CSC447

Concepts of Programming Languages

Folds

Instructor: James Riely

Map-Reduce Frameworks

Count the appearance of each word in a set of documents 

def map(name: String, contents: String) = 
  for w <- contents do
    emit (w, 1)

def reduce(word: String, partialCounts: Iterator) = 
  var sum = 0
  for pc <- partialCounts do
    sum = sum + pc
  emit (word, sum)
An example of MapReduce

Parallel Execution

Map-reduce execution
From here

Sum computing forward


def sum (xs:List[Int], z:Int = 0) : Int =  xs match 
  case Nil   => z
  case y::ys => sum (ys, z + y)

val xs = List(11,21,31)
sum (xs)
          

sum(11::21::31::Nil)
--> sum(11::21::31::Nil, 0)
--> sum(21::31::Nil, 11)
--> sum(31::Nil, 32)
--> sum(Nil, 63)
-->
-->
-->
--> 63 = (((0 + 11) + 21) + 31)

Sum computing backward


def sum (xs:List[Int], z:Int = 0) : Int =  xs match 
  case Nil   => z
  case y::ys => y + sum (ys, z)

val xs = List(11,21,31)
sum (xs)
          

sum(11::21::31::Nil)
--> sum(11::21::31::Nil, 0)
--> 11 + sum(21::31::Nil, 0)
--> 11 + (21 + sum(31::Nil, 0))
--> 11 + (21 + (31 + sum(Nil, 0)))
--> 11 + (21 + (31 + 0))
--> 11 + (21 + 31)
--> 11 + 52
--> 63 = (11 + (21 + (31 + 0)))

Sum to foldLeft

Computing forward 

def sum       (xs:List[Int], z:Int)                     : Int =  
  xs match 
    case Nil   => z
    case y::ys => sum       (ys, z + y)

val xs = List(11,21,31)
sum       (xs, 0)
          

res1: Int = 63

Sum to foldLeft

abstract away the + operation 

def foldLeft  (xs:List[Int], z:Int, f:((Int,Int)=>Int)) : Int = 
  xs match 
    case Nil   => z
    case y::ys => foldLeft  (ys, f(z,y), f)

val xs = List(11,21,31)
foldLeft  (xs, 0, _+_)
          

res1: Int = 63

Changing the return type

Fold into string 

def foldLeft  (xs:List[Int], z:String, f:(String,Int)=>String) : String = 
  xs match 
    case Nil   => z
    case y::ys => foldLeft  (ys, f(z, y), f)

val xs = List(11,21,31)
foldLeft  (xs, "", _ + "0x%02x".format(_))
            

res1: String = 0x0b0x150x1f

Changing the parameter type

Fold a List of Lists 

def foldLeft  (xs:List[List[Int]], z:Int, f:(Int,List[Int])=>Int) : Int = 
  xs match 
    case Nil   => z
    case y::ys => foldLeft  (ys, f(z, y), f)

val xss = List(List(11,21,31),List(),List(41,51))
foldLeft  (xss, 0, _ + _.length)
            

res1: Int = 5

Type parameters

Abstracting the type 

def foldLeft  [Z,X] (xs:List[X], z:Z, f:((Z,X)=>Z)) : Z =
  xs match 
    case Nil   => z
    case y::ys => foldLeft  (ys, f(z,y), f)

val xs = List(11,21,31)
foldLeft  (xs, "!", (z:String,x:Int) => z + "0x%02x".format(x))
          

res1: String = !0x0b0x150x1f

Fold Right

Computing backward 

def foldRight [X,Z] (xs:List[X], z:Z, f:((X,Z)=>Z)) : Z =
  xs match 
    case Nil   => z
    case y::ys => f (y, foldRight (ys, z, f))

val xs = List(11,21,31)
foldRight (xs, "!", (x:Int,z:String) => "0x%02x".format(x) + z)
          

res1: String = 0x0b0x150x1f!

Left v Right

  • foldLeft is tail recursive:
    • return foldLeft (ys, f(z, y))
      • apply f to the head and the accumulated result
      • recursive call on the tail
    • base case used with first element
  • foldRight is recursive into an argument:
    • return f (y, foldRight (ys, z))
      • recursive call on the tail
      • apply f to the head and result of recursion
    • base case used with last element

Builtin folds

  • Scala List class has fold methods 
    
xss.foldLeft (0) ((z,xs)=>z + xs.length)
    vs our own foldLeft function 
    
foldLeft (xss, 0, (z,xs)=>z + xs.length)
  • Operator forms: 
    
 xss.foldLeft(0)(f) == (0 /: xss)(f)
xss.foldRight(0)(f) == (xss :\ 0)(f)

Left v Right


def foldLeft [Z,X] (xs:List[X], z:Z, f:((Z,X)=>Z)) : Z =  xs match 
  case Nil   => z
  case y::ys => foldLeft (ys, f(z,y), f)
          

def foldRight [X,Z] (xs:List[X], z:Z, f:((X,Z)=>Z)) : Z =  xs match 
  case Nil   => z
  case y::ys => f (y, foldRight (ys, z, f))
          

val xs = List(a, b, c)
foldLeft (xs, z, f) === f( f( f(z,a),b),c) === (z /: xs)(f)
foldRight(xs, z, f) === f(a, f(b, f(c,z))) === (xs :\ z)(f) 
          
                   (z /: xs)(f)           (xs :\ z)(f)
                      f                       f
                     / \                     / \
                    f   c                   a   f
                   / \                         / \
                  f   b                       b   f
                 / \                             / \
                z   a                           c   z

Folds are universal


def sum         (xs: List[Int])              = xs.foldLeft(0)(_+_)
def prod        (xs: List[Int])              = xs.foldLeft(1)(_*_)
def or          (xs: List[Boolean])          = xs.foldLeft(false)(_||_)
def and         (xs: List[Boolean])          = xs.foldLeft(true)(_&&_)
def append  [X] (xs: List[X])(ys: List[X])   = xs.foldRight(ys)(_::_)
def flatten [X] (xs: List[List[X]])          = xs.foldLeft(Nil:List[X])(_:::_)
def length  [X] (xs: List[X])                = xs.foldLeft(0)((z,x)=>z+1)
def reverse [X] (xs: List[X])                = xs.foldRight(Nil:List[X])((x,zs)=>zs:::List(x))
def map   [X,Y] (xs: List[X], f: X=>Y)       = xs.foldRight(Nil:List[Y])(f(_)::_)
def filter  [X] (xs: List[X], f: X=>Boolean) = xs.foldRight(Nil:List[X])((x,zs)=>if f(x) then x::zs else zs)

Here are lots and lots of foldleft examples

A tutorial on the universality and expressiveness of fold