S*

Lecture 9

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Review of Homework 8

Write an IO program that reverses text
 A B C D

-- Designed by Eric J. Bomgardner

module Main where

main = do
theInput <- readFile "input.txt"
writeFile "output.txt" (reverse theInput)

The Maybe data type

data Maybe a = Nothing | Just a

Use Maybe when the output isn't guaranteed

Prelude> import Data.List

Prelude> :t elemIndex
elemIndex :: Eq a => a -> [a] -> Maybe Int

Prelude Data.List> elemIndex 'b' "abc"
Just 1

Prelude Data.List> elemIndex 'z' "abc"
Nothing

Context

Maybe wraps things into a context.

• Just 1 vs 1

• Just "hello" vs "hello"

• Just 'a' vs 'a'

• Nothing vs ERROR!

Functor

A Functor generalizes the map function

It implements fmap

fmap :: (a -> b) -> f a -> f b

Naturally, a list derives the Functor Typeclass

Prelude> fmap (+1) [1,2,3]
[2,3,4]

What else is a Functor?

Maybe is a Functor

*Main> fmap (+1) (Just 1)
Just 2

IO is a Functor

*Main> fmap (++"!") getLine
hi
"hi!"

The Functor Laws

Additionally, these rules are necessary to define a Functor.

First law:

fmap id F = F

Second law:

fmap (f . g) F = fmap f (fmap g F)

Applicative Typeclass

Prelude> import Control.Applicative

Prelude> (*) <\$> Just 2 <*> Just 8
Just 16

We won't go into much detail about the Applicative Typeclass

There is a great introduction here

Monoid

A Monoid is a function (•) that satisfies the following:

1. ∀ a,b ∈ S: a•b ∈ S

The Set (S) is closed under the binary function (•).
2. ∀ a,b,c ∈ S: (a•b)•c = a•(b•c)

The binary function is associative
3. ∃ e∈S: ∀ a∈S: e•a = a•e = a

e is the identity element

(A Monoid is a Group without the requirement of an inverse) (Or a Category with a single object.)

Examples of Monoids

+ , 0 is a Monoid.

Which of the following are Monoids?

• * , 1           , S={0,1,2,...}
• || , False , S={True, False}
• ++ , []       , S={"", "a", "ab",...}
• ,           , S={A,B,...}
• ÷ , 1           , S={1,2,...}

Try it out

In Haskell, the binary function is called mappend and its identity element is mempty

Prelude> import Data.Monoid

Prelude Data.Monoid> [1,2,3] `mappend` [4,5,6]
[1,2,3,4,5,6]

Prelude Data.Monoid> [1,2,3] `mappend` mempty
[1,2,3]

Prelude Data.Monoid> mempty `mappend` [1,2,3]
[1,2,3]

As you can see above, lists are Monoids.

Quick Reminder

If you have a type a and a function a -> b

then you can get b.

Prelude> odd 3
True

Maybe Continued

If we instead have m a and a function a -> m b

How would we get m b?

For example, given Maybe a and a -> Maybe b

How would we get Maybe b?

The answer we desire looks like this

(>>=) :: (Monad m) => m a -> (a -> m b) -> m b

>>= is called a bind

Necessity of bind

Let's say we had the function f below

f :: Int -> Maybe Int
f n = Just (n+1)

Prelude> f 1
Just 2

What if we wanted to pass in Just 1?

Prelude> f (Just 1)
ERROR!

bind(>>=)

Prelude> :t (>>=)
(>>=) :: Monad m => m a -> (a -> m b) -> m b

Prelude> Just 1 >>= f
Just 2

A Monad is Typeclass with some extra rules.

The two important functions: return and >>=

return :: a -> m a

(>>=) :: m a -> (a -> m b) -> m b

The >> function

This function simply glues together Monads.

Prelude> print "foo" >>= \_ -> print "bar"
"foo"
"bar"

Prelude> print "foo" >> print "bar"
"foo"
"bar"

The do notation is basically an interweaving of >>

main = do
print "foo"
print "bar"

Minimize do usage

Write this piece of code without using do

main = do
putStrLn "Enter name:"
name <- getLine
putStrLn ("Hi " ++ name)

Try it out by yourself

main = putStrLn "Enter name:" >> getLine >>= putStrLn.("Hi " ++)

For something to truly be a Monad,

it must also obey the following laws: *

• Left Identity

return a >>= f   ≡   f a
• Right Identity

m >>= return   ≡   m
• Associativity

(m >>= f) >>= g   ≡   m >>= (\x -> f x >>= g)

These laws are somewhat similar to those of Monoids.

Here's another way to write the three laws

 do x' <- return x f x' ≡ do f x do x <- m return x ≡ do m do y <- do x <- m f x g y ≡ do x <- m y <- f x g y

It's based off the Kleisli composition operator *

(This is for your own curiosity.)

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c

• Left Identity

return >=> g   ≡   g
• Right Identity

f >=> return   ≡   f
• Associativity

(f >=> g) >=> h   ≡   f >=> (g >=> h)

Why such specific rules?

These Monad laws form a mathematical category.

A monad in X is just a monoid in the category of endofunctors of X, with product × replaced by composition of endofunctors and unit set by the identity endofunctor.

-- Mac Lane

Monads are like burritos

-- Mark Dominus

Homework

1. Fill out this week's form
2. Think about final project ideas.
• Building a SuDoKu solver
• Building a personal finance software
• Adding a library to Hackage
• Using Haskell syntax as formal documentation
• ...
• etc.