What is "lifting" in Scala?

There are a few usages:

PartialFunction

Remember a PartialFunction[A, B] is a function defined for some subset of the domain A (as specified by the isDefinedAt method). You can "lift" a PartialFunction[A, B] into a Function[A, Option[B]]. That is, a function defined over the whole of A but whose values are of type Option[B]

This is done by the explicit invocation of the method lift on PartialFunction.

scala> val pf: PartialFunction[Int, Boolean] = { case i if i > 0 => i % 2 == 0}
pf: PartialFunction[Int,Boolean] = <function1>

scala> pf.lift
res1: Int => Option[Boolean] = <function1>

scala> res1(-1)
res2: Option[Boolean] = None

scala> res1(1)
res3: Option[Boolean] = Some(false)

Methods

You can "lift" a method invocation into a function. This is called eta-expansion (thanks to Ben James for this). So for example:

scala> def times2(i: Int) = i * 2
times2: (i: Int)Int

We lift a method into a function by applying the underscore

scala> val f = times2 _
f: Int => Int = <function1>

scala> f(4)
res0: Int = 8

Note the fundamental difference between methods and functions. res0 is an instance (i.e. it is a value) of the (function) type (Int => Int)

Functors

A functor (as defined by scalaz) is some "container" (I use the term extremely loosely), F such that, if we have an F[A] and a function A => B, then we can get our hands on an F[B] (think, for example, F = List and the map method)

We can encode this property as follows:

trait Functor[F[_]] { 
  def map[A, B](fa: F[A])(f: A => B): F[B]
}

This is isomorphic to being able to "lift" the function A => B into the domain of the functor. That is:

def lift[F[_]: Functor, A, B](f: A => B): F[A] => F[B]

That is, if F is a functor, and we have a function A => B, we have a function F[A] => F[B]. You might try and implement the lift method - it's pretty trivial.

Monad Transformers

As hcoopz says below (and I've just realized that this would have saved me from writing a ton of unnecessary code), the term "lift" also has a meaning within Monad Transformers. Recall that a monad transformers are a way of "stacking" monads on top of each other (monads do not compose).

So for example, suppose you have a function which returns an IO[Stream[A]]. This can be converted to the monad transformer StreamT[IO, A]. Now you may wish to "lift" some other value an IO[B] perhaps to that it is also a StreamT. You could either write this:

StreamT.fromStream(iob map (b => Stream(b)))

Or this:

iob.liftM[StreamT]

this begs the question: why do I want to convert an IO[B] into a StreamT[IO, B]?. The answer would be "to take advantage of composition possibilities". Let's say you have a function f: (A, B) => C

lazy val f: (A, B) => C = ???
val cs = 
  for {
    a <- as                //as is a StreamT[IO, A]
    b <- bs.liftM[StreamT] //bs was just an IO[B]
  }
  yield f(a, b)

cs.toStream //is a Stream[IO[C]], cs was a StreamT[IO, C]

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Note any collection that extends PartialFunction[Int, A] (as pointed out by oxbow_lakes) may be lifted; thus for instance

Seq(1,2,3).lift
Int => Option[Int] = <function1>

which turns a partial function into a total function where values not defined in the collection are mapped onto None,

Seq(1,2,3).lift(2)
Option[Int] = Some(3)

Seq(1,2,3).lift(22)
Option[Int] = None

Moreover,

Seq(1,2,3).lift(2).getOrElse(-1)
Int = 3

Seq(1,2,3).lift(22).getOrElse(-1)
Int = -1

This shows a neat approach to avoid index out of bounds exceptions.

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Another usage of lifting that I've come across in papers (not necessarily Scala-related ones) is overloading a function from f: A -> B with f: List[A] -> List[B] (or sets, multisets, ...). This is often used to simplify formalisations because it then doesn't matter whether f is applied to an individual element or to multiple elements.

This kind of overloading is often done declaratively, e.g.,

f: List[A] -> List[B]
f(xs) = f(xs(1)), f(xs(2)), ..., f(xs(n))

or

f: Set[A] -> Set[B]
f(xs) = \bigcup_{i = 1}^n f(xs(i))

or imperatively, e.g.,

f: List[A] -> List[B]
f(xs) = xs map f

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There is also unlifting, which is the inverse process to lifting.

If lifting is defined as

turning a partial function PartialFunction[A, B] into a total function A => Option[B]

then unlifting is

turning a total function A => Option[B] into a partial function PartialFunction[A, B]

Scala standard library defines Function.unlift as

def unlift[T, R](f: (T) ⇒ Option[R]): PartialFunction[T, R]

For example, play-json library provides unlift to help with construction of JSON serialisers:

import play.api.libs.json._
import play.api.libs.functional.syntax._

case class Location(lat: Double, long: Double)

implicit val locationWrites: Writes[Location] = (
  (JsPath \ "lat").write[Double] and
  (JsPath \ "long").write[Double]
)(unlift(Location.unapply))

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