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  1. Introduction to Haskell and ForSyDe Axel JantschMarch 2001

  2. The Origin of Functional Languages • Alonzo Church, The Calculi of Lambda Conversion, 1941. (1932-33) • Church-Rosser theorem, 1936. • Unique normal form for l expressions (“value” of an expression); • Normal-order reduction yields normal form; • Fixed-point theorem: Every l expression e has a fixed-point e’ such that e e’ = e’. • Every recursive function can be written non-recursively. • Church’s thesis: “Effectively computable functions from positive integers to positive integers are just those definable in the l calculus.” • Turing 1937: “Turing-computability is equivalent to l-definability.” Axel Jantsch, ESDlab, KTH

  3. Influential Functional Languages • McCarthy 1960: LISP: • Applicative-order reduction with conditionals; • Lists and higher-order operations on lists; • Garbage collection; • Use of S-expressions (abstract syntax) to represent both data and program; • Landin 1964-1966: Iswim • Backus 1978: FP • Gordon et al. 1978: ML; Milner 1984: SML: • I/O facility with side effects; • Sophisticated module system; • Exception mechanism; • Innovative typing system (Hindley1969, Milner 1978): • Strongly and statically typed; • Use of inference to determine the type of every expression; • Allows polymorphic functions and data structures; • User defined concrete and abstract types; • Hudak and Wadler 1988: Haskell (after Haskell B. Curry); Axel Jantsch, ESDlab, KTH

  4. Imperative Languages • Turing 1930s: Turing machine; • Fortran 1950s • E.W.Dijkstra, "GOTO statement considered harmful", ACM Comm. vol.11, nr.3, 1968. • Structural programming, Pascal, 1970s • Object Oriented Concepts, 1980s: • Abstract data types • Inheritance • Polymorphic functions Axel Jantsch, ESDlab, KTH

  5. Bibliography • Paul Hudak, "Conception, Evolution, and Application of Functional Programming Languages", ACM Computing Surveys, vol. 21, no. 3, pp. 359-411, September 1989. • The Haskell 98 Report, ed: Simon Peyton Jones, John Hughes, February 1999. • Simon Thompson, Haskell - The Craft of Functional Programming, Addison-Wesley, 2 edition, 1999. • Paul Hudak and Joseph H. Fasel, A Gentle Introduction to Haskell, 1999. • Haskel Home Page: http://haskell.org Axel Jantsch, ESDlab, KTH

  6. Haskell and Skeleton Libraries • Haskell is a general purpose, functional programming language; • Library for perfectly synchronous process networks; • Library for data flow process networks; Axel Jantsch, ESDlab, KTH

  7. Haskell Features - 1 - • Declarative: The program is a list of equations which have to be satisfied; no predefined execution order; • Referential transparancy: y=f(x) • Single assignment: “Variables” are assigned only once; • Higher order functions: Functions are first class citizens and can be used as arguments; • Formal semantics: denotational • Strong type system Axel Jantsch, ESDlab, KTH

  8. Features of Haskell - 2 - • Higher Order Functions map :: (a -> a) -> [a] -> [a] map f [] = [] map f (x:xs)= f x : map f xs mult :: Int -> Int -> Int mult x y = x*y double :: Int -> Int double = mult 2 mapDouble :: [a] -> [a] mapDouble = map double • Lazy Evaluation (non-strict) stupid x y = 2*x stupid 3 (12/3) = 6 stupid 3 (5/0) = 6 -- lazy stupid 3 (5/0) = error -- strict Axel Jantsch, ESDlab, KTH

  9. Features of Haskell - 3 - • Data abstraction module Store (Store, initial, -- Store value, -- Store->Var->Int update --Store->Var->Int->Store ) where • Type system: • strong, static, polymorphic, inference based; • Pattern matching and equations mystery x y | x==0 = y | otherwise = x mystery 0 y = y mystery x y = x Axel Jantsch, ESDlab, KTH

  10. Features of Haskell - 4 - • Polymorphism • Parametric polymorphism map :: (a -> a) -> [a] -> [a] map f [] = [] map f (x:xs)= f x : map f xs • Ad hoc polymorphism (overloading) + :: Int -> Int -> Int + :: Float -> Float -> Float + :: String -> String -> String • Type constraints class Num a where (+) :: a -> a -> a -- read as "a type a belongs to the class -- Num if there is an overloaded -- function + defined on it instance Num Int where x + y = addInt x y addPair :: Num a=> (a,a) -> a addPair (x,y) = x + y Axel Jantsch, ESDlab, KTH

  11. Haskell Example Quicksort in Haskell: qsort :: [Int] -> [Int] qsort [ ] = [ ] qsort (x:xs) = (qsort lt_x) ++ [x] ++ (qsort greq_x) where lt_x = [y | y <- xs, y < x] greq_x = [y | y <- xs, y >= x] Axel Jantsch, ESDlab, KTH

  12. Data and Types Examples: 5 :: Integer 'a' :: Char inc :: Integer -> Integer [1,2,3] :: [Integer] ('b',4) :: (Char,Integer) Build-in types: Int - integer Integer - integer Bool - boolean (True, False) Float - floating point String - string of characters Char - character Axel Jantsch, ESDlab, KTH

  13. Function Types • Many functions are polymorphic, i.e. they operate on different types: length :: [a] -> Integer length [] = 0 length (x:xs) = 1 + length xs head :: [a] -> a head (x:xs) = x tail :: [a] -> [a] tail (x:xs) = xs Axel Jantsch, ESDlab, KTH

  14. Partial Evaluation of Functions mymap = map inc mymap :: [Integer] -> [Integer] mymap [1,2,3] => [2,3,4] monitor :: Show a => String -> a -> a monitorF1 = monitor “file1” add :: Integer -> Integer -> Integer add x y = x+y inc :: Integer -> Integer inc = add 1 map :: (a -> b) -> [a] -> [b] map f [] = [] map f (x:xs) = f x : map f xs Axel Jantsch, ESDlab, KTH

  15. User Defined Types • Type synonyms: type String = [Char] type IntPair = (Int,Int) • Type declarations: data Bool = False | True data Color = Red | Green | Blue deriving (Show,Eq) data Tval a = Abst | Prst a deriving (Eq, Show) data Signal a = NullS | a :- Signal a deriving (Eq, Show) Axel Jantsch, ESDlab, KTH

  16. Lists, Strings, and Tuples l1 = [1,2,3] ++ [4,5,6] => [1,2,3,4,5,6] l2 = 1 : [2,3,4,5,6] => [1,2,3,4,5,6] st1 = [‘a’,’b’,’c’] ++ “efg” => “abcefg” s1 :: Signal Int s1 = signal [1,2,3,4] s2 = s1 +: s1 => 1 :- 2 :- 3 :- 4 :- 1 :- 2 :- 3 :- 4 :- NullS s3 = 0 :- s1 => 0 :- 1 :- 2 :- 3 :- 4 :- NullS s4 :: Signal (Int,Int) s4 = signal [(1,2), (3,4)] => (1,2) :- (3,4) :- NullS type TInt = TVal Int ts1 :: Signal Tint ts1 = signal [Abst, Prst 1, Prst 2] => Abst :- (Prst 1) :- (Prst 2) :- NullS Axel Jantsch, ESDlab, KTH

  17. Infix Operators mapSY :: (a -> b) -> Signal a -> Signal b p1 = mapSY inc s1 = signal [1,2,3] p1 s1 => 2 :- 3 :- 4 :- NullS p2 = mapSY (*2) p2 s1 => 2 :- 4 :- 6 :- NullS p3 = p2 . p1 p3 s1 => 4 :- 6 :- 8 :- NullS • List concatenation: (++) :: [a] -> [a] -> [a] [] ++ ys = ys (x:xs) ++ ys = x : (xs ++ ys) • Function composition: (.) :: (b -> c) -> (a -> b) -> (a -> c) (f . g) x = f (g x) Axel Jantsch, ESDlab, KTH

  18. Lazy Evaluation • Only values which are needed are computed. il1 = [1,2..] il2 = [1,3..] take 5 il1 => [1,2,3,4,5] take 3 il2 => [1,3,5] take 5 (drop 5 il1) => [6,7,8,9,10] s1 = signal [1..] takeS 3 s1 => 1 :- 2 :- 3 :- NullS p1 = mapSY inc p2 = mapSY (*2) p3 = p2 . p1 takeS 5 (p3 s1) => 4 :- 6 :- 8 :- 10 :- 12 :- NullS takeS 3 (drop 98 (p3 s1)) => 200 :- 202 :- 204 :- NullS Axel Jantsch, ESDlab, KTH

  19. Pattern Matching • As-patterns f (x:xs) = x : x : xs f s@(x:xs) = x : s • Wild cards: head (x:_) = x tail (_:xs) = xs • Case expressions: take m ys = case (m,ys) of (0,_) -> [] (_,[]) -> [] (n,x:xs) -> x : take (n-1) xs take 0 _ = [] take _ [] = [] take n (x:xs) = x : take (n-1) xs • Guards: sign x | x > 0 = 1 | x== 0 = 0 | x < 0 = -1 sign x | x > 0 = 1 | x== 0 = 0 | otherwise = -1 Axel Jantsch, ESDlab, KTH

  20. Synchronous Library - State-less Skeletons mapSY :: (a -> b) Signal a -> Signal b p1 = mapSY (*2) p1 (1:-2:-3:-NullS) => 2 :- 4 :- 6 :- NullS zipWithSY :: (a -> b -> c) -> Signal a -> Signal b -> Signal c p2 = zipWithSY (*) p2 (1:-2:-3:-NullS) (1:-2:-3:-NullS) => 1 :- 4 :- 9 :- NullS zipWith3SY :: (a -> b -> c -> d) -> Signal a -> Signal b -> Signal c -> Signal d zipWith4SY :: (a -> b -> c -> d -> e) -> Signal a -> Signal b -> Signal c -> Signal d -> Signal e zipSY :: Signal a -> Signal b -> Signal (a,b) zipSY (1:-2:-3:-NullS) (4:-5:-6:-NullS) => (1,4):-(2,5):-(3,6):-NullS) unzipSY :: Signal (a,b) -> (Signal a, Signal b) unzipSY (zipSY (1:-2:-3:-NullS) (4:-5:-6:-NullS)) => ((1:-2:-3:-NullS), (4:-5:-6:-NullS)) Axel Jantsch, ESDlab, KTH

  21. Synchronous Library - State Skeletons scanlSY :: (a -> b -> c) -> a -> Signal b -> Signal a scanlSY f mem NullS = NullS scanlSY f mem (x:-xs) = f mem x :- (scanlSY f newmem xs) where newmem = f mem x scanl2SY :: (a -> b -> c -> d) -> a -> Signal b -> Signal c -> Signal a p1 = scanlSY cntrFn 0 s1 = signal (concat (repeat [True,False])) s2 = p1 s1 takeS 7 s2 => 1 :- 1 :- 2 :- 2 :- 3 :- 3 :- 0 :- NullS cntrFn :: Int -> Bool -> Int cntrFn y x = case (x) of True -> ((y+1) `mod` 4) False -> y Axel Jantsch, ESDlab, KTH

  22. Synchronous Library - State Machines mooreSY :: (a -> b -> c) -> (a -> c) -> a -> Signal b -> Signal a mooreSY nextstate output init = mapSY output . (scanlSY nextstate init) mealySY :: (a -> b -> c) -> (a -> b -> c) -> a -> Signal b -> Signal a mooreSY nextstate output init signal = zipWithSY output (init :- (scanlSY nextstate init signal)) signal p2 = mealySY cntrFn outFn 0 s3 = p2 s1 take 7 s3 => “one” :- “one” :- “two” :- “two” :- “three” :- “three” :- “zero” :- NullS outFn :: Int -> String outFn 0 = “zero” outFn 1 = “one” outFn 2 = “two” outFn 3 = “three” Axel Jantsch, ESDlab, KTH

  23. Synchronous Library - Timed Signals hdTrans :: PacketEv -> PacketEv hdTrans Abs = Abs hdTrans p@((Prst UHd),_) = p hdTrans ((Prst EHd1),b) = ((Prst EHd2),b) hdTrans ((Prst EHd2),b) = ((Prst EHd1),b) p = mapSY hdTrans s1 = (Abst :- (Prst (UHd, [0,1])):-(Prst (EHd1,[])) :- (Prst (EHd2,[]))) p s1 => Abst :- (Prst (UHd, [0,1])):-(Prst (EHd2,[])) :- (Prst (EHd1,[])) type Value = Integer type ValEvent = TVal Value s1 :: Signal ValEvent s1 = signal [Abst, (Prst 1), Abst, (Prst 2)] data Header = UHd | EHd1 | EHd2 type Body = [Int] type Packet = (Header,Body) type PacketEv = TVal Packet Axel Jantsch, ESDlab, KTH

  24. Data Flow Library • Data flow skeletons do not operate on timed signals. • Data flow skeletons allow combinatorial functions which consume several input and output tokens. mapDF :: (a -> b) -> Signal a -> Signal b map12DF :: (a -> (b,b)) -> Signal a -> Signal b map21DF :: ((a,a) -> b) -> Signal a -> Signal b map22DF :: ((a,a) -> (b,b)) -> Signal a -> Signal b mapGen :: Int -> ([a] -> [b]) -> Signal a -> Signal b mapUpDF :: (a -> [b]) -> Signal a -> Signal b mapDownDF :: Int -> ([a] -> b) -> Signal a -> Signal b Axel Jantsch, ESDlab, KTH

  25. Stochastic Processes and Skeletons • sigmaUn generates a signal of uniformly distributed integers within a given range. sigmaUn :: Int -> (Int, Int) -> Signal Int takeS 5 (sigmaUn 3 (0,9)) => 5 :- 5 :- 4 :- 7 :- 3 :- NullS • selMapDF is a stochastic variant of mapDF; it selects one out of two functions to be applied on the input based on a probability distribution. selMapDF :: Int -> (a -> b) -> (a -> b) -> Signal a -> Signal b selScanlDF :: Int -> (a -> b -> a) -> (a -> b -> a) -> a -> Signal b -> Signal a Axel Jantsch, ESDlab, KTH

  26. How to Model with Skeletons ? 1. Draw a process graph. 2. Decide the types of processes: no internal state -> map skeleton internal state same as output -> scanl skeleton output depends only on state -> moore skeleton output depends on input and state -> mealy 3. Define data types of signals and states. 4. Decide on skeleton variants (# inputs, # outputs,…) 5. Design the combinatorial functions Axel Jantsch, ESDlab, KTH

  27. sc sd sa p2 p1 se sb p3 How to Model with Skeletons ? • 6. Compose the processes into a system. p1 :: Signal a -> Signal b -> Signal c p2 :: Signal c -> (Signal d, Signal e) p3 :: Signal e -> Signal b system sa = sd where sd = fst (p2 sc) sc = p1 sa sb sb = p3 se se = snd (p2 sc) Axel Jantsch, ESDlab, KTH