Download presentation

Presentation is loading. Please wait.

Published bySamson Gragg Modified over 2 years ago

1
cs776(Prasad)L7fold1 Fold Operations Abstracting Repetitions

2
cs776(Prasad)L7fold2 Motivation : Appending lists lappend (link) –generalization of binary append to appending an arbitrary number of lists fun lappend [] = [] | lappend (s::ss) = s @ (lappend ss) lappend : ’ a list list -> ’ a list

3
cs776(Prasad)L7fold3 Properties (Identities) (map f) o lappend = lappend o (map (map f)) filter p (xs @ ys) = (filter p xs) @ (filter p ys) (filter p) o lappend = lappend o (map (filter p))

4
cs776(Prasad)L7fold4 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
cs776(Prasad)L7fold5 Foldr (SML97) foldr f a [x1,x2,…,xn] = f( x1, f(x2, …, f(xn,a)...)) fun foldr f a [] = a | foldr f a (x::xs) = f (x, foldr f a xs) foldr : ( ’ a* ’ b -> ’ b) -> ’ b -> ’ a list -> ’b foldr (op +) 10 [1,2,3] = 16

6
cs776(Prasad)L7fold6 Examples foldr (op +) a [x1,x2,x3] = ( x1 + ( x2 + ( x3 + a) ) ) ) foldr (op ::) [1,0] [4,3,2] = [4,3,2,1,0]

7
cs776(Prasad)L7fold7 (cont’d) fold in OldML ( not suited for partial eval.) fold : ( ’ a* ’ b -> ’ b) -> ’ a list -> ’ b -> ’ b foldr in Bird and Wadler (Curried func.) reduce in Reade foldr : ( ’ a -> ’ b -> ’ b) -> ’ b -> ’ a list -> ’ b

8
cs776(Prasad)L7fold8 Foldl (SML97) foldl f a [x1,x2,…,xn] = f(xn,f(…,f(x2,f(x1,a))…)) fun foldl f a [] = a | foldl f a (x::xs) = foldl f (f(x,a)) xs foldl : ( ’ a* ’ b -> ’ b) -> ’ b -> ’ a list -> ’ b foldl (op * ) 10 [1,2,3] = 60

9
cs776(Prasad)L7fold9 Examples foldl (op *) a [x1,x2,…,xn] = (xn * (…*(x2 * (x1 * a))...)) foldl (op @) [0] [[1],[2],[3],[4]] = [4,3,2,1,0] foldl (op ::) [1,0] [4,3,2] = [2,3,4,1,0]

10
cs776(Prasad)L7fold10 (cont’d) revfold in OldML ( not suited for partial eval.) revfold : ( ’ b* ’ a -> ’ b) -> ’ a list -> ’ b -> ’ b foldl in Bird and Wadler (Curried func.) accumulate in Reade foldl : ( ’ b -> ’ a -> ’ b) -> ’ b -> ’ a list -> ’ b

11
cs776(Prasad)L7fold11 Examples fun pack ds = foldl (fn (d,v)=> d+v*10) 0 ds pack [1,2,3,4] = 1234 fun packNot ds = foldl (fn (d,v)=> d*10+v) 0 ds packNot [1,2,3,4] = 100 fun packNott ds = foldr (fn (d,v)=> d+v*10) 0 ds packNott [1,2,3,4] = 4321

12
cs776(Prasad)L7fold12 fun myId lis = foldr (fn (x,xs) => x::xs) [] lis myId [1,2,3,4] = [1,2,3,4] fun myRev lis = foldl (fn (x,xs) => x::xs) [] lis myRev [1,2,3,4] = [4,3,2,1]

13
cs776(Prasad)L7fold13 fun filter p lis = foldr (fn (x,xs) => if p x then x::xs else xs) [] lis; filter (fn x => x = 2) [1,2,3,2] fun filterNeg p lis = foldl (fn (x,xs) => if p x then xs else xs@[x]) [] lis; filterNeg (fn x => true) [“a”,”b”,”a”]

14
cs776(Prasad)L7fold14 fun takewhile p lis = foldr (fn (x,xs) => if p x then x::xs else []) [] lis; takewhile (fn x => x = 2) [2,2,3,1,2] fun dropwhile p lis = foldl (fn (x,xs) => if null xs andalso p x then xs else xs@[x]) [] lis; dropwhile (fn x => true) [1,2,3]

15
cs776(Prasad)L7fold15 Generalizing Operators without identity element E.g., max, min, etc for which basis clause (for []) cannot be defined. foldl_1 fun foldl_1 f (x::xs) = foldl f x xs; foldr_1 fun foldr_1 f (x::[]) = x foldr_1 | foldr_1 f (x::y::ys) = foldr_1 f x (foldr_1 f (y::ys))

16
cs776(Prasad)L7fold16 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 @) [] [[1,2],[3,4],[5]] = [1,2,3,4,5] foldl (op @) [] (rev [[1,2],[3,4],[5]]) = [1,2,3,4,5] foldl (op @) [] [[1,2],[3,4],[5]] = [5,3,4,1,2]

17
cs776(Prasad)L7fold17 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)=> b andalso v) true [false] = false foldl (fn(b,v)=> b andalso v) true [false] = false

18
cs776(Prasad)L7fold18 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] =1+foldr (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 Comparing foldl and foldr.

19
cs776(Prasad)L7fold19 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 @ [(f (last t) (last xs))] end;

Similar presentations

OK

Introduction A function is called higher-order if it takes a function as an argument or returns a function as a result. twice :: (a a) a a twice.

Introduction A function is called higher-order if it takes a function as an argument or returns a function as a result. twice :: (a a) a a twice.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on relations and functions for class 11th english Ppt on machine translation services View my ppt online reader Ppt on save environment in hindi Ppt on iso-osi reference model Ppt on steve jobs biography pdf Ppt on airbag in cars Ppt on aircraft landing gear system information Ppt on tunnel diode circuit Ppt on natural vegetation and wildlife of north america