{-# LANGUAGE TypeSynonymInstances, FlexibleInstances, MultiParamTypeClasses, TypeFamilies #-}
{-# LANGUAGE OverlappingInstances, IncoherentInstances #-}
{- |
Module : Data.PolyToMonoid
Copyright : Copyright (C) 2010 Kevin Jardine
License : BSD3
Maintainer : Kevin Jardine
Stability : provisional
Portability: non-portable (depends on GHC extensions)
Creates polyvariadic functions that map their parameters into a given monoid.
-}
module Data.PolyToMonoid (
-- * Introduction
-- $introduction
-- * Concept
-- $usage
-- * Examples
-- $examples
-- * Extensions
-- $extensions
Monoidable(toMonoid),
PolyVariadic(ptm),
CPolyVariadic(ctm),
Terminate(..)
) where
import Data.Monoid
{- $introduction
Haskell lists contain an indefinite number of elements of a single type.
It is sometimes useful to create list-like functions that accept an indefinite
number of parameters of multiple types.
PolyToMonoid provides two such functions, 'ptm' and 'ctm', as well as a typeclass 'Monoidable'.
The only precondition is that the
parameters of 'ptm' and 'ctm' can be mapped to an underlying monoid using
the 'toMonoid' function provided by 'Monoidable'.
-}
{- $usage
To understand how the polyToMonoid functions work, consider a function @list@ that maps its parameters to
a list:
> list p1 p2 ... pN =
> [p1] ++ [p2] ++ ... ++ [pN]
(which is the same as @[p1,p2, ..., pN]@ )
@list@ can be generalised to any monoid conceptually as:
> polyToMonoid p1 p2 ... pN =
> (toMonoid p1) `mappend` (toMonoid p2) `mappend` ... `mappend` (toMonoid pN)
Remember that a monoid, defined in "Data.Monoid", is any set of elements
with an identity element @mempty@ and an associative operator @mappend@ .
As any list type is automatically a monoid with @mempty = []@ and @mappend = (++)@,
the @list@ function defined above is just a specific version of @polyToMonoid@.
The main difficulty with defining a @polyToMonoid@ function is communicating to Haskell what
underlying monoid to use.
Through Haskell type magic, this can be done with a simple type annotation.
Specifically, you can pass @mempty@ as the first element of the function and annotate it with the type of the
monoid it belongs to. Think of the @mempty@ value as like the initial value of a fold.
This library provides two variants of a polyToMonoid function. The first, 'ptm',
simply takes a list of arguments starting with mempty and returns the monoid result.
The second variant, 'ctm', is composible. In effect, it returns a function that consumes the next parameter.
You can feed the 'ctm' result to a second termination function 'trm' to get the actual result.
-}
{- $examples
We can first tell Haskell how to map a set of types to a list of strings using 'Monoidable':
> instance Show a => Monoidable a [String] where
> toMonoid a = [show a]
and then tell 'ptm' to use the @[String]@ monoid:
> ptm (mempty :: [String]) True "alpha" [(5 :: Int)]
The result returned would be:
> ["True","\"alpha]\"","[5]"]
In this case, 'Monoidable' tells the 'ptm' function to accept a wide variety of types
(anything with a @show@ function) when using the @[String]@ monoid.
The first parameter of 'ptm', @(mempty :: [String])@ tells it to map its parameters into the
@[String]@ monoid.
Unlike 'ptm', the 'ctm' function in effect returns a partial function ready to consume
the next parameter rather than a monoid result.
It is therefore more composable, at the cost of requiring a second termination function 'trm'
to return the actual monoid result.
> ctmfirstbit = ctm mempty :: [String]) True "alpha"
> ctmsecondbit = ctmfirstbit [(5 :: Int)]
> finalresult = trm ctmsecondbit
The result returned would be the same as above:
> ["True","\"alpha]\"","[5]"]
Monoids, of course, do not have to be lists.
Here's a second example which multiplies together numbers of several types:
> instance Monoid Double where
> mappend = (*)
> mempty = (1.0) :: Double
> instance Monoidable Int Double where
> toMonoid = fromIntegral
> instance Monoidable Double Double where
> toMonoid = id
> ptm (mempty :: Double) (5 :: Int) (2.3 :: Double) (3 :: Int)
In this case, 'ptm' accepts parameters that are either ints or doubles, converts them to doubles,
and then multiplies them together.
You can use the composibility of 'ctm' to define a @productOf@ function:
> productOf = ctm (mempty :: Double)
> trm $ productOf (5 :: Int) (2.3 :: Double) (3 :: Int)
As before the 'trm' function is required to terminate the 'ctm' calculation and deliver the final result.
Note that since 'ptm' returns its result immediately, it is not possible to use it to
define other functions using Haskell. You can use it in CPP defines, however. For example:
> #define productOf ptm (mempty :: Double)
> productOf (5 :: Int) (2.3 :: Double) (3 :: Int)
-}
{- $extensions
You will probably need to enable the following extensions to use this library:
> TypeSynonymInstances, FlexibleInstances, MultiParamTypeClasses
-}
-- | Define instances of Monoidable to tell Haskell
-- how to convert your parameters into values in the underlying
-- monoid
class Monoid m => Monoidable a m where
toMonoid :: a -> m
squish :: Monoidable a m => m -> a -> m
squish m a = (m `mappend` (toMonoid a))
{-|
Conceptually, ptm is defined as:
> ptm (mempty :: MyMonoid) p1 p2 ... pN =
> (toMonoid p1) `mappend` (toMonoid p2) `mappend` ... `mappend` (toMonoid pN)
-}
class Monoid m => PolyVariadic m r where
ptm :: m -> r
{-|
ctm is a composable variant of ptm.
To actually get its value, use the terminator function trm.
-}
class Monoid m => CPolyVariadic m r where
ctm :: m -> r
data Terminate m =
Terminate {
trm :: m -- ^ Use the terminator function trm to get the value of a ctm calculation.
}
instance (Monoid m', m' ~ m) => CPolyVariadic m (Terminate m') where
ctm acc = Terminate acc
instance (Monoidable a m, CPolyVariadic m r) => CPolyVariadic m (a->r) where
-- ctm m a = \b -> m `mappend` (ctm (toMonoid a) b)
ctm acc = \a -> ctm (squish acc a)
instance (Monoid m', m' ~ m) => PolyVariadic m m' where
ptm acc = acc
instance (Monoidable a m, PolyVariadic m r) => PolyVariadic m (a->r) where
ptm acc = \a -> ptm (squish acc a)