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 (+) 0 xs multl xs = foldr (*) 1 xs andl xs = foldr (&&) True xs orl 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 (+) 0 multl = foldr (*) 1 andl = foldr (&&) True orl = 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 of 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