cantor-pairing-0.2.0.1: Convert data to and from a natural number representation

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,Cantor)


# Recursive example

This should work nicely even with simple inductive types:

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


# 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,Cantor,Finite)


# 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

# 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 where 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.

Bundled Patterns

 pattern Countable :: Cardinality pattern Finite :: Integer -> Cardinality
Instances
 Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor MethodsshowList :: [Cardinality] -> ShowS # Source # Instance detailsDefined in Cantor Associated Typestype Rep Cardinality :: Type -> Type # Methods type Rep Cardinality Source # Instance detailsDefined in Cantor type Rep Cardinality

class Cantor a where Source #

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 :: 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
 Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Cantor () Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> () Source #fromCantor :: () -> Integer Source # Source # Instance detailsDefined in Cantor Methods Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor [a] Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> [a] Source #fromCantor :: [a] -> Integer Source # Cantor a => Cantor (Maybe a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Min a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Max a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (First a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Last a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Option a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Identity a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Sum a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Product a) Source # Instance detailsDefined in Cantor Methods Cantor a => Cantor (Seq a) Source # Instance detailsDefined in Cantor Methods (Ord a, Finite a) => Cantor (Set a) Source # Instance detailsDefined in Cantor Methods (Finite a, Cantor b) => Cantor (a -> b) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> a -> b Source #fromCantor :: (a -> b) -> Integer Source # (Cantor a, Cantor b) => Cantor (Either a b) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> Either a b Source #fromCantor :: Either a b -> Integer Source # (Cantor a, Cantor b) => Cantor (a, b) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b) Source #fromCantor :: (a, b) -> Integer Source # (Cantor a, Cantor b) => Cantor (Arg a b) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> Arg a b Source #fromCantor :: Arg a b -> Integer Source # Cantor (Proxy a) Source # Instance detailsDefined in Cantor Methods (Cantor a, Cantor b, Cantor c) => Cantor (a, b, c) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b, c) Source #fromCantor :: (a, b, c) -> Integer Source # Cantor a => Cantor (Const a b) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> Const a b Source #fromCantor :: Const a b -> Integer Source # (Cantor a, Cantor b, Cantor c, Cantor d) => Cantor (a, b, c, d) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b, c, d) Source #fromCantor :: (a, b, c, d) -> Integer Source # (Cantor a, Cantor b, Cantor c, Cantor d, Cantor e) => Cantor (a, b, c, d, e) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b, c, d, e) Source #fromCantor :: (a, b, c, d, e) -> Integer Source # (Cantor a, Cantor b, Cantor c, Cantor d, Cantor e, Cantor f) => Cantor (a, b, c, d, e, f) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b, c, d, e, f) Source #fromCantor :: (a, b, c, d, e, f) -> Integer Source # (Cantor a, Cantor b, Cantor c, Cantor d, Cantor e, Cantor f, Cantor g) => Cantor (a, b, c, d, e, f, g) Source # Instance detailsDefined in Cantor MethodstoCantor :: Integer -> (a, b, c, d, e, f, g) Source #fromCantor :: (a, b, c, d, e, f, g) -> Integer Source #

class Cantor a => Finite a Source #

The Finite typeclass simply entails that the Cardinality of the set is finite.

Instances
 Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite () Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Maybe a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Min a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Max a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (First a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Last a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Option a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Identity a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Sum a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Product a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Ord a, Finite a) => Finite (Set a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b) => Finite (a -> b) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b) => Finite (Either a b) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b) => Finite (a, b) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b) => Finite (Arg a b) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite (Proxy a) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b, Finite c) => Finite (a, b, c) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge Finite a => Finite (Const a b) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b, Finite c, Finite d) => Finite (a, b, c, d) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b, Finite c, Finite d, Finite e) => Finite (a, b, c, d, e) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b, Finite c, Finite d, Finite e, Finite f) => Finite (a, b, c, d, e, f) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge (Finite a, Finite b, Finite c, Finite d, Finite e, Finite f, Finite g) => Finite (a, b, c, d, e, f, g) Source # Instance detailsDefined in Cantor MethodsfCardinality' :: Huge

fCardinality :: forall a. Finite a => Integer Source #

Cardinality of a finite type.