Recursion and Induction: Higher Order Functions Greg Plaxton - - PowerPoint PPT Presentation

Recursion and Induction: Higher Order Functions

Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin

Higher Order Functions

• A higher order function is a function that takes a function as an

argument

• In this lecture we will see a number of examples of useful higher order

functions

• We will see that some of the functions we defined in previous lectures

are more easily defined using a suitable higher order function

Theory in Programming Practice, Plaxton, Spring 2005

Remarks on Infix and Prefix Operators

• We use the term operator to mean a function of two arguments
• An infix operator, such as +, is written between its two arguments,

whereas a prefix operator, such as max, precedes its arguments

• As we have remarked in an earlier lecture, an operator can be converted

from prefix to infix by surrounding it with backquotes – Example: div 5 3 is the same as 5 ‘div‘ 3

• An operator can be converted from infix to prefix by parenthesizing it

– Example: (+) x y is the same as x + y

Theory in Programming Practice, Plaxton, Spring 2005

An Example of a Higher Order Function

• We previously developed the functions suml and multl that operate

similarly on the argument list – For the empty list, each function produces a specific value (0 for suml, 1 for multl) – For a nonempty list, say x:xs, the item x and the function value for xs are combined using a specific operator (+ for suml, * for multl)

• This suggests that we can code a generic function that has three

arguments – The specific value to be returned for the empty list – The operator to be used for combining – The list on which the function is to be applied

Theory in Programming Practice, Plaxton, Spring 2005

Function foldr

• Here is the definition of function foldr

foldr f z [] = z foldr f z (x:xs) = f x (foldr f z xs)

• Various functions can be easily defined using foldr

suml xs = foldr (+) xs multl xs = foldr (*) 1 xs andl xs = foldr (&&) True xs

• rl

xs = foldr (||) False xs eql xs = foldr (==) True xs flatten xs = foldr (++) [] xs

Theory in Programming Practice, Plaxton, Spring 2005

Remark

• There is an even nicer way to define functions such as suml and multl;

just omit xs from both sides of the function definition suml = foldr (+) multl = foldr (*) 1 andl = foldr (&&) True

• rl

= foldr (||) False eql = foldr (==) True flatten = foldr (++) []

• In this course, we will not describe the justification for this type of

definition

Theory in Programming Practice, Plaxton, Spring 2005

The Type of Function foldr

• Let’s ask hugs to tell us the type of function foldr

Main> :t foldr foldr :: (a -> b -> b) -> b -> [a] -> b

• In all of the examples given previously, the types a and b were the same

(e.g., the operator (+) takes two arguments of the same type)

• In the next slide we will give an example in which types a and b differ

Theory in Programming Practice, Plaxton, Spring 2005

Another foldr Example

• The function ev takes an integer and a boolean as arguments and

returns a boolean ev x b = (even x) && b

• The function evenl determines if all integers of a given list are even

evenl = foldr ev True

Theory in Programming Practice, Plaxton, Spring 2005

Function fold

• Function fold is a simpler version of foldr that applies to nonempty

lists only fold f [x] = x fold f (x:xs) = f x (fold f xs)

• We now define the function maxl in terms of fold

maxl = fold max

Theory in Programming Practice, Plaxton, Spring 2005

Function map

• Another useful higher order function is map

– The arguments are a function f and a list of elements on which f can be applied – The return value is the list obtained by applying f to each element

• f the given list
• Here is the definition of map

map f [] = [] map f (x:xs) = (f x) : (map f xs)

Theory in Programming Practice, Plaxton, Spring 2005

Function map

• Here are some examples involving map

Main> map not [True,False] [False,True] Main> map even [2,4,5] [True,True,False] Main> map chCase "jmisra" "JMISRA" Main> map len ["Jayadev","Misra"] [7,5]

• What is the type of map?
• Characterize the value of the following expression

andl (map even xs)

Theory in Programming Practice, Plaxton, Spring 2005

Function filter

• Another useful higher order function is filter

– The arguments are a predicate p and a list xs – The return value is the list containing the elements of xs for which p holds

• Here is the definition of filter

filter p [] = [] filter p (x:xs) | p x = x: (filter p xs) | otherwise = (filter p xs)

Theory in Programming Practice, Plaxton, Spring 2005

The Function filter

• Here are some examples involving filter

Main> filter even [2,3,4] [2,4] Main> filter digit [’a’,’9’,’b’,’0’,’c’] "90" Main> filter upper "Jayadev Misra" "JM" Main> filter digit "Jayadev Misra" ""

• What is the type of filter?

Theory in Programming Practice, Plaxton, Spring 2005