Merge pull request #1097 from akka/wip-channels-∂π

add Typed Channels, see #2929
This commit is contained in:
Roland Kuhn 2013-02-01 06:37:41 -08:00
commit e06076553c
19 changed files with 2295 additions and 15 deletions

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/**
* Copyright (C) 2009-2013 Typesafe Inc. <http://www.typesafe.com>
*/
package docs.channels
import akka.testkit.AkkaSpec
import akka.channels._
import akka.actor.Actor
import scala.concurrent.forkjoin.ThreadLocalRandom
import scala.concurrent.Future
import scala.concurrent.duration._
import akka.util.Timeout
import akka.testkit.TestProbe
object ChannelDocSpec {
//#motivation0
trait Request
case class Command(msg: String) extends Request
trait Reply
case object CommandSuccess extends Reply
case class CommandFailure(msg: String) extends Reply
//#motivation0
//#child
case class Stats(b: Request)
case object GetChild
case class ChildRef(child: ChannelRef[(Request, Reply) :+: TNil])
class Child extends Actor
with Channels[(Stats, Nothing) :+: TNil, (Request, Reply) :+: TNil] {
channel[Request] { (x, snd)
parentChannel <-!- Stats(x)
snd <-!- CommandSuccess
}
}
class Parent extends Actor
with Channels[TNil, (Stats, Nothing) :+: (GetChild.type, ChildRef) :+: TNil] {
val child = createChild(new Child)
channel[GetChild.type] { (_, snd) ChildRef(child) -!-> snd }
channel[Stats] { (x, _)
// collect some stats
}
}
//#child
}
class ChannelDocSpec extends AkkaSpec {
import ChannelDocSpec._
class MsgA
class MsgB
class MsgC
class MsgD
"demonstrate why Typed Channels" in {
def someActor = testActor
//#motivation1
val requestProcessor = someActor
requestProcessor ! Command
//#motivation1
expectMsg(Command)
/*
//#motivation2
val requestProcessor = new ChannelRef[(Request, Reply) :+: TNil](someActor)
requestProcessor <-!- Command // this does not compile
//#motivation2
*/
type Example = //
//#motivation-types
(MsgA, MsgB) :+: (MsgC, MsgD) :+: TNil
//#motivation-types
}
// only compile test
"demonstrate channels creation" ignore {
//#declaring-channels
class AC extends Actor with Channels[TNil, (Request, Reply) :+: TNil] {
channel[Request] { (req, snd)
req match {
case Command("ping") snd <-!- CommandSuccess
case _
}
}
}
//#declaring-channels
//#declaring-subchannels
class ACSub extends Actor with Channels[TNil, (Request, Reply) :+: TNil] {
channel[Command] { (cmd, snd) snd <-!- CommandSuccess }
channel[Request] { (req, snd)
if (ThreadLocalRandom.current.nextBoolean) snd <-!- CommandSuccess
else snd <-!- CommandFailure("no luck")
}
}
//#declaring-subchannels
}
// only compile test
"demonstrating message sending" ignore {
import scala.concurrent.ExecutionContext.Implicits.global
//#sending
implicit val dummySender: ChannelRef[(Any, Nothing) :+: TNil] = ???
implicit val timeout: Timeout = ??? // for the ask operations
val channelA: ChannelRef[(MsgA, MsgB) :+: TNil] = ???
val channelB: ChannelRef[(MsgB, MsgC) :+: TNil] = ???
val channelC: ChannelRef[(MsgC, MsgD) :+: TNil] = ???
val a = new MsgA
val fA = Future { new MsgA }
channelA <-!- a // send a to channelA
a -!-> channelA // same thing as above
channelA <-!- fA // eventually send the futures value to channelA
fA -!-> channelA // same thing as above
// ask the actor; return type given in full for illustration
val fB: Future[WrappedMessage[(MsgB, Nothing) :+: TNil, MsgB]] = channelA <-?- a
val fBunwrapped: Future[MsgB] = fB.lub
a -?-> channelA // same thing as above
channelA <-?- fA // eventually ask the actor, return the future
fA -?-> channelA // same thing as above
// chaining works as well
a -?-> channelA -?-> channelB -!-> channelC
//#sending
}
"demonstrate message forwarding" in {
//#forwarding
import scala.reflect.runtime.universe.TypeTag
class Latch[T1: TypeTag, T2: TypeTag](target: ChannelRef[(T1, T2) :+: TNil])
extends Actor with Channels[TNil, (Request, Reply) :+: (T1, T2) :+: TNil] {
implicit val timeout = Timeout(5.seconds)
//#become
channel[Request] {
case (Command("close"), snd)
channel[T1] { (t, s) t -?-> target -!-> s }
snd <-!- CommandSuccess
case (Command("open"), snd)
channel[T1] { (_, _) }
snd <-!- CommandSuccess
}
//#become
channel[T1] { (t, snd) t -?-> target -!-> snd }
}
//#forwarding
val probe = TestProbe()
val _target = new ChannelRef[(String, Int) :+: TNil](probe.ref)
val _self = new ChannelRef[(Any, Nothing) :+: TNil](testActor)
//#usage
implicit val selfChannel: ChannelRef[(Any, Nothing) :+: TNil] = _self
val target: ChannelRef[(String, Int) :+: TNil] = _target // some actor
// type given just for demonstration purposes
val latch: ChannelRef[(Request, Reply) :+: (String, Int) :+: TNil] =
ChannelExt(system).actorOf(new Latch(target), "latch")
"hello" -!-> latch
//#processing
probe.expectMsg("hello")
probe.reply(5)
//#processing
expectMsg(5) // this is a TestKit-based example
Command("open") -!-> latch
expectMsg(CommandSuccess)
"world" -!-> latch
//#processing
probe.expectNoMsg(500.millis)
//#processing
expectNoMsg(500.millis)
//#usage
}
"demonstrate child creation" in {
implicit val selfChannel = new ChannelRef[(Any, Nothing) :+: TNil](testActor)
//#child
//
// then it is used somewhat like this:
//
val parent = ChannelExt(system).actorOf(new Parent, "parent")
parent <-!- GetChild
val child = expectMsgType[ChildRef].child // this assumes TestKit context
child <-!- Command("hey there")
expectMsg(CommandSuccess)
//#child
}
}

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@ -8,6 +8,7 @@ Scala API
actors
typed-actors
typed-channels
logging
event-bus
scheduler

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.. _typed-channels:
#############################
Typed Channels (EXPERIMENTAL)
#############################
.. note::
*This is a preview of the upcoming Typed Channels support, its API may change
during development up to the released version where the EXPERIMENTAL label is
removed.*
Motivation
==========
Actors derive great strength from their strong encapsulation, which enables
internal restarts as well as changing behavior and also composition. The last
one is enabled by being able to inject an actor into a message exchange
transparently, because all either side ever sees is an :class:`ActorRef`. The
straight-forward way to implement this encapsulation is to keep the actor
references untyped, and before the advent of macros in Scala 2.10 this was the
only tractable way.
As a motivation for change consider the following simple example:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#motivation0
.. includecode:: code/docs/channels/ChannelDocSpec.scala#motivation1
This is an error which is quite common, and the reason is that the compiler
does not catch it and cannot warn about it. Now if there were some type
restrictions on which messages the ``commandProcessor`` can process, that would
be a different story:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#motivation2
The :class:`ChannelRef` wraps a normal untyped :class:`ActorRef`, but it
expresses a type constraint, namely that this channel accepts only messages of
type :class:`Request`, to which it may reply with messages of type
:class:`Reply`. The types do not express any guarantees on how many messages
will be exchanged, whether they will be received or processed, or whether a
reply will actually be sent. They only restrict those actions which are known
to be doomed already at compile time. In this case the second line would flag
an error, since the companion object ``Command`` is not an instance of type
:class:`Request`.
While this example looks pretty simple, the implications are profound. In order
to be useful, the system must be as reliable as you would expect a type system
to be. This means that unless you step outside of it (i.e. doing the
equivalent of ``.asInstanceOf[_]``) you shall be protected, failures shall be
recognized and flagged. There are a number of challenges included in this
requirement, which are discussed in `The Design Background`_ below.
Terminology
===========
.. describe:: type Channel[I, O] = (I, O)
A Channel is a pair of an input type and and output type. The input type is
the type of message accepted by the channel, the output type is the possible
reply type and may be ``Nothing`` to signify that no reply is sent. The input
type cannot be ``Nothing``.
.. describe:: type ChannelList
A ChannelList is an ordered collection of Channels, without further
restriction on the input or output types of these. This means that a single
input type may be associated with multiple output types within the same
ChannelList.
.. describe:: type TNil <: ChannelList
The empty ChannelList.
.. describe:: type :+:[Channel, ChannelList] <: ChannelList
This binary type constructor is used to build up lists of Channels, for which
infix notation will be most convenient:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#motivation-types
.. describe:: class ChannelRef[T <: ChannelList]
A ChannelRef is what is referred to above as the channel reference, it bears
the ChannelList which describes all input and output types and their relation
for the referenced actor. It also contains the underlying :class:`ActorRef`.
.. describe:: trait Channels[P <: ChannelList, C <: ChannelList]
A mixin for the :class:`Actor` trait which is parameterized in the channel
requirements this actor has for its parent (P) and its selfChannel (C).
.. describe:: selfChannel
An ``Actor with Channels[P, C]`` has a ``selfChannel`` of type
``ChannelRef[C]``. This is the same type of channel reference which is
obtained by creating an instance of this actor.
.. describe:: type ReplyChannels[T <: ChannelList] <: ChannelList
Within an ``Actor with Channels[_, _]`` which takes a fully generic channel,
i.e. a type argument ``T <: ChannelList`` which is part of its selfChannel
type, this channels reply types are not known. The definition of this
channel uses the ReplyChannels type to abstractly refer to this unknown set
of channels in order to forward a reply from a ``ChannelRef[T]`` back to the
original sender. This operations type-safety is ensured at the senders site
by way of the ping-pong analysis described above.
.. describe:: class WrappedMessage[T <: ChannelList, LUB]
Scalas type system cannot directly express type unions. Asking an actor with
a given input type may result in multiple possible reply types, hence the
:class:`Future` holding this reply will contain the value wrapped inside a
container which carries this type (only at compile-time). The type parameter
LUB is the least upper bound of all input channels contained in the
ChannelList T.
Sending Messages across Channels
================================
Sending messages is best demonstrated in a quick overview of the basic operations:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#sending
The first line is included so that the code compiles, since all message sends
including ``!`` will check the implicitly found selfChannel for compatibility
with the target channels reply types. In this case we want to demonstrate just
the syntax of sending, hence the dummy sender which accepts everything and
replies never.
Presupposing three channel references of chainable types, an input value ``a``
and a Future holding such a value, we demonstrate the two basic operations
which are well known from untyped actors: tell/! and ask/?. The type of the
Future returned by the ask operation may seem surprising at first, but as the
last line demonstrates, it is built in a way which makes building actor chains
very simple. What the last line does is the following:
* it asks channelA, which returns a Future
* a callback is installed on the Future which will use the reply value of
channelA and ask channelB with it, returning another Future
* a callback is installed on that Future to send the reply value of channelB to
channelC, returning a Future with that previously sent value (using ``andThen``)
This example also motivates the introduction of the “turned-around” syntax
where messages flow more naturally from left to right, instead of the standard
object-oriented view of having the tell method operate on the ActorRef given to
the left.
This example informally introduced what is more precisely specified in the
following subsection.
The Rules
---------
Operations on typed channels are composable and obey a few simple rules:
* the message to be sent can be one of three things:
* a :class:`Future[_]`, in which case the contained value will be sent once
available; the value will be unwrapped if it is a :class:`WrappedMessage[_, _]`
* a :class:`WrappedMessage[_, _]`, which will be unwrapped (i.e. only the
value is sent)
* everything else is sent as is
* the operators are fully symmetric, i.e. ``-!->`` and ``<-!-`` do the same
thing provided the arguments also switch places
* sending with ``-?->`` or ``<-?-`` always returns a
``Future[WrappedMessage[_, _]]`` representing all possible reply channels,
even if there is only one (use ``.lub`` to get a :class:`Future[_]` with the
most precise single type for the value)
* sending a :class:`Future[_]` with ``-!->`` or ``<-!-`` returns a new
:class:`Future[_]` which will be completed with the value after it has been
sent; sending a strict value returns that value
Declaring an Actor with Channels
================================
The declaration of an Actor with Channels is done like this:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#declaring-channels
It should be noted that it is impossible to declare channels which are not part
of the channel list given as the second type argument to the :class:`Channels`
trait. It is also checked—albeit at runtime—that when the actors construction
is complete (i.e. its constructor and ``preStart`` hook have run) every channel
listed in the selfChannel type parameter has been declared. This can in general
not be done at compile time, both due to the possibility of overriding
subclasses as well as the problem that the compiler cannot determine whether a
``channel[]`` statement will be called in the course of execution due to
external inputs (e.g. if conditionally executed).
It should also be noted that the type of ``req`` in this example is
``Request``, hence it would be a compile-time error to try to match against the
``Command`` companion object. The ``snd`` reference is the sender channel
reference, which in this example is of type
``ChannelRef[(Reply, UnknownDoNotWriteMeDown) :+: TNil]``, meaning that sending
back a reply which is not of type ``Reply`` would be a compile-time error.
The last thing to note is that an actor is not obliged to reply to an incoming
message, even if that was successfully delivered to it: it might not be
appropriate, or it might be impossible, the actor might have failed before
executing the replying message send, etc. And as always, the ``snd`` reference
may be used more than once, and even stored away for later. It must not leave
the actor within it was created, however, because that would defeat the
ping-pong check; this is the reason for the curious name of the fabricated
reply type ``UnknownDoNotWriteMeDown``, if you find yourself declaring that
type as part of a message or similar you know that you are cheating.
Declaration of Subchannels
--------------------------
It can be convenient to carve out subchannels for special treatment like so:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#declaring-subchannels
This means that all ``Command`` requests will be positively answered while all
others may or may not be lucky. This dispatching between the two declarations
does not depend on their order but is solely done based on which type is more
specific.
Forwarding Messages
-------------------
Forwarding messages has been hinted at in the last sample already, but here is
a more complete sample actor:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#forwarding
:exclude: become
This actor declares a single-Channel parametric type which it forwards to a
target actor, handing replies back to the original sender using the ask/pipe
pattern.
.. note::
It is important not to forget the ``TypeTag`` context bound for all type
arguments which are used in channel declarations, otherwise the not very
helpful error “Predef is not an enclosing class” will haunt you.
Changing Behavior at Runtime
----------------------------
The actor from the previous example gets a lot more interesting when
implementing its control channel:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#forwarding
This shows all elements of the toolkit in action: calling ``channel[T1]`` again
during the lifetime of the actor will alter its behavior on that channel. In
this case a latch or gate is modeled which when closed will permit the message
flow through and when not will drop the messages to the floor.
Creating Actors with Channels
-----------------------------
Creating top-level actors with channels is done using the ``ChannelExt`` extension:
.. includecode:: code/docs/channels/ChannelDocSpec.scala
:include: usage
:exclude: processing
Inside an actor with channels children are created using the ``createChild`` method:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#child
In this example we create a simple child actor which responds to requests, but
also keeps its parent informed about what it is doing. The parent channel
within the child is thus declared to accept :class:`Stats` messages, and the
parent must consequently declare such a channel in order to be able to create
such a child. The parents job then is to create the child, make it available
to the outside via properly typed messages and collect the statistics coming in
from the child.
Implementation Restrictions
---------------------------
The erasure-based dispatch of incoming messages requires all channels which are
declared to have unique JVM type representations, i.e. it is not possible to
have two channel declarations with types ``List[A]`` and ``List[B]`` because
both would at runtime only be known as ``List[_]``.
The specific dispatch mechanism also require the declaration of all channels or
subchannels during the actors construction, independent of whether they shall
later change behavior or not. Changing behavior for a subchannel is only
possible if that subchannel was declared up-front.
TypeTags are currently (Scala 2.10.0) not serializable, hence narrowing of
:class:`ActorRef` does not work for remote references.
The Design Background
=====================
This section outlines the most prominent challenges encountered during the
development of Typed Channels and the rationale for their solutions. It is not
necessary to understand this material in order to use Typed Channels, but it
may be useful to explain why certain things are as they are.
The Type Pollution Problem
--------------------------
What if an actor accepts two different types of messages? It might be a main
communications channel which is forwarded to worker actors for performing some
long-running and/or dangerous task, plus an administrative channel for the
routing of requests. Or it might be a generic message throttler which accepts a
generic channel for passing it through (which delay where appropriate) and a
management channel for setting the throttling rate. In the second case it is
especially easy to see that those two channels will probably not be related,
their types will not be derived from a meaningful common supertype; instead the
least upper bound will probably be :class:`AnyRef`. If a typed channel
reference only had the capability to express a single type, this type would
then be no restriction anymore.
One solution to this is to never expose references describe more than one
channel at a time. But where would these references come from? It would be very
difficult to make this construction process type-safe, and it would also be an
inconvenient restriction, since message ordering guarantees only apply for the
same senderreceive pair, and if there are relations between the messages sent
on multiple channels those would need more boilerplate code to realize than if
all interaction were possible through a single reference.
The other solution thus is to express multiple channel types by a single
channel reference, which requires the implementation of type lists and
computations on these. And as we will see below it also requires the
specification of possibly multiple reply channels per input type, hence a type
map. The implementation chosen uses type lists like this:
.. includecode:: code/docs/channels/ChannelDocSpec.scala#motivation-types
This type expresses two channels: type ``A`` may stimulate replies of type
``B``, while type ``C`` may evoke replies of type ``D``. The type operator
``:+:`` is a binary type which form a list of these channel definitions, and
like every good list it ends with an empty terminator ``TNil``.
The Reply Problem
-----------------
Akka actors have the power to reply to any message they receive, which is also
a message send and shall also be covered by typed channels. Since the sending
actor is the one which will also receive the reply, this needs to be verified.
The solution to this problem is that in addition to the ``self`` reference,
which is implicitly picked up as the sender for untyped actor interactions,
there is also a ``selfChannel`` which describes the typed channels handled by
this actor. Thus at the call site of the message send it must be verified that
this actor can actually handle the reply for that given message send.
The Sender Ping-Pong Problem
----------------------------
After successfully sending a message to an actor over a typed channel, that
actor will have a reference to the messages sender, because normal Akka
message processing rules apply. For this sender reference there must exist a
typed channel reference which describes the possible reply types which are
applicable for each of the incoming message channels. We will see below how
this reference is provided in the code, the problem we want to highlight here
is a different one: the nature of any sender reference is that it is highly
dynamic, the compiler cannot possibly know who sent the message we are
currently processing.
But this does not mean that all hope is lost: the solution is to do *all*
type-checking at the call site of the message send. The receiving actor just
needs to declare its channel descriptions in its own type, and channel
references are derived at construction from this type (implying the existence
of a typed ``actorOf``). Then the actor knows for each received message type
which the allowed reply types are. The typed channel for the sender reference
hence has the reply types for the current input channel as its own input types,
but what should the reply types be? This is the ping-pong problem:
* ActorA sends MsgA to ActorB
* ActorB replies with MsgB
* ActorA replies with MsgC
Every “reply” uses the sender channel, which is dynamic and hence only known
partially. But ActorB did not know who sent the message it just replied to and
hence it cannot check that it can process the possible replies following that
message send. Only ActorA could have known, because it knows its own channels
as well as ActorBs channels completely. The solution is thus to recursively
verify the message send, following all reply channels until all possible
message types to be sent have been verified. This sounds horribly complex, but
the algorithm for doing so actually has a worst-case complexity of O(N) where N
is the number of input channels of ActorA or ActorB, whoever has fewer.
The Parent Problem
------------------
There is one other actor reference which is available to every actor: its
parent. Since the childparent relationship is established permanently when the
child is created by the parent, this problem is easily solvable by encoding the
requirements of the child for its parent channel in its type signature having
the typed variant of ``actorOf`` verify this against the ``selfChannel``.
Anecdotally, since the guardian actor does not care at all about message sent
to it, top-level type channel actors must declare their parent channel to be
empty.
The Exposure/Restriction Problem
--------------------------------
An actor may provide more than one service, either itself or by proxy, each
with their own set of channels. Only having references for the full set of
channels leads to a too wide spread of capabilities: in the example of the
message rate throttling actor its management channel is only meant to be used
by the actor which inserted it, not by the two actors between it was inserted.
Hence the manager will have to create a channel reference which excludes the
management channels before handing out the reference to other actors.
Another variant of this problem is an actor which handles a channel whose input
type is a supertype for a number of derived channels. It should be allowed to
use the “superchannel” in place of any of the subchannels, but not the other
way around. The intuitive approach would be to model this by making the channel
reference contravariant in its channel types and define those channel types
accordingly. This does not work nicely, however, because Scalas type system is
not well-suited to modeling such calculations on unordered type lists; it might
be possible but its implementation would be forbiddingly complex.
Therefore this topic gained traction as macros became available: being able to
write down type calculations using standard collections and their
transformations reduces the implementation to a handful of lines. The “narrow”
operation implemented this way allows all narrowing of input channels and
widening of output channels down to ``(Nothing, Any)`` (which is to say:
removal).
The Forwarding Problem
----------------------
One important feature of actors mentioned above is their composability which is
enabled by being able to forward or delegate messages. It is the nature of this
process that the sending party is not aware of the true destination of the
message, it only sees the façade in front of it. Above we have seen that the
sender ping-pong problem requires all verification to be performed at the
senders end, but if the sender does not know the final recipient, how can it
check that the message exchange is type-safe?
The forwarding party—the middle-man—is also not in the position to make this
call, since all it has is the incomplete sender channel which is lacking reply
type information. The problem which arises lies precisely in these reply
sequences: the ping-pong scheme was verified against the middle-man, and if the
final recipient would reply to the forwarded request, that sender reference
would belong to a different channel and there is no single location in the
source code where all these pieces are known at compile time.
The solution to this problem is not to allow forwarding in the normal untyped
:class:`ActorRef` sense. Replies must always be sent by the recipient of the
original message in order for the type checks at the sender site to be
effective. Since forwarding is an important communication pattern among actors,
support for it is thus provided in the form of the :meth:`ask` pattern combined
with the :meth:`pipe` pattern, which both are not add-ons but fully integrated
operations among typed channels.
The JVM Erasure Problem
-----------------------
When an actor with typed channels receives a message, this message needs to be
dispatched internally to the right channel, so that the right sender channel
can be presented and so on. This dispatch needs to work with the information
contained in the message, which due to the erasure of generic type information
is an incomplete image of the true channel types. Those full types exist only
at compile-time and reifying them into TypeTags at runtime for every message
send would be prohibitively expensive. This means that channels which erase to
the same JVM type cannot coexist within the same actor, message would not be
routable reliably in that case.
The Actor Lookup Problem
------------------------
Everything up to this point has assumed that channel references are passed from
their point of creation to their point of use directly and in the regime of
strong, unerased types. This can also happen between actors by embedding them
in case classes with proper type information. But one particular useful feature
of Akka actors is that they have a stable identity by which they can be found,
a unique name. This name is represented as a :class:`String` and naturally does
not bear any type information concerning the actors channels. Thus, when
looking up an actor with ``system.actorFor(...)`` you will only get an untyped
:class:`ActorRef` and not a channel reference. This :class:`ActorRef` can of
course manually be wrapped in a channel reference bearing the desired channels,
but this is not a type-safe operation.
The solution in this case must be a runtime check. There is an operation to
“narrow” an :class:`ActorRef` to a channel reference of given type, which
behind the scenes will send a message to the designated actor with a TypeTag
representing the requested channels. The actor will check these against its own
TypeTag and reply with the verification result. This check uses the same code
as the compile-time “narrow” operation introduced above.
How to read The Types
=====================
In case of errors in your code the compiler will try to inform you in the most
precise way it can, and that will then contain types like this::
akka.channels.:+:[(com.example.Request, com.example.Reply),
akka.channels.:+:[(com.example.Command, Nothing), TNil]]
These types look unwieldy because of two things: they use fully qualified names
for all the types (thankfully using the ``()`` sugar for :class:`Tuple2`), and
they do not employ infix notation. That same type there might look like this in
your source code::
(Request, Reply) :+: (Command, Nothing) :+: TNil
As soon as someone finds the time, it would be nice if the IDEs learned to
print types making use of the files import statements and infix notation.