Lifting

Lifting is a concept which allows you to transform a function into a corresponding function within another (usually more general) setting.

Lifting in general

We usually start with a (covariant) functor, for simplicity we will consider the Pair functor first. Haskell doesn't allow a type Pair a = (a, a) to be a functor instance, so we define our own Pair type instead.

data Pair a = Pair a a deriving Show
instance Functor Pair where
    fmap f (Pair x y) = Pair (f x) (f y)

If you look at the type of fmap (Functor f => (a -> b) -> (f a -> f b)), you will notice that fmap already is a lifting operation: It transforms a function between simple types into a function between pairs of these types.

lift :: (a -> b) -> Pair a -> Pair b
lift = fmap

plus2 :: Pair Int -> Pair Int
plus2 = lift (+2)
-- plus2 (Pair 2 3) ---> Pair 4 5

Note, however, that not all functions between Pair a and Pair b can constructed as a lifted function (e.g. \(x, _) -> (x, 0) can't).

A functor can only lift functions of exactly one variable, but we want to lift other functions, too:

lift0 :: a -> Pair a
lift0 x = Pair x x

lift2 :: (a -> b -> r) -> (Pair a -> Pair b -> Pair r)
lift2 f (Pair x1 x2) (Pair y1 y2) = Pair (f x1 y1) (f x2 y2)

plus :: Pair Int -> Pair Int -> Pair Int
plus = lift2 (+)
-- plus (Pair 1 2) (Pair 3 4) ---> Pair 4 6

In a similar way, we can define lifting operations for all containers that have "a fixed size", for example for the functions from Double to any value ((->) Double), which might be thought of as values that are varying over time (given as Double). The function \t -> if t < 2.0 then 0 else 2 would then represent a value which switches at time 2.0 from 0 to 2. Using lifting, such functions can be manipulated in a very high-level way. In fact, this kind of lifting operation is already defined. Control.Monad.Reader (see MonadReader) provides a Functor, Monad, MonadFix and MonadReader instance for the type (->) r. The liftM (see below) functions of this monad are precisely the lifting operations we are searching for.

If the containers don't have fixed size, it's not always clear how to make lifting operations for them. The [] - type could be lifted using the zipWith-family of functions or using liftM from the list monad, for example.

Monad lifting

Lifting is often used together with monads. The members of the liftM-family take a function and perform the corresponding computation within the monad.

return  :: (Monad m) => a -> m a
liftM   :: (Monad m) => (a1 -> r) -> m a1 -> m r
liftM2  :: (Monad m) => (a1 -> a2 -> r) -> m a1 -> m a2 -> m r

Consider for example the list monad (MonadList). It performs a nondeterministic calculation, returning all possible results. liftM2 just turns a deterministic function into a nondeterministic one:

plus :: [Int] -> [Int] -> [Int]
plus = liftM2 (+)
-- plus [1,2,3] [3,6,9] ---> [4,7,10, 5,8,11, 6,9,12]
-- plus [1..] []        ---> _|_ (i.e., keeps on calculating forever)
-- plus [] [1..]        ---> []

Lifting becomes especially interesting when there are more levels you can lift between. Control.Monad.Trans (see Monad transformers) defines a class

class MonadTrans t where
    lift :: Monad m => m a -> t m a -- lifts a value from the inner monad m to the transformed monad t m
                                    -- could be called lift0

lift takes the side effects of a monadic computation within the inner monad m and lifts them into the transformed monad t m. We can easily define functions which lift functions between inner monads to functions between transformed monads. Then we can perform three different lifting operations: liftM can be used both to transform a pure function into a function between inner monads and to a function between transformed monads, and finally lift transforms from the inner monad to the transformed monad. Because of the purity of Haskell, we can only lift "up".

Arrow lifting

Until now, we have only considered lifting from functions to other functions. John Hughes' arrows (see Understanding arrows) are a generalization of computation that aren't functions anymore. An arrow a b c stands for a computation which transforms values of type b to values of type c. The basic primitive arr, aka pure,

arr :: (Arrow a) => b -> c -> a b c

is also a lifting operation.

Applicative Functor

This should only provide a definition what lifting means (in the usual cases, not in the arrow case). It's not a suggestion for an implementation. I start with the (simplest?) basic operations zipL, which combines to containers into a single one and zeroL, which gives a standard container for ().

class Functor f => Liftable f where
    zipL :: f a -> f b -> f (a, b)
    zeroL :: f ()
    
liftL :: Liftable f => (a -> b) -> (f a -> f b)
liftL = fmap

liftL2 :: Liftable f => (a -> b -> c) -> (f a -> f b -> f c)
liftL2 f x y = fmap (uncurry f) $ zipL x y

liftL3 :: Liftable f => (a -> b -> c -> d) -> (f a -> f b -> f c -> f d)
liftL3 f x y z = fmap (uncurry . uncurry $ f) $ zipL (zipL x y) z

liftL0 :: Liftable f => a -> f a
liftL0 x = fmap (const x) zeroL 

appL :: Liftable f => f (a -> b) -> f a -> f b
appL = liftL2 ($)

Now, for example every Monad can be made an instance of Liftable:

{-# OPTIONS -fglasgow-exts #-}
{-# OPTIONS -fallow-undecidable-instances #-}
import Control.Monad

instance (Functor m, Monad m) => Liftable m where 
    zipL  = liftM2 (\x y -> (x,y))
    zeroL = return ()

We need to postulate a few laws so that the definitions make sense. (Are they complete and/or minimal?)

assoc :: ((a, b), c) -> (a, (b, c))
assoc ~(~(x, y), z) = (x, (y, z))

{-
Identity:
fmap snd $ zipL zeroL x === x
fmap fst $ zipL x zeroL === x

Associativity:
fmap assoc $ zipL (zipL x y) $ z === zipL x $ zipL y z
-}

Applicative functors were introduced by several people under different names:

• Ross Paterson called them Sequence
• Conor McBride called them Idiom
• The same kind of structure is used in the UU Parsing-Combinators.