Singletons and You

Justin Le (

Lambdaconf 2017, May 27, 2017


Slide available at

GHC extensions (potentially) used:

{-# LANGUAGE GADTs                #-}
{-# LANGUAGE KindSignatures       #-}
{-# LANGUAGE LambdaCase           #-}
{-# LANGUAGE RankNTypes           #-}
{-# LANGUAGE RankNTypes           #-}
{-# LANGUAGE ScopedTypeVariables  #-}
{-# LANGUAGE TemplateHaskell      #-}
{-# LANGUAGE TypeFamilies         #-}
{-# LANGUAGE TypeInType           #-}
{-# LANGUAGE TypeOperators        #-}
{-# LANGUAGE UndecidableInstances #-}

import Data.Kind        -- to get type Type = *
import Data.Singletons

Safety with Phantom Types

data DoorState = Opened | Closed | Locked
  deriving (Show, Eq)

data Door (s :: DoorState) = UnsafeMkDoor

-- alternatively
data Door :: DoorState -> Type where
    UnsafeMkDoor :: Door s

Other similar examples

  • State machines (socket connections, file handles, opened/closed)
  • Refinement types
  • “Tagged” types (santized/unsantized strings)

Phantom types in action

closeDoor :: Door 'Opened -> Door 'Closed
closeDoor UnsafeMkDoor = UnsafeMkDoor
openDoor  :: Door 'Closed -> Door 'Opened
openDoor UnsafeMkDoor = UnsafeMkDoor

Phantom types in action

doorStatus :: Door s -> DoorState
doorStatus = -- ????

We have a problem.

doorStatus :: Door s -> DoorState
doorStatus UnsafeMkDoor = -- s ???

More Problems

initalizeDoor :: DoorStatus -> Door s
initializeDoor = \case
    Opened -> UnsafeMkDoor
    Closed -> UnsafeMkDoor
    Locked -> UnsafeMkDoor

Neat, but does this work?

More Problems

ghci> :t initializeDoor Opened :: Door 'Closed
initializeDoor Opened :: Door 'Closed


The Fundamental Issue in Haskell

  • In Haskell, types only exist at compile-time. They are erased at runtime.
  • This is a good thing for performance! Types incur no runtime overhead!
  • But it makes functions like doorStatus fundamentally unwritable without fancy typeclasses.
  • …or does it?

The Singleton Pattern

data SingDS :: DoorStatus -> Type where
    SOpened :: SingDS 'Opened
    SClosed :: SingDS 'Closed
    SLocked :: SingDS 'Locked

Creates three constructors:

SOpened :: SingDS 'Opened
SClosed :: SingDS 'Closed
SLocked :: SingDS 'Locked

The Singleton Pattern

  • A singleton is a type that has exactly one inhabited value.
  • There is only one value of type SingDS 'Opened, and only one value of type SingDS 'Closed.
  • The constructor that a SingDS s uses reveals to us what s is.

The Singleton Pattern

With our new singletons, we can essentially pattern match on types:

showSingDS :: SingDS s -> String
showSingDS = \case
    SOpened -> "Opened"
    SClosed -> "Closed"
    SLocked -> "Locked"

Alone like this, it’s a bit boring. We didn’t need GADTs for this.

Door Status

doorStatus' :: SingDS s -> Door s -> DoorState
doorStatus' = \case
    SOpened -> \_ -> "Door is opened"
    SClosed -> \_ -> "Door is closed"
    SLocked -> \_ -> "Door is locked"
  • GADT-ness allows us to enforce that the s in SingDS s is the same as the s in our Door.
  • Singleton property means that SingDS s has a one-to-one correspondence with its constructors.
  • Pattern matching on that single constructor reveals to us the type of Door.

Implicit Passing

class SingDSI s where
    singDS :: SingDSI s

instance SingDSI 'Opened where
    singDS = SOpened
instance SingDSI 'Closed where
    singDS = SClosed
instance SingDSI 'Locked where
    singDS = SLocked
doorStatus :: SingDSI s => Door s -> DoorState
doorStatus = doorStatus' singDS
ghci> doorStatus (UnsafeMkDoor :: Door 'Locked)
Door is locked!

Initialize Door

initializeDoor' :: SingDS s -> Door s
initializeDoor' _ _ = UnsafeMkDoor
ghci> :t initializeDoor' SOpened
initializeDoor SOpened :: Door 'Opened
ghci> :t initializeDoor' SClosed
initializeDoor SClosed :: Door 'Closed

Initialize Door

Implicit passing style:

initializeDoor :: SingDSI s => Door s
initializeDoor = initializeDoor' singDS

SingDS vs. SingDSI

  • Really, SingDS s -> is the same as SingDSI s =>
  • The two are the same way of providing the same information to the compiler, and at runtime.
  • We can use the two styles interchangebly.
  • One is explicitly passing the type, the other is explicitly passing the type.

Ditching the phantom

Sometimes we don’t care about what the status of our door is, and we want the type system to relax.

This is essentially the same as saying that the status of our door is a runtime property that we do not want to (or sometimes can’t) check at compile-time.

Ditching the phantom

data SomeDoor :: Type where
    MkSomeDoor :: SingDS s => Door s -> SomeDoor
ghci> let myDoor = MkSomeDoor (initializeDoor SOpened)
ghci> :t myDoor
myDoor :: SomeDoor
ghci> case myDoor of MkSomeDoor d -> doorStatus d
Door is opened.

Runtime-deferred types

initializeSomeDoor :: DoorStatus -> SomeDoor
initializeSomeDoor = \case
    Opened -> SomeDoor (initialiseDoor' SOpened)
    Closed -> SomeDoor (initialiseDoor' SClosed)
    Locked -> SomeDoor (initialiseDoor' SLocked)
ghci> let myDoor = initializeSomeDoor Locked
ghci> :t myDoor
myDoor :: SomeDoor
ghci> case myDoor of MkSomeDoor d -> doorStatus d
Door is locked.

The Singletons Library

The singletons library provides a unified framework for creating and working with singletons for different types (not just DoorStatus), and also for functions on those types.

The singletons way

$(singletons [d|
  data DoorState = Opened | Closed | Locked
    deriving (Show, Eq)

This creates three types and three constructors:

-- not the actual code, but essentially what happens
data Sing :: DoorState -> Type where
    SOpened :: Sing 'Opened
    SClosed :: Sing 'Closed
    SLocked :: Sing 'Locked

Sing is a poly-kinded type constructor (family):

The singletons way

And also

instance SingI 'Opened where
    sing = SOpened
instance SingI 'Closed where
    sing = SClosed
instance SingI 'Locked where
    sing = SLocked

(SingI is a poly-kinded typeclass)


STrue :: Sing 'True
SJust SFalse :: Sing ('Just 'True)
SOpened `SCons` SClosed `SCons` SNil :: Sing '[ 'Opened, 'Closed ]
ghci> sing :: Sing 'True'

Other stuff created from the library

Some other convenient features:

ghci> fromSing SOpened
ghci> let s = toSing Opened
ghci> :t s
s :: SomeSing DoorStatus
ghci> case s of
        SomeSing SOpened -> "Opened."
        SomeSing SClosed -> "SClosed."
        SomeSing SLocked -> "SLocked."

Non-trivial type logic

knock :: Door s -> IO ()
knock = -- ??

We want to allow the user to knock on a closed or locked door, but not an opened door.

We can do this simple case using pattern matching, but it’s not always feasible or scalable. We want to define a type relationship that can be used by potentially many functions.

Singletons to the Rescue

$(singletons [d|
  canKnock :: DoorState -> Bool
  canKnock Opened = False
  canKnock Closed = True
  canKnock Locked = True
knock :: (CanKnock s ~ True) => Door s -> IO ()
knock _ = putStrLn "knock knock!"
ghci> knock (initializeDoor SOpened)
Compile Error!!!!
ghci> knock (initializeDoor SClosed)
knock knock!

Singletons to the Rescue

tryKnock' :: Sing s -> Door s -> IO ()
tryKnock' s = case sCanKnock s of
    STrue  -> knock
    SFalse -> \_ -> putStrLn "Cannot knock door!"

tryKnock :: SingI s => Door s -> IO ()
tryKnock = tryKnock' sing
ghci> tryKnock (initializeDoor SOpened)
Cannot knock door!
ghci> tryKnock (initializeDoor SClosed)
knock knock!


$(singletons [d|
  data N = Z | S N

data Vec :: N -> Type -> Type where
    VNil :: Vec Z a
    (:*) :: a -> Vec n a -> Vec (S n) a

infixr 5 :*

The Types demand it

    :: Sing n
    -> a
    -> Vec n a
replicateV' = \case
    SZ   -> \_ -> VNil
    SS n -> \x -> x :* replicateV' n x
    :: SingI n
    => a
    -> Vec n a
replicateV = replicateV' sing

Thank you!