Download presentation

Presentation is loading. Please wait.

Published byMarissa Phillips Modified over 4 years ago

1
Functional Programming Lecture 11 - higher order functions

2
Rule of Cancellation If f :: t 1 -> t 2 -> … -> t n -> t and f is applied to arguments e 1 :: t 1, … e k :: t k, where k<n then the resulting type is given by cancelling out the types t 1 to t k t 1 -> t 2 -> … -> t k -> t k+1 … -> t n -> t i.e. (f e 1 … e k ) :: t k+1 -> t k+2 … -> t n Example: max :: Int -> Int -> Int So, max 2 :: Int -> Int

3
Example: max :: Int -> Int -> Int So, max 2 :: Int -> Int (max 2) is just a function. It is the function that takes the maximum of 2 and another number. given: max x y = if x>=y then x else y then: (max 2) y = if 2>=y then 2 else y This is an example of partial evaluation. (The function max has been partially evaluated, to give another function, max 2.)

4
Function Application and -> Associativity Function application is left associative. i.e. f x y = (f x) y f x y = f (x y) !! E.g. max :: Int -> Int -> Int max 2 3 = (max 2) 3 = max (2 3) (doesnt make sense)

5
Function Application and -> Associativity Function space symbol -> is right associative. i.e. a -> b -> c means a -> (b -> c) not (a -> b) -> c f :: a -> (b -> c) g :: (a -> b) -> c a b f c (a->b) g c E.g. max :: Int -> (Int -> Int) max 2 :: Int -> Int

6
Partial evaluation When you partially evaluate a function, the new, resulting function must be enclosed in (.. ). The new function is also called an operator section. Examples (+2)the function which adds 2 to its argument. (2+)the function which adds 2 to its argument. (>2)the function which compares its argument with 2. (3:)the function the puts 3 at the front of a list. (++\n)the function which puts a newline at the end of a string.

7
An aside Displaying strings with newline characters. \n is the newline character. If you type in > show test\ntest the result is test\ntest To display a string, and interpret the special symbols, use putStr :: String -> IO String. If you type > putStr test\ntest the result is test test ::IO ()

8
Anonymous functions Functions do not have to have names. E.g. \m -> n+m This defines a function with one argument that returns the sum of n and that argument. The arguments - given between \ and -> The result - given after the ->. \ stands for the Greek letter.

9
Using a Lambda-defined function Define a function comp2 which applies a function f to each of its 2 arguments before applying function g to the results. f g f comp2 :: (a -> b) -> (b -> b -> c) -> (a -> a -> c) f g resulting function comp2 f g = \x y -> g (f x) (f y) e.g. comp2 sq add 3 4 => add (sq 3) (sq 4) => 9+16 => 25

10
Curry and Uncurry A function which takes its arguments one at a time is said to be a curried function, or in curried form. (Named after Haskell Curry, a logician who worked on combinatory logic and showed relationships to - calculus.) E.g. max :: Int -> Int -> Int is in curried form. x y Most of the functions in this course so far have been curried. A function which takes all its arguments bundled together as a tuple is said to an uncurried function, or in uncurried form. E.g. maxUC :: (Int,Int) -> Int is in uncurried form. (x,y)

11
There is an obvious relationship between the curried and uncurried versions. Namely, max x y = maxUC (x,y) We can turn an uncurried function into a curried one, and vice-versa. Suppose g is an uncurried function, g :: (a,b) ->c. Then (curry g) :: a -> b -> c. curry :: ((a,b) ->c) -> (a -> b -> c) curry g x y = g (x,y) Recall: function application is left assoc. i.e. (curry g) x y = g (x,y)

12
Similarly, suppose f is an curried function, f :: a -> b ->c. Then (uncurry f) :: (a,b) -> c. uncurry :: (a -> b -> c) -> ((a,b) ->c) uncurry f (x,y) = f x y So, (uncurry f) (x,y) = f x y curry and uncurry are defined in the standard prelude. E.g. Prelude> uncurry (+) (2,1) 3

13
Finally, we would expect that curry and uncurry are inverses: E.g. curry (uncurry f) = f uncurry (curry g) = g It works for specific cases: uncurry max = maxUC curry maxUC = max So, curry (uncurry max) = curry maxUC = max uncurry (curry maxUC) = uncurry max = maxUC But to prove it for an arbitrary function, requires induction.

Similar presentations

OK

Winter 2016CISC101 - Prof. McLeod1 CISC101 Reminders Quiz 3 this week – last section on Friday. Assignment 4 is posted. Data mining: –Designing functions.

Winter 2016CISC101 - Prof. McLeod1 CISC101 Reminders Quiz 3 this week – last section on Friday. Assignment 4 is posted. Data mining: –Designing functions.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google