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1. 1. Functions Sebastian RettigEvery function in haskell consumes exactly Every function in haskell consumes exactly one parameter and returns aa value. one parameter and returns value.
2. 2. Functional Programming● No Variables● Functions only, eventually stored in Modules – Behavior do not change, once defined – → Function called with same parameter calculates always the same result● Function definitions (Match Cases)● Recursion (Memory)
3. 3. Haskell Features● Pure Functional Programming Language● Lazy Evaluation● Pattern Matching and Guards● List Comprehension● Type Polymorphism
4. 4. Nice to remember (1) Typeclasses:● define properties of the types● like an interface – Eq can be compared – Ord can be ordered (>, <, >=, <=) (extending Eq) – Show can be shown as string – Read opposite of Show – Enum sequentially ordered types (can be enumerated and usable in List-Ranges [a..e])
5. 5. Nice to remember (2) Typeclass-Membership:1. derive from existing Memberships of used types data Vector2D = Vector Float Float deriving (Show, Eq)2. implement Membership by your own instance Show Vector2D where show Vector a b = “x: ” ++ [a] ++ “ ; y: ” ++ [b]
6. 6. Curried Functions (1)● lets look at the Function header again, e.g: max :: (Ord a) => a -> a -> a 1. we have a Typeclass restriction for Ord 2. we have 2 Parameters of Types, who have a membership for the Ord Typeclass 3. we have a return value of the same type● But, why is the syntax not separating the parameters better from the result set?
7. 7. Curried Functions (2)● every function in haskell consumes exactly one parameter and returns a value● so we could write the function header instead of: max :: (Ord a) => a -> a -> a● also in the following way: max :: (Ord a) => a -> (a -> a) – max is a function which consumes a parameter and returns a function – these function consumes a parameter and returns a value (the max of both values)● → Partial Application
8. 8. Curried Functions (3)● some examples: – :t max returns max :: (Ord a) => a -> a -> a – max 4 5 returns 5 – (max 4) 5 returns 5 – :t max 4 returns max 4 :: (Num a, Ord a) => a -> a – let foo = max 4 foo 5 returns 5 – :t max 4 5 returns max 4 5 :: (Num a, Ord a) => a
9. 9. Curried Functions (4)● we can also partial apply infix functions (/10) 2 returns 0.2 (10/) 2 returns 5 (++ “bar”) “foo” returns “foobar” (“bar” ++) “foo” returns “barfoo” :t (`elem` [a..z]) (`elem` [a..z]) :: Char -> Bool :t (a `elem`) (`elem` [a..z]) :: [Char] -> Bool :t elem elem :: Eq a => a -> [a] -> Bool
10. 10. Pointless Style● lets say, we have the following function: maxWithFour :: (Num a, Ord a) => a -> a maxWithFour x = max 4 x● what is the result of max 4 again? :t max 4 returns max 4 :: (Num a, Ord a) => a -> a● max 4 returns already a function which consumes a parameter, so we can simplify our function: maxWithFour :: (Num a, Ord a) => a -> a maxWithFour = max 4● but how can we achieve that with the following function? fn x = ceiling (negate (tan (cos (max 50 x))))
11. 11. Function Application (1)● used with (\$)● has the following header: :t (\$) (\$) :: (a -> b) -> a -> b● compared with SPACE (normal function application) – (\$) has lowest priority – (\$) is right associative f \$ g \$ a b = (f (g (a b))) – SPACE has highest priority – SPACE is left associative f a b c = (((f a) b) c)
12. 12. Function Application (2)● what can we do with (\$) ?● helps us to avoid parentheses: – instead of: sum (map sqrt [1..20]) – we can write: sum \$ map sqrt [1..20]● allows us also to wrap a value in a function: map (\$ 3) [(4+), (10*), (^2), sqrt] returns [7.0,30.0,9.0,1.7320508075688772]
13. 13. Function Composition (1)● Definition:● used with (.)● has the following header: :t (.) (.) :: (b -> c) -> (a -> b) -> a -> c● composing two functions produces a new function● good for on-the-fly pass of a function to the next function
14. 14. Function Composition (2)● what can we do with (.) ?● helps us also to avoid parentheses● → unwraps the parameter out of parentheses – instead of: map (xs -> negate (sum (tail xs))) [[1..5], [3..6],[1..7]] – we can write: map (negate . sum . tail) [[1..5],[3..6], [1..7]]
15. 15. Composition vs. Application (1)● IMPORTANT to know the difference between (\$) and (.)● best to compare both headers again: – (\$) :: (a -> b) -> a -> b – (.) :: (b -> c) -> (a -> b) -> a -> c● combination of both allows us parenthesis-less writing● shorter code and mostly easier to read● → possibility to write pointless-style functions with more then one function call
16. 16. Composition vs. Application (2)● example with parameter: – foo xs = sum (filter (> 10) (map (*2) xs)) – foo xs = sum \$ filter (> 10) \$ map (*2 ) xs● example without parameter (pointless-style): – foo xs = sum (filter (> 10) (map (*2) xs)) – foo = sum . filter (> 10) . map (*2)
17. 17. Useful Types● Maybe: data Maybe a = Nothing | Just a – contains maybe a type a or Nothing – good for handling e.g. Hashmap lookup ● Nothing if key not exist, else Value of type a● Either: data Either a b = Left a | Right b deriving (Eq, Ord, Read, Show) – can contain type a or type b – Left for e.g. Error Messages, Right for value – like an extended Maybe with information for Nothing
18. 18. Functor Typeclass (1) class Functor f where fmap :: (a -> b) -> f a -> f b● for things that can be mapped over● !!! Functor needs Types with 1 Typeparameter !!!● fmap gets a function and a type and maps the function over the type variable● Instance for List: – Remember: List has 1 Typeparameter [a], but is just Syntactic Sugar for ([] a) instance Functor [] where fmap = map – Example: fmap (*2) [2,3,4] returns [4,6,8]
19. 19. Functor Typeclass (2) class Functor f where fmap :: (a -> b) -> f a -> f b● Instance for Maybe instance Functor Maybe where fmap g (Just x) = Just (g x) fmap g Nothing = Nothing● Example: – fmap (+3) Nothing returns Nothing – fmap (+3) (Just 4) returns (Just 7)
20. 20. Functor Typeclass (3) class Functor f where fmap :: (a -> b) -> f a -> f b● Instance for Either instance Functor (Either a) where fmap f (Right x) = Right (f x) fmap f (Left x) = Left x● Either is a type with 2 Typeparameters!● → we have to permanently include the first parameter in the instance (curried function, partial application)● → fmap do not change the 1st Typeparameter, only the 2nd● → Left Type-Constructor of Either is often used for Error Messages
21. 21. Functor Typeclass (4) class Functor f where fmap :: (a -> b) -> f a -> f b● Example: – fmap (+3) (Left 4) returns (Left 4) – fmap (+3) (Right 4) returns (Right 7)● what happens, if we try to do that? fmap (+) (Just 4)● lets look at the type: :t fmap (+) (Just 4) fmap (+) (Just 4) :: Num a => Maybe (a -> a)● partial application, BUT we can not use the Functor instance on the result!● → we need an extension → Applicative Functors