Free monads in the real world

After finishing my master’s degree, I applied to several companies I was interested in. During one of the selection processes, the interviewer asked me to do the following exercise:

“Write a stack-based interpreted language (byte code) that includes: literals, arithmetic operations, variables, and control flow primitives. As a bonus, add asynchronous primitives such as fork and await.”

Fortunately, I was already familiar with the assignment because I implemented a statically typed programming language a year ago. Consequently, I decided to take this as a chance and do something different than the rest of the candidates. Spoiler: free monads!

Free monads

The concept of free monad comes from the field of category theory (CT). The formal definition requires a strong background in CT. So, to keep this pragmatic, we are going to consider the following definition:

For any functor $f$, there exists a uniquely determined monad called the free monad of the functor $f$.

Probably this definition is still too general, so let’s see how this is translated into Haskell. In particular, we are going to base our work on the encoding of free monads of the haskell package free by Edward Kmett et al. The definition is as follows:

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data Free f a = Pure a | Free (f (Free f a))

Free f is simply a stack of layers f on top of a value a.

The interesting part about Free f is that Free f is a monad as long as f is a functor.

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instance Functor f => Monad (Free f) where
  return = pure
  Pure a >>= f = f a
  Free m >>= f = Free ((>>= f) <$> m)

This will allow us to compose free actions by embedding new layers on the bottom of the stack of layers, which is essential to model domain specific languages.

In practise, the user builds a free monad stacking layers of their domain specific language modelled as a functor. Later, it executes iter (or its corresponding applicative and monadic version) to interpret and collapse each layer into a single (effectful) value.

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iter :: Functor f => (f a -> a) -> Free f a -> a
iterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a 
iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a 

In the next section, we will see how to put that into practise.

Solving the assignment

When modeling an embedded domain specific language (eDSL) using free monads, you usually divide the problem in two parts:

  1. Define your language.
  2. Define the interpreter for your language.

Defining our eDSL

First, we need to define our language. We are free to model any kind of language, but the type that represents our language must be a functor. Otherwise, our language will not be able to be composed using bind (=<<).

Here is the sum type representing our byte code language:

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type ByteCode = Free ByteCodeF

data ByteCodeF next
  = Lit Value next
  | Load Var next
  | Write Var next
  | BinaryOp OpCode next
  | Loop (ByteCode Bool) (ByteCode ()) next
  | Ret (Value -> next)
  | NewChan (Channel -> next)
  | Send Channel next
  | Recv Channel next
  | Fork (ByteCode ()) (Future () -> next)
  | Await (Future ()) next
  deriving stock (Functor)

data Value
  = B Bool
  | I Integer
  deriving stock (Eq, Show, Ord)

newtype Chan a = Chan { getChan :: MVar a}
  deriving newtype (Eq)

data Future a where
  Future :: Exception e => Async (Either e a) -> Future a
  • Lit instruction to instantiate literals (integer and boolean values).
  • Load/Write instructions to load and write variables, respectively.
  • BinaryOp instruction to apply binary operations (+, *, <) on the top values of the stack.
  • Loop instruction to execute a set of instructions iteratively based on a certain condition.
  • Ret instruction to stop the program and return the value on top of the stack.
  • Fork/Await instructions to start and await an asynchronous program, respectively.
  • NewChan/Send/Recv instructions to create a new asynchronous channel, send and receive a value through the channel, respectively. This channel allow to independent asynchronous programs to communicate.

The careful reader may have notice that our type ByteCodeF has a type parameter next :: Type. This type parameter is required for ByteCodeF to be a functor. We can think of next as the next instruction in our program. In fact, when we interpret the ByteCode free monad, next will be replaced by the (evaluated) next instruction of our program.

Next, for each constructor of our ByteCodeF, we need a function that lifts ByteCodeF into ByteCode. For example, here is the definition of that function for Lit:

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lit :: Value -> ByteCode ()
lit v = Free (Lit v (Pure ()))

Implementing this for each constructor is tedious and error prone. This problem can be solved with metaprogramming. In particular, with template haskell (for an introduction to template haskell see my blog post). Fortunately, this is already implemented in Control.Monad.Free.TH.

Therefore, we are going to use makeFree to automatically derive all these free monadic actions

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$(makeFree ''ByteCodeF)

which generates the following code

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lit      :: Value                         -> ByteCode ()
load     :: Var                           -> ByteCode ()
write    :: Var                           -> ByteCode ()
loop     :: ByteCode Value -> ByteCode () -> ByteCode ()
newChan  ::                                  ByteCode Channel
send     :: Channel                       -> ByteCode ()
recv     :: Channel                       -> ByteCode ()
fork     :: ByteCode ()                   -> ByteCode (Async ())
await    :: Async ()                      -> ByteCode ()

The actual signatures use MonadFree which is an mtl-style class that allow us to compose FreeT with other monad transformers. We were able to monomorphize the return value MonadFree f m => m ~ Free ByteCodeF thanks to the instance Functor f => MonadFree f (Free f).

Once we have the basic building blocks of our language, we can start building more complex primitives using monadic composition!

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loopN :: Integer -> ByteCode ()
loopN until = do

  litI until
  write n
  litI 1
  write i
  loop (i < n) (i ++)

  where
  
      (<) i n = do
      load n
      load i
      lessThan
      ret
  
      (++) i = do
      litI 1
      load i
      add
      write i

Interpreting our eDSL

Now that we have the building blocks to construct our domain specific programs, it is only remaining to implement an interpreter to evaluate this embedded DSL in our host language Haskell.

The interpreter is usually implemented using iterM:

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iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a
iterM _   (Pure x) = return x
iterM phi (Free f) = phi (iterM phi <$> f)

iterM allow us to collapse our Free f into a monadic value m a by providing a function f (m a) -> m a (usually called an algebra) that operates on a single layer of our free monad. If you are familiar with recursion schemes, iterm is a specialization of cataA.

This may sound confusing at first, but it is easier than it looks like. So, let’s see how it would look like in our example. First of all, we need to choose the monadic value m. For our interpreted language, we choose m ~ Interpreter

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newtype Interpreter a = Interpreter 
  { runInterpreter :: StateT Ctx (ExceptT Err IO) a }

data Ctx = Ctx
  { stack :: [Value],
    variables :: Map Var Value
  }

data Err
  = VariableNotFound Var
  | StackIsEmpty
  | BinaryOpExpectedTwoOperands
  | AsyncException Text
  | WhoNeedsTypes

where Ctx is the current context of our program i.e. the state of the stack and the memory registers.

Then, we specialize iterM to our example

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--       <---------------- algebra ----------------->
iterm :: (ByteCodeF (Interpreter a) -> Interpreter a) -> ByteCode a -> Interpreter a

The last step and usually the most difficult one is to implement the algebra of our DSL

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algebra :: ByteCodeF (Interpreter a) -> Interpreter a
algebra = \case
  Ret f -> popI >>= f
  Lit i k -> pushI i >> k
  Load var k -> loadI var >>= pushI >> k
  Write var k -> popI >>= storeI var >> k
  BinaryOp op k ->
    do
      catchError
        (liftA2 (applyOp op) popI popI >>= either throwError pushI)
        ( \case
            StackIsEmpty -> throwError BinaryOpExpectedTwoOperands
            e -> throwError e
        )
      >> k
  Loop cond expr k ->
    fix $ \rec -> do
      b <- interpret cond
      if b
        then interpret expr >> rec
        else k
  NewChan f ->
    liftIO getChan >>= f
  Send chan k ->
    popI >>= (liftIO . sendChan chan) >> k
  Recv chan k ->
    liftIO (recvChan chan) >>= pushI >> k
  Fork branch k ->
    future branch >>= k
  Await (Future async') k -> do
    ea <- liftIO $ waitCatch async'
    case ea of
      Left (SomeException ex) -> throwError (AsyncException (T.pack $ show ex))
      Right (Left ex) -> throwError (AsyncException (T.pack $ show ex))
      Right _r -> k

Finally, we combine iterM and algebra to obtain

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interpret :: ByteCode a -> Interpreter a
interpret = iterM algebra

which allow us to interpret our embedded language in our host language. The result of interpret can be composed with other effectful programs and it can also be evaluated using runByteCode:

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runByteCode :: ByteCode a -> Either Err a
runByteCode = unsafePerformIO . runExceptT . flip evalStateT emptyCtx . runInterpreter . interpret

All together

So far, we have built the following components:

  1. An embedded stack-based language in Haskell with primitives like: lit, load, write, loop
  2. An interpreter for that language: interpret and runByteCode.

Now that we have all the ingredients to create embedded stack-based programs and interpret them, we can put this into practise.

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program :: ByteCode Value
program = do
  chan1 <- newChan
  chan2 <- newChan
  _ <- fork $ do
    loopN 100000
    litI 1
    send chan1
  _ <- fork $ do
    loopN 100000
    litI 1
    send chan2
  loopN 10
  recv chan1
  recv chan2
  add
  ret

main :: IO ()
main = case runByteCode program of
  Left err -> throw err
  Right res -> putStrLn $ "Result = " <> show res

The following program creates two asynchronous tasks that after a period of time, return the integer 1 through an asynchronous channel. The main program waits for these two asynchronous tasks to finish and outputs the sum of the results of the asynchronous tasks.

The source code from the examples can be found here.

Conclusion

In this post, we have introduced free monads and how they can be used to implement embedded domain specific languages. In particular, we have seen how to embed a stack-based language in Haskell. To see other examples of domain specific languages, we refer the reader to the examples of the free package.

This post only covered a half of the free package. In the next post of this series, we’ll explore the other half: church encoding, applicative free, cofree…