6 TypeClasses

6 TypeClasses - cs242 Parametric polymorphism Single...

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10/6/10 1 Kathleen Fisher cs242 Thanks to Simon Peyton Jones for some of these slides. Reading: “Concepts in Programming Languages”, New Chapter 7, Type Classes Parametric polymorphism Single algorithm may be given many types Type variable may be replaced by any type if f::t t then f::Int Int , f::Bool Bool , . .. Overloading A single symbol may refer to more than one algorithm. Each algorithm may have different type. Choice of algorithm determined by type context. + has types Int Int Int and Float Float Float , but not t t t for arbitrary t . Many useful functions are not parametric. Can member work for any type? No! Only for types w for that support equality . Can sort work for any type? No! Only for types w that support ordering . member :: [w] -> w -> Bool sort :: [w] -> [w] Many useful functions are not parametric. Can serialize work for any type? No! Only for types w that support serialization . Can sumOfSquares work for any type? No! Only for types that support numeric operations . serialize:: w -> String sumOfSquares:: [w] -> w Allow functions containing overloaded symbols to de±ne multiple functions: But consider: This approach has not been widely used because of exponential growth in number of versions. square x = x * x -- legal -- Defines two versions: -- Int -> Int and Float -> Float squares (x,y,z) = (square x, square y, square z) -- There are 8 possible versions! First Approach Basic operations such as + and * can be overloaded, but not functions de±ned in terms of them. Standard ML uses this approach. Not satisfactory : Why should the language be able to de±ne overloaded operations, but not the programmer? 3 * 3 -- legal 3.14 * 3.14 -- legal square x = x * x -- Int -> Int square 3 -- legal square 3.14 -- illegal Second Approach
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10/6/10 2 Equality defned only For types that admit equality : types not containing Function or abstract types. Overload equality like arithmetic ops + and * in SML. But then we can’t defne Functions using ‘==‘ : Approach adopted in frst version oF SML. 3 * 3 == 9 -- legal ‘a’ == ‘b’ -- legal \x->x == \y->y+1 -- illegal member [] y = False member (x:xs) y = (x==y) || member xs y member [1,2,3] 3 -- ok if default is Int member “Haskell” ‘k’ -- illegal ±irst Approach Make equality Fully polymorphic. Type oF member Function: Miranda used this approach. Equality applied to a Function yields a runtime error . Equality applied to an abstract type compares the underlying representation, which violates abstraction principles . (==) :: a -> a -> Bool member :: [a] -> a -> Bool Second Approach Make equality polymorphic in a limited way: where a (==) is a type variable ranging only over types that admit equality. Now we can type the member Function:
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6 TypeClasses - cs242 Parametric polymorphism Single...

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