Abstract Binding Trees

On reddit, Paul Chiusano asked me:

I have a somewhat related question - do you know if it is possible / easy to implement one or both typechecking algorithms using a representation of terms / types where polymorphic recursion is used to enforce proper variable scoping, a la bound.

The answer to this questions is no, I haven't thought about this at all. Except when I'm using a proper proof assistant, I don't bother with using types to track scoping. In fact, I tend to view term representations with highly-structured syntax as strong evidence for layering violations: the parser shouldn't know too much about the binding or type structure of a program, because conceptually binding structure is determined after we produce the syntax tree. This is made evident when you look at languages like ML or Haskell, which have features which make it impossible to know which names are in scope prior to doing significant amounts of work (because of import in Haskell, or open in ML).

That said, the problem which motivates types-for-binding is a real one: how can we make substitution safe, once and for all?

The technique I favor is what Bob Harper has dubbed "abstract binding trees", and documented in Chapter 1 of his book. I think it originates in Nuprl, and is also closely related to Martin-Löf's "system of arities" (see Chapter 3 of Programming in Martin-Löf's Type Theory).

To understand abstract binding trees, it helps to start thinking about ordinary abstract syntax trees. ASTs are really just constants $c$, or applications of a term constructor $f$ to some sequence of terms $f(t_1, \ldots t_n)$. (If you are a lisp programmer, you can think of these as s-expressions, and Prolog programmers can think of these as the ground elements of the Herbrand universe.)

Then, to get ABTs, we extend ASTs with two constructors, one for variables $x$, and one for abstractions $x.\;t$. Abstractions are purely binders, with no reduction rules associated with them, and abstract binding trees are simply equated up to alpha-equivalence.

By programming this idea up, we can simply parameterize over the set of term constructors, and have everything to do with scoping and substitution be programmed once and for all. So we end up in a situation where everything Just Works, and all the types are satisfyingly uncomplicated. Furthermore, the implementation is simple enough to fit into a blog post. (All the code in this post in in this gist.)

I'll do it in Ocaml, and so we'll need to define some signatures that Haskell programmers get out of the box. First, we need a signature for functorial types. This signature is satisfied by any parametric datatype with a map functional. It also needs to send identities to identities and respect composition, but we can't represent that unless we're in Agda or Coq.

 module type FUNCTOR = sig
   type 'a t
   val map : ('a -> 'b) -> 'a t -> 'b t
 end

Next, we'll introduce a type of monoids, with a unit and combining operation. I'll just use a record here, since that's a lot lighter weight than ML's modules.

 type 'a monoid = {unit : 'a ; join : 'a -> 'a -> 'a}

With these preliminaries out of the way, we can move on towareds giving the meat of the representation. First, I'll give a type of variables (and sets of variables). This time, instead of being abstract, I'll just be concrete and use strings as the representation of variables. This is partly because this is what I actually do, but partly also for the rhetorical effect of emphasising that an often-maligned representation technique is A-ok.

 
 type var = string
 module V = Set.Make(struct type t = var let compare = compare end)

Now, I'll give the signature for abstract binding trees.

 module type ABT = sig
   type 'a signature
   type 'a f = Var of var | Abs of var * 'a | Tm of 'a signature
   val map : ('a -> 'b) -> 'a f -> 'b f
 
   type t
   val into : t f -> t
   val out  : t -> t f
 
   val freevars : t -> V.t
   val var : V.elt -> t
   val abs : V.elt * t -> t
   val tm : t signature -> t
 
   val subst : t -> var -> t -> t
 end

Here, the type 'a signature tells us what the term constructors we have are. We augment this with the 'a f type, which adds a new term former for variables Var of var, as well as a term constructor Abs of var * 'a for scoping a variable. Then we have an abstract type of terms t, which we can wrap and unwrap one level using the into and out functions.

We can also have ask for the free variables of a type with freevars, and we have some convenience constructors for variables, abstractions and terms, as well as the subst function which does the alpha-equivalence respecting substitution. We could also put in other operations (such as equality up to alpha-equivalence), but substitution suffices to illustrate the technique.

We'll also say just what a signature is, now. Basically, it's any functor which can collect up the elements of a monoid. This idea is from Björn Bringert and Aarne Ranta's paper A Pattern for Almost-Compositional Functions.

 
 module type SIGNATURE = sig
   include FUNCTOR
 
   val join : 'a monoid -> 'a t -> 'a 
 end

We are finally in a position to give the implementation of ABTs. It's an Ocaml functor taking a signature F, and returning an instance of the ABT signature where the 'a signature type is identified with the type of the argument.

module Abt(F : SIGNATURE) : ABT with type 'a signature := 'a F.t =
struct
  type 'a f =
    | Var of var
    | Abs of var * 'a
    | Tm of 'a F.t

  let map f = function
    | Var x -> Var x
    | Abs(x, e) -> Abs(x, f e)
    | Tm t -> Tm (F.map f t)

As you can see, the type 'a f is just the same as in the signature, and the map function just lifts the signature type's map operation to the type augmented with binding structure.

  type t = In of V.t * t f

The concrete representation of terms is just the obvious recursive type built over the signature, augmented with a set of free variables at each node. This makes computing the free variables very easy:

  let freevars (In(vs, _)) = vs

In addition, unrolling the type t simply throws away some variables and returns the subterm:

  let out (In(_, t)) = t 

On the other hand, building the embedding function with smart constructors is a little more work. First, we note that sets of variables form a monoid, with the empty set as the unit and the set union as the join operation.

  let m = {unit = V.empty; join = V.union}

We can use this to define the smart constructors. The var function simply says that its free variables are its one variable itself, and in abs(z,e), the free variables are the free variables of e minus z. For the Tm t case, we use the join operation that we required the signature to possess.

  let var x = In(V.singleton x, Var x)
  let abs(z, e) = In(V.remove z (freevars e), Abs(z, e))
  let tm t = In(F.join m (F.map freevars t), Tm t)

Then we can define into just by using these constructors.

  let into = function
    | Var x -> var x
    | Abs(x, e) -> abs(x, e)
    | Tm t -> tm t 

Now all that remains is to implement capture-avoiding substitution. First, we give a fresh function which returns a new renaming of v which is not in a set of names vs. It does this by sticking primes onto v until it finds one that isn't in vs. It's possible to do something smarter, but it serves for a blog post.

  let rec fresh vs v =
    if V.mem v vs then fresh vs (v ^ "'") else v 

We also need a variable renaming function. rename x y e renames x to y in the body e, and just walks down the term structurally. If it hits an abstractor, it will stop if the name is shadowed, and otherwise continue on down. Here, we rely on the functoriality of the signature to handle the Tm case.

  let rec rename x y (In(fvs, t)) =
    match t with
    | Var z -> if x = z then var y else var z
    | Abs(z, e) -> if x = z then abs(z, e) else abs(z, rename x y e)
    | Tm v -> tm (F.map (rename x y) v)

We can now give the definition of capture-avoiding substitution. At each abstraction, it does the usual thing of choosing a fresh identifierand renaming. It's easy to do this more efficiently (for instance with a parallel substitution), but it's not worth bothering with until it becomes a bottleneck.

  let rec subst t x body = 
    match out body with
    | Var z when x = z -> t
    | Var z            -> var z 
    | Abs(x, e) ->
      let x' = fresh (V.union (freevars t) (freevars body)) x in
      let e' = subst t x (rename x x' e) in 
      abs(x', e')
    | Tm body -> tm (F.map (subst t x) body)
end

I'll finish off with a simple example of how we might use it. Let's proceed by giving a syntax for the simply-typed lambda calculus, in a bidirectional style. Note that the signature type 'a t simply has one constructor for each of the term formers of the lambda calculus, and it has no binding information in it at all – all binding is handled through the ABT interface. So the constructor for functions Lam of 'a, has a single subterm — we'll use typechecking to ensure that the subterm is of the form Abs(z, e).

module Lambda =
struct
  type tp = Base | Arrow of tp * tp 
  type 'a t = Lam of 'a | App of 'a * 'a | Let of 'a * 'a | Annot of tp * 'a
  let map f = function
    | Lam x      -> Lam (f x)
    | App (x, y) -> App(f x, f y)
    | Let (x, y) -> Let(f x, f y)
    | Annot(t, x) -> Annot(t, f x)

  let join m = function
    | Lam x -> x
    | App(x, y) -> m.join x y
    | Let(x, y) -> m.join x y
    | Annot(_, x) -> x
end

With this signature in hand, we can create our syntax:

module Syntax = Abt(Lambda)

Now, here's a simple bidirectional typechecker for this calculus.

module Bidir = struct
  open Lambda
  open Syntax

Contexts are just lists of variables and their types.

  type ctx = (var * tp) list

In a bidirectional typechecker, every term is either used to synthesize a type, or is checked against a type. Here, we give predicates to check that.

  let is_synth = function 
    | Tm (Lam _) | Tm (Let (_, _)) -> false
    | _ -> true
  let is_check e = not(is_synth e)

Now, we give the unabs function, which deconstructs an abstraction and fails with an assertion error if it isn't handed an abstraction.

  let unabs e =
    match out e with
    | Abs(x, e) -> (x, e)
    | _ -> assert false

The reasoning can be seen in the definition of the check ctx e tp function, which checks that term e has type tp in context ctx.

  let rec check ctx e tp =
    match out e, tp with
    | Tm (Lam t), Arrow(tp1, tp') ->
      let (x, e') = unabs t in
      check ((x, tp1) :: ctx) e' tp'

In particular, in the lambda case, we check that the outer constructor is Lam, and then unabstract the body to get the binder and subterm. This should always succeed (or else we've been passed an ill-formed binding tree). It might be possible to tighten up the definition of signatures, but since ABTs are constructed by the parser, errors are likely to be few (and easily detected by Quickcheck-y techniques).

    | Tm (Lam _), _               -> failwith "expected arrow type"
    | Tm (Let(e', t)), _ ->
      let (x, e'') = unabs t in 
      let tp1 = synth ctx e' in 
      check ((x, tp1) :: ctx) e'' tp

The let case works similarly to the lambda case.

    | body, _ when is_synth body ->
      if tp = synth ctx e then () else failwith "Type mismatch"
    | _ -> assert false

If we know a term is synthesizing, we can ask for its type, and then see if it's equal to the type we are checking against. Otherwise, we have ill-formed syntax.

  and synth ctx e =
    match out e with
    | Var x -> (try List.assoc x ctx with Not_found -> failwith "unbound variable")

If we want to synthesize a type for a variable, we just look it up in the context.

    | Tm(Annot(tp, e)) -> let () = check ctx e tp in tp

If we have a type-annotated term, we can use that to synthesize a type for it.

    | Tm(App(f, e))  ->
      (match synth ctx f with
       | Arrow(tp, tp') -> let () = check ctx e tp in tp'
       | _ -> failwith "Applying a non-function!")

To infer a type for an application, we infer a function type for the head, and then check the argument against the argument type of the function, returning the result type of the function.

    | body when is_check body -> failwith "Cannot synthesize type for checking term"
    | _ -> assert false
end

As you can see, this is more or less what you'd write without any binding machinery – there are a few more annoying Tm constructors, but that's about it. The key point is that the type of substitution is pure, without the need for a context monad, name supply, or similar. That leaves us free to engineer the context monad as needed for the typechecker, without having to work around the requirements of the term representation.