| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
Cantor
Description
Cantor pairing gives us an isomorphism between a single natural number and pairs of natural numbers. This package provides a modern API to this functionality using GHC generics, allowing the encoding of arbitrary combinations of finite or countably infinite types in natural number form.
As a user, all you need to do is derive generic and get the instances for free.
Example
import GHC.Generics
import Cantor
data MyType = MyType {
value1 :: [ Maybe Bool ]
, value2 :: Integer
} deriving (Generic)
instance Cantor MyType
Recursive example
This should work nicely even with simple inductive types:
data Tree a = Leaf | Branch (Tree a) a (Tree a) deriving (Generic) instance Cantor a => Cantor (Tree a)
Finite example
If your type is finite, you can specify this by deriving the Finite typeclass, which is a subclass of Cantor:
data Color = Red | Green | Blue deriving (Generic) instance Cantor Color instance Finite Color
Mutually-recursive types
If you have mutually-recursive types, unfortunately you'll need to manually specify the cardinality for now, but you can still get the to/from encodings for free:
data Foo = FooNil | Foo Bool Bar deriving (Generic,Show) data Bar = BarNil | Bar Bool Foo deriving (Generic,Show) instance Cantor Foo where cardinality = Countable instance Cantor Bar
Synopsis
- cantorEnumeration :: Cantor a => [a]
- data Cardinality = Countable
- pattern Finite :: Integer -> Cardinality
- class Cantor a where
- cardinality :: Cardinality
- toCantor :: Integer -> a
- fromCantor :: a -> Integer
- class Cantor a => Finite a
- fCardinality :: forall a. Finite a => Integer
Documentation
cantorEnumeration :: Cantor a => [a] Source #
Enumerates all values of a type by mapping toCantor over the naturals or finite subset of naturals with the correct cardinality.
>>>take 5 cantorEnumeration :: [Data.IntSet.IntSet][fromList [],fromList [0],fromList [1],fromList [0,1],fromList [2]]
data Cardinality Source #
Cardinality can be either Finite or Countable. Countable cardinality entails that a type has the same cardinality as the natural numbers. Note that not all infinite types are countable: for example, Natural -> Natural is an infinite type, but it is not countably infinite; the basic intuition is that there is no possible way to enumerate all values of type Natural -> Natural without "skipping" almost all of them. This is in contrast to the naturals, where despite their being infinite, we can trivially (by definition, in fact!) enumerate all of them without skipping any.
Constructors
| Countable |
Instances
| Eq Cardinality Source # | |
Defined in Cantor | |
| Ord Cardinality Source # | |
Defined in Cantor Methods compare :: Cardinality -> Cardinality -> Ordering # (<) :: Cardinality -> Cardinality -> Bool # (<=) :: Cardinality -> Cardinality -> Bool # (>) :: Cardinality -> Cardinality -> Bool # (>=) :: Cardinality -> Cardinality -> Bool # max :: Cardinality -> Cardinality -> Cardinality # min :: Cardinality -> Cardinality -> Cardinality # | |
| Show Cardinality Source # | |
Defined in Cantor Methods showsPrec :: Int -> Cardinality -> ShowS # show :: Cardinality -> String # showList :: [Cardinality] -> ShowS # | |
| Generic Cardinality Source # | |
| type Rep Cardinality Source # | |
Defined in Cantor | |
pattern Finite :: Integer -> Cardinality Source #
Finite cardinality.
The Cantor class gives a way to convert a type to and from the natural numbers, as well as specifies the cardinality of the type.
Minimal complete definition
Nothing
Methods
cardinality :: Cardinality Source #
cardinality :: GCantor a (Rep a) => Cardinality Source #
toCantor :: Integer -> a Source #
toCantor :: (Generic a, GCantor a (Rep a)) => Integer -> a Source #
fromCantor :: a -> Integer Source #
fromCantor :: (Generic a, GCantor a (Rep a)) => a -> Integer Source #
Instances
class Cantor a => Finite a Source #
The Finite typeclass simply entails that the Cardinality of the set is finite.
Instances
fCardinality :: forall a. Finite a => Integer Source #
Cardinality of a finite type.