So far, we have seen that typeclasses allow us to implement functions (for example,
(+) in the
Num typeclass) separately for various types.This is known as ad hoc polymorphism, in contrast to the parametric polymorphism provided by type variables.
Another common Haskell idiom is to create a typeclass for functions with a specific structure and then let a value’s type determine which function with that structure to apply.
For example, consider the
Semigroup typeclass (already implemented in Haskell).
The primary function provided by the
Semigroup typeclass is
<>. Besides the type signature
a -> a -> a, the class definition does not tell us anything about this function. However, the documentation tells us that instances of
Semigroup should satisfy the following associativity law.
Just like an implementation of
(+) for a type being made an instance of
Num should conform to the laws of addition, an implementation of
<> for a type being made an instance of
Semigroup should be associative.
Many types have a natural associative operation. For example, concatenation is the natural associative operation on lists, and lists have a
Concatenation is associative since
x ++ (y ++ z) = (x ++ y) ++ z, and therefore this instance conforms to the
Some types, however, do not have a natural associative operation. For example, or and and are both associative functions on booleans. Since there is not a natural associative operation on the
Bool type, it does not make sense to give
Semigroup instance. Instead, we can wrap a boolean value in a new type for each associative function we want to use.
newtype is basically equivalent to
data, but can only be used when there is one constructor with one field. In return, it incurs no runtime cost over the wrapped type.See here for the subtle differences between
We now have two new types,
All. We can recover the wrapped boolean by pattern matching, and we can write separate
Semigroup instances for each.
Any values are combined using or and
All values are combined using and.
- Integers, like booleans, do not have a natural associative operation. However, both addition and multiplication are associative operations on integers. Write
Semigroupinstances for the
Intin the starter code, mirroring the implementations of
The power of using typeclasses to represent operations with a given structure becomes apparent when we want to write generic functions using the structures they represent. Consider the
stimes function, which is also implemented in the
class Semigroup a where (<>) :: a -> a -> a -- Integral is a typeclass containing Int and Integer stimes :: Integral b => b -> a -> a stimes n x -- not the actual default implementation | n <= 0 = error "positive multiplier expected" -- runtime error | n = 1 = x | otherwise = x <> stimes (n - 1) x -- 1 more function defined in terms of <>
It computes the following.
The default given above has \(O(n)\) performance, which is less efficient than the best general implementation. Using the associative property of
<>, provide a more efficient implementation,
stimes', with time complexity \(O(\log n)\).
While \(O(\log n)\) is the best possible performance in general, specific instances can perform even better. Extend your instances for
Productwith \(O(1)\) implementations of
Here is starter code which you should complete for your solutions.
Send an email to email@example.com with either:
(Preferred) A link to a Gitlab / Github repository with your code.
A .zip file with your code.