Peano numbers are a simple way of representing the natural numbers using only a zero value and a successor function. In Haskell it is easy to create a type of Peano number values, but since unary representation is inefficient, they are more often used to do type arithmetic due to their simplicity.
1.1 Peano number values
data Peano = Zero | Succ Peano
Here's a simple addition function:
add Zero b = b add (Succ a) b = Succ (add a b)
See an example implementation.
1.2 Peano number types
data Zero data Succ a
Arithmetic can be done using fundeps:
class Add a b ab | a b -> ab instance Add Zero b b instance (Add a b ab) => Add (Succ a) b (Succ ab)
Note that classes express relationships between types, rather than functions from type to type. Accordingly, with the instance declarations above one can add another fundep to the class declaration to get subtraction for free:
class Add a b ab | a b -> ab, a ab -> b instance Add Zero b b instance (Add a b ab) => Add (Succ a) b (Succ ab)
See type arithmetic for other functions and encodings.
2.1 Lazy counting
The lazy nature of Peano numbers rehabilitates the use of list functions which count list elements. As described in Things to avoid one should not write
length xs == 0
genericLength xs == Zero -- or genericLength xs == (0::Peano)
length xs == length ys
is harder to make lazy without Peano numbers.With them (and an appropriate
genericLength xs == (genericLength ys :: Peano)
traverses only as many list nodes as are in the shorter list.
2.2 Equation solving
Lazy Peano numbers can also be used within "Tying the Knot" techniques. There is a package for determining the right order for solving equations, where an equation is only solved if only one of its variables is still indeterminate.
3 See also
See example implementations: