From Expression to Free

We've been defining the function, fold, the same way for a while. However, there's another way to write the function. To help you understand the upcoming code, we'll rewrite it in this new way:

fold :: forall f a. Functor f => (f a -> a) -> Expression f -> a
fold f (In t) = f (map (fold f) t)                                          {-
... which can be rewritten using infix notation                             -}
fold f (In t) = f ((fold f) <$> t)                                          {-
... which can be rewritten to not pass `f` through recursive calls          -}
fold f = go where
  go (In t) = f (go <$> t)                                                  {-
... which can be rewritten to use "case _ of" to pattern match              -}
fold f = go where
  go in_t = case in_t of
    In t -> f (go <$> t)

With that out of the way, let's compare Expression to Free. We can see that Expression is really just a variant of the Free monad without the Pure constructor.

newtype Expression f
  -- no pure here...
  = In     (f (Expression f  ))

newtype Free       f a
  = Pure a
  | Impure (f (Free       f a))

How would we rewrite our solution from before to use Free instead of Expression? Value is replaced with Pure.

To see an example of this for just Value and Add (Multiply is excluded) and to understand its code, refer to the below code and table before viewing ADT8.purs.

-- when Value and Add were both `f`
fold    f = go where
  go in_t = case in_t of
    In     t -> f (go <$> t)

-- when Value is simply Pure now
fold    f = go where
  go free = case free of
    Pure   a -> a              -- Value
    Impure t -> f (go <$> t)   -- Add
xgrommx's codeOur code
Free ExprF a
Expression (Coproduct Value ExprF) a
iter k gofold algebra expression
Left f
Right a

A Quick Overview of Some of Purescript's Free API

Purescript's Free monad is implemented in the "reflection without remorse" style, which adds complexity to the implementation. Thus, rather than redirecting you there, we'll explain the general idea of what the code is doing.


The Free monad has its own way of injecting a value into it called liftF. It can be understood like this:

-- Before
someValue :: forall a. a -> Expression SomeFunctor
someValue a =
  In (SomeFunctor a) -- or `inj (SomFunctor a)`

-- After...
liftF :: a -> Free SomeFunctor a
liftF = Impure (SomeFunctor a)


LiftF is useful, but it won't let us compile the examples we will show next because it expects the a to be any a. In cases like AddF/MultiplyF, sometimes that a has to be Free AddF/Free MultiplyF instead of just the Int type. In such cases, we can use wrap:

-- Using `liftF`
add :: Int -> Int -> Free AddF Int
add x y = liftF $ AddF x y

-- the above 'add' definition does not allow us to create
-- nested adds

-- Using `wrap`
add :: Free AddF Int -> Free AddF Int -> Free AddF Int
add x y = wrap $ AddF x y

Other Functions

foldFree interprets the Free into some other monad (e.g. Effect, etc.).

runFree computes a "pure" program described in Free (by 'pure,' we meant that Free only stores the instructions the program would execute, but does not run them).

Compilation Instructions

You can run the upcoming code examples using these instructions:

spago run -m Free.AddAndMultiply