Safe Haskell | None |
---|---|

Language | Haskell2010 |

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

This should work nicely even with simple inductive types:

# Recursive example

data Tree a = Leaf | Branch (Tree a) a (Tree a) deriving (Generic) instance Cantor a => Cantor (Tree a)

If your type is finite, you can specify this by deriving the `Finite`

typeclass, which is a subclass of `Cantor`

:

# Finite example

data Color = Red | Green | Blue deriving (Generic) instance Cantor Color instance Finite Color

## Synopsis

- cantorEnumeration :: Cantor a => [a]
- data Cardinality
- class Cantor a where
- cardinality :: Cardinality
- toCantor :: Integer -> a
- fromCantor :: a -> Integer

- class Cantor a => Finite a where

# 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.

If the cardinality of the type is large and finite, (e.g., `IntSet`

), you will need to try fixing the amount of items you want instead like `toCantor `

IntSet $ [ 0 .. 10 ]`. This is unfortunately necessary because even though the list is computed lazily in `

cantorEnumeration`, its *size* is not, and the size of `

IntSet` is a *very* large number which is not feasible to compute even on a modern system (it has more than 200k terabytes of digits!). Note that if you defer to using even larger types like `

Integer` which have true non-finite cardinality instead of finite approximations like `

Int@, you will naturally tend to avoid this problem.

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.

## Instances

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.

Nothing

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 where Source #

The `Finite`

typeclass simply entails that the `Cardinality`

of the set is finite.

Nothing