CS321 Functional Programming 2 Dr John A Sharp 10 credits Tuesday 10am Robert Recorde Room Thursday 11am Robert Recorde

1. CS321 Functional Programming 2 Dr John A Sharp 10 credits Tuesday 10am Robert Recorde Room Thursday 11am Robert Recorde Room © JAS 2005

2. Assessment 80% written examination in May/June 20% coursework probably two assignments – roughly weeks 4 and 7 © JAS 2005

3. Syllabus (Provisional) • Type Classes • Programming with Streams • Lazy Data Structures • Memoization • Lambda Calculus • Type Checking and Inference • Implementation Approaches © JAS 2005

4. Recommended Reading Course will not follow any specific text S Thompson, Haskell: The Craft of Functional Programming, Second Edition, Addison-Wesley, 1999 P Hudak, The Haskell School of Expression – Learning Functional Programming through Multimedia, Cambridge University press, 2000 R Bird, Introduction to Functional Programming using Haskell, Second Edition, Prentice-Hall, 1998 A J T Davie, An Introduction to Functional Programming using Haskell, Cambridge University Press, 1992 www.haskell.org © JAS 2005

5. Notes will be handed out in sections mainly copies of PowerPoint slides some Haskell programs They will also be available on a course web page www.cs.swan.ac.uk/~csjohn/cs321/main.html © JAS 2005

6. Assumptions from CS221 Functional Programming 1 • Familiar with principles of Functional Programming • Able to write and run simple Haskell programs/scripts • Familiar with concepts of types and higher-order functions • Able to define own structured types • Familiar with basic λ calculus © JAS 2005

7. Type Classes Monomorphic functions add1 :: Int -> Int add1 x = x + 1 Parametric polymorphic functions map :: (a -> b) -> [a] -> [b] map f [] = [] map f (x:xs) = f x : (map f xs) © JAS 2005

8. Ad-hoc polymorphic functions add :: Int -> Int -> Int add x y = x + y add :: Float -> Float -> Float add :: Double -> Double -> Double add :: Num a => a -> a -> a add x y = x + y © JAS 2005

9. This sort of type expression is sometimes referred to as a “Qualified Type”. Num a is termed a predicate which limits the types for which a type expression is valid. It is also termed the context for the type. Multiple predicates can be defined contrived :: (Num a, Eq b)=>a->b->b->a contrived x y z = if y == z then x + x else x + x + x © JAS 2005

10. For completeness we could specify an empty context for any polymorphic function map :: () => (a -> b) -> [a] -> [b] map f [] = [] map f (x:xs) = f x : (map f xs) © JAS 2005

11. The predicate restricts the set of types for which the expression is valid. Num a should be read as “for all types a that are members of the class Num. Members of a class (which are types) are also called instances of a class. © JAS 2005

12. Type classes allow us to group together types which have common properties (or more accurately common operations that can be applied to them). An obvious example is the types for which equality can be defined. An appropriate Class is defined in the standard prelude:- class Eq a where (==), (/=) :: a -> a -> Bool x /= y = not ( x == y ) © JAS 2005

13. class Eq a where header introducing name of class and parameters (a) (==), (/=) :: a -> a -> Bool signature listing functions applicable to instances of class and their type (==) and (/=) are termed member functions x /= y = not ( x == y ) default definitions of member functions that can be overridden © JAS 2005

14. Having defined the concept of an equality class the next step is to define various instances of the class. The following examples are again taken from the standard prelude. instance Eq Int where (==) = primEqInt --primEqint is a primitive Haskell function instance Eq Bool where True == True = True False == False = True _ == _ = False -- defined by pattern matching © JAS 2005

15. instance Eq Char where c == d = ord c == ord d -- note the second == is defined on integers instance (Eq a,Eq b) => Eq(a,b) where (x,y) == (u,v) = x==u && y==v -- pairwise equality instance Eq a => Eq [a] where [] == [] = True [] == (y:ys) = False (x:xs)== [] = False (x:xs)== (y:ys) = x==y && xs==ys -- extend to equality on lists © JAS 2005

16. These declarations allow us to obtain definitions for equality on an infinite family of types involving pairs, and lists of (pairs and lists of) Int, Bool, Char. The general format of an instance declaration is instancecontext=>predicatewhere definitions of member functions We can define instances of Eq for user defined types. © JAS 2005

17. Consider a definition of Sets data Set a = Set [a] To make the type Set a a member of the class Eq we define an instance of Eq. Note that we do not use the standard (==) on the type a as we do not require the elements of the Sets to be in the same order. instance Eq a => Eq (Set a) where Set xs == Set ys = xs 'subset' ys && ys 'subset xs where xs 'subset' ys = all ('elem' ys) xs © JAS 2005

18. Some types (such as functions) can not be defined to be members of the class Eq (for obvious reasons, I hope). The error message that results from a mistaken attempt to test for equality can be obscure. It is possible to make a test for equality on functions type check ok and produce a run-time error message using the standard error function defined in the standard prelude. instance Eq (a->b) where (==) = error "== not defined on fns" © JAS 2005

19. If the functions are defined to operate on a finite set of elements then a form of equality can be defined. instance Eq a => Eq (Bool->a) where f == g = f False == g False && f True == g True It is possible to have instances for both Eq (a->b) and Eq (Bool->a) as long as both are not required at the same time in type checking some expression. An a © JAS 2005

20. Inheritance, Derived Classes, Superclasses, and Subclasses In general a class declaration has the form classcontext=>Class a1 .. anwhere type declarations for member functions default declarations of member functions where Class is the name of a new type class which takes n arguments (a1 .. an ). As with instances the context must be satisfied in order to construct any instance of the Class. The predicates in the context part of the declaration are called the superclasses of Class. Class is a subclass of the classes in the context. © JAS 2005

21. Consider a definition of a Class Ord whose instances have both strict (<), (>) and non-strict (<=), (>=) versions of an ordering defined on their elements. class Eq a => Ord a where (<),(<=),(>),(>=) :: a->a->Bool max,min :: a->a->a x < y = x <= y && x /= y x >= y = y <= x x > y = y < x max x y | x >= y = x | y >= x = y min x y | x <= y = x | y <= x = y © JAS 2005

22. Why define Eq as a superclass of Ord? • the default definition of (<) relies on the use of (/=) taken from the class Eq. Therefore every instance of Ord must be an instance of Eq. • given the definition of non-strict ordering (<=) it is always possible to define (==) and hence (/=) using x==y = x<=y && y<= x so there will be no loss of generality in requiring Eq to be a superclass of Ord. © JAS 2005

23. It is possible for some types to require a type to be an instance of class Ord in order to define it to be an instance of the class Eq. For example consider an alternative way of defining equality on Sets. instance Ord (Set a) => Eq (Set a) where x == y = x <= y && y <= x instance Eq a => Ord (Set a) where Set xs <= Set ys = all ('elem' ys) xs © JAS 2005

24. Eq Ord Enum Integral Fractional RealFloat Num Real RealFrac Floating Monad Read Functor Show MonadPlus All Prelude Types All but IO, (->) IO, [], Maybe IO, [], Maybe IO, [], Maybe Numeric Class Hierarchy (), Bool, Char, Int, Integer, Float, Double, Ordering All but IO, (->), IOError Int, Integer All but IO, (->) Int, Integer, Float, Double Int, Integer, Float, Double Float, Double Bounded Int, Char, Bool, (), Ordering, tuples Float, Double Float, Double Float, Double © JAS 2005

25. Operations defined in these classes (red means must be provided) Eq == /= Ord < <= > >= max min compare Num + - * negate abs signum fromInteger Real toRational Enum succ pred toEnum fromEnum enumFrom enumFromThen enumFromTo enumFromThenTo Integral quot rem div mod quotRem divMod toInteger Fractional / recip fromRational RealFrac properFraction truncate round ceiling floor © JAS 2005

26. Floating pi exp log sqrt ** logBase sin cos tan asin acos atan sinh cosh tanh asinh acosh atanh RealFloat floatRadix floatDigits floatRange decodeFloat encodeFloat exponent significand scaleFloat isNaN isInfinite isDenormalized isIEEE isNegativeZero atan2 Show showsPrec showList show Read readsPrec readList Bounded minBound maxBound Monad >>= >> return fail Functor fmap © JAS 2005

27. By introducing type classes Haskell has enabled programmers to use numerical types in a manner similar to the way they are used to in traditional imperative languages. But what else are type classes useful for? They can provide a mechanism for adding your own numerical types eg complex numbers They can provide a mechanism for a sort of object-oriented programming © JAS 2005

28. Imperative O-O Classes = Data + Methods Objects contain values which can be changed by methods Classes inherit data fields and methods FP Classes = Methods Methods/Functions can be applied to variables whose type is an Instance of a Class Classes inherit methods/functions © JAS 2005

29. Implementing Type Classes – An Approach using Dictionaries A function with a qualified type context=>type is implemented by a function which takes an extra argument for every predicate in the context. When the function is used each of these parameters is filled by a 'dictionary' which gives the values of each of the member functions in the appropriate class. For example, for the class Eq each dictionary has at least two elements containing the definitions of the functions for (==) and (/=). © JAS 2005

30. We will write (#n d) to select nth element of dictionary {dict} to denote a specific dictionary ( contents not displayed) dnnn for a dictionary variable representing an unknown dictionary d:: Class Inst to denote that d is dictionary for the instance Inst of a class Class © JAS 2005

31. The member functions of the class Eq thus behave as if defined (==) d1 = (#1 d1) (/=) d1 = (#2 d1) The dictionary for Eq Int contains two entries: d1 :: Eq Int primEqInt defNeq d1 © JAS 2005

32. Evaluating 2 == 3 => (==) d1 2 3 => (#1 d1) 2 3 => primEqInt 2 3 => False Evaluating 2 /= 3 => (/=) d1 2 3 => (#2 d1) 2 3 => defNeq d1 23 => not ((==) d1 2 3) => not ((#1 d1) 2 3) => not (primEqInt 2 3) => not False => True © JAS 2005

33. Now consider a more complex example We have seen definitions of (==) in the instances of the class Eq defined for pairs and lists. eqPair d (x,y) (u,v) = (==) (#3 d) x u && (==) (#4 d) y v eqList d [] [] = True eqList d [] (y:ys) = False eqList d (x:xs) [] = False eqList d (x:xs) (y:ys) = (==) (#3 d) x y && (==) d xs ys © JAS 2005

34. The dictionary structure for Eq (Int, [Int]) is d3 :: Eq (Int, [Int]) eqPair d3 defNeq d3 d2 :: Eq [Int] eqList d3 defNeq d3 d1 :: Eq Int defNeq d3 primEqInt © JAS 2005

35. (2,[1]) == (2,[1.3]) => eqPair d3 (==) d3 (#1 d3) (2,[1]) (2,[1,3]) => (==) primEqInt (#1 d1) True (#3 d3) d1 2 2 && (==) eqList d2 (#1 d2) (#4 d3) d2 [1] [1,3] => (==) primEqInt (#1 d1) True (#3 d2) d1 1 1 && eqList d2 (==) (#1 d2) False d2 [] [3] © JAS 2005

36. Dictionaries for superclasses can be defined in a similar way as instance dictionaries. For example for the Ord class which has Eq as a context class Eq => Ord a where (<),(<=),(>),(>=) :: a->a->Bool max,min :: a->a->Bool the dictionary would contain the following (<) (<=) (>) Eq a (>=) max min defLessThan d x y = (<=) d x y && (/=) (#7 d) x y © JAS 2005

37. Combining Classes A dictionary consists of three (possibly empty) components: • Superclass dictionaries • Instance specific dictionaries • Implementation of class members instances of Eq have no superclass dictionaries Eq Int has no instance specific dictionary Classes with no member functions can be used as abbreviations for lists of predicates class C a where cee :: a -> a class D a where dee :: a -> a class (C a, D a) => CandD a © JAS 2005

38. Contexts and Derived Types eg1 x = [x] == [x] || x == x Since the(==)operation is applied to both lists and an argument xif x::athen we would seem to require instancesEq aandEq [a]giving a type signature (Eq [a], Eq a) => a -> Bool with translation eg1 d1 d2 x = (==) d1 [x] [x] || (==) d2 x x © JAS 2005

39. However, givend1::Eq[a]we can always findEq aby taking the third element ofd1((#3 d1)::Eq a). Since it is more efficient to select an element from a dictionary than to complicate both type and translation with extra parameters the type ofeg1is (by default) derived as Eq [a] => a -> Bool with translation eg1 d1 x = (==) d1 [x] [x] ||(==) (#3 d1) x x © JAS 2005

40. If you definitely required the tuple context then this can be produced by explicitly defining the type and context ofeg1 eg2 = (\x y-> x ==x || y==y) The type derived for this is (Eq b, Eq a) => a -> b -> Bool with translation \d1 d2 x y -> (==) d2 x x || (==) d1 y y © JAS 2005

41. If you wished to ensure that only one dictionary parameter is used you could explicitly type is using Eq (a,b) => a -> b -> Bool with translation \d1 x y -> (==) (#3 d1) x x || (==) (#4 d1) y y © JAS 2005