg]
3. Lists not arrays
4. Recursion not iteration
5. Higher-order functions
New idioms, total procedural abstraction

Rewriting
fun square x = x * x;
fun sos (x,y) = (square x) + (square y);
sos (3,4)
==> (square 3) + (square 4) [Def’n of sos]
==> 3*3 + (square 4) [Def’n of square]
==> 9 + (square 4) [Def’n of *]
==> 9 + 4*4 [Def’n of square]
==> 9 + 16 [Def’n of *]
==> 25 [Def’n of +]
Language of expressions only, no statements.
fun test (x) = if x>20 then "big" else "small"
test (sos (3,4))
==> test(25)
==> if 25>20 then "big" else "small"

Canonical Value
Canonical value. A canonical value is one which cannot be rewritten
further.
For example, 2+3 is not canonical, it evaluates to 5; 5 is a canonical
value.
See canonical in the “The on-line hacker Jargon File,” version 4.4.7, 29
Dec 2003.

History of Functional Languages
1959 LISP: List processing, John McCarthy
1975 Scheme: MIT
1977 FP: John Backus
1980 Hope: Burstall, McQueen, Sannella
1984 COMMON LISP: Guy Steele
1985 ML: meta-language (of LCF), Robin Milner
1986 Miranda: Turner
1990 Haskell: Hudak & Wadler editors

xkcd—a webcomic of romance, sarcasm, math, and language by
Randall Munroe

Functional Languages
Lazy: don’t evaluate the function (constructor) arguments until needed
(call-by-name), e.g., Haskell. Permits inﬁnite data structures.
Eager: call-by-value, e.g., ML

ML and Haskell
Similar to ML: functional, strongly-typed, algebraic data types,
type inferencing
Differences: no references, exception handling, or side effects of
any kind; lazy evaluation, list comprehensions

Introduction to Haskell
1. Haskell (1.0) 1990
2. By 1997 four iterations of language design (1.4)

Salient Features of Haskell
1. Strongly-typed, lazy, functional language
2. Polymorphic types, type inference
3. Algebraic type deﬁnitions
4. Pattern matching function deﬁnitions
5. System of classes
6. Interactive

Introduction to Functional Languages

Information about Haskell
O’Sullivan, Goerzen, Stewart Real World Haskell
Hutton, Graham, Programming in Haskell.
O’Donnell et al., Discrete Mathematics Using a Computer.

Introduction to SML
1. Robin Milner, Turing Award winner 1991
2. Metalanguage (ML) in Logic of Computable Functions (LCF)
1980s
3. Actually general purpose language
4. SML deﬁnition 1990, revised 1997.
5. AT&T (D. McQueen), Princeton (A. Appel) implementation

Salient Features of SML
1. Strongly-typed, eager, functional language
2. Polymorphic types, type inference
3. Algebraic type deﬁnitions
4. Pattern matching function deﬁnitions
5. Exception handling
6. Module (signatures/structures) system
7. Interactive

Information about ML
Ullman, Jeffrey D. Elements of ML Programming. Second edition.
Prentice-Hall, Upper Saddle River, New Jersey, 1998. 0-13-790387-1.
Paulson, Lawrence C. ML for the Working Programmer. Second
edition. Cambridge University Press, Cambridge, England, 1996.
ISBN 0-521-56543-X.

Haskell
Similar to ML: functional, strongly-typed, algebraic data types,
type inferencing
Differences: no references, exception handling, or side effects of
any kind; lazy evaluation, list comprehensions
fac n = if n==0 then 1 else n * fac (n-1)
data Tree = Leaf | Node (Tree , String, Tree)
size (Leaf) = 1
size (Node (l,_,r)) = size (l) + size (r)
squares = [ n*n | n <- [0..] ]
pascal = iterate (row ->zipWith (+) ([0]++row) (row

Haskell List Comprehension
[e | x1 <- l1, ..., xm <- lm, P1, ..., Pn]
e is an expression, xi is a variable, li is a list, Pi is a predicate
[ xˆ2 | x <- [ 1..10 ], even x]
[ xˆ2 | x <- [ 2,4..10 ] ]
[ x+y | x <- [1..3], y <- [1..4] ]
perms [] = [[]]
perms x = [a:y | a<-x, y<-perms (x [a]) ]
quicksort [] = []
quicksort (s:xs) =
quicksort[x|x<-xs,x<s]++[s]++quicksort[x|x<-xs,x>=

Patterns
Patterns are a very natural way of expression complex problems.
Consider the code to re-balance red-black trees. This is usually quite
complex to express in a programming language. But with patterns it
can be more concise. Notice that constructors of user-deﬁned types
(line RBTree) as well as pre-deﬁned types (like list) can be used in
patterns.

data Color = R | B deriving (Show, Read)
data RBTree a = Empty | T Color (RBTree a) a (RBTr
deriving (Show, Read)
balance :: RBTree a -> a -> RBTree a -> RBTree a
balance (T R a x b) y (T R c z d)=T R (T B a x b)
balance (T R (T R a x b) y c) z d=T R (T B a x b)
balance (T R a x (T R b y c)) z d=T R (T B a x b)
balance a x (T R b y (T R c z d))=T R (T B a x b)
balance a x (T R (T R b y c) z d)=T R (T B a x b)
balance a x b = T B a x b

Functions
Prelude> : m Text.Show.Functions
Prelude Text.Show.Functions > x->x+1
<function >
Prelude Text.Show.Functions > :t x->x+(1::Int)
x->x+(1::Int) :: Int -> Int
Prelude Text.Show.Functions > :t x->x+1
x->x+1 :: (Num a) => a -> a

Partial Application
Any curried function may be called with fewer arguments than it was
deﬁned for. The result is a function of the remaining arguments.
If f is a function Int->Bool->Int->Bool, then
f :: Int->Bool->Int->Bool
f 2 :: Bool->Int->Bool
f 2 True :: Int->Bool
f 2 True 3 :: Bool
Higher-order functions after lists.

Haskell Fold
“A tutorial on the universality and expressiveness of fold” by Graham
Hutton.

Fold
foldr z[x1,x2,...,xn] = x1 (x2 (...(xn z)...))
foldr f z [x1, x2, ..., xn] = x1 f (x2 f (...(xn f z)...))

Fold
foldl z[x1,x2,...,xn] = (...((z x1) x2)...) xn
foldl f z [x1, x2, ..., xn] = (...((z f x1) f x2) ... ) f xn

Haskell Fold
foldr :: (b -> a -> a) -> a -> [b] ->
foldr f z [] = z
foldr f z (x:xs) = f x (foldr f z xs)
foldl :: (a -> b -> a) -> a -> [b] ->
foldl f z [] = z
foldl f z (x:xs) = foldl f (f z x) xs
foldl’ :: (a -> b -> a) -> a -> [b] -> a
foldl’ f z0 xs = foldr f’ id xs z0
where f’ x k z = k $! f z x
[Real World Haskell says never use foldl instead use foldl’.]

Haskell Fold
Evaluates its ﬁrst argument to head normal form, and then returns its
second argument as the result.
seq :: a -> b -> b
Strict (call-by-value) application, deﬁned in terms of ’seq’.
($!) :: (a -> b) -> a -> b
f $! x = x ‘seq‘ f x

Haskell Fold
One important thing to note in the presence of lazy, or
normal-order evaluation, is that foldr will immediately return
the application of f to the recursive case of folding over the
rest of the list. Thus, if f is able to produce some part of its
result without reference to the recursive case, and the rest of
the result is never demanded, then the recursion will stop.
This allows right folds to operate on infinite lists. By contrast,
foldl will immediately call itself with new parameters until it
reaches the end of the list. This tail recursion can be
efficiently compiled as a loop, but can’t deal with infinite lists
at all – it will recurse forever in an infinite loop.

Haskell Fold
Another technical point to be aware of in the case of left
folds in a normal-order evaluation language is that the new
initial parameter is not being evaluated before the recursive
call is made. This can lead to stack overflows when one
reaches the end of the list and tries to evaluate the resulting
gigantic expression. For this reason, such languages often
provide a stricter variant of left folding which forces the
evaluation of the initial parameter before making the
recursive call, in Haskell, this is the foldl’ (note the
apostrophe) function in the Data.List library. Combined with
the speed of tail recursion, such folds are very efficient when
lazy evaluation of the final result is impossible or
undesirable.

Haskell Fold
sum’ = foldl (+) 0
product’ = foldl (*) 1
and’ = foldl (&&) True
or’ = foldl (||) False
concat’ = foldl (++) []
composel = foldl (.) id
composer = foldr (.) id
length = foldl (const (+1)) 0
list_identity = foldr (:) []
reverse’ = foldl (flip (:)) []
unions = foldl Set.union Set.empty

Haskell Fold
reverse = foldl ( xs x -> xs ++ [x]) []
map f = foldl ( xs x -> f x : xs) []
filter p = foldl ( xs x -> if p x then x:xs el

Haskell Fold
If this is your pattern
g [] = v
g (x:xs) = f x (g xs)
then
g = foldr f v

Haskell Data Structures
data Bool = False | True
data Color = Red | Green | Blue
deriving Show
data Day = Mon|Tue|Wed|Thu|Fri|Sat|Sun
deriving (Show,Eq,Ord)
Types and constructors capitalized.
Show allows Haskell to print data structures.

Haskell Data Structures
Constructors can take arguments.
data Shape = Circle Float | Rectangle Float Floa
deriving Show
area (Circle r) = pi*r*r
area (Rectangle s1 s2) = s1*s2

Haskell Classes
next :: (Enum a, Bounded a, Eq a) => a -> a
next x | x == maxBound = minBound
| otherwise = succ x

Haskell Classes
data Triangle = Triangle
data Square = Square
data Octagon = Octagon
class Shape s where
sides :: s -> Integer
instance Shape Triangle where
sides _ = 3
instance Shape Square where
sides _ = 4
instance Shape Octagon where
sides _ = 8

Haskell Types
Type constructors can type types as parameters.
data Maybe a = Nothing | Just a
maybe :: b -> (a -> b) -> Maybe a -> b
maybe n _ Nothing = n
maybe _ f (Just x) = f x
data Either a b = Left a | Right b
either :: (a -> c) -> (b -> c) -> Either a b ->
either f _ (Left x) = f x
either _ g (Right y) = g y

Haskell Lists
Data types can be recursive, as in lists:
data Nat = Nil | Succ Nat
data IList = Nil | Cons Integer IList
data PolyList a = Nil | Cons a (PolyList a)

Haskell Trees
See Hudak PPT, Ch7.
data SimpleTree = SimLeaf | SimBranch SimpleTree
data IntegerTree = IntLeaf Integer |
IntBranch IntegerTree IntegerTree
data InternalTree a = ILeaf |
IBranch a (InternalTree a) (InternalTree a)
data Tree a = Leaf a | Branch (Tree a) (Tree a)
data FancyTree a b = FLeaf a |
FBranch b (FancyTree a b) (FancyTree a b)
data GTree = GTree [GTree]
data GPTree a = GPTree a [GPTree a]

Nested Types
data List a = NilL | ConsL a (List a)
data Nest a = NilN | ConsN a (Nest (a,a))
data Bush a = NilB | ConsB a (Bush (Bush a))
data Node a = Node2 a a | Node3 a a a
data Tree a = Leaf a | Succ (Tree (Node a))
Bird and Meertens, Nested Datatypes, 1998.
Hinze, Finger Trees.

Haskell
input stream --> program --> output stream
Real World
[Char] --> program --> [Char]
Haskell World
module Main where
main = do
input <- getContents
putStr $ unlines $ f $ lines input
countWords :: String -> String
countWords = unlines . format . count . words
count :: [String] -> [(String,Int)]
count = map (ws->(head ws, length ws))
. groupBy (==)
. sort

Haskell
input stream --> program --> output stream
Real World
[Char] --> program --> [Char]
Haskell World
module Main where
main = interact countWords
countWords :: String -> String
countWords = unlines . format . count . words
count :: [String] -> [(String,Int)]
count = map (ws->(head ws, length ws))
. groupBy (==)
. sort

Introduction to Functional Languages

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Introduction to Functional Languages