Haskell in Harsh Words

Table of Contents

The Fuck is Haskell?

A pure, lazy, functional programming language. Basically, it is something that happens when like-minded cool people come together.

Meaning of the above bullshit words?

Functional

Pure

Lazy

Haskell, the new cool guy in campus

Types

Abstraction

What can I do with it?

Haskell: A Functional Programming Langauge

What does it mean for Haskell to be a functional programming language? Well, it means that Haskell follows the principle of functional programming — a programming paradigm where functions are the basic building blocks of computation.

Def: A function is a mapping that takes one or more arguments and produces a single result.

Properties of Haskell

First Steps

Haskell comes with a large number of built-in functions, which are defined in a library file called the standard prelude.

GHCi Command

Function Format

Just as in lambda calculus haskell follows a fixed pattern for functions. Here are some examples to elaborate:

Examples

-- Program 1
main = print (a ++ b ++ c)
a = [((2**3) * 4)]
b = [(2*3) + (4*5)]
c = [2 + (3 * (4**5))]


-- Program 2
main = print (n)
n = (a `div` (length xs))   where     a = 10     xs = [1 .. 5]


-- Program 3
main = print (a [1 .. 5])
a xs = sum (drop (length xs - 1) (take (length xs) xs))
main = putStrLn "hello, world"

Types and class

Types

Classes

Functions

Let us now introduce some really cool implementation techniques in haskell with respect to defining functions.

-- Let's see an example
even :: a -> Bool
even n = (n `mod` 2 == 0)

-- Conditional using "if else"
signum n = if n > 0 then 1        else if n < 0 then -1        else 0

-- Conditional using "such that"
signum n | n > 0 = 1     | n < 0 = -1     | otherwise  = 0
-- Define functions using '_'
len [] = 0
len (_:xs) = 1 + len xs

initials :: String -> String -> String  
initials firstname lastname = [f] ++ ". " ++ [l] ++ "."    where (f:_) = firstname  (l:_) = lastname

-- Defining using '_' fundamentally
test :: Int -> Int
test 0 = 1
test 1 = 2
test _ = 0
\x -> x + 1
-- For example,
add :: Int -> Int -> Int
add x y = x + y
-- and
odds n = map f [0 .. n - 1]  where      f x = x * 2 + 1

-- can be written as 
add :: Int -> (Int -> Int)
add = \x -> (\y -> x + y)
-- and
odds n = map (\x -> x * 2 + 1) [0 .. n - 1]

Currying Functions

Let $x = f(a, b, c)$ then, we shall have the following.

$$f :: (\text{Type}(a),\ \text{Type}(b),\ \text{Type}(c)) \to \text{Type}(x)$$

Upon currying, this gets translated to the below expression.

$$f :: \text{Type}(a) \to (\text{Type}(b) \to (\text{Type}(c) \to (\text{Type}(x))))\\ \text{or, } f :: \text{Type}(a) \to \text{Type}(b) \to \text{Type}(c) \to \text{Type}(x)$$

This is because, $x = f(a, b, c)$ becomes $h = g(a), i = h(b), x = i(c)$ or if called in sequence $x = g(a)(b)(c)$.

fst :: (a, b) -> a
zip :: [a] -> [b] -> [(a, b)]
id :: a -> a
take :: Int -> [a] -> [a]
head :: [a] -> a

Operator Sections

-- Operator Sections
w = 1 + 2
-- is same as
x = (+) 1 2
-- is same as
y = (1+) 2
-- or
z = (+2) 1

This is very useful to construct certain functions.

List Comprehension

The idea is based on set construction from other sets.

xy = [(x, y) | x <- [1..3], y <- [x..3]]

-- The above is an example of dependent generators.
-- Definitions of x and y serve as generators.
-- Check out the usability of .. operator.

concat :: [[a]] -> [a]
concat xss = [x | xs <- xss, x <- xs]

-- Guards are possible for example as bellow.
evens = [x | x <- [1..10], even x]

Recursion

How to think recursively?

Recursion

-- Example of a recursion
zip :: [a] -> [b] -> [(a, b)]
zip [] _ = []
zip _ [] = []
zip (x:xs) (y:ys) = (x, y) : zip xs ys

-- Append Definition
(++) :: [a] -> [a] -> [a]
[] ++ ys = ys
(x:xs) ++ ys = x : (xs ++ ys)

Memoization

The following memoize function takes a function of type Int -> a and returns a memoized version of the same function. The trick is to turn a function into a value because, in Haskell, functions are not memoized but values are.

import Data.Function (fix)

memoize :: (Int -> a) -> (Int -> a)
memoize f = (map f [0 ..] !!)

fib :: (Int -> Integer) -> Int -> Integer
fib f 0 = 0
fib f 1 = 1
fib f n = f (n - 1) + f (n - 2)

fibMemo :: Int -> Integer
fibMemo = fix (memoize . fib)

Higher Order Function

A function is called a higher order if it takes a function as an argument or returns a function as a result.

twice :: (a -> a) -> a -> a
twice f x = f (f x)
-- Example
map :: (a -> b) -> [a] -> [b]
map f xs = [f x | x <- xs]

-- Alternate
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = f x : map f xs
-- Example
filter :: (a -> Bool) -> [a] -> [a]
filter f xs = [x | x <- xs, f x]

-- Alternatively
filter f [] = []
filter f (x:xs) | f x = x : filter f xs      | otherwise = filter f xs

Foldr

A number of functions on lists can be defined using the following simple pattern of recursion.

f [] = v
f (x:xs) = x ⊕ f xs

Here $f$ maps the empty list to some value $v$ and any non-empty list to some function $⊕$ applied to its head and $f$ of its tail.

The higher-order library function foldr (fold-right) encapsulates this simple pattern of recursion with the function $ $$⊕$ and the value $v$ as arguments.

foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f v [] = v
foldr f v (x:xs) = f x (foldr f v xs)
-- Generic Examples
sum = foldr (+) 0
product = foldr (*) 1
or = foldr (||) False
and = foldr (&&) True

-- Other examples
length = foldr (\ _ n -> 1 + n) 0
reverse = foldr (\ x xs -> xs ++ [x]) []
(++ ys) = foldr (:) ys

Composition

(.) :: (b -> c) -> (a -> b) -> (a -> c)
f . g = \x -> f (g x)

odd :: Int -> Bool
odd = not . even

Other nice library functions

all :: (a -> Bool) -> [a] -> Bool
all f xs = and [f x | x <- xs]

any :: (a -> Bool) -> [a] -> Bool
any f xs = or [f x | x <- xs]

takeWhile :: (a -> Bool) -> [a] -> [a]
takeWhile f [] = []
takeWhile f (x:xs) | f x = x : takeWhile f xs         | otherwise  = []

dropWhile :: (a -> Bool) -> [a] -> [a]
dropWhile f [] = []
dropWhile f (x:xs) | f x = dropWhile f xs         | otherwise  = x:xs

Moreover, curried functions are higher-order functions that returns a function.

Where and Let Clauses

-- Let
f = let x = 1; y = 2 in (x + y)

-- Where
f = x + y where x = 1; y = 1

Type and Class Declaration

type Pos = (Int, Int)
type Trans = Pos -> Pos

type Tree = (Int, [Tree])

A completely new type can be defined by specifying its values in context-free formulation.

data Bool = False | True
data Answer = Yes | No | Unknown

answers :: [Answer]
answers = [Yes, No, Unknown]

flip :: Answer -> Answer
flip Yes = No
flip No = Yes
flip Unknown = Unknown
Circle :: Float -> Shape
Rect :: Float -> Float -> Shape

data Shape = Circle Float | Rect Float Float

square :: Float -> Shape
square n = Rect n n

area :: Shape -> Float
area (Circle r) = pi * r^2
area (Rect x y) = x * y

Maybe, Nothing, Just

data Maybe a = Nothing | Just a

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)

This is basically a cool syntactic encapsulation to prevent crashing. This is used to often deal with exceptions and create failsafe programs.

Recursive Types

data Nat = Zero | Succ Nat

-- we have Zero :: Nat
-- and Succ :: Nat -> Nat
data Expr = Val Int       | Add Expr Expr       | Mul Expr Expr

eval :: Expr -> Int
eval (Val n) = n
eval (Add x y) = eval x + eval y
eval (Mul x y) = eval x * eval y

Interactive Programming

This is a problem because Haskell is designed to have no side effects and thus only to create batch programs.

Interactive programs, on the other hand, necessarily require side effects.

Solution

The standard library provides a number of actions including the following three primitives.

-- reads character from the keyboard 
-- and echoes it o the screen
-- returning the character as the result value
getChar :: IO Char

-- takes a character as a input and writes it to the screen 
-- and returns no result value
putChar :: Char -> IO ()

-- simply return value without performing any interaction
-- this is basically a pure to impure conversion
return :: a -> IO a

Sequencing

act :: IO (Char, Char)
act = do x <- getChar    getChar    y <- getChar    return (x, y)

Derived Primitives

getLine :: IO String
getLine = do x <- getChar    if x == '\n' then return []    else      do xs <- getLine        return (x:xs)
putStr :: String -> IO ()
putStr [] = return ()
putStr (x:xs) = do putChar x          putStr xs
-- Just mentioned here so that 
-- one can try out problems with input

import Control.Arrow ((>>>))

main :: IO ()
main = interact $     lines >>> head >>> read >>> solve >>> (++ "\n")

solve :: Int -> String


-- Type 2
import Control.Arrow ((>>>))

main :: IO ()
main =
  interact $   words >>> map read >>> solve >>> show >>> (++ "\n")

solve :: [Integer] -> Integer

Simple I/O Operations

These are in-built.

putChar :: Char -> IO()
putStr :: String -> IO ()
putStrLn :: String -> IO ()
print :: Show a => a -> IO ()

getChar :: IO Char
getLine :: IO String
getContents :: IO String
interact :: (String -> String) -> IO ()

show :: Show a => a -> String
read :: Read a => String -> a
main = do    putStrLn "enter value for x: "     input1 <- getLine     putStrLn "enter value for y: "      input2 <- getLine      let x = (read input1 :: Int)     let y = (read input2 :: Int)     print (x + y)

Lazy Evaluation

$$\text{lazy evaluation} = \text{outermost evaluation}\ +\ \text{shared arguments}$$

Why is it important?

Notes on Functional Programming

Haskell is very expressive language. It makes you think abstractly and forces to model and define the problem mathematically.

The main drawback is that it is hard to reason about efficiency.