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Welcome! I'm Pepijn. I study natural language, programming languages and logical languages. Basically, all the language...


Linear Types for Erlang/OTP (1)

This is the first post in a series in which I will investigate linear types for Erlang/OTP. In these posts, I will discuss the the generic server architecture implemented in Erlang’s gen_server and supervisor trees.

Whenever possible, I will implement similar programs directly in Wadler’s Classical Processes. Quite often, however, the type system for CP will not be expressive enough to type such programs. In these cases, I will implement the types and programs using Agda as my meta-language—as a sort-of template CP.

Prelude: Linear Types

As mentioned I will occasionally resort to using Agda as my meta-language. In this post, I will use Agda to talk about linear types—and in order to do that, I need to tell Agda what linear types are.

Below I define a small Agda module which describes linear types. Feel free to inspect it. Briefly, it defines the structure of linear types as close to the syntax of linear logic as we can get.

-- Prelude. Click `+` to unfold.

-- As we can have both positive and negative atoms, we
-- defined a type `Pol`, which will represent the polarities
-- of primitive types.
data Pol : Set where
  pos : Pol
  neg : Pol

-- As in software foundations, we defined identifiers as a
-- simple wrapper type around the natural numbers.
data Id : Set where
  id :   Id

-- The types are defined, using as much as is possible of the
-- original syntax of linear logic. The only except are the
-- exponentials: ? is a reserved symbol, so we use ¡ and ¿.
data Type : Set where
  var     : Pol  Id  Type
        : Type
  𝟘   𝟙   : Type
  _&_ _⅋_ : Type  Type  Type
  _⊕_ _⊗_ : Type  Type  Type
  ¡_  ¿_  : Type -> Type
  μ   ν   : (Id  Type)  Type

-- While we would like to be able to treat both positive
-- and negative occurances of variables as a single case
-- in proofs, writing `var pos X` is tedious. Therefore,
-- we defined the syntax (+ A) and (- A) for variables.
pattern +_ A = var pos A
pattern -_ A = var neg A
{-# DISPLAY var pos A = + A #-}
{-# DISPLAY var neg A = - A #-}

-- We want to be able to write both (μ F) and (μ[ X ] A).
-- Therefore, we define the primitive type constructor as
-- μ, and attach a syntax macro which allows it to bind an
-- argument X. This way we never have to write μ (λ X → A)).
syntax μ  X  A) = μ[ X ] A
syntax ν  X  A) = ν[ X ] A

-- We define negation as a recursive function over the
-- structure of types -- as you would expect, basically.
~_ : Type  Type
~ (+ A)   = - A           ; ~ (- A)   = + A
~ 𝟘       =              ; ~        = 𝟘
~ 𝟙       =              ; ~        = 𝟙
~ (A  B) = ~ A & ~ B     ; ~ (A & B) = ~ A  ~ B
~ (A  B) = ~ A  ~ B     ; ~ (A  B) = ~ A  ~ B
~ (¡ A )  = ¿ ~ A         ; ~ (¿ A )  = ¡ ~ A
~ (μ F)   = ν[ X ](~ F X) ; ~ (ν F)   = μ[ X ](~ F X)

For instance, we can use the definitions in this module to define the lollipop.

_⊸_ : Type  Type  Type
A  B = ~ A  B

Another thing that we can do—which will be extremely useful later on—is to define the structure of “named” sums. The implementation of CP supports these, and I use them quite heavily to implement datatypes.

This is also where the use of Agda as a meta-language becomes apparent. I cannot describe these types within CP, but I can describe them as a template on top of CP, and use Agda to feel reassured that what I’m doing is relatively sane.

We define named sums as a fold over a finite list of names, applying some type-level function F to each name, and joining the resulting types with plus (). Then we define a syntax macro which hides the binder in the function F.

syntax ⊕-Fin  i  A) = ⊕[ i ] A

⊕-Fin :  {n} (F : Fin (2 + n)  Type)  Type
⊕-Fin {n} F = foldr₁ _⊕_ (map F (allFin (2 + n)))

We define a similar construction for with (&).

syntax &-Fin  i  A) = &[ i ] A

&-Fin :  {n} (F : Fin (2 + n)  Type)  Type
&-Fin {n} F = foldr₁ _&_ (map F (allFin (2 + n)))

This is as good a time as any to mention that this post is both literate Agda and literate CP.

The Road to the Generic Server: The Basic Server

In the first few posts, I will let myself be guided by Joe Armstrong’s Road to the Generic Server1, in which he explains what he considers to be the most important part of Erlang—the generic server.

Below I’ve reproduced (without any permission) a small Erlang module from Joe’s book. This module implements the first generic server: it exports the function start/2, which starts a server which behaves according to some request handler, and the function rpc/2, which—as it’s name suggests—sends procedure calls to the server. In this first attempt, the server merely receives requests, and handles them using the given request handler.

module(server1).
export([start/2, rpc/2]).

start(Name, Mod) ->
  register(Name, spawn(fun() -> loop(Name, Mod, Mod:init()) end)).

rpc(Name, Request) ->
  Name ! {self(), Request},
  receive
    {Name, Response} -> Response
  end.

loop(Name, Mod, State) ->
  receive
    {From, Request} ->
      {Response, State1} = Mod:handle(Request, State),
      From ! {Name, Response},
      loop(Name, Mod, State1}
  end.

Joe then writes a small request handler for server1. This handler can handle two types of requests: add and whereis. For example, the request {add,joe,at_home} will tell the server that joe is currently at home—this overwrites any previous knowledge the server might’ve had about Joe’s location. The request {whereis,joe} will ask the server where Joe is.

This behaviour is implemented in the function handle/2. The functions add and whereis then communicate with the name_server by sending remote proceduce calls using rpc/2.

module(name_server).
export([init/0, add/2, whereis/1, handle/2]).
import(server1, [rpc/2]).

%% client routines
add(Name, Place) -> rpc(name_server, {add, Name, Place}).
whereis(Name)    -> rpc(name_server, {whereis, Name}).

%% callback routines
init() -> dict::new().

handle({add, Name, Place}, Dict) -> {ok, dict:store(Name, Place, Dict)};
handle({whereis, Name}, Dict)    -> {dict:find(Name, Dict), Dict}.

The point is this: We specified a server and its client as a sequential function—handle—and a pair of remote procedure calls. We didn’t have to be aware of any underlying concurreny.

If we want to use the name_server module, we simply start a server—the only telltale sign that some measure of concurrency is involved—and then use the exposed remote procedure calls.

1> server1:start(name_server, name_server).
true
2> name_server:add(joe, "at home").
ok
3> name_server:whereis(joe).
{ok, "at home"}

A generic server in Classical Processes?

Before we start to think about how to translate server1 to CP, it may be useful to consider which parts are essential, and which parts are merely useful. For instance, starting the server and binding it to a global name—using register/2 might be incredible useful. However, such instances of dynamic name binding are tedious to formalise. Instead, we choose to read this code more like an assignment.

main() ->
  name_server = spawn(fun() -> loop(...) end).
  ...

This allows us to treat the spawning of a server as a top-level cut, and the PID assigned to name_server as a channel.

This means that the main problem is implementing loop/2. If we briefly consider the arguments of loop, we see that it should receive some initial state Init, and some request handler Mod—the Name argument is no longer required. The type of the initial state can be whatever we need. However, the type of the request handler should be a sum of “handler functions”…

Handler := +{ ... callback types ...  }

The server that it returns should be able to handle various requests, which should correspond to the requests handled by Mod

Server := +{ ... similar callback types without state ... } * Server

This constraint may be somewhat tricky—if not impossible—to formalise using the types of CP… so it may be easier to start out by implementing the name_server.

A name server in Classical Processes

Of course, if our exercise in implementing the name_server is to have any value, we must use the same structure that we would use generically. That is to say, we should construct it using a “sequential” request handler and a loop function. This is what we will do… in a while…

Names, places, stuff…

CP doesn’t exactly come with a rich standard library, which means we’ll have to implement the datatypes we need. First, we implement two datatypes to represent names and places. We’ll represent both as named sums: all possible names are joe and helen, and all possible places are at_home and at_work—that second one at least seems general enough to me.

In addition to these basic types, we define the processes which communicate these values along a channel. And because we’ll often use places in non-linear manners—because it’s convenient, not because it’s needed—we’ll have their processes communicate !-places instead.

type  Name      = +{joe:1, helen:1}.
type  Place     = +{at_home:1, at_work:1}.

check Joe(n)    |- n : Name.
def   Joe(n)    = n/joe.n[].0.
check Helen(n)  |- n : Name.
def   Helen(n)  = n/helen.n[].0.

check AtHome(p) |- p : !Place.
def   AtHome(p) = !p(p).p/at_home.p[].0.
check AtWork(p) |- p : !Place.
def   AtWork(p) = !p(p).p/at_work.p[].0.

Secondly, we define dictionaries. That is to say, finite maps between names and (maybe) places. We really don’t want to get into defining generic maps right now, trust me.

type Maybe(A) = +{nothing:1, just:A}.
type Dict     = Name -o Maybe(Place).

To use such dictionaries, we define a couple functions:

  • Empty(d) sends an empty dictionary along d;
  • Store(d,n,p,d1) receives a dictionary d, a name n, and a place p, and then sends a dictionary along d1 which maps n to p and all other names according to d; and,
  • Find(d,n,p) receives a dictionary d and a name n, and sends the place associated with n according to d along p.

What you’d expect them to do, basically. Their definitions have been folded because they’re tedious and the post isn’t really concerned with how to implement dictionaries—nor would they help you if we really wanted to implement a generic dictionary. Feel free to inspect them, though.

check Empty(d) |- d : !Dict.
def   Empty(d)
      = !d(d).d(n).case n
        { joe   : n().d/nothing.d[].0
        ; helen : n().d/nothing.d[].0
        }.
check Store(d,n,p,d1) |- d : ~!Dict, n : ~Name, p : ~!Place, d1 : !Dict.
def   Store(d,n,p,d1)
      = case n
        { joe   : n().!d1(d1).d1(n1).case n1
          { joe   : n1().?p[p].d1/just.d1<->p
          ; helen : n1().?d[d].d[d].( d/helen.d[].0 | d1<->d )
          }
        ; helen : n().!d1(d1).d1(n1).case n1
          { joe   : n1().?d[d].d[d].( d/helen.d[].0 | d1<->d )
          ; helen : n1().?p[p].d1/just.d1<->p
          }
        }.
check Find(d,n,p) |- d : ~!Dict, n : ~Name, p : !Maybe(Place).
def   Find(d,n,p)
      = case n
        { joe   : ?d[d].d[d].( d/joe.d[].0
        | n().case d
          { nothing : d().!p(p).p/nothing.p[].0
          ; just    : case d
            { at_home : d().!p(p).p/just.p/at_home.p[].0
            ; at_work : d().!p(p).p/just.p/at_work.p[].0
            }})
        ; helen : ?d[d].d[d].( d/helen.d[].0
        | n().case d
          { nothing : d().!p(p).p/nothing.p[].0
          ; just    : case d
            { at_home : d().!p(p).p/just.p/at_home.p[].0
            ; at_work : d().!p(p).p/just.p/at_work.p[].0
            }})}.

While we’re at it, we should probably tell Agda about these types.

Modules and their types

We can implement the type of the name_server… euh… module as… well, as follows: it should respond to either add or whereis. In the first case, it should receive a name, a place and a dict, and it’ll reply with nothing, but update the dictionary. In the second case, it’ll receive just a name n and a dict—and it’ll reply with whatever location it finds in the dict under n, but it’ll leave it dict unchanged.

type  Module =
      &{ add     : (Name * !Place * !Dict) -o (1 * !Dict)
       , whereis : (Name * !Dict) -o (!Maybe(Place) * !Dict)
       }.

We can implement a name_server module which does what it ought to do in CP.

check Module(h) |- h : !Module.
def   Module(h) = !h(h).case h
      { add     : h(d).d(p).p(n).h[x].( x[].0 | Store(d,n,p,h) )
      ; whereis : h(d).d(n).h[p].( Find(d,n,p) | h<->d )
      }.

There is a generic pattern to module types: they form a named “with”, in which the types of each option follow a specific pattern, but are allowed to vary in some parameters with the name.

Module :  {n} (A R : Fin (2 + n)  Type) (S : Type)  Type
Module A R S = &[ i ] (A i  S  R i  S)

We can use this type to derive the type for the name_server module. First, we defined a datatype for requests. We can either define this as a datatype, and then provide a bijection with a finite type, or we can simply define it as some syntactic sugar on top of finite types. We’ll choose the second option here.

Request = Fin 2
pattern add     = zero
pattern whereis = suc zero
pattern end     = suc (suc ())

Once we have a type for requests, we can define the A, R and S parameters of Module.

A : Request  Type
A add     = Name  ¡ Place
A whereis = Name
A end

R : Request  Type
R add     = 𝟙
R whereis = ¡ Maybe Place
R end

S : Type
S = ¡ Dict

Now we can show that Module A R S indeed derives the correct type.

NameModule = Module A R S
-->
test_NameModule : NameModule 
  {- add     -} (Name  ¡ Place  ¡ Dict  𝟙              ¡ Dict) &
  {- whereis -} (Name            ¡ Dict  ¡ Maybe Place  ¡ Dict)
test_NameModule = refl

Servers and loops

The server type is similar to that of the module, but different in subtle ways. First of all, a module only needs to handle a single request. However, a server—after handling a request—should be ready to receive the next one.

Therefore, server types are greatest fixed points: they will handle a request, and return a new channel on which they can receive requests. Ideally, this wouldn’t need to terminate—we can hold the server channel for a while, and then return it to the environment when we are done interacting.2 However, for practical purposes, we’ll have the server respond to a request to halt, in addition to the name_server requests add and whereis.

type  Server = nu X.
      &{ halt    : 1
       , add     : (!Name * !Place) -o (1 * X)
       , whereis : !Name -o (!Maybe(Place) * X)
       }.

Now, our mission is to construct a server using a module and an initial state—to define loop. This—it turns out—is not possible in CP. The reason for this is the definition of the -rule.

As it is defined in Talking Bananas3, corec is only well-type if the context consists solely of the channels involved in the recursion. This is crucial to preserve linearity, since channels involved in a recursion may be used an arbitrary number of times. However, it should not be harmful to allow a context of unrestricted channels (and perhaps fixed points) to be used during the recursion. We define to be the variant which allows this.

It turns out that such a rule is exactly what we need to define loop. The loop is a fold over all requests the server will receive, with the corec threading the state through the server. The freedom that we gained in is exactly what we needed to be able to respond to requests—albeit always non-linearily.4

check Loop(d,h,s) |- d : ~!Dict, h : ~!Module, s : Server.
def   Loop(d,h,s) =
      corec s(d).case s
      { halt    : s[].0
      ; add     : s(p).p(n).
                  ?h[h].h/add.h[h].(h[h].(h[h].(?n[n].h<->n|h<->p)|h<->d)|
                  h(r).s[s].(s<->r|s<->h))
      ; whereis : s(n).
                  ?h[h].h/whereis.h[h].(
                  h[h].(?n[n].h<->n|h<->d)|
                  h(r).s[s].(s<->r|s<->h))
      }.

We can look at an example execution of such a server. For instance, how would we encode the example from way-up-above in the bit about Erlang? I mean this one:

1> server1:start(name_server, name_server).
true
2> name_server:add(joe, "at home").
ok
3> name_server:whereis(joe).
{ok, "at home"}

Well, the anwer is, we’d encode it as follow. We spawn an empty dictionary and a module, and use them to start a server. We then send two add requests to the server, storing the locations of joe and helen, and finally request the location of joe.

check   new [d:!Dict]   ( Empty(d)    |
        new [h:!Module] ( Module(h)   |
        new [s:Server]  ( Loop(d,h,s) |
        rec s.
            s/add.
            s[s].( s[s].(
                !s(s).s/joe.s[].0 |
                !s(s).s/at_home.s[].0 ) |
                s(r).r().
        rec s.
            s/add.
            s[s].( s[s].(
                !s(s).s/helen.s[].0 |
                !s(s).s/at_work.s[].0 ) |
                s(r).r().
        rec s.
            s/whereis.
            s[s].( !s(s).s/joe.s[].0 |
            s(r).
        rec s.
            s/halt.
            s().
        p<->r
        ) ) ) ) ) ) |- p : !Maybe(Place).

This normalises, eventually, to the channel that just sends at_home. Exactly as we’d want.

check   !p(p).p/just.p/at_home.p[].0
        |- p : !Maybe(Place).

So: what is the type of a generic server? It’s very similar to that of a module—a named with where the types of the different options are allowed to vary with the name.

Server :  {n} (A R : Fin (2 + n)  Type)  Type
Server A R = ν[ X ] &[ i ] (A i  R i  + X)

We can check this by showing that the server type for the name_server derived from Server corresponds to the type that we wrote down in our CP implementation. And indeed it does.

NameServer = Server A R
-->
test_NameServer : NameServer  ν[ X ] (
  {- add     -} (Name  ¡ Place  𝟙              + X ) &
  {- whereis -} (Name            ¡ Maybe Place  + X))
test_NameServer = refl

This makes the type of loop fairly easy to define: it’s a function from a module to a server, where the both respect the same names, and their argument and response types vary with the name in the same way.

Loop :  {n} (A R : Fin (2 + n)  Type) (S : Type)  Type
Loop A R S = Module A R S  Server A R

  1. Joe Armstrong, Programming Erlang: Software for a Concurrent World. The Pragmatic Bookshelf, 2007. Chapter 16.

  2. This has a strong feeling of locks to it. This makes sense, as CP is a synchronous π-calculus. If we move to an asynchronous π-calculus, we might be able to give semantics for a server that feel a little less… lock-y.

  3. Sam Lindley and Garrett J. Morris, Talking Bananas: Structural Recursion for Session Types. In the proceedings of ICFP 2016.

  4. It may be sane to include fixed points in such contexts as well.

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