(Introduced in 5.0)

\textit{Note: Effect handlers in OCaml 5.0 should be considered experimental.
Effect handlers are exposed in the standard library as a thin wrapper
around their implementations in the runtime. They are not supported as a
language feature with new syntax. You can rely on them to build non-local
control-flow abstractions such as user-level threading that do not expose
the effect handler primitives to the user. Expect breaking changes in the
future.}

Effect handlers are a mechanism for modular programming with user-defined
effects. Effect handlers allow the programmers to describe
\textit{computations} that \textit{perform} effectful \textit{operations},
whose meaning is described by \textit{handlers} that enclose the computations.
Effect handlers are a generalization of exception handlers and enable non-local
control-flow mechanisms such as resumable exceptions, lightweight threads,
coroutines, generators and asynchronous I/O to be composably expressed. In this
tutorial, we shall see how some of these mechanisms can be built using effect
handlers.

\subsection{s:effects-basics}{Basics}

To understand the basics, let us define an effect (that is, an operation) that
takes an integer argument and returns an integer result. We name this effect
"Xchg".

\begin{caml_example*}{verbatim}
open Effect
open Effect.Deep

type _ Effect.t += Xchg: int -> int t
let comp1 () = perform (Xchg 0) + perform (Xchg 1)
\end{caml_example*}

We declare the exchange effect "Xchg" by extending the pre-defined extensible
variant type "Effect.t" with a new constructor "Xchg: int -> int t". The
declaration may be intuitively read as ``the "Xchg" effect takes an integer
parameter, and when this effect is performed, it returns an integer''. The
computation "comp1" performs the effect twice using the "perform" primitive and
returns their sum.

We can handle the "Xchg" effect by implementing a handler that always returns
the successor of the offered value:

\begin{caml_example}{verbatim}
try_with comp1 ()
{ effc = fun (type a) (eff: a t) ->
    match eff with
    | Xchg n -> Some (fun (k: (a, _) continuation) ->
        continue k (n+1))
    | _ -> None }
\end{caml_example}

"try_with" runs the computation "comp1 ()" under an effect handler that handles
the "Xchg" effect. As mentioned earlier, effect handlers are a generalization
of exception handlers. Similar to exception handlers, when the computation
performs the "Xchg" effect, the control jumps to the corresponding handler.
However, unlike exception handlers, the handler is also provided with the
delimited continuation "k", which represents the suspended computation between
the point of "perform" and this handler.

The handler uses the "continue" primitive to resume the suspended computation
with the successor of the offered value. In this example, the computation
"comp1" performs "Xchg 0" and "Xchg 1" and receives the values "1" and "2"
from the handler respectively. Hence, the whole expression evaluates to "3".

It is useful to note that the we must use locally abstract type "(type a)" in
the effect handler. The type "Effect.t" is a GADT, and the effect declarations
may have different type parameters for different effects. The type parameter
"a" in the type "a Effect.t" represents the type of the value returned when
performing the effect. From the fact that "eff" has type "a Effect.t" and from
the fact that "Xchg n" has type "int Effect.t", the type-checker deduces that
"a" must be "int", which is why we are allowed to pass the integer value "n+1"
as an argument to "continue k".

Another point to note is that the catch-all case ``"| _ -> None"'' is necessary
when handling effects. This case may be intuitively read as ``forward the
unhandled effects to the outer handler''.

In this example, we use the \emph{deep} version of the effect handlers here as
opposed to the \emph{shallow} version. A deep handler monitors a computation
until the computation terminates (either normally or via an exception), and
handles all of the effects performed (in sequence) by the computation. In
contrast, a shallow handler monitors a computation until either the computation
terminates or the computation performs one effect, and it handles this single
effect only. In situations where they are applicable, deep handlers are usually
preferred. An example that utilises shallow handlers is discussed later
in~\ref{s:effects-shallow}.

\subsection{s:effects-concurrency}{Concurrency}

The expressive power of effect handlers comes from the delimited continuation.
While the previous example immediately resumed the computation, the computation
may be resumed later, running some other computation in the interim. Let us
extend the previous example and implement message-passing concurrency between
two concurrent computations using the "Xchg" effect. We call these concurrent
computations \textit{tasks}.

A task either is in a suspended state or is completed. We represent the task
status as follows:

\begin{caml_example*}{verbatim}
type 'a status =
  Complete of 'a
| Suspended of {msg: int; cont: (int, 'a status) continuation}
\end{caml_example*}

A task either is complete, with a result of type "'a", or is suspended with the
message "msg" to send and the continuation "cont". The type "(int,'a status)
continuation" says that the suspended computation expects an "int" value to
resume and returns a "'a status" value when resumed.

Next, we define a "step" function that executes one step of computation until
it completes or suspends:

\begin{caml_example*}{verbatim}
let step (f : unit -> 'a) () : 'a status =
  match_with f ()
  { retc = (fun v -> Complete v);
    exnc = raise;
    effc = fun (type a) (eff: a t) ->
      match eff with
      | Xchg msg -> Some (fun (cont: (a, _) continuation) ->
          Suspended {msg; cont})
      | _ -> None }
\end{caml_example*}

The argument to the "step" function, "f", is a computation that can perform an
"Xchg" effect and returns a result of type "'a". The "step" function itself
returns a "'a status" value.

In the "step" function, we use the "match_with" primitive. Like "try_with",
"match_with" primitive installs an effect handler. However, unlike "try_with",
where only the effect case "effc" is provided, "match_with" expects the
handlers for the value ("retc") and exceptional ("exnc") return cases. In fact,
"try_with" can be defined using "match_with" as follows: "let try_with f v
{effc} = match_with f v {retc = Fun.id; exnc = raise; effc}".

In the "step" function,

\begin{itemize}
  \item Case "retc": If the computation returns with a value "v", we return
    "Complete v".
  \item Case "exnc": If the computation raises an exception, then the handler
    raises the same exception.
  \item Case "effc": If the computation performs the effect "Xchg msg" with the
    continuation "cont", then we return "Suspended{msg;cont}". Thus, in this
    case, the continuation cont is not immediately invoked by the handler;
    instead, it is stored in a data structure for later use.
\end{itemize}

Since the "step" function handles the "Xchg" effect, "step f" is a computation
that does not perform the "Xchg" effect. It may however perform other effects.
Moreover, since we are using deep handlers, the continuation "cont" stored in
the status does not perform the "Xchg" effect.

We can now write a simple scheduler that runs a pair of tasks to completion:

\begin{caml_example*}{verbatim}
let rec run_both a b =
  match a (), b () with
  | Complete va, Complete vb -> (va, vb)
  | Suspended {msg = m1; cont = k1},
    Suspended {msg = m2; cont = k2} ->
      run_both (fun () -> continue k1 m2)
               (fun () -> continue k2 m1)
  | _ -> failwith "Improper synchronization"
\end{caml_example*}

Both of the tasks may run to completion, or both may offer to exchange a
message. In the latter case, each computation receives the value offered by the
other computation. The situation where one computation offers an exchange while
the other computation terminates is regarded as a programmer error, and causes
the handler to raise an exception

We can now define a second computation that also exchanges two messages:

\begin{caml_example*}{verbatim}
let comp2 () = perform (Xchg 21) * perform (Xchg 21)
\end{caml_example*}

Finally, we can run the two computations together:

\begin{caml_example}{verbatim}
run_both (step comp1) (step comp2)
\end{caml_example}

The computation "comp1" offers the values "0" and "1" and in exchange receives
the values "21" and "21", which it adds, producing "42". The computation
"comp2" offers the values "21" and "21" and in exchange receives the values "0"
and "1", which it multiplies, producing "0". The communication between the two
computations is programmed entirely inside "run_both". Indeed, the definitions
of "comp1" and "comp2", alone, do not assign any meaning to the "Xchg" effect.

\subsection{s:effects-user-threads}{User-level threads}

Let us extend the previous example for an arbitrary number of tasks. Many
languages such as GHC Haskell and Go provide user-level threads as a primitive
feature implemented in the runtime system. With effect handlers, user-level
threads and their schedulers can be implemented in OCaml itself. Typically,
user-level threading systems provide a "fork" primitive to spawn off a new
concurrent task and a "yield" primitive to yield control to some other task.
Correspondingly, we shall declare two effects as follows:

\begin{caml_example*}{verbatim}
type _ Effect.t += Fork : (unit -> unit) -> unit t
                 | Yield : unit t
\end{caml_example*}

The "Fork" effect takes a thunk (a suspended computation, represented as a
function of type "unit -> unit") and returns a unit to the performer. The
"Yield" effect is unparameterized and returns a unit when performed. Let us
consider that a task performing an "Xchg" effect may match with any other task
also offering to exchange a value.

We shall also define helper functions that simply perform these effects:

\begin{caml_example*}{verbatim}
let fork f = perform (Fork f)
let yield () = perform Yield
let xchg v = perform (Xchg v)
\end{caml_example*}

A top-level "run" function defines the scheduler:

\begin{caml_example*}{verbatim}
(* A concurrent round-robin scheduler *)
let run (main : unit -> unit) : unit =
  let exchanger = ref None in (* waiting exchanger *)
  let run_q = Queue.create () in (* scheduler queue *)
  let enqueue k v =
    let task () = continue k v in
    Queue.push task run_q
  in
  let dequeue () =
    if Queue.is_empty run_q then () (* done *)
    else begin
      let task = Queue.pop run_q in
      task ()
    end
  in
  let rec spawn (f : unit -> unit) : unit =
    match_with f () {
      retc = dequeue;
      exnc = (fun e ->
        print_endline (Printexc.to_string e);
        dequeue ());
      effc = fun (type a) (eff : a t) ->
        match eff with
        | Yield -> Some (fun (k : (a, unit) continuation) ->
            enqueue k (); dequeue ())
        | Fork f -> Some (fun (k : (a, unit) continuation) ->
            enqueue k (); spawn f)
        | Xchg n -> Some (fun (k : (int, unit) continuation) ->
            begin match !exchanger with
            | Some (n', k') ->
                exchanger := None; enqueue k' n; continue k n'
            | None -> exchanger := Some (n, k); dequeue ()
            end)
        | _ -> None
    }
  in
  spawn main
\end{caml_example*}

We use a mutable queue "run_q" to hold the scheduler queue. The FIFO queue
enables round-robin scheduling of tasks in the scheduler. "enqueue" inserts
tasks into the queue, and "dequeue" extracts tasks from the queue and runs
them. The reference cell "exchanger" holds a (suspended) task offering to
exchange a value. At any time, there is either zero or one suspended task that
is offering an exchange.

The heavy lifting is done by the "spawn" function. The "spawn" function runs
the given computation "f" in an effect handler. If "f" returns with a value
(case "retc"), we dequeue and run the next task from the scheduler queue. If
the computation "f" raises an exception (case "exnc"), we print the exception
and run the next task from the scheduler.

The computation "f" may also perform effects. If "f" performs the "Yield"
effect, the current task is suspended (inserted into the queue of ready tasks),
and the next task from the scheduler queue is run. If the effect is "Fork f",
then the current task is suspended, and the new task "f" is executed
immediately via a tail call to "spawn f". Note that this choice to run the new
task first is arbitrary. We could very well have chosen instead to insert the
task for "f" into the ready queue and resumed "k" immediately.

If the effect is "Xchg", then we first check whether there is a task waiting to
exchange. If so, we enqueue the waiting task with the current value being
offered and immediately resume the current task with the value being offered.
If not, we make the current task the waiting exchanger, and run the next task
from the scheduler queue.

Note that this scheduler code is not perfect -- it can leak resources. We shall
explain and fix this in the next section~\ref{s:effects-discontinue}.

Now we can write a concurrent program that utilises the newly defined
operations:

\begin{caml_example}{verbatim}
open Printf

let _ = run (fun _ ->
  fork (fun _ ->
    printf "[t1] Sending 0\n";
    let v = xchg 0 in
    printf "[t1] received %d\n" v);
  fork (fun _ ->
    printf "[t2] Sending 1\n";
    let v = xchg 1 in
    printf "[t2] received %d\n" v))
\end{caml_example}

Observe that the messages from the two tasks are interleaved. Notice also that
the snippet above makes no reference to the effect handlers and is in direct
style (no monadic operations). This example illustrates that, with effect
handlers, the user code in a concurrent program can remain in simple direct
style, and the use of effect handlers can be fully contained within the
concurrency library implementation.

\subsubsection{s:effects-discontinue}{Resuming with an exception}

In addition to resuming a continuation with a value, effect handlers also
permit resuming by raising an effect at the point of perform. This is done with
the help of the "discontinue" primitive. The "discontinue" primitive helps
ensure that resources are always eventually deallocated, even in the presence
of effects.

For example, consider the dequeue operation in the previous example reproduced
below:

\begin{caml_example*}{verbatim}
let[@ellipsis] run_q = Queue.create ()
let dequeue () =
  if Queue.is_empty run_q then () (* done *)
  else (Queue.pop run_q) ()
\end{caml_example*}

If the scheduler queue is empty, dequeue considers that the scheduler is done
and returns to the caller. However, there may still be a task waiting to
exchange a value (stored in the reference cell "exchanger"), which remains
blocked forever! If the blocked task holds onto resources, these resources are
leaked. For example, consider the following task:

\begin{caml_example*}{verbatim}
let leaky_task () =
  fork (fun _ ->
    let oc = open_out "secret.txt" in
    Fun.protect ~finally:(fun _ -> close_out oc) (fun _ ->
      output_value oc (xchg 0)))
\end{caml_example*}

The task writes the received message to the file "secret.txt". It uses
"Fun.protect" to ensure that the output channel "oc" is closed on both normal
and exceptional return cases. Unfortunately, this is not sufficient. If the
exchange effect "xchg 0" cannot be matched with an exchange effect performed by
some other thread, then this task remains blocked forever. Thus, the output
channel "oc" is never closed.

To avoid this problem, one must adhere to a simple discipline:
\emph{\textbf{every continuation must be eventually either continued or
discontinued}}. Here, we use "discontinue" to ensure that the blocked task does
not remain blocked forever. By discontinuing this task, we force it to
terminate (with an exception):

\begin{caml}
\begin{camlinput}
exception Improper_synchronization

let dequeue () =
  if Queue.is_empty run_q then begin
    match !exchanger with
    | None -> () (* done *)
    | Some (n, k) ->
        exchanger := None;
        discontinue k Improper_synchronization
  end else (Queue.pop run_q) ()
\end{camlinput}
\end{caml}

When the scheduler queue is empty and there is a blocked exchanger thread, the
dequeue function discontinues the blocked thread with an
"Improper_synchronization" exception. This exception is raised at the blocked
"xchg" function call, which causes the "finally" block to be run and closes the
output channel "oc". From the point of view of the user, it seems as though the
function call "xchg 0" raises the exception "Improper_synchronization".

\subsection{s:effects-sequence}{Control inversion}

When it comes to performing traversals on a data structure, there are two
fundamental ways depending on whether the producer or the consumer has the
control over the traversal. For example, in "List.iter f l", the producer
"List.iter" has the control and pushes the element to the consumer "f" who
processes them. On the other hand, the \stdmoduleref{Seq} module provides a
mechanism similar to delayed lists where the consumer controls the traversal.
For example, "Seq.forever Random.bool" returns an infinite sequence of random
bits where every bit is produced (on demand) when queried by the consumer.

Naturally, producers such as "List.iter" are easier to write in the former
style. The latter style is ergonomically better for the consumer since it is
preferable and more natural to be in control. To have the best of both worlds,
we would like to write a producer in the former style and automatically convert
it to the latter style. The conversion can be written \emph{once and for all}
as a library function, thanks to effect handlers. Let us name this function
"invert". We will first look at how to use the "invert" function before looking
at its implementation details. The type of this function is given below:

\begin{caml_example*}{signature}
val invert : iter:(('a -> unit) -> unit) -> 'a Seq.t
\end{caml_example*}

\begin{caml_eval}
let invert (type a) ~(iter : (a -> unit) -> unit) : a Seq.t =
  let module M = struct
    type _ Effect.t += Yield : a -> unit t
  end in
  let yield v = perform (M.Yield v) in
  fun () -> match_with iter yield
  { retc = (fun _ -> Seq.Nil);
    exnc = raise;
    effc = fun (type b) (eff : b Effect.t) ->
      match eff with
      | M.Yield v -> Some (fun (k: (b,_) continuation) ->
          Seq.Cons (v, continue k))
      | _ -> None }
\end{caml_eval}

The "invert" function takes an "iter" function (a producer that pushes elements
to the consumer) and returns a sequence (where the consumer has the control).
For example,

\begin{caml_example}{verbatim}
let lst_iter = Fun.flip List.iter [1;2;3]
\end{caml_example}

is an "iter" function with type "(int -> unit) -> unit". The expression
"lst_iter f" pushes the elements 1, 2 and 3 to the consumer "f". For example,

\begin{caml_example}{verbatim}
lst_iter (fun i -> Printf.printf "%d\n" i)
\end{caml_example}

The expression "invert lst_iter" returns a sequence that allows the consumer to
traverse the list on demand. For example,

\begin{caml_example}{verbatim}
let s = invert ~iter:lst_iter
let next = Seq.to_dispenser s;;
next();;
next();;
next();;
next();;
\end{caml_example}

We can use the same "invert" function on any "iter" function. For example,

\begin{caml_example}{verbatim}
let s = invert ~iter:(Fun.flip String.iter "OCaml")
let next = Seq.to_dispenser s;;
next();;
next();;
next();;
next();;
next();;
next();;
\end{caml_example}

\subsubsection{s:effects-sequence-implementation}{Implementing control inversion}

The implementation of the "invert" function is given below:

\begin{caml_example*}{verbatim}
let invert (type a) ~(iter : (a -> unit) -> unit) : a Seq.t =
  let module M = struct
    type _ Effect.t += Yield : a -> unit t
  end in
  let yield v = perform (M.Yield v) in
  fun () -> match_with iter yield
  { retc = (fun _ -> Seq.Nil);
    exnc = raise;
    effc = fun (type b) (eff : b Effect.t) ->
      match eff with
      | M.Yield v -> Some (fun (k: (b,_) continuation) ->
          Seq.Cons (v, continue k))
      | _ -> None }
\end{caml_example*}


The "invert" function declares an effect "Yield" that takes the element to be
yielded as a parameter. The "yield" function performs the "Yield" effect. The
lambda abstraction "fun () -> ..." delays all action until the first element of
the sequence is demanded. Once this happens, the computation "iter yield" is
executed under an effect handler. Every time the "iter" function pushes an
element to the "yield" function, the computation is interrupted by the "Yield"
effect. The "Yield" effect is handled by returning the value
"Seq.Cons(v,continue k)" to the consumer. The consumer gets the element "v" as
well as the suspended computation, which in the consumer's eyes is just the
tail of sequence.

When the consumer demands the next element from the sequence (by applying it to
"()"), the continuation "k" is resumed. This allows the computation "iter
yield" to make progress, until it either yields another element or terminates
normally. In the latter case, the value "Seq.Nil" is returned, indicating to
the consumer that the iteration is over.

It is important to note that the sequence returned by the "invert" function is
\emph{ephemeral} (as defined by the \stdmoduleref{Seq} module) i.e., the
sequence must be used at most once. Additionally, the sequence must be fully
consumed (i.e., used at least once) so as to ensure that the captured
continuation is used linearly.

\subsection{s:effects-semantics}{Semantics}

In this section, we shall see the semantics of effect handlers with the help of
examples.

\subsubsection{s:effects-nesting}{Nesting handlers}

Like exception handlers, effect handlers can be nested.

\begin{caml_example*}{verbatim}
type _ Effect.t += E : int t
                 | F : string t

let foo () = perform F

let bar () =
  try_with foo ()
  { effc = fun (type a) (eff: a t) ->
      match eff with
      | E -> Some (fun (k: (a,_) continuation) ->
          failwith "impossible")
      | _ -> None }

let baz () =
  try_with bar ()
  { effc = fun (type a) (eff: a t) ->
      match eff with
      | F -> Some (fun (k: (a,_) continuation) ->
          continue k "Hello, world!")
      | _ -> None }
\end{caml_example*}

In this example, the computation "foo" performs "F", the inner handler handles
only "E" and the outer handler handles "F". The call to "baz" returns "Hello,
world!".

\begin{caml_example}{verbatim}
baz ()
\end{caml_example}

\subsubsection{s:effects-fibers}{Fibers}

It is useful to know a little bit about the implementation of effect handlers
to appreciate the design choices and their performance characteristics. Effect
handlers are implemented with the help of runtime-managed, dynamically growing
segments of stack called \textit{fibers}. The program stack in OCaml is a
linked list of such fibers.

A new fiber is allocated for evaluating the computation enclosed by an effect
handler. The fiber is freed when the computation returns to the caller either
normally by returning a value or by raising an exception.

At the point of "perform" in "foo" in the previous example, the program stack
looks like this:

\begin{caml}
\begin{camlinput}
+-----+   +-----+   +-----+
|     |   |     |   |     |
| baz |<--| bar |<--| foo |
|     |   |     |   |     |
|     |   |     |   |     |
+-----+   +-----+   +-----+ <- stack_pointer
\end{camlinput}
\end{caml}

The two links correspond to the two effect handlers in the program. When the
effect "F" is handled in "baz", the program state looks as follows:

\begin{caml}
\begin{camlinput}
+-----+                   +-----+   +-----+
|     |                   |     |   |     |   +-+
| baz |                   | bar |<--| foo |<--|k|
|     |                   |     |   |     |   +-+
+-----+ <- stack_pointer  +-----+   +-----+
\end{camlinput}
\end{caml}

The delimited continuation "k" is an object on the heap that refers to the
segment of the stack that corresponds to the suspended computation. Capturing a
continuation does not involve copying stack frames. When the continuation is
resumed, the stack is restored to the previous state by linking together the
segment pointed to by "k" to the current stack. Since neither continuation
capture nor resumption requires copying stack frames, suspending the execution
using "perform" and resuming it using either "continue" or "discontinue" are
fast.

\subsubsection{s:effects-unhandled}{Unhandled effects}

Unlike languages such as Eff and Koka, effect handlers in OCaml do not provide
\textit{effect safety}; the compiler does not statically ensure that all the
effects performed by the program are handled. If effects do not have a matching
handler, then an "Effect.Unhandled" exception is raised at the point of the
corresponding "perform". For example, in the previous example, "bar" does not
handle the effect "F". Hence, we will get an "Effect.Unhandled F" exception
when we run "bar".

\begin{caml_example}{verbatim}
try bar () with Effect.Unhandled F -> "Saw Effect.Unhandled exception"
\end{caml_example}

\subsubsection{s:effects-linearity}{Linear continuations}

As discussed earlier~\ref{s:effects-discontinue}, the delimited continuations
in OCaml must be used linearly -- \emph{\textbf{every captured continuation
must be resumed either with a "continue" or "discontinue" exactly once}}.
Attempting to use a continuation more than once raises a
"Continuation_already_resumed" exception. For example:

\begin{caml_example}{verbatim}
try_with perform (Xchg 0)
{ effc = fun (type a) (eff : a t) ->
    match eff with
    | Xchg n -> Some (fun (k: (a, _) continuation) ->
        continue k 21 + continue k 21)
    | _ -> None }
\end{caml_example}

The primary motivation for adding effect handlers to OCaml is to enable
concurrent programming. One-shot continuations are sufficient for almost all
concurrent programming needs. They are also much cheaper to implement
compared to multi-shot continuations since they do not require stack frames to
be copied. Moreover, OCaml programs may also manipulate linear resources such
as sockets and file descriptors. The linearity discipline is easily broken if
the continuations are allowed to resume more than once. It would be quite hard
to debug such linearity violations on resources due to the lack of static
checks for linearity and the non-local nature of control flow. Hence, OCaml
does not support multi-shot continuations.

While the ``at most once resumption'' property of continuations is ensured with
a dynamic check, there is no check to ensure that the continuations are resumed
``at least once''. It is left to the user to ensure that the captured
continuations are resumed at least once. Not resuming continuations will leak
the memory allocated for the fibers as well as any resources that the suspended
computation may hold.

One may install a finaliser on the captured continuation to ensure that the
resources are freed:

\begin{caml}
\begin{camlinput}
exception Unwind
Gc.finalise (fun k ->
  try ignore (discontinue k Unwind) with _ -> ()) k
\end{camlinput}
\end{caml}

In this case, if "k" becomes unreachable, then the finaliser ensures that the
continuation stack is unwound by discontinuing with an "Unwind" exception,
allowing the computation to free up resources. However, the runtime cost of
finalisers is much more than the cost of capturing a continuation. Hence, it is
recommended that the user take care of resuming the continuation exactly once
rather than relying on the finaliser.

\subsection{s:effects-shallow}{Shallow handlers}

The examples that we have seen so far have used \textit{deep} handlers. A deep
handler handles all the effects performed (in sequence) by the computation.
Whenever a continuation is captured in a deep handler, the captured continuation
also includes the handler. This means that, when the continuation is resumed,
the effect handler is automatically re-installed, and will handle the effect(s)
that the computation may perform in the future.

OCaml also provides \textit{shallow} handlers. Compared to deep handlers, a
shallow handler handles only the first effect performed by the computation. The
continuation captured in a shallow handler does not include the handler. This
means that, when the continuation is resumed, the handler is no longer present.
For this reason, when the continuation is resumed, the user is expected to
provide a new effect handler (possibly a different one) to handle the next
effect that the computation may perform.

Shallow handlers make it easier to express certain kinds of programs. Let us
implement a shallow handler that enforces a particular sequence of effects (a
protocol) on a computation. For this example, let us consider that the
computation may perform the following effects:

\begin{caml_example*}{verbatim}
type _ Effect.t += Send : int -> unit Effect.t
                 | Recv : int Effect.t
\end{caml_example*}

Let us assume that we want to enforce a protocol that only permits an
alternating sequence of "Send" and "Recv" effects that conform to the regular
expression "(Send;Recv)*;Send?". Hence, the sequence of effects "[]" (the empty
sequence), "[Send]", "[Send;Recv]", "[Send;Recv;Send]", etc., are allowed, but
not "[Recv]", "[Send;Send]", "[Send;Recv;Recv]", etc. The key observation here
is that the set of effects handled evolves over time. We can enforce this
protocol quite naturally using shallow handlers as shown below:

\begin{caml_example*}{verbatim}
open Effect.Shallow

let run (comp: unit -> unit) : unit =
  let rec loop_send : type a. (a,unit) continuation -> a -> unit = fun k v ->
    continue_with k v
      { retc = Fun.id;
        exnc = raise;
        effc = fun (type b) (eff : b Effect.t) ->
          match eff with
          | Send n -> Some (fun (k: (b,_) continuation) ->
              loop_recv n k ())
          | Recv -> failwith "protocol violation"
          | _ -> None }
  and loop_recv : type a. int -> (a,unit) continuation -> a -> unit = fun n k v ->
    continue_with k v
      { retc = Fun.id;
        exnc = raise;
        effc = fun (type b) (eff : b Effect.t) ->
          match eff with
          | Recv -> Some (fun (k: (b,_) continuation) ->
              loop_send k n)
          | Send v -> failwith "protocol violation"
          | _ -> None }
  in
  loop_send (fiber comp) ()
\end{caml_example*}

The "run" function executes the computation "comp" ensuring that it can only
perform an alternating sequence of "Send" and "Recv" effects. The shallow
handler uses a different set of primitives compared to the deep handler. The
primitive "fiber" (on the last line) takes an "'a -> 'b" function and returns a
"('a,'b) Effect.Shallow.continuation". The expression "continue_with k v h"
resumes the continuation "k" with value "v" under the handler "h".

The mutually recursive functions "loop_send" and "loop_recv" resume the given
continuation "k" with value "v" under different handlers. The "loop_send"
function handles the "Send" effect and tail calls the "loop_recv" function. If
the computation performs the "Recv" effect, then "loop_send" aborts the
computation by raising an exception. Similarly, the "loop_recv" function
handles the "Recv" effect and tail calls the "loop_send" function. If the
computation performs the "Send" effect, then "loop_recv" aborts the
computation. Given that the continuation captured in the shallow handler do not
include the handler, there is only ever one handler installed in the dynamic
scope of the computation "comp".

The computation is initially executed by the "loop_send" function (see last
line in the code above) which ensures that the first effect that the
computation is allowed to perform is the "Send" effect. Note that the
computation is free to perform effects other than "Send" and "Recv", which may
be handled by an outer handler.

We can see that the "run" function will permit a computation that follows the
protocol:

\begin{caml_example}{verbatim}
run (fun () ->
  printf "Send 42\n";
  perform (Send 42);
  printf "Recv: %d\n" (perform Recv);
  printf "Send 43\n";
  perform (Send 43);
  printf "Recv: %d\n" (perform Recv))
\end{caml_example}

and aborts those that do not:

\begin{caml_example}{verbatim}
run (fun () ->
  Printf.printf "Send 0\n";
  perform (Send 0);
  Printf.printf "Send 1\n";
  perform (Send 1) (* protocol violation *))
\end{caml_example}

We may implement the same example using deep handlers using reference cells
(easy, but unsatisfying) or without them (harder). We leave this as an exercise
to the reader.
