cantor-pairing-0.1.0.0: Convert data to and from a natural number representation

Safe HaskellNone
LanguageHaskell2010

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

A warning: this package will work with recursive types, but you *must* manually specify the cardinality. This unfortunately is necessary due to GHC generics marking all fields as recursive, regardless of whether or not they actually are. Still, it's straightforward to manually specify the cardinality:

Recursive example

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

instance Cantor a => Cantor (Tree a) where
  cardinality = Countable

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

Documentation

cantorEnumeration :: Cantor a => [a] Source #

Enumerates all values of a type by mapping toCantor over the naturals.

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

Finite Integer 
Countable 
Instances
Eq Cardinality Source # 
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Defined in Cantor

Ord Cardinality Source # 
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Defined in Cantor

Show Cardinality Source # 
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Generic Cardinality Source # 
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Associated Types

type Rep Cardinality :: Type -> Type #

type Rep Cardinality Source # 
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Defined in Cantor

type Rep Cardinality = D1 (MetaData "Cardinality" "Cantor" "cantor-pairing-0.1.0.0-9psbWPZlnGKBdY6UCFstnv" False) (C1 (MetaCons "Finite" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Integer)) :+: C1 (MetaCons "Countable" PrefixI False) (U1 :: Type -> Type))

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 :: Cardinality Source #

cardinality :: GCantor (Rep a) => Cardinality Source #

toCantor :: Integer -> a Source #

toCantor :: (Generic a, GCantor (Rep a)) => Integer -> a Source #

fromCantor :: a -> Integer Source #

fromCantor :: (Generic a, GCantor (Rep a)) => a -> Integer Source #

Instances
Cantor Bool Source # 
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Defined in Cantor

Cantor Char Source # 
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Defined in Cantor

Cantor Int Source # 
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Defined in Cantor

Cantor Int8 Source # 
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Cantor Int16 Source # 
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Cantor Int32 Source # 
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Cantor Int64 Source # 
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Cantor Integer Source # 
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Cantor Natural Source # 
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Cantor Word8 Source # 
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Defined in Cantor

Cantor Word16 Source # 
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Cantor Word32 Source # 
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Defined in Cantor

Cantor Word64 Source # 
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Cantor () Source # 
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Cantor Void Source # 
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Cantor a => Cantor [a] Source # 
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Cantor a => Cantor (Maybe a) Source # 
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Cantor a => Cantor (Min a) Source # 
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Defined in Cantor

Cantor a => Cantor (Max a) Source # 
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Defined in Cantor

Cantor a => Cantor (First a) Source # 
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Defined in Cantor

Cantor a => Cantor (Last a) Source # 
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Cantor a => Cantor (Option a) Source # 
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Cantor a => Cantor (Identity a) Source # 
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Cantor a => Cantor (Sum a) Source # 
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Cantor a => Cantor (Product a) Source # 
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(Finite a, Cantor b) => Cantor (a -> b) Source # 
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Defined in Cantor

(Cantor a, Cantor b) => Cantor (Either a b) Source # 
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(Cantor a, Cantor b) => Cantor (a, b) Source # 
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(Cantor a, Cantor b) => Cantor (Arg a b) Source # 
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Cantor (Proxy a) Source # 
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(Cantor a, Cantor b, Cantor c) => Cantor (a, b, c) Source # 
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Defined in Cantor

Methods

cardinality :: Cardinality Source #

toCantor :: Integer -> (a, b, c) Source #

fromCantor :: (a, b, c) -> Integer Source #

Cantor a => Cantor (Const a b) Source # 
Instance details

Defined in Cantor

(Cantor a, Cantor b, Cantor c, Cantor d) => Cantor (a, b, c, d) Source # 
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Defined in Cantor

Methods

cardinality :: Cardinality Source #

toCantor :: 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 details

Defined in Cantor

Methods

cardinality :: Cardinality Source #

toCantor :: 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 details

Defined in Cantor

Methods

cardinality :: Cardinality Source #

toCantor :: 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 details

Defined in Cantor

Methods

cardinality :: Cardinality Source #

toCantor :: Integer -> (a, b, c, d, e, f, g) Source #

fromCantor :: (a, b, c, d, e, f, g) -> Integer Source #

class Cantor a => Finite a where Source #

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

Minimal complete definition

Nothing

Instances
Finite Bool Source # 
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Defined in Cantor

Finite Char Source # 
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Defined in Cantor

Finite Int Source # 
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Defined in Cantor

Finite Int8 Source # 
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Defined in Cantor

Finite Int16 Source # 
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Defined in Cantor

Finite Int32 Source # 
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Defined in Cantor

Finite Int64 Source # 
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Defined in Cantor

Finite Word8 Source # 
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Defined in Cantor

Finite Word16 Source # 
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Defined in Cantor

Finite Word32 Source # 
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Finite Word64 Source # 
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Finite () Source # 
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Finite Void Source # 
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Finite a => Finite (Maybe a) Source # 
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Defined in Cantor

Finite a => Finite (Min a) Source # 
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Defined in Cantor

Finite a => Finite (Max a) Source # 
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Defined in Cantor

Finite a => Finite (First a) Source # 
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Defined in Cantor

Finite a => Finite (Last a) Source # 
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Defined in Cantor

Finite a => Finite (Option a) Source # 
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Defined in Cantor

Finite a => Finite (Identity a) Source # 
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Defined in Cantor

Finite a => Finite (Sum a) Source # 
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Defined in Cantor

Finite a => Finite (Product a) Source # 
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Defined in Cantor

(Finite a, Finite b) => Finite (a -> b) Source # 
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Defined in Cantor

(Finite a, Finite b) => Finite (Either a b) Source # 
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Defined in Cantor

(Finite a, Finite b) => Finite (a, b) Source # 
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Defined in Cantor

(Finite a, Finite b) => Finite (Arg a b) Source # 
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Defined in Cantor

Finite (Proxy a) Source # 
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Defined in Cantor

(Finite a, Finite b, Finite c) => Finite (a, b, c) Source # 
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Defined in Cantor

Finite a => Finite (Const a b) Source # 
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Defined in Cantor

(Finite a, Finite b, Finite c, Finite d) => Finite (a, b, c, d) Source # 
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Defined in Cantor

(Finite a, Finite b, Finite c, Finite d, Finite e) => Finite (a, b, c, d, e) Source # 
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Defined in Cantor

(Finite a, Finite b, Finite c, Finite d, Finite e, Finite f) => Finite (a, b, c, d, e, f) Source # 
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Defined in Cantor

(Finite a, Finite b, Finite c, Finite d, Finite e, Finite f, Finite g) => Finite (a, b, c, d, e, f, g) Source # 
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Defined in Cantor