Background Functions (1C) Young Won Lim 12/6/17

Transcription

1 Background Functions (1C)

2 Copright (c) Young W. Lim. Permission is granted to cop, distribute and/or modi this document under the terms o the GNU Free Documentation License, Version 1.2 or an later version published b the Free Sotare Foundation; ith no Invariant Sections, no Front-Cover Tets, and no Back-Cover Tets. A cop o the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to ounglim@hotmail.com. This document as produced b using OpenOice.

3 Based on Haskell in 5 steps Functions 3

4 Function Deinition Function Deinition I. square = * - unction tpe is inerred not eicient Tpe Inerence Function Deinition II. square :: Double -> Double square = * unction tpe declaration unction deinition unction tpe declaration unction deinition Functions 4

5 Tpe Declaration Tpe Declaration the declaration o an identiier's tpe identiier name :: tpe name... identiier names (including unction identiiers) must alas begin ith a loer-case letter tpe names in Haskell alas begin ith a capital letter Functions 5

6 Function Tpes and Tpe Classes Function Deinition I. square = * unction deinition = Function Deinition II. square :: Double -> Double square = * unction deinition unction tpe declaration = tpe class a set o tpes unction tpe 1 unction tpe 2 unction tpe n Requirements Subclasses Functions 6

7 Curr & Uncurr :: a -> b -> c the curried orm og :: (a, b) -> c = curr g g = uncurr = g (,) the curried orm is usuall more convenient because it allos partial application. the curried orm :: a -> b -> c curring uncurring g :: (a, b) -> c all unctions are considered curried g (, ) all unctions take just one argument Functions 7

8 Functions : First-class Data Tpes unctions are irst-class data tpes Haskell treats unctions as regular data, just like integers, or loating-point values, or other tpes. a unction can take other unctions as parameters a unction takes a parameter and produces another unction (curried unction) :: a -> b -> c a (b->c) :: a -> b -> c ( ) :: a -> (b -> c) returns a unction o tpe b -> c b g c g g :: b -> c Functions 8

9 Curring Eamples Functions 9

10 Polmorphic Functions speciic tpes vs. arbitrar tpes a polmorphic unctions an abstract tpe each tpe variable is generall a loer-case letter. Eample) A translate unction takes a unction and a distance d returns a ne unction g that is "translated" d units to the right Functions 10

11 Polmorphic Function Eamples translate :: (Double -> Double) -> Double -> (Double -> Double) Double Double translate d = g here g = ( d) Double d translate translate :: (Double -> a) -> Double -> (Double -> a) Double g Double Double d Double Functions 11

12 Curring Curring recursivel transorms a unction that takes multiple arguments into a unction that takes just a single argument and returns another unction i an arguments are still needed. :: a -> b -> c :: a -> b -> c :: a -> b -> c a (b->c) ( ) :: a -> (b -> c) g g :: b -> c b g c Functions 12

13 Partiall Applied Functions, ( ) :: a -> b -> c -> d -> e = ( ) g1 :: b -> c -> d -> e g1 = Functions 13

14 Partiall Applied Functions ( ), ( ) ( ) g2 :: c -> d -> e g2 = ( ) g3 :: d -> e g3 = Functions 14

15 Partiall Applied Functions g1, g2, g3 ( ) = g1 g1 ( ) = g2 g2 ( ) = g3 g3 Functions 15

16 Returning Functions returns g1 unction g1 g1 returns g2 unction g1 g1 g1 g2 g2 returns g3 unction g2 g2 g2 g3 Functions 16

17 Curring Eamples :: a -> b -> c -> d -> e :: a -> (b -> (c -> (d -> e))) a (b -> (c -> (d -> e))) (((( ) ) ) ) b g1 (c -> (d -> e)) c (d -> e) g2 d g3 e Functions 17

18 Curring Eamples :: a -> b -> c -> d -> e :: a -> (b -> (c -> (d -> e))) a (b -> (c -> (d -> e))) (((( ) ) ) ) b g1 (c -> (d -> e)) c g1 (d -> e) g2 d g3 e Functions 18

19 Curring Eamples :: a -> b -> c -> d -> e :: a -> (b -> (c -> (d -> e))) a (b -> (c -> (d -> e))) (((( ) ) ) ) b g1 (c -> (d -> e)) c (d -> e) g2 d g2 g3 e Functions 19

20 Curring Eamples :: a -> b -> c -> d -> e :: a -> (b -> (c -> (d -> e))) a (b -> (c -> (d -> e))) (((( ) ) ) ) b g1 (c -> (d -> e)) c (d -> e) g2 d g3 e Functions 20

21 Curring Eamples mult :: Int -> Int -> Int -> Int (((mult ) ) ) :: a -> (b -> (c -> d)) ((( ) ) ) Int Int Int Int Functions 21

22 Partial Applications mult :: Int -> Int -> Int -> Int :: Int -> (Int -> (Int -> Int)) mult mult a 1 = g1 mult a 1 a 2 = g2 :: Int -> (Int -> (Int -> Int)) :: Int -> (Int -> Int) g1 :: Int -> (Int -> Int) g1 :: Int -> Int g2 :: Int -> Int g2 mult a 1 a 2 a 3 constants Functions 22

23 Returning Functions mult :: Int -> Int -> Int -> Int mult Int (Int -> (Int -> Int)) mult a 1 Int g1 (Int -> Int) mult a 1 a 2 Int g2 Int mult a 1 a 2 a 3 Functions 23

24 Curring Eamples mult :: Int -> Int -> Int -> Int mult mult a 1 mult a 1 a 2 mult a 1 a 2 a 3 Functions 24

25 Reerences [1] tp://tp.geoino.tuien.ac.at/navratil/haskelltutorial.pd [2]