eighty-twenty

In this post, I’ll build a monad library for Racket, using its generic interfaces feature in place of Haskell’s type classes.

There have been lots of demonstrations of monads in dynamically-typed languages before; notably, Oleg’s article on monads in Scheme. However, I haven’t yet (Update: see below) seen one that smoothly allows for return-type polymorphism.

This post introduces (1) coercions between monad representations and (2) “undecided” quasi-monadic placeholder values. Taken together, these give a Haskell-like feel for monads in a dynamically-typed language.

The techniques are illustrated using Racket, but aren’t specific to Racket.

The Challenge

How can we write monadic programs like this, where monad types are implicit, just as they are in comparable Haskell programs?

(run-io (do (mdisplay "Enter a number: ")
            n <- mread
            all-n <- (return (for/list [(i n)] i))
            evens <- (return (do i <- all-n
                                 #:guard (even? i)
                                 (return i)))
            (return evens)))

The overall computation uses do-notation at IO type, while the evens list uses do-notation at List type.

The problem

Simple monomorphic monads are easy. Here’s the List monad in Racket:

(define (fail)      (list))
(define (return x)  (list x))
(define (bind xs f) (append-map f xs))

Let’s add some simple do-notation sugar:

(define-syntax do
  (syntax-rules (<-)
    [(_ mexp)                 mexp]
    [(_ var <- mexp rest ...) (bind mexp (λ (var)   (do rest ...)))]
    [(_ mexp rest ...)        (bind mexp (λ (dummy) (do rest ...)))]
    [(_ #:guard exp rest ...) (if exp (do rest ...) (fail))]))

Now we can write list comprehensions. For example, let’s select the odd numbers from a list, and multiply each by two:

>>> (do i <- '(1 2 3 4 5)
        #:guard (odd? i)
        (return (* i 2)))
'(2 6 10)

Let’s define another monad! How about for Racket’s lazy streams?

(define (fail)     empty-stream)
(define (return x) (stream-cons x empty-stream))
(define (bind s f) (if (stream-empty? s)
                       empty-stream
                       (let walk ((items (f (stream-first s))))
                         (if (stream-empty? items)
                             (bind (stream-rest s) f)
                             (stream-cons (stream-first items)
                                          (walk (stream-rest items)))))))

This is a fine definition, as far as it goes, but we’ve just run into the first problem:

Monads behave differently at different types

Simply defining fail, return and bind as functions won’t work. We need some kind of dynamic method dispatch. Haskell does this by dispatching using dictionary passing.

But there’s a deeper problem, too:

Monads don’t learn their types until it’s too late

How should we choose which dictionary to use? It’s easy with bind, because the specific monad being bound is always given as bind’s first argument. The bind implementation can be a method on the monad’s class.

This doesn’t work for return or fail. When they’re called, they don’t know what kind of monad they should produce. That only becomes clear later, when their results flow into a bind.

Haskell uses the type system. We have to do it at runtime, on a case-by-case basis.

A solution

To address these problems, we explicitly represent the dictionary we need:

(struct monad-class (failer    ;; -> (M a)
                     returner  ;; a -> (M a)
                     binder    ;; (M a) (a -> (M b)) -> (M b)
                     coercer)) ;; N (M a) -> (N a)

The failer, returner and binder methods correspond to the monad operations we met above, but coercer is new. It lets us dynamically reinterpret a monad at a different monadic type. We’ll see its use below.

Now we can define our List monad class:

(define List
  (monad-class (λ () '())
               (λ (x) (list x))
               (λ (xs f) (append-map (λ (x) (coerce List (f x))) xs))
               not-coercable))

The not-coercable function is for use when it’s not possible to reinterpret the type of a monad, which is the overwhelmingly common case. It checks to make sure we’re already at the expected type, and raises an exception if we’re not.

(define (not-coercable N m)
  (if (eq? (monad->monad-class m) N)
      m
      (error 'coerce)))

We need some way of learning which dictionary to use for a given monad instance. It’s here that Racket’s generics come in:

(define-generics monad
  (monad->monad-class monad)
  #:defaults ([null? (define (monad->monad-class m) List)]
              [pair? (define (monad->monad-class m) List)]))

This declares a new generic interface, gen:monad, with a single method, monad->monad-class. It supplies implementations of gen:monad for the builtin types null? and pair?, in each case returning the List monad-class when asked.

Now we can implement bind and coerce, not just for a single monad, but for all monads:

(define (bind ma a->mb)
  (define binder (monad-class-binder (monad->monad-class ma)))
  (binder ma a->mb))

(define (coerce N ma)
  (define coercer (monad-class-coercer (monad->monad-class ma)))
  (coercer N ma))

They use monad->monad-class to find the relevant dictionary, extract the right method from the dictionary, and delegate to it.

Note that bind dispatches on its first argument, while coerce dispatches on its second argument: monad instances get to decide how they are bound and how (or if!) they are to be interpreted at other monad types.

Next, we introduce neutral quasi-monad instances for when return and fail don’t know which monad type to use. Because we might bind such a quasi-monad before it collapses into a particular type, we need to define a quasi-monad to represent binds in superposition, too.

(struct return (value)
        #:methods gen:monad [(define (monad->monad-class m) Return)])
(struct fail ()
        #:methods gen:monad [(define (monad->monad-class m) Fail)])
(struct suspended-bind (ma a->mb)
        #:methods gen:monad [(define (monad->monad-class m) Bind)])

The Return method dictionary looks very similar to the implementation of the identity monad, with the addition of a return-struct box around the carried value:

(define Return
  (monad-class (λ () (error 'fail))
               (λ (x) (return x))
               (λ (m f) (f (return-value m)))
               (λ (N m) ((monad-class-returner N) (return-value m)))))

There are two interesting things to note here:

1. The bind implementation is a direct statement of one of the monad laws,

   return x >>= f ≡ f x
   

2. The coerce implementation rebuilds the monad using the return implementation from the target monad.

The Return coerce implementation gives us the equivalent of return-type polymorphism, letting us avoid sprinkling monad type annotations throughout our code.

Fail and Bind are implemented similarly: ¹

(define Fail
  (monad-class 'invalid
               'invalid
               (λ (ma a->mb) (suspended-bind ma a->mb))
               (λ (N m) ((monad-class-failer N)))))

The coerce method for an undetermined failure calls the new monad-class’s failer method to produce the correct representation for the final monad.

(define Bind
  (monad-class 'invalid
               'invalid
               (λ (ma a->mb) (suspended-bind ma a->mb))
               (λ (N m) (bind (coerce N (suspended-bind-ma m))
                              (suspended-bind-a->mb m)))))

Likewise, coerce for a suspended-bind re-binds its parameters, after first coercing them into the now-known monad.

You might have noticed a call to coerce in the bind implementation for List above:

(λ (xs f) (append-map (λ (x) (coerce List (f x))) xs))

This is necessary because f might simply (return 1), which is one of these neutral quasi-monad instances that needs to be told what kind of thing it should be before it can be used. On the other hand, f might have returned (list 1), in which case List’s coercer method will notice that no conversion needs to be done.

The IO Monad

Racket is an eager, imperative language, and so doesn’t need an IO monad the way Haskell does. But we can define one anyway, to see what it would be like.

It’s also useful as a template for implementing unusual control structures for monadic embedded domain-specific languages.

Values in the IO type denote sequences of delayed actions. We’ll represent these using two structure types, one for return and one for bind:

(struct io-return (thunk)
        #:methods gen:monad [(define (monad->monad-class m) IO)])
(struct io-chain (io k)
        #:methods gen:monad [(define (monad->monad-class m) IO)])

(define IO
  (monad-class (λ () (error 'fail))
               (λ (x) (io-return (λ () x)))
               (λ (ma a->mb) (io-chain ma a->mb))
               not-coercable))

Lifting Racket’s primitive input/output procedures into our IO type is straightforward:

(define (mdisplay x) (io-return (λ () (display x))))
(define mnewline     (io-return newline))
(define mread        (io-return read))

Finally, we need a way to actually execute a built-up chain of actions:

(define (run-io io)
  (match (coerce IO io)
    [(io-return thunk) (thunk)]
    [(io-chain io k) (run-io (k (run-io io)))]))

Revisiting the Challenge Problem

We now have all the pieces we need to run the challenge program given above.

The challenge comes from the end of Oleg’s article on monadic programming in Scheme. His example uses many different types of monad together, using the monomorphic technique he developed:

(IO::run
 (IO::>> (put-line "Enter a number: ")
         (IO::>>= (read-int)
                  (λ (n)
                    (IO::>>= (IO::return (for/list [(i n)] i))
                             (λ (all-n)
                               (IO::>>=
                                (IO::return
                                 (List:>>= all-n
                                           (λ (i)
                                             (if (even? i)
                                                 (List::return i)
                                                 (List::fail "odd")))))
                                (λ (evens) (IO::return evens)))))))))

With our polymorphic monads, we are able to use the generic do-notation macro defined above, and write instead

(run-io (do (mdisplay "Enter a number: ")
            n <- mread
            all-n <- (return (for/list [(i n)] i))
            evens <- (return (do i <- all-n
                                 #:guard (even? i)
                                 (return i)))
            (return evens)))

which compares favourably to the Haskell

main = do putStr "Enter a number: "
          n <- getInt
          allN <- return [0 .. n-1]
          evens <- return $ do i <- allN
                               guard $ i `mod` 2 == 0
                               return i
          return evens

Implementation

A somewhat fleshed-out implementation of these ideas is available here:

• It uses prop:procedure, making monad-classes directly callable and allowing (coerce List foo) to be written as (List foo).

• It uses the double-dispatch pattern to let monad instances flexibly adapt themselves to each other in coercions. For example, lists, streams and sets can all be considered interconvertible.

• It uses Racket’s built-in exception type to represent pending failures instead of the simple structure sketched above. This way, failures may carry stack traces and an informative error message.

• It includes a simple State monad, with sget and sput operations.

Related Work

I based this work on some experiments I ran back in 2006 in writing a monadic library for Scheme. Racket offers much better support for object-oriented/generic programming (among many other things!) than vanilla Scheme does, which makes for smoother integration between my monad library and the rest of the language.

Ben Wolfson pointed me to his monad library for clojure in a reddit thread about this post. It seems to be using a very similar approach, with “neutral” carriers for monadic values that stay neutral until a later point in the program. The main difference seems to be that with my approach, no “run” step is needed for some monads, since monadic values carry their bind implementation with them. Ben’s library supplies the bind implementation at “run” time instead.

Update, 20150127: Paul Khuong pointed me to his Scheme monad implementation based around delimited continuations in this twitter conversation. It doesn’t have dynamic-dispatch for automatically determining the appropriate bind implementation, but I suspect it could be added; perhaps in his join operator, which makes an interesting contrast to the technique explored in this post. The really nice thing about the delimited continuation approach to monads is that the return is implicit. You never mention return explicitly in monadic code. That makes a big difference, because return-type polymorphism is only needed to handle explicit returns! By placing the return right next to the appropriate run operation, the delimited continuation technique neatly sidesteps the problem.

Conclusions

Haskell’s return is generic: it injects a value into any monad at all. Which specific monad is used depends on the context. This is impossible to directly implement in general without static analysis.

Lacking such static analyses in dynamically-typed languages, achieving the equivalent of Haskell’s return is challenging, even though implementing particular monomorphic variations is straightforward.

The trick is to make return produce a kind of superposition which doesn’t collapse to a specific monad type until it touches a monad that already knows what type it is.

In this post, I’ve shown how this kind of monad “coercion” works well to give the kind of return-type polymorphism we need to make monads feel in Racket similar to the way they do in Haskell.