Ëåêöèÿ 6. Ðåêóðñèâíûå ôóíêöèè â Haskell 1/10 Ïåðåæîãèí À.Ñ. 8 îêòÿáðÿ 2012 ã.

Ëåêöèÿ 6. Ðåêóðñèâíûå ôóíêöèè â Haskell
Ïåðåæîãèí À.Ñ.
8 îêòÿáðÿ 2012 ã.
1/10
Ðåêóðñèâíûå ôóíêöèè. Ââåäåíèå
Ìíîãèå ôóíêöèè â Haskell îïðåäåëÿþòñÿ â òåðìèíàõ äðóãèõ ôóíêöèé.
 ïðèìåðå ôóíêöèÿ âû÷èñëåíèÿ ôàêòîðèàëà ïðåäñòàâëÿåòñÿ ÷åðåç
ôóíêöèþ âû÷èñëåíèÿ ïðîèçâåäåíèÿ ÷èñåë èç ñïèñêà îò 1 äî n.
factorial :: Int -> Int
factorial n = product [1..n]
Âûðàæåíèå âû÷èñëÿåòñÿ ïî øàãàì
=
=
=
=
factorial 4
product [1..4]
product [1,2,3,4]
1*2*3*4
24
2/10
Ðåêóðñèâíûå ôóíêöèè
 Haskell ôóíêöèè ìîãóò áûòü îïðåäåëåíû â òåðìèíàõ ñàìèõ ñåáÿ.
Òàêèå ôóíêöèè íàçûâàþòñÿ ðåêóðñèâíûå.
factorial 0 = 1
factorial (n+1) = (n+1)*factorial n
=
=
=
=
=
=
=
factorial 3
3 * factorial 2
3 * (2 * factorial 1)
3 * (2 * (1 *(factorial 0)))
3 * (2 * (1 *1))
3 * (2 * 1)
3 * 2
6
3/10
Ðåêóðñèâíûå ôóíêöèè
Çàìå÷àíèÿ:
factorial 0 = 1
ÿâëÿåòñÿ áàçîé ðåêóðñèè, òàê êàê 1 íåéòðàëüíûé ýëåìåíò äëÿ
óìíîæåíèÿ
ðåêóðñèâíîå îïðåäåëåíèå ôóíêöèè ôàêòîðèàëà ïðè
îòðèöàòåëüíûõ öåëûõ ÷èñëàõ äàåò ðàñõîäèìîñòü è ïåðåïîëíåíèå
ñòåêà
Óäîáñòâî èñïîëüçîâàíèÿ ðåêóðñèè
íåêîòîðûå ôóíêöèè áîëåå êîìïàêòíî îáúÿâëÿþòñÿ â òåðìèíàõ
ðåêóðñèâíûõ ôóíêöèé
ñâîéñòâà ðåêóðñèâíûõ ôóíêöèé äîêàçûâàþòñÿ ñ ïîìîùüþ
ìåòîäà ìàòåìàòè÷åñêîé èíäóêöèè
4/10
Ðåêóðñèÿ ñïèñêîâ
Ðåêóðñèÿ èñïîëüçóåòñÿ íå òîëüêî äëÿ ÷èñåë, íî òàê æå ïðèìåíÿåòñÿ
äëÿ ñïèñêîâ
product
:: [Int] -> Int
product []
= 1
product (n:ns) = n * product ns
Ôóíêöèÿ product ïîäñòàâëÿåò âìåñòî ïóñòîãî ñïèñêà åäèíèöó è
ïðîèçâîäèò ïåðåìíîæåíèå ãîëîâû íà ïðîèçâåäåíèå ýëåìåíòîâ õâîñòà.
=
=
=
=
=
product [2,3,4]
2 * product [3,4]
2 * ( 3 * product [4])
2 * ( 3 * ( 4 * product [])))
2 * ( 3 * ( 4 * 1)))
24
5/10
Ðåêóðñèÿ ñïèñêîâ
Ðåêóðñèÿ èñïîëüçóåòñÿ íå òîëüêî äëÿ ÷èñåë, íî òàê æå ïðèìåíÿåòñÿ
äëÿ ñïèñêîâ
product
:: [Int] -> Int
product []
= 1
product (n:ns) = n * product ns
Ôóíêöèÿ product ïîäñòàâëÿåò âìåñòî ïóñòîãî ñïèñêà åäèíèöó è
ïðîèçâîäèò ïåðåìíîæåíèå ãîëîâû íà ïðîèçâåäåíèå ýëåìåíòîâ õâîñòà.
=
=
=
=
=
product [2,3,4]
2 * product [3,4]
2 * ( 3 * product [4])
2 * ( 3 * ( 4 * product [])))
2 * ( 3 * ( 4 * 1)))
24
6/10
Ðåêóðñèÿ ñïèñêîâ
Èñïîëüçóÿ ïîäîáíóþ ñõåìó ðåêóðñèè, ìîæíî îïðåäåëèòü ôóíêöèþ
äëèíû ñïèñêà
length :: [a] -> Int
length [] = 0
length (_:xs) = 1 + length xs
Ïðèìåð. Ôóíêöèÿ îáðàùåíèÿ ïîðÿäêà â ñïèñêå
reverse
:: [a] -> [a]
reverse []
= []
reverse (x:xs) = reverse xs ++ [x]
7/10
Ðåêóðñèÿ â ñëó÷àå íåñêîëüêèõ àðãóìåíòîâ
Ôóíêöèè íåñêîëüêèõ àðãóìåíòîâ ìîãóò èñïîëüçîâàòü ðåêóðñèâíîå
îáúÿâëåíèå.
Ïðèìåð. Ôóíêöèÿ zipping
zip
::
zip []
_
=
zip _
[]
=
zip (x:xs) (y:ys) =
[a] -> [b] -> [(a,b)]
[]
[]
(x,y) : zip xs ys
Ïðèìåð. Ôóíêöèÿ drop
drop
::
drop 0
xs
=
drop (n+1) []
=
drop (n+1) (_:xs) =
Int -> [a] -> [a]
xs
[]
drop n xs
Ïðèìåð. Ôóíêöèÿ (++) ñîåäèíåíèÿ äâóõ ñïèñêîâ
drop
::
drop 0
xs
=
drop (n+1) []
=
drop (n+1) (_:xs) =
Int -> [a] -> [a]
xs
[]
drop n xs
8/10
Àëãîðèòì áûñòðîé ñîðòèðîâêè
Àëãîðèòì áûñòðîé ñîðòèðîâêè ñïèñêà öåëûõ ÷èñåë ìîæåò áûòü
âûïîëíåí ñ ïîìîùüþ äâóõ ïðàâèë:
Ïóñòîé ñïèñîê îòñîðòèðîâàí
Ñïèñîê ñ ýëåìåíòàìè: ñîðòèðóåì ýëåìåíòû õâîñòà, êîòîðûå
ìåíüøå ãîëîâû, è ñîðòèðóåì ýëåìåíòû õâîñòà, êîòîðûå áîëüøå
ãîëîâû. Äîáàâëÿåì ðåçóëüòèðóþùèå ñïèñêè ñ îäíîé è äðóãîé
ñòîðîíû îò ãîëîâû.
qsort
:: [Int] -> [Int]
qsort []
= []
qsort (x:xs) =
qsort smaller ++ [x] ++ qsort larger
where
smaller = [a | a <- xs, a <= x]
larger = [b | b <- xs, b > x]
9/10
Êîíòðîëüíûå çàäàíèÿ
Îïðåäåëèòü ôóíêöèè
Ëîãè÷åñêîå and
and :: [Bool] -> Bool
Ëîãè÷åñêîå or
or :: [Bool] -> Bool
Âûáîð n-ýëåìåíòà èç ñïèñêà
(!!) :: [a] -> Int -> Bool
Ôóíêöèÿ ïðèíàäëåæíîñòè ýëåìåíòà ñïèñêó
elem :: Eq a => a -> [a] -> Bool
Ðåêóðñèâíàÿ ôóíêöèÿ
msort :: [Int] -> [Int]
Ñïèñîê ñîðòèðîâàí, åñëè äëèíà <= 1.
Îñòàëüíûå ñïèñêè ñîðòèðóþòñÿ êàê 2 ÷àñòè è îáúåäèíÿþòñÿ â
ðåçóëüòèðóþùèé ñïèñîê.
10/10