Fold Operations

1
Fold Operations

• Abstracting Repetitions

2
Motivation Appending lists

• lappend (link)
• generalization of binary append to appending an
  arbitrary number of lists
• fun lappend
• lappend (sss) s _at_ (lappend ss)
• lappend a list list -gt a list

3
Properties (Identities)

• (map f) o lappend
• lappend o (map (map f))
• filter p (xs _at_ ys)
• (filter p xs) _at_ (filter p ys)
• (filter p) o lappend
• lappend o (map (filter p))

4
Fold Operators

• Generalize binary operators to n-ary functions
  (list functions).
• Abstract specific patterns of recursion / looping
  constructs.
• Potential for optimization of special forms.
• foldl, accumulate, revfold,
• foldr, reduce, fold,
• foldl1
• foldr1

5
Foldr (SML97)

• foldr f a x1,x2,,xn
• f( x1 , f(x2, , f(xn,a)...))
• fun foldr f a a
• foldr f a (xxs)
• f (x, foldr f a xs)
• foldr (ab -gt b) -gtb -gt
• a list -gt b
• foldr (op ) 10 1,2,3 16

6
Examples

• foldr (op ) a x1,x2,x3
• ( x1 ( x2 ( x3 a) ) ) )
• foldr (op ) 1,0 4,3,2
• 4,3,2,1,0

7
(contd)

• fold in OldML (not suited for partial eval.)
• fold (ab -gt b) -gt
• a list -gtb -gt b
• foldr in Bird and Wadler (Curried func.)
• reduce in Reade
• foldr (a -gt b -gt b) -gt b -gt
• a list -gt b

8
Foldl (SML97)

• foldl f a x1,x2,,xn
• f(xn,f(,f(x2,f(x1,a))))
• fun foldl f a a
• foldl f a (xxs)
• foldl f (f(x,a)) xs
• foldl (ab -gt b) -gtb -gt
• a list -gt b
• foldl (op ) 10 1,2,3 60

9
Examples

• foldl (op ) a x1,x2,,xn
• (xn ((x2 (x1 a))...))
• foldl (op _at_) 0 1,2,3,4
• 4,3,2,1,0
• foldl (op ) 1,0 4,3,2
• 2,3,4,1,0

10
(contd)

• revfold in OldML (not suited for partial eval.)
• revfold (ba -gt b) -gt
• a list -gtb -gt b
• foldl in Bird and Wadler (Curried func.)
• accumulate in Reade
• foldl (b -gt a -gt b) -gt b -gt
• a list -gtb

11
Examples

• fun pack ds
• foldl (fn (d,v)gt dv10) 0 ds
• pack 1,2,3,4 1234
• fun packNot ds
• foldl (fn (d,v)gt d10v) 0 ds
• packNot 1,2,3,4 100
• fun packNott ds
• foldr (fn (d,v)gt dv10) 0 ds
• packNott 1,2,3,4 4321

12

• fun myId lis
• foldr (fn (x,xs) gt xxs) lis
• myId 1,2,3,4 1,2,3,4
• fun myRev lis
• foldl (fn (x,xs) gt xxs) lis
• myRev 1,2,3,4 4,3,2,1

13

• fun filter p lis
• foldr (fn (x,xs) gt if p x
• then xxs else xs)
• lis
• filter (fn x gt x 2) 1,2,3,2
• fun filterNeg p lis
• foldl (fn (x,xs) gt if p x
• then xs else xs_at_x)
• lis
• filterNeg (fn x gt true) a,b,a

14

• fun takewhile p lis
• foldr (fn (x,xs) gt if p x
• then xxs else )
• lis
• takewhile (fn x gt x 2) 2,2,3,1,2
• fun dropwhile p lis
• foldl (fn (x,xs) gt
• if null xs andalso p x
• then xs else xs_at_x)
• lis
• dropwhile (fn x gt true) 1,2,3

15
Generalizing Operators without identity element

• E.g., max, min, etc for which basis clause (for
  ) cannot be defined.
• fun foldl_1 f (xxs)
• foldl f x xs
• fun foldr_1 f (x) x
• foldr_1 f (xyys)
• f x (foldr_1 f (yys))

16
Laws Identities

• If f is a binary function that is associative and
  a is an identity w.r.t. f, then
• foldr f a xs
• foldl f a (rev xs)
• foldr (op _at_) 1,2,3,4,5
• 1,2,3,4,5
• foldl (op _at_) (rev 1,2,3,4,5)
• 1,2,3,4,5
• foldl (op _at_) 1,2,3,4,5
• 5,3,4,1,2

17
Laws Identities

• If f is a binary function that is commutative
  and a is an identity w.r.t. f, then
• foldr f a xs foldl f a xs
• foldr (op ) 1 1,2,3 6
• foldl (op ) 1 1,2,3 6
• foldr (fn(b,v)gt b andalso v) true false
• false
• foldl (fn(b,v)gt b andalso v) true false
• false

18
Comparing foldl and foldr.

• foldl (op ) 0 1,2,3
• foldl (op ) 1 2,3
• foldl (op ) 3 3
• foldl (op ) 6
• 6
• (foldl Efficient)
• (Tail Recursive)
• foldr (op ) 0 1,2,3
• 1foldr (op ) 0 2,3
• 3 foldr (op ) 0 3
• 6 foldr (op ) 0
• 6

• foldl and t t,f,t
• foldl and t f,t
• foldl and f t
• foldl and f
• false
• (foldr Efficient)
• (Short-circuit evaluation)
• foldr and t t,f,t
• and t
• (foldr and t f,t
• and f
• (foldr and t t)
• false

19
scan (processing prefixes)

• scan f a x1,x2,,xn
• a,(f a x1),(f (f a x1) x2),,
• (f (f (f a x1) x2) xn)
• fun scan f a a
• scan f a xs
• let
• val t (scan f a (init xs))
• in
• t _at_ (f (last t) (last xs))
• end