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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
All hail functions. Children of the idea behind lambda calculus. Use functions just like any other sort of values.
Evaluate not execute.
Pure
Immutability is the key.
Fuck all side-effects.
Deterministic as fuck.
Benefits: equational reasoning, parallelism, happiness
Lazy
Infinity? ez.
Compositional programming — you feel like Mozart.
Disadvantage: Wot is time? Wot is space?
Haskell, the new cool guy in campus
Types
Statically typed: run time errors $\rightarrow$﻿ compile-time errors
Expressive: the code is the documentation
Expressive: brings clarity into coding
Abstraction
Eat, Code, Sleep, Repeat: Haskell has polymorphism, higher-order functions and type classes — so fuck repetition.
Think about the big picture and don't cry about some stupid exception.
What can I do with it?
Program Correctness (QuickCheck)
Fail safe programming (Cardano)
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.
Consice programs
Powerful type system
List comprehension
Recursive functions
Higher-order functions
Effectful functions
Generic functions
Lazy evaluation
Equational reasoning
First Steps
Haskell comes with a large number of built-in functions, which are defined in a library file called the standard prelude.
tail <list>
take <num> <list>
drop <num> <list>
sum <list>
reverse <list>
product <list>
GHCi Command Function Format
Just as in lambda calculus haskell follows a fixed pattern for functions. Here are some examples to elaborate:
$f(x)$﻿ as f x
$f(x, y)$﻿ as f x y
$f(g(x))$﻿ as f (g x)
$f(x, g(y))$﻿ as f x (g y)
$f(x)g(y)$﻿ as f x * g y
Function application has the highest priority than all other operators.
x f y is a syntactic sugar for f x y
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
They are a collection of related values.
f :: A -> B and e :: A then, f e :: B
Bool – logical values
Char – single characters
String – strings of characters
Int – fixed-precision integers
Integer – arbitrary-precision integers
Float – single-precision floating-point numbers
Double – double-precision floating-point numbers
[[’a’,’b’],[’c’,’d’,’e’]] :: [[Char]]
List types — ["One","Two","Three"] :: [String]
Tuple types — ("Yes",True,’a’) :: (String,Bool,Char)
Function types — not :: Bool -> Bool and add :: (Int,Int) -> Int
Polymorphic types — length :: [a] -> Int and zip :: [a] -> [b] -> [(a, b)]
Classes
They are collections of types that support certain overloaded operations called methods.
Eq — (==) :: Eq a => a -> a -> Bool
Ord — <, >, min, max with (<) :: Ord a => a -> a -> Bool
Show — show :: a -> String
Num — e.g., (+), (-), negate, abs, signum with (+) :: Num a => a -> a -> a
Integral — div :: Int a => a -> a -> a and mod :: Int a => a -> a -> a, also Int and Integer types are instances of this class.
Fractional — Float and Double are instances of this class. We also have methods such as / and recip.
Functions
Let us now introduce some really cool implementation techniques in haskell with respect to defining functions.
Conditionals
-- 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
Pattern Matching
-- 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
Lambda Functions
\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?
Name the function
Define its type
Enumerate the cases
Define the base cases
List the "ingredients"
Define the transition (non-trivial cases)
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)
Common programming idioms can be encoded as functions within the language itself.
Domain specific languages can be defined as collections of higher order functions.
Algebraic properties of higher-order functions can be used to reason about numbers.
-- 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) = xf 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
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
We will use types to describe impure actions involving side effects (IO a).
IO Char is the type of actions that return a character.
IO () is the type of purely side effecting actions that return no result value.
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?
No unnecessary evaluation
Ensures termination when possible
Supports programming with infinite structures
Allows programs to be modular (separates control from data)
Notes on Functional Programming
Haskell is very expressive language. It makes you think abstractly and forces to model and define the problem mathematically.
Focus on what to compute (definitions) rather than how
Power of abstraction and modularity
Equational reasoning
How powerful types are
The main drawback is that it is hard to reason about efficiency.