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FP Workshop

HackerHub Intro to FP Workshop

Ethan He yfhe@ucdavis.edu

Goal

Create functions from functions.

• composition
• higher order functions

Different in Style

If we want a function that add 1 to an array and return the result

def add1(xs):
  res = xs
  for i in range(len(res)):
    res[i] = res[i] + 1
  return res

with some functional thinking

Py with list comprehension

def add1(xs):
  return [i + 1 for i in xs]

Functional JS

function add1(xs) {
  return xs.map(x => x + 1);
}

Still lots of noise in the function definition

• Why do we have to explicitly construct a new list?
• Why do the map function is associated with the list xs?
• Why we still need to write a arrow function even we know map function applies to each element in xs?
• Why so much parentheses?

add1 xs = map (+1) xs

-- better
add1 = map (+1)

A Quick Comparison With Quick Sort

Python Example

From geeks for geeks.

def partition(arr, low, high):
    i = (low-1)         # index of smaller element
    pivot = arr[high]     # pivot
  
    for j in range(low, high):
  
        # If current element is smaller than or
        # equal to pivot
        if arr[j] <= pivot:
  
            # increment index of smaller element
            i = i+1
            arr[i], arr[j] = arr[j], arr[i]
  
    arr[i+1], arr[high] = arr[high], arr[i+1]
    return (i+1)
  
# The main function that implements QuickSort
# arr[] --> Array to be sorted,
# low  --> Starting index,
# high  --> Ending index
  
# Function to do Quick sort

  
def quickSort(arr, low, high):
    if len(arr) == 1:
        return arr
    if low < high:
  
        # pi is partitioning index, arr[p] is now
        # at right place
        pi = partition(arr, low, high)
  
        # Separately sort elements before
        # partition and after partition
        quickSort(arr, low, pi-1)
        quickSort(arr, pi+1, high)

Easier with List Comprehension

NOTE: not exactly the same, but intuitive

def quicksort(xs):
    if len(xs) == 0:
        return []

    pivot = xs[0]
    xs = xs[1:]
    left = [x for x in xs if x <= pivot]
    right = [x for x in xs if x > pivot]

    res = quicksort(left)
    res.append(pivot)
    res += quicksort(right)
    return res

Haskell Example

quicksort :: Ord a => [a] -> [a]
quicksort [] = []
quicksort (x : xs) = quicksort left ++ [x] ++ quicksort right
  where
    left = [y | y <- xs, y < x]
    right = [y | y <- xs, y >= x]

Use of ghc/ghci

ghc: compiler ghci: The interactive shell of haskell, just like IDLE for python

How to compile

ghc main.hs

Useful command in ghci

• :t or :type check the type signature of a expression.
• :l load a local file (Module)

For more detailed doc, see links at end of slides.

Basic Syntax

We will use haskell for this workshop

Variables

define a variable in the following way

x = 5 -- a number
xs = [1..5] -- a list
ys = [1, 2, 3, 4, 5]
str = "abc" -- a string
ch = 'a' -- a char
isCorrect = True -- a bool
tuple = (ch, x)

• all variables are constant, so immutable
• don't declare variable outside function

Functions

There are two ways to define a function

1. regular function
2. lambda expression

-- regular way
sum a b = a + b
-- lambda
sum = \a b -> a + b 
-- the identifier sum and assignment op = is optional

We don't usually give a name to lambda functions since they are mostly used when we only need this function once in code (often called a callback function in engineering)

Types

We don't have to write the type explicitly in haskell, the compiler (or linter in your editor) will infer the types itself.

x = 5 :: Int

sum :: Int -> Int -> Int
sum a b = a + b

sum' :: Num a => a -> a -> a
sum' a b = a + b

The type signature is inspired by the math notation

$$ sum : Int \times Int \rightarrow Int $$

The meaning of this type signature is: the function sum takes two integers and maps to integer.

Currying

$$ sum : Int \times Int \rightarrow Int \\ sum : Int \rightarrow Int \rightarrow Int $$

Think about this: If a function takes two parameter, but you just pass one in, what will it turn?

Take binary addition (+) as a example, suppose (+) :: Num a => a -> a -> a , what is x ?

add1 = (1+)
x = add1 2

Function Composition

is the same idea as in math, suppose we have two functions

$$ g : A \rightarrow B \\ f : B \rightarrow C \\ h = f \circ g $$

is same as

g :: a -> b
f :: b -> c
h = f . g

You should know the type of function h by now.

Higher-order-functions

functions who takes in a functions as parameter and/or returns a function. Some well-known examples that exits in may languages

• map :: (a -> b) -> [a] -> [b]
• reduce :: (a -> a -> a) -> [a] -> a
• all :: (a -> Bool) -> [a] -> Bool
• takeWhile :: (a -> Bool) -> [a] -> [a]
• zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]

the composition operator is also a binary higher-order function

(.) :: (b -> c) -> (a -> b) -> a -> c

Example in Computer Graphics

Suppose we have a list of numbers representing a shape. Each pixel on the screen is represented by 9 values (vertex, color, normal):

[
  x, y, z, r, g, b, u, v, w,
  x, y, z, r, g, b, u, v, w,
  x, y, z, r, g, b, u, v, w,
  ...
]

A triangle is formed by three vertices, i.e.

[
  [x1, y2, z1],
  [x2, y2, z2],
  [x3, y3, z3]
]

How can we write a function that takes a 1D list of values and returns a list of triangles?

Nested Loops

def getTriangles(data):
  count = 0;
  vertex = []
  vertices = []
  for value in data:
    vertex.append(value)
    if len(vertex) == 9:
      vertices.append(vertex[:3])
      vertex = []
  ...

Composition

with the chunksOf function

getTriangles :: [a] -> [[[a]]]
getTriangles = chunksOf 3 . map (take 3) . chunksOf 9

Fold

fold is the key function in functional programming, ideally we can create any function with only foldr

foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b

-- or for simplicity
foldr :: (a -> b -> b) -> b -> [a] -> b

To understand this function, we can take list as an example. In haskell, the list is just a data connected with operator (:)

(:) :: a -> [a] -> [a]

so

[] == []
[5] == 5 : []
[4, 5] == 4 : (5 : [])

Since (:) is bust a binary operator, we can replace it with any other binary operator, which gives the definition of "Foldable".

Examples

map' :: (a -> b) -> [a] -> [b]
map' f = foldr f' []
  where
    f' x acc = f x : acc

reduce :: (a -> a -> a) -> [a] -> a
reduce _ [] = error "TypeError: reduce of empty sequence with no initial value" -- error msg from py
reduce f (x : xs) = foldr f x xs

-- here advanced linter would suggest usr and to replace foldr (&&) True
-- for this project, ignore
-- for future usage of haskell, you should use and
all' :: (a -> Bool) -> [a] -> Bool
all' f xs = foldr (&&) True ys
  where
    ys = map f xs

Challenge

Make a interesting and useful function library based on foldr .

• you may only use foldr and operators in haskell Prelude
• once you make the first two functions with foldr, you can start composing them to get new functions
• To Submit, fork this repository, write your name and email on top of README.md.
  • write your library in MyLibrary.hs
  • create and write a documentation doc.md for your library.
  • To test your functions, use the interactive shell ghci MyLibrary.hs
  • Submit a pull request to this repo when finished.

Useful Links

• GHC User's Guide
• Foldl vs Foldr
• A Gentle Introduction to Haskell
• Official Doc