Copyright | (c) Global Access Internet Services GmbH |
---|---|

License | BSD3 |

Maintainer | Pavlo Kerestey <pavlo@kerestey.net> |

Safe Haskell | Safe |

Language | Haskell2010 |

A Haskell implementation of setoid - a set equipped with an equivalence relation. Setoid is a useful data structure when equivalence is chosen not to be equality. This allows to influence the membership of the elements in a setoid. When equality is all one needs - using sets is a better option.

Here we have chosen to use a specific variant of equivalence of transforming the elements to comparable intermediaries. Although it does not make every equivalence relation possible, it is a practial choice for a lot of computations.

## Usage

When manipulating collections of objects in the real world, we often use lists/arrays. Sometimes we need to represent some properties of the relation between the elements though, and the lists do not provide such possibility. This library not only provides the guarantee that a setoid is correct by construction, but also that the manipulations will not change its structure.

We use it to run computations over time series of sampling data, collections of users (who are unique by username or email) - to keep the same structure as the one which would be used in the database with unique indexes.

To implement equivalence we chose to use a data class `EquivalenceBy`

which provides a method of mapping an element to an intermediary,
which is then used for comparison and ultimately lead to a choice
of the members.

The type of a setoid is `Setoid e a` where `a`

is the member type and
e is the type of equivalence intermediary. To chose the members of the
setoid we compare the e(quivalences) of the elements with each other.

The definition of `EquivalenceBy e a` is

class EquivalenceBy e a where eqRel :: a -> e

To give a simple example of how the library could be used we will combine apples and oranges to a Setoid of fruit names by color. We want one fruit per colour as a result and don't care if its apple or an orange.

```
import Data.Setoid (Setoid)
import qualified Data.Setoid as Setoid
data Colour = Red | Green | Blue deriving (Eq,Ord)
instance EquivalenceBy Colour (Colour,String) where
eqRel = fst
apples, organges, fruits :: Setoid Int (Int,String)
apples = Setoid.fromList [(Green,"golden delicious"), (Orange,"honeycrunch")]
oranges = Setoid.fromList [(Orange,"seville"), (Red,"blood orange")]
fruits = apples
````union`

oranges
-- > [(Green,"golden delicious"), (Orange,"seville"), (Red,"blood orange")]

One can see the benefit of using a `Setoid`

instead of Data.List
because with the latter, we would have to use `nubBy`

every
time the data is transformed.

When performing a `union`

, our implementation would use `max`

between
two equivalent elements to resolve the conflict. Bear in mind, that
the elements, though equivalent, might not be equal. In the example
above, ordering of ` "seville" `

is bigger than ` "golden delicious" `

thus ` (Orange, "seville") `

is chosen in the result.

### Friends of friends and computation on union

For another example, lets get all the users of two different services F and G. We are not interested in the different details, but want the instance of the users to be unique.

type Email = String data User = User { email :: Email, contacts :: Int } deriving (Eq,Show) instance EquivalenceBy Email User where eqRel u = email u usersF, usersG, allUsers :: Setoid Email User usersF <- getUsers F usersG <- getUsers G allUsers = Setoid.unionWith mergeContactDetails usersF usersG mergeContactDetails :: User -> User -> User mergeContactDetails a b = User (email a) (contacts a + contacts b) -- ... --

We assume that here are equivalent elements in both setoids - in this
case they have the same email adress. Thus we use `unionWith`

to merge
the other details of the contact. Here, we could also do computations
and, for example, sum the number of friends/contacts from bothe
services.

Here is also one of the shortcommings of the library. mergeContactDetails choses the email of the first argument. Sinse in the context of unionWith, the emails of the first and the second users are the same. It is not nice from the perspective of the function itself though.

` Setoid.size allUsers `

Would give us the amount of all unique users
in both services together.

## Future Work

- There is an unproven hypothesis about a relation between setoids and Quotient Sets. It seems, that a `Setoid (a,b) (a,b,c)` is equivalent to a `QuotientSet a (Setoid b (a,b,c))`. This means that every QuotientSet can actually be represented as a setoid.
- Performance is another issue. Current implementation uses the `newtype Setoid x y = Setoid (Map x y)` which may be inefficient.

- data Setoid e a
- class EquivalenceBy e a where
- (=~=) :: Eq e => Setoid e a -> Setoid e a -> Bool
- (\\) :: Ord e => Setoid e a -> Setoid e a -> Setoid e a
- (∪) :: (Ord e, Ord a) => Setoid e a -> Setoid e a -> Setoid e a
- empty :: Setoid e a
- ø :: Setoid e a
- singleton :: EquivalenceBy e a => a -> Setoid e a
- union :: (Ord e, Ord a) => Setoid e a -> Setoid e a -> Setoid e a
- unions :: (Ord e, Ord a) => [Setoid e a] -> Setoid e a
- unionWith :: Ord e => (a -> a -> a) -> Setoid e a -> Setoid e a -> Setoid e a
- difference :: Ord e => Setoid e a -> Setoid e a -> Setoid e a
- filter :: Ord e => (a -> Bool) -> Setoid e a -> Setoid e a
- null :: Setoid e a -> Bool
- size :: Setoid e a -> Int
- member :: (EquivalenceBy e a, Ord e) => a -> Setoid e a -> Bool
- equivalence :: Eq e => Setoid e a -> Setoid e a -> Bool
- map :: (EquivalenceBy eb b, Ord eb, Ord b) => (a -> b) -> Setoid ea a -> Setoid eb b
- mapResolve :: (EquivalenceBy eb b, Ord eb) => (b -> b -> b) -> (a -> b) -> Setoid ea a -> Setoid eb b
- mapM :: (Monad m, EquivalenceBy eb b, Ord eb, Ord b) => (a -> m b) -> Setoid ea a -> m (Setoid eb b)
- fromList :: (EquivalenceBy e a, Ord e, Ord a) => [a] -> Setoid e a
- fromListWith :: (EquivalenceBy e a, Ord e) => (a -> a -> a) -> [a] -> Setoid e a
- toList :: Setoid e a -> [a]

# Type

# Class

class EquivalenceBy e a where Source #

Equivalence class. It reduces the data to the part which is then being tested for equality in a Setoid.

# Operators

# Construction

singleton :: EquivalenceBy e a => a -> Setoid e a Source #

A Setoid with a single element

union :: (Ord e, Ord a) => Setoid e a -> Setoid e a -> Setoid e a Source #

Combine two Setoids resolving conflicts with `max`

by
default. This makes the union operation commutative and
associative.

unions :: (Ord e, Ord a) => [Setoid e a] -> Setoid e a Source #

Union several Setoids into one. This uses de default union variant

unionWith :: Ord e => (a -> a -> a) -> Setoid e a -> Setoid e a -> Setoid e a Source #

A generalized variant of union which accepts a function that will be used when two equivalent elements are found an the conflict needs to be resolved. Note that the elements are not necessarily equal

# Difference

difference :: Ord e => Setoid e a -> Setoid e a -> Setoid e a Source #

Difference of two setoids. Return elements of the first setoid not existing in the second setoid.

# Filter

filter :: Ord e => (a -> Bool) -> Setoid e a -> Setoid e a Source #

Filter a setoid. Return a setoid with elements that statisfy the predicate

# Query

member :: (EquivalenceBy e a, Ord e) => a -> Setoid e a -> Bool Source #

Test if an element is a member of a setoid

equivalence :: Eq e => Setoid e a -> Setoid e a -> Bool Source #

Test if two Setoids are equivalent i.e. if all the elements are equivalent

# Traversal

## map

map :: (EquivalenceBy eb b, Ord eb, Ord b) => (a -> b) -> Setoid ea a -> Setoid eb b Source #

Map a function over elements of a setoid. It resolves conflict in the result by chosing the maximum one

:: (EquivalenceBy eb b, Ord eb) | |

=> (b -> b -> b) | conflict resolution function |

-> (a -> b) | map function |

-> Setoid ea a | input |

-> Setoid eb b | result |

Generalized version of map, allowing to use custom function to resolve a conflict if two equivalent elements are found in the result

mapM :: (Monad m, EquivalenceBy eb b, Ord eb, Ord b) => (a -> m b) -> Setoid ea a -> m (Setoid eb b) Source #

Monadic variant of a map

# Conversion

fromList :: (EquivalenceBy e a, Ord e, Ord a) => [a] -> Setoid e a Source #

A default variant of fromList using `max`

to resolve a conflict
if two equivalent elements are found. Therefore it depends on Ord
instance of the element

fromListWith :: (EquivalenceBy e a, Ord e) => (a -> a -> a) -> [a] -> Setoid e a Source #

A generalized version of fromList, which will use a supplied funtion if two equivalent elements are found in the input list