CITS3211 FUNCTIONAL PROGRAMMING. 6. Folding operators
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1 CITS3211 FUNCTIONAL PROGRAMMING 6. Folding operators Summary: This lecture discusses an important group of higher order functions known as the folding operators. Almost any recursive function over lists (or other recursive datatypes) can be defined non recursively using a folding operator. R.L. While, Modified by R. Davies
2 Patterns of recursion map and filter capture two common patterns of recursion over lists map applies a function to each element of a list filter uses a predicate to select elements from a list Both map and filter treat the elements on a list independently Both map and filter always return a list as their result filter returns a list with the same type as its argument, whereas map can return a list with a different type map returns a list with the same length as its argument, whereas filter can return a shorter list map and filter are often used together, as in list comprehensions But most functions don t return a list as their result! Another common (and even more general) pattern of recursion is exhibited by functions like sum, product, or, and, length, concat, insertsort, etc even map and filter themselves! CITS3211 Functional Programming 2 6. Folding operators
3 foldr foldr :: (a > b > b) > b > [a] > b foldr f b [] = b foldr f b (x : xs) = f x (foldr f b xs) fold from the right foldr op b [] = b foldr op b [x1] = op x1 b = x1 `op` b foldr op b [x1, x2] = op x1 (op x2 b) = x1 `op` (x2 `op` b) foldr op b [x1, x2, x3] = op x1 (op x2 (op x3 b)) = x1 `op` (x2 `op` (x3 `op` b)) etc CITS3211 Functional Programming 3 6. Folding operators
4 foldl foldl :: (b > a > b) > b > [a] > b foldl f b [] foldl f b (x : xs) = b = foldl f (f b x) xs fold from the left foldl op b [] = b foldl op b [x1] = op b x1 = b `op` x1 foldl op b [x1, x2] = op (op b x1) x2 = (b `op` x1) `op` x2 foldl op b [x1, x2, x3] = op (op (op b x1) x2) x3 = ((b `op` x1) `op` x2) `op` x3 etc CITS3211 Functional Programming 4 6. Folding operators
5 Packaged recursion sum = foldr (+) 0 product = foldr (*) 1 or = foldr ( ) False and = foldr (&&) True length = foldr (const (+ 1)) 0 concat = foldr (++) [] insertsort = foldr insert [] where insert x xs =... map f = foldr ((:). f) [] filter p = foldr f [] where f x b p x = x : b not (p x) = b CITS3211 Functional Programming 5 6. Folding operators
6 foldr1 and foldl1 These two functions assume that the argument list is not empty note the more restricted types foldr1 :: (a > a > a) > [a] > a foldr1 f [x] = x foldr1 f (x : xs) = f x (foldr1 f xs) foldr1 f [] = error foldr1 has an empty list foldl1 :: (a > a > a) > [a] > a foldl1 f (x : xs) = foldl f x xs foldl1 f [] = error foldl1 has an empty list e.g. maximum, minimum, unwords CITS3211 Functional Programming 6 6. Folding operators
7 Higher order functions over non list datatypes Higher order functions can be defined over any recursive datatype Consider the type Bintree data Bintree a = Empty Leaf a Node a (Bintree a) (Bintree a) Mapping a function over a Bintree involves applying a function to each element of type a mapt :: (a > b) > Bintree a > Bintree b mapt f Empty = Empty mapt f (Leaf x) = Leaf (f x) mapt f (Node x l r) = Node (f x) (mapt f l) (mapt f r) e.g. mapplus, takefst Folding a Bintree with a function involves replacing each terminal with a value of the result type and folding up each internal node foldt :: b > (a > b) > (a > b > b > b) > Bintree a > b foldt b f g Empty = b foldt b f g (Leaf x) = f x foldt b f g (Node x l r) = g x (foldt b f g l) (foldt b f g r) e.g. size, frontier mapt f = foldt Empty (Leaf. f) (Node. f) CITS3211 Functional Programming 7 6. Folding operators
8 mapt and foldt The structure of these definitions corresponds closely to the structure of the datatype (as described in the data declaration) There is always one equation for each constructor foldt has one argument for each constructor (plus the tree itself) the type of each argument is determined by the type of the corresponding constructor mapt has one argument for each constructor with non recursive sub parts (plus the tree itself) the type of each argument is determined by the types of the sub parts of the corresponding constructor cf foldr and map CITS3211 Functional Programming 8 6. Folding operators
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