Effectful OCaml with Objects and Variants

This is a framework for generic composable effectful asynchronous programming basically using only objects and polymorphic variants.

Features:

• Ability to write "monad generic code".
• Functors, Applicatives, Monads, ...
• Extensible environment or reader.
• Laziness via η-expansion.
• Inferred checked exception handing.
• Asynchronous computations.
• Extensible with user defined effects.
• Plays well with type inference allowing relatively concise code.
• Interoperable with existing monadic libraries.

More specifically, this uses a tagless-final approach 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.

Contents

• Introduction
  • Basics
  • Extensible environment
  • Projections
  • Interoperability
  • Traversals
• Limitations

Introduction

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.

Basics

To begin, we just open the Rea module, which brings a lot of generic combinators to scope:

open Rea

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 pure and 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))

The pure aka 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 map', pair', and pure' seen above are defined by classes of the same name and each of those classes takes two type parameters ('R, 'D):

  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 composition.

Why do the class names have an apostrophe at the end? IOW, why map' instead of just 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 Rea module with reduced risk of naming collisions with other libraries.

Recall that we used two combinators pure and lift'2 from the framework in the eager function. Obviously pure corresponds to or requires the pure' class. lift'2 requires both of the map' and pair' classes. OCaml conveniently infers the required combination for us.

We can simplify the type further. The combination of map', pair', and pure' is well known. That combination is the signature of the so called applicative functor. The Rea library defines the abbreviation applicative' (as 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 the ('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 type:

  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:

• 'R represents a higher-kinded abstract type constructor that is specific to a particular representation of effects.

• 'e is the type of errors or failures that may be signaled during interpretation of the effect reader.

• 'a is the type of answers or results of the effect reader in case it does not signal an error.

• 'D is 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 interpreter.

The ('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 'R type constructor to the 'e and 'a arguments.

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 identification type '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 applicative' effect capabilities.

For example, we can use the identity monad implementation provided by the Rea framework:

  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.of_rea and 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.

The run function used above just passes the interpreter to the computation. One could also just write eager 10 Identity.monad.

We can also use the self tail recursive (and Js_of_ocaml safe) interpreter also provided by the Rea framework:

  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 eta'0 function 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))

The inert and 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 -> ...

while the 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 inert computations:

  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.

end

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.

Extensible environment

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]

Like in

Code reuse through polymorphic variants by Jacques Garrigue

we use polymorphic variants with open recursion for the AST representation.

We will also use open recursion in the eval functions:

  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))

The eta'1 combinator is another way to write η-expanded functions. It takes a function and returns a two parameter function. The >>= aka bind aka let* is for monadic bind and <*> aka pair aka let+ ... and+ ... is for applicative pairing of computation.

Errors are reported with the fail combinator. We use polymorphic variants for errors.

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 eval.

The sync' class is a combination of monad' and errors' and errors' is a combination offail' and tryin'. In other words, it provides both the basic monadic capabilities for sequencing and the ability to signal and handle errors.

end

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 bindings property:

  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

The 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 bindings class.

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)

The let+ binding operator is the map operation of functors. The get 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 eval function:

  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)

Notice the bindings as part of the 'D dictionary of capabilities.

end

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 eval function:

  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 't),
• the errors ([> ...]),
• the value type (as 'v), and
• the capability dictionaries (as 'D).

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 standard 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 nothing and returns (). 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))

The 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 nothing.

Normally one cannot assume that a computation started with Tailrec.spawn completes immediately. In this case we had nothing asynchronous in the implementation and nothing asynchronous running in the background.

end

This concludes the example. We now know about the extensible environment as well as about error handling.

Projections

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

The mapping_env combinator allows us to get the outer environment o and 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 sync' capabilities by delegating to the given object. In other words, we project the sync' capability out of the environment o.

The following definition shows a cleaned up type for the closed eval:

  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)

The bindings capability no longer appears in the environment type 'D and we can run it with just the base Tailrec interpreter:

  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.

end

This concludes the example. We now know more about the extensible environment.

Interoperability

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 down the callcc' class:

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 callcc:

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 Cont.t type 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. The special

  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 types.

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

based on Cont. As an example, we can use it with the previously defined Fibonacci computations:

  let () =
    assert (55 = Cont.run (of_rea (run (new monad_callcc) (Fib.inert 10))))

And 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 framework.

end

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.

Traversals

Wonder what the last example will be about?

The answer is always traverse. — Rúnar

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 above 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 pure functions:

• map -> map_er
• find -> find_er
• fold -> fold_er
• ...

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 subexpressions

  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"))))

Using 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 Traverse.to_exists work?

Well, what 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 we call 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 for is_free.

This concludes the example and the introduction.

end

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!

Limitations

The main drawbacks of this approach come from the 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 difficulty of handling effects locally).

Unfortunately the library reference manual is rather unfinished at the moment.