# set-monad

Set monad

LTS Haskell 9.6: | 0.2.0.0 |

Stackage Nightly 2017-09-26: | 0.2.0.0 |

Latest on Hackage: | 0.2.0.0 |

**George Giorgidze**

**giorgidze@gmail.com**

#### Module documentation for 0.2.0.0

- Data
- Data.Set

The `set-monad`

library exports the `Set`

abstract data type and
set-manipulating functions. These functions behave exactly as their namesakes
from the `Data.Set`

module of the `containers`

library. In addition, the
`set-monad`

library extends `Data.Set`

by providing `Functor`

, `Applicative`

,
`Alternative`

, `Foldable`

, `Monad`

, and `MonadPlus`

instances for sets.

In other words, you can use the `set-monad`

library as a drop-in replacement
for the `Data.Set`

module of the `containers`

library and, in addition, you
will also get the aforementioned instances which are not available in the
`containers`

package.

It is not possible to directly implement instances for the aforementioned
standard Haskell type classes for the `Set`

data type from the `containers`

library. This is because the key operations `map`

and `union`

, are constrained
with `Ord`

as follows.

```
map :: (Ord a, Ord b) => (a -> b) -> Set a -> Set b
union :: (Ord a) => Set a -> Set a -> Set a
```

The `set-monad`

library provides the type class instances by wrapping the
constrained `Set`

type into a data type that has unconstrained constructors
corresponding to monadic combinators. The data type constructors that
represent monadic combinators are evaluated with a constrained run function.
This elevates the need to use the constraints in the instance definitions
(this is what prevents a direct definition). The wrapping and unwrapping
happens internally in the library and does not affect its interface.

For details, see the rather compact definitions of the `run`

function and
type class instances. The left identity and associativity monad laws play a
crucial role in the definition of the `run`

function. The rest of the code
should be self explanatory.

The technique is not new. This library was inspired by [1]. To my knowledge, the original, systematic presentation of the idea to represent monadic combinators as data is given in [2]. There is also a Haskell library that provides a generic infrastructure for the aforementioned wrapping and unwrapping [3].

The `set-monad`

library is particularly useful for writing set-oriented code
using the do and/or monad comprehension notations. For example, the
following definitions now type check.

```
s1 :: Set (Int,Int)
s1 = do a <- fromList [1 .. 4]
b <- fromList [1 .. 4]
return (a,b)
```

```
-- with -XMonadComprehensions
s2 :: Set (Int,Int)
s2 = [ (a,b) | (a,b) <- s1, even a, even b ]
```

```
s3 :: Set Int
s3 = fmap (+1) (fromList [1 .. 4])
```

As noted in [1], the implementation technique can be used for monadic libraries and EDSLs with restricted types (compiled EDSLs often restrict the types that they can handle). Haskell's standard monad type class can be used for restricted monad instances. There is no need to resort to GHC extensions that rebind the standard monadic combinators with the library or EDSL specific ones.

`[`

1`]`

CSDL Blog: The home of applied functional programming at KU. Monad
Reification in Haskell and the Sunroof Javascript compiler.
http://www.ittc.ku.edu/csdlblog/?p=88

`[`

2`]`

Chuan-kai Lin. 2006. Programming monads operationally with Unimo. In
Proceedings of the eleventh ACM SIGPLAN International Conference on Functional
Programming (ICFP '06). ACM.

`[`

3`]`

Heinrich Apfelmus. The operational package.
http://hackage.haskell.org/package/operational