This is a framework for generic composable effectful asynchronous programming
basically using only objects and polymorphic variants.
Ability to write
"monad generic code".
Functors, Applicatives, Monads, ...
Extensible environment or reader.
Laziness via η-expansion.
Inferred checked exception handing.
Extensible with user defined effects.
Plays well with type inference allowing relatively concise code.
Interoperable with existing monadic libraries.
More specifically, this uses a
with the signatures specified using object types and a
higher-kinded type encoding. Programs can
be written against abstract generic interfaces allowing the same program to be
executed with multiple interpreters. Interpreters are implemented as objects and
passed via the reader pattern.
Polymorphic variants can be used for errors. The end result is a system that
provides a variety of effects (environment, checked exceptions, asynchronicity,
...) à la carte.
This library requires OCaml version 4.08 or later for
binding operators and some other
convenience features, but it seems a library like this could have already been
written for OCaml 3 (released in 2000). All the elements of this approach have
been known at least since 2014.
The following subsections introduce some aspects of the Rea framework via simple
examples. The code snippets are also extracted as a
test to ensure that they are accurate.
To begin, we just
Rea module, which brings a lot of generic
combinators to scope:
Let's look at a rather familiar example, the implementation of a naïve
exponential time Fibonacci function as an effectful computation. We use such a
trivial example in order to focus on some of the basics of the Rea framework. It
is assumed that the reader has basic familiarity with monads and the like. If
not, rest assured, there is
no shortage of material introducing monads
on the Internet.
module Fib = struct
And here is a naïvely written eager Fibonacci function implementation using the
lift'2 combinators from the framework:
let rec eager n = if n <= 1 then pure n else lift'2 ( + ) (eager (n - 2)) (eager (n - 1))
return combinator should already be familiar to you. The
lift'2 combinator (sometimes called
map2) combinator takes an ordinary
function of two plain value arguments and returns a function that works on
effectful computations. In this case, the
+ operator is used to combine the
results of the two recursive Fibonacci computations as a computation.
The below definition shows the type inferred for
eager (after renaming some
type variables to match what is used in the Rea framework):
let _ = (eager : int -> (< map' : 'e 'a 'b. ('b -> 'a) -> ('R, 'e, 'b, 'D) er -> ('R, 'e, 'a) s ; pair' : 'e 'a 'b. ('R, 'e, 'a, 'D) er -> ('R, 'e, 'b, 'D) er -> ('R, 'e, 'a * 'b) s ; pure' : 'e 'a. 'a -> ('R, 'e, 'a) s ; .. > as 'D) -> ('R, 'e, int) s)
The above undoubtedly looks rather complicated. We can simplify it by using type
abbreviations (class types) that are used in the definition of the framework
itself. In particular, each of the methods
above are defined by classes of the same name and each of those classes takes
two type parameters
let _ = (eager : int -> (< ('R, 'D) map' ; ('R, 'D) pair' ; ('R, 'D) pure' ; .. > as 'D) -> ('R, 'e, int) s)
The curious recursive use of the
'D type parameter makes sure that all of the
individual elements of the combined type agree on the type of the whole
Why do the class names have an apostrophe at the end? IOW, why
map' instead of
map? Well, there are lots of such a classes, multiple are often used in
type signatures, and they tend to have fairly common names. The apostrophe is
there to allow the classes to be conveniently at the top level of the
module with reduced risk of naming collisions with other libraries.
Recall that we used two combinators
lift'2 from the framework in
eager function. Obviously
pure corresponds to or requires the
lift'2 requires both of the
pair' classes. OCaml
conveniently infers the required combination for us.
We can simplify the type further. The combination of
pure' is well known. That combination is the signature of the so called
applicative functor. The Rea library defines the abbreviation
a class) for the combination:
let _ = (eager : int -> (('R, 'D) #applicative' as 'D) -> ('R, 'e, int) s)
We are not quite done yet. Ignore the
int ->. The remaining part of the type
is of the form
'D -> ('R, 'e, 'a) s. This is the type of "effect readers" of
('R, 'e, 'a, 'D) er type which is actually the main abstraction of the Rea
framework. Combinators in the Rea framework take effect readers as arguments and
return effect readers as results. So, we end up with the following equivalent
let _ = (eager : int -> ('R, 'e, int, (('R, 'D) #applicative' as 'D)) er)
Let's discuss the general form of the effect reader type
('R, 'e, 'a, 'D) er.
The type has four type arguments:
'Rrepresents a higher-kinded abstract type constructor that is specific to
a particular representation of effects.
'eis the type of errors or failures that may be signaled during
interpretation of the effect reader.
'ais the type of answers or results of the effect reader in case it does
not signal an error.
'Dis the type of the dictionary of capabilities or of the environment
required by the effect reader. In other words, it is the type of the effect
('R, +'e, +'a) s type we saw earlier is the abstract effect signature
type. It represents the application
('e, 'a) 'R of the higher-kinded
constructor to the
Now, let's interpret the effect reader type of our Fibonacci function:
('R, 'e, int, (('R, 'D) #applicative' as 'D)) er
First of all, we can see that it returns an
int in case it does not fail with
an error. Speaking of which, the type of errors is
'e, which, due to
parametricity, means that it cannot produce errors. In other words, we (and the
OCaml type system) statically know that it cannot fail. The representation
'R is also parametric, which means that it does not
require a specific representation. The dictionary type
'D is also parametric
and must be a subtype of
applicative'. In other words, we can run the effect
reader with any interpreter that implements the
For example, we can use the identity monad implementation provided by the Rea
let () = assert (55 = Identity.of_rea (run Identity.monad (eager 10)))
The identity monad does not perform any effects per se and the values computed
during interpretation are not wrapped. In other words, with the identity monad,
('R, 'e, 'a) s is equivalent to
'a. This equivalence is witnessed by a pair
of identity functions
Identity.to_rea. By itself, the
identity monad cannot support the error handling effects of the Rea framework
nor it can support asynchronous computations. It may seem like a useless
interpreter, but it actually has many interesting applications.
run function used above just passes the interpreter to the computation.
One could also just write
eager 10 Identity.monad.
let () = assert (`Ok 55 = Tailrec.run Tailrec.sync (eager 10))
The tail recursive interpreter provides both a synchronous, as seen above, and
an asynchronous interpreter and also supports error handling. The Rea framework
also provides a number of other interpreter implementations, that we could use
here, but let's move on.
At the beginning we mentioned that the
eager function is written naïvely. The
problem with it is that it is eager: as soon as the first argument is passed to
it, a whole tree of suspended computations is built.
Now, we saw that the result of
eager n is actually a function — namely
an effect reader. Because it is a function, we can use (eta) η-expansion to make
it lazy. For this purpose the Rea framework provides a simple
that takes a thunk and returns a single parameter function. Using
eta'0 we can
write an η-expanded Fibonacci function as follows:
let rec inert n = eta'0 @@ fun () -> if n <= 1 then pure n else lift'2 ( + ) (inert (n - 2)) (inert (n - 1))
eager functions have exactly the same types. The difference is
that the η-expanded
inert function returns in O(1) time with a function
inert n = fun d -> ...
eager function builds a complete computation tree
eager 0 = pure 0 eager 1 = pure 1 eager 2 = lift'2 (+) (pure 0) (pure 1) eager 3 = lift'2 (+) (pure 1) (lift'2 (+) (pure 0) (pure 1)) eager 4 = lift'2 (+) (lift'2 (+) (pure 0) (pure 1)) (lift'2 (+) (pure 1) (lift'2 (+) (pure 0) (pure 1))) ...
taking exponential time and space.
Of course, when either one of the effect readers is interpreted, it will take
exponential time due to the naïve exponential Fibonacci algorithm. Again, the
difference is that one of the computations is generated lazily on demand while
the other is generated eagerly.
Just like with the previous
eager, we can use multiple interpreters to run
let () = assert (55 = Identity.of_rea (run Identity.monad (inert 10))) let () = assert (`Ok 55 = Tailrec.run Tailrec.sync (inert 10))
This concludes the Fibonacci example.
We now have a basic understanding of effect readers. They are just functions
that take a dictionary of capabilities aka an interpreter as an argument.
Let's continue with an example demonstrating the extensible environment of the
Rea approach. To do so, let's implement a very rudimentary interpreter using a
modular approach. Although the techniques in this section scale to more
interesting language processors, we will keep the example very minimal.
First we'll implement a simple arithmetic language:
module Num = struct
For later use, we'll define a structural type matching the arithmetic language:
type 't t = [`Num of int | `Uop of [`Neg] * 't | `Bop of [`Add | `Mul] * 't * 't]
we use polymorphic variants with open recursion for the AST representation.
We will also use open recursion in the
let uop = function | `Neg -> ( ~-) let bop = function | `Add -> ( + ) | `Mul -> ( * ) let eval eval = eta'1 @@ function | `Num _ as v -> pure v | `Uop (op, x) -> ( eval x >>= function | `Num x -> pure @@ `Num (uop op x) | x -> fail @@ `Error_attempt_to_apply_uop (op, x)) | `Bop (op, l, r) -> ( eval l <*> eval r >>= function | `Num l, `Num r -> pure @@ `Num (bop op l r) | l, r -> fail @@ `Error_attempt_to_apply_bop (op, l, r))
eta'1 combinator is another way to write η-expanded functions. It takes a
function and returns a two parameter function. The
is for monadic bind and
let+ ... and+ ... is for
applicative pairing of computation.
Errors are reported with the
fail combinator. We use polymorphic variants for
After some cleaning up, the type inferred for
eval is roughly equivalent to
the following definition:
let _ = (eval : ('t -> ( 'R, ([> `Error_attempt_to_apply_bop of ([< `Add | `Mul] as 'bop) * ([> `Num of int] as 'v) * 'v | `Error_attempt_to_apply_uop of ([< `Neg] as 'uop) * 'v ] as 'e), 'v, (< ('R, 'D) sync' ; .. > as 'D) ) er) -> [< 't t] -> ('R, 'e, [> `Num of int], 'D) er)
Notice that the above type shows both of the errors that might arise from
sync' class is a combination of
errors' is a
tryin'. In other words, it provides both the basic
monadic capabilities for sequencing and the ability to signal and handle errors.
Let's then move on to implement (lambda) λ-expressions:
module Lam = struct
Like with the arithmetic language we define a type for the language:
module Id = String type 't t = [`Lam of Id.t * 't | `App of 't * 't | `Var of Id.t]
Deviating from Garrigue's example, we'll use an environment of bindings
module Bindings = Map.Make (Id)
that maps variables to values. Recall that the arithmetic language above knows
nothing about environments. To pass around the environment we'll use the
extensible environment of the Rea framework. For that we define a new class
['v] bindings that exposes a
class ['v] bindings : object method bindings : 'v Bindings.t Prop.t end = object val mutable v : 'v Bindings.t = Bindings.empty method bindings = Prop.make (fun () -> v) (fun x -> v <- x) end
Prop.t type, whose values are introduced by
Prop.make, is provided by
the Rea framework to concisely expose a mutable instance variable as a readable
and functionally updatable property. To be clear, it is not actually possible to
observably mutate the
v instance variable outside of the
For easy access to the
bindings method we define a trivial extractor:
let bindings d = d#bindings
Now we are ready to write the open
eval function for λ-expressions:
let eval eval = eta'1 @@ function | `Lam (i, e) -> let+ bs = get bindings in `Fun (bs, i, e) | `App (f, x) -> ( eval f >>= function | `Fun (bs, i, e) -> let* v = eval x in setting bindings (Bindings.add i v bs) (eval e) | f -> fail @@ `Error_attempt_to_apply f) | `Var i -> ( get_as bindings (Bindings.find_opt i) >>= function | None -> fail @@ `Error_unbound_var i | Some v -> pure v)
let+ binding operator is the
map operation of functors. The
combinator reads the value of a property extracted from the environment. The
setting combinator runs a computation with the value of a property
functionally updated to the given value. The
get_as combinator reads a
property and also maps it through the given function.
The following definition shows a cleaned up type for the
let _ = (eval : ('t -> ( 'R, ([> `Error_attempt_to_apply of ([> `Fun of 'v Bindings.t * Id.t * 't] as 'v) | `Error_unbound_var of Id.t ] as 'e), 'v, (< ('R, 'D) sync' ; 'v bindings ; .. > as 'D) ) er) -> [< 't t] -> ('R, 'e, 'v, 'D) er)
bindings as part of the
'D dictionary of capabilities.
Moving on to compose the full interpreter
module Full = struct
from the above parts, we write a recursive
eval function that dispatches to
one of the above
eval functions depending on the input:
let rec eval = function | #Num.t as e -> Num.eval eval e | #Lam.t as e -> Lam.eval eval e
Again, here is a cleaned up type for the
let _ = (eval : ([< 't Num.t | 't Lam.t] as 't) -> ( 'R, [> `Error_attempt_to_apply of ([> `Fun of 'v Lam.Bindings.t * Lam.Id.t * 't | `Num of int] as 'v) | `Error_attempt_to_apply_bop of [`Add | `Mul] * 'v * 'v | `Error_attempt_to_apply_uop of [`Neg] * 'v | `Error_unbound_var of Lam.Id.t ], 'v, (< ('R, 'D) sync' ; 'v Lam.bindings ; .. > as 'D) ) er)
Notice how the type combines
the arithmetic and λ-expressions (as
the errors (
the value type (as
the capability dictionaries (as
To actually run
eval we will need an effect interpreter that provides the
sync' capabilities as well as
bindings. There is an interpreter for the
result type that we can use for the
sync' capabilities. For the
bindings we can just use
bindings. Here is how:
let () = assert ( Error (`Error_unbound_var "y") = StdRea.Result.of_rea (run (object inherit [_] StdRea.Result.monad_errors inherit [_] Lam.bindings end) (eval (`App (`Lam ("x", `Bop (`Add, `Num 2, `Var "y")), `Num 1)))))
Another interpreter that comes bundled with the Rea framework that provides
sync' is the
Tailrec interpreter we used previously. So, we could also use
Tailrec as follows:
let () = assert ( `Ok (`Num 3) = Tailrec.run (object inherit [_] Tailrec.sync inherit [_] Lam.bindings end) (eval (`App (`Lam ("x", `Bop (`Add, `Num 2, `Var "x")), `Num 1))))
We could also use the asynchronous version of the
Tailrec interpreter. To
ensure that neither errors nor results are implicitly ignored, the
Tailrec.spawn function requires that the computation throws
(). We need to wrap the computation with handlers:
let () = let result = ref @@ Ok (`Num 0) in Tailrec.spawn (object inherit [_] Tailrec.async inherit [_] Lam.bindings end) (eval (`App (`Lam ("x", `Bop (`Add, `Num 2, `Var "x")), `Num 1)) |> tryin (fun e -> pure (result := Error e)) (fun v -> pure (result := Ok v))); assert (!result = Ok (`Num 3))
tryin combinator allows us to handle errors and continue with results.
Since we handle all of the errors, the error type for the whole computation
becomes parametric and can be unified with
Normally one cannot assume that a computation started with
completes immediately. In this case we had nothing asynchronous in the
implementation and nothing asynchronous running in the background.
This concludes the example. We now know about the extensible environment as well
as about error handling.
In the previous section we wrote a simple modular interpreter that used the
extensible environment of Rea to pass along the bindings capability. Adding to
the extensible environment is easy — you just declare what you want. In a
more complex program different parts of the program might require different
capabilities. The top-level of the program then ends up having to know about
about all of those capabilities. This is a modularity problem.
Imagine using a library using the framework. A new version of the library comes
along and your program no longer compiles just because the library now
internally needs a different set of capabilities compared to the previous
version. That is not good. We need a way to handle effects locally. Ideally we'd
like to be able to say that a program requires a specific set of capabilities to
run and also requires the freedom to extend the environment with some other
capabilities (i.e. that it "lacks" or is "disjoint" from the additional
capabilities). This way the environment could be efficiently extended using some
form of polymorphic record extension. Unfortunately OCaml's objects do not offer
such a form of polymorphism. What can we do?
OCaml does offer half of what we need: we can declare that we need at least some
specific capabilities from the environment. What we can do then is to project
those capabilities out of the environment and build our own scoped environment.
module Scoped = struct
Let's see how that is done. Here is an
eval function that does not require
bindings from the environment:
let eval e = Full.eval e |> mapping_env @@ fun o -> object inherit [_, _, _] sync'of o inherit [_] Lam.bindings end
mapping_env combinator allows us to get the outer environment
substitute our own. For that we use the
sync'of class above. It is given an
object that must be of some subtype of
sync'. It then provides the
capabilities by delegating to the given object. In other words, we project the
sync' capability out of the environment
The following definition shows a cleaned up type for the closed
let _ = (eval : ([< 't Num.t | 't Lam.t] as 't) -> ( 'R, [> `Error_attempt_to_apply of ([> `Fun of 'v Lam.Bindings.t * Lam.Id.t * 't | `Num of int] as 'v) | `Error_attempt_to_apply_bop of [`Add | `Mul] * 'v * 'v | `Error_attempt_to_apply_uop of [`Neg] * 'v | `Error_unbound_var of Lam.Id.t ], 'v, (('R, 'D) #sync' as 'D) ) er)
bindings capability no longer appears in the environment type
'D and we
can run it with just the base
let () = assert ( `Ok (`Num 42) = Tailrec.run Tailrec.sync (eval (`App (`Lam ("x", `Bop (`Add, `Num 2, `Var "x")), `Num 40))))
Being able to scope effects in this fashion is important for modularity.
Unfortunately doing so is not free as each projection adds some delegation
overhead to the effect invocations. Also, what we dealth with above is the easy
case where we modularized a first-order function. Higher-order functions where
the caller supplied functions also need to use the environment require wrapping
the user supplied functions replacing the environment back to what it was.
This concludes the example. We now know more about the extensible environment.
Let's suppose next that we have an existing monadic library that we need to
interoperate with. For the sake of argument, let's assume the following
continuation monad implementation:
module Cont : sig type 'a t val return : 'a -> 'a t val bind : 'a t -> ('a -> 'b t) -> 'b t val callcc : (('a -> 'b t) -> 'a t) -> 'a t val run : 'a t -> 'a end = struct type 'a t = ('a -> unit) -> unit let return x k = k x let bind xK xyK k = xK (fun x -> (xyK x) k) let callcc kxK k = kxK (fun x _ -> k x) k let run xK = let result = ref None in xK (fun x -> result := Some x); Option.get !result end
First we notice that we ran into a bit of a snag. Although Rea provides a number
of related effects, at the time of writing, no
callcc effect is provided.
Fortunately nothing prevents users from extending the framework. We just write
class virtual ['R, 'D] callcc' = object method virtual callcc' : 'e 'f 'a 'b. (('a -> ('R, 'f, 'b, 'D) er) -> ('R, 'e, 'a, 'D) er) -> ('R, 'e, 'a) s end
And the generic effect reader combinator
let callcc f (d: (_, _) #callcc') = d#callcc' f
Now, how do we relate
'a Cont.t with the Rea framework? What we need to do is
to embed it into the abstract
('R, 'e, 's) s effect signature type. We create
an abstract representation type
r corresponding to the
constructor and an injection projection pair:
module ContRea = struct type r external to_rea : 'a Cont.t -> (r, 'e, 'a) s = "%identity" external of_rea : (r, 'e, 'a) s -> 'a Cont.t = "%identity"
Rea probably should provide a functor for the above, but it currently doesn't.
external fn : s -> t = "%identity"
construct tells OCaml that
fn is an external function of type
s -> t that is
actually the identity function. Because both the above type
r and the type
(_, _, _) s from Rea are abstract, the above two coercions are safe — as
long as we don't provide other incompatible coercions between the abstract
Now we can write down an interpreter class
class ['D] monad_callcc = object (d: 'D) inherit [r, 'D] monad'd method pure' x = to_rea (Cont.return x) method bind' x f = to_rea (Cont.bind (of_rea (x d)) (fun x -> of_rea (f x d))) inherit [r, 'D] callcc' method callcc' f = to_rea (Cont.callcc (fun k -> of_rea (f (fun x _ -> to_rea (k x)) d))) end
Cont. As an example, we can use it with the previously defined
let () = assert (55 = Cont.run (of_rea (run (new monad_callcc) (Fib.inert 10))))
callcc is also available:
let () = assert ( 101 = Cont.run (of_rea (run (new monad_callcc) (callcc (fun k -> k 101)))))
All the generic combinators for monads and simpler functors, and the extensible
environment are now available when constructing
Cont computations via the Rea
We now know that we can write effect interpreters using existing monadic
libraries that run their computations embedded into the Rea framework and that
we can also introduce new effects into the framework.
Wonder what the last example will be about?
Indeed, mapping with an effect reader,
map_er, or "traverse" as it is often
called, tends to be the answer to many problems. So, let's explore the topic a
bit. We'll build on the earlier modular interpreter example.
module Answer = struct
Although we could use a modular approach and build traversal functions modularly
for expressions, that's not really the point here. So, let's just work on the
full AST. Here is a generic traversal function for the structural AST:
let map_er' nE o1E o2E iE eE = eta'1 @@ function | `Num x -> map_er'1 nE x >>- fun x -> `Num x | `Uop x -> map_er'2 o1E eE x >>- fun x -> `Uop x | `Bop x -> map_er'3 o2E eE eE x >>- fun x -> `Bop x | `Lam x -> map_er'2 iE eE x >>- fun x -> `Lam x | `App x -> map_er'2 eE eE x >>- fun x -> `App x | `Var x -> map_er'1 iE x >>- fun x -> `Var x
Instead of just traversing over the direct subexpressions of an expression, the
map_er' also allows traversing over the numbers, unary and binary
operators, and identifiers. The constructors of the datatype are traversed in a
systematic way using a family of canned
map_er'n functions for each arity of a
tuple the constructors are carrying.
Why call it
map_er rather than
traverse? Well, mapping over a datatype with
an effect constructor is just one of many useful generalizations of ordinary
Systematic naming hopefully makes the API easier to learn.
I find it convenient to implement traversal for a structural type in the above
manner as it allows one to easily specialize to the basic traversal over
let map_er eE = map_er' pure pure pure pure eE
and also use the generic traverse for other purposes.
The type of the
map_er over subexpressions is seen in the below definition:
type 't t = [ 't Num.t | 't Lam.t ] let _ = (map_er : ('s -> ('R, 'e, 't, (('R, 'D) #applicative' as 'D)) er) -> [< 's t] -> ('R, 'e, [> 't t], 'D) er)
What can one do with traversals? Well, a lot of things.
For example, here is a function that recursively traverses an expression to find
out whether a given variable is free or not in a given expression:
let rec is_free i' = function | `Var i -> i = i' | `Lam (i, _) when i = i' -> false | e -> Traverse.to_exists map_er (is_free i') e let () = assert (is_free "y" (`App (`Lam ("x", `Var "x"), `Var "y"))) let () = assert (not (is_free "x" (`App (`Lam ("x", `Var "x"), `Var "y"))))
map_er we can treat all the "basic" cases of the datatype generically
and focus on the two cases that are interesting with respect to bindings.
But we kind of got ahead of ourselves. How does
map_er does is that it maps every element of a datatype to an
effect and then combines those effects to reconstruct the shape of the datatype.
If, for example, we use the previously mentioned
Identity monad to run the
effects, then we basically get a
map function for the datatype.
Effects are not limited to simply returning a value of the answer type. We
already previously saw the
fail effect, which doesn't return an answer.
Another example is the
Constant functor, which, as its name suggests, carries
along a constant value of some type and the answer type is a phantom type. The
Constant functor can be augmented to an applicative over a monoid. For
example, we can use disjunction. And that is what
Traverse.to_exists does. It
maps the traversed elements to boolean constants as returned by the given
predicate. Then those booleans are combined using disjunction.
But this is all quite common knowledge. Why are we discussing this? Well, recall
that the Rea framework provides a form of laziness via η-expansion. That same
laziness also works with traversals over monoids, among other things. So, when
Traverse.to_exists map_er (is_free i') e, instead of first eagerly
building the computation for the whole expression tree
e, as one might expect
to be the case in a strict language like OCaml, the computation is built on
demand and actually stops roughly as soon as the first free variable occurrence
has been encountered. So, although the use of objects and whatnot brings quite a
bit of overhead, at least we actually get the desired asymptotic time complexity
This concludes the example and the introduction.
We now know about most aspects of the Rea framework. Perhaps the best way to
learn more is to take a brief look at the
reference manual and
start using the framework. Have fun!
The main drawbacks of this approach come from the deficiencies and limitations
of OCaml's objects:
OCaml does not aggressively optimize (statically known) method invocations.
This means that every effect invocation has some overhead.
OCaml's object system does not support adding methods to or removing methods
from objects (i.e. polymorphic record extension). This means that effects
cannot be easily handled locally.
On the other hand, this approach arguably has a rather straightforward
implementation and is convenient to use (modulo the inability to handle effects
Unfortunately the library
reference manual is
rather unfinished at the moment.