package um-abt

um-abt

Table of Contents

• um-abt

  • Overview

    • Documentation

    • Aims

    • Features

  • Synopsis

    • An ABT for the λ-calculus

    • Unification over λ-calculus terms

  • Additional References

Overview

um-abt is OCaml library implementing abstract binding trees (ABTs) exhibiting the properties defined in Robert Harper's Practical Foundations for Programming Labguages (PFPL).

Documentation

• API Documentation

• Synposis

• Examples

Aims

This library aims for the following qualities:

1. It should be correct.

2. It should be well tested, to ensure its correctness.

3. It should be easy to use.

4. It should be well documented.

Features

This ABT library has two distinctive (afaik) features:

1. The library augments the binding functionality of the ABT approach with general syntactic (perhaps later also equational) unification. We might therefore describe this library as an implentation of unifiable abstract binding trees (or UABTs). ABTs provide a generalized a reusable system for variable binding for any language implemented in its terms. UABTs provide -- in addition -- a generalized, reusable system for (first-order, syntactic) unification for any language implemented in its terms.

   Unification is lovely and not used nearly enough, imo.

2. It implements variable binding via (what we might call) binding by reference; i.e., variable binding is implemented by way of "immutable" reference cells. I suspect the advantages of this approach to include:

   • a trivial algorithm for computing ɑ-equivalence

   • neutralization of the usual problem of renaming bound variables

   • the representation is easy to read, in contrast with De Bruijn indices, and, even more important, there's no tedious bookkeeping required during substitution.

   • the representation is trivial to inspect and manipulate, in contrast with HOAS

   Note that I have not done any rigorous analysis or other work to test these suspicions. Feedback or correction on these points would be welcome.

   I also suspect this approach lacks the safety and formal elegance of HOAS or NbE. The approach used here is also dependent on OCaml's definition of physical equality to identify ref cells, and on MLs ability to ensure that the references are immutable via type abstraction.

Synopsis

The following short examples help illustrate use of the library. For more extensive examples, see test/example/example.ml.

An ABT for the λ-calculus

Here is a short example showing a naive implementation of the simply typed lambda calculus using um-abt.

ABTs representing the syntax of a language are produced by applying the Abt.Make functor to a module implementing the Operator specification.

The generated ABT will have the following form, where module O : Operator:

type t = private
  | Var of Abt.Var.t
  | Bnd of Abt.Var.binding * t
  | Opr of t O.t

Most of the values required by the Operator specification can be derived using ppx_deriving. So all that is usually required is to define a datatype representing the operators and their arities.

After the ABT is generated However, it is recommended that one also define constructors making it more convenient and safer to construct terms of the language:

module Syntax = struct

  (* Define the usual operators, but without the variables, since we get those free *)
  module O = struct
      type 'a t =
      | App of 'a * 'a
      | Lam of 'a
      [@@deriving eq, map, fold]

      let to_string : string t -> string = function
      | App (l, m) -> Printf.sprintf "(%s %s)" l m
      | Lam abs    -> Printf.sprintf "(λ%s)" abs
  end

  (* Generate the syntax, which will include a type [t] of the ABTs over the operators **)
  include Abt.Make (O)

  (* Define some constructors to ensure correct construction *)

  let app m n : t =
    (* [op] lifts an operator into an ABT *)
    op (App (m, n))

  let lam x m : t =
    (* ["x" #. scope] binds all free variables named "x" in the [scope] *)
    op (Lam (x #. m))
end

The private annotation indicates that you can use pattern matching to deconstruct the ABT, but you cannot construct new values without using the supplied combinators. This ensures essential invariants are preserved. E.g., it is impossible to construct a binding in which the expected variables are not bound in the term in scope.

For a more perspicuous view of our produce, let's define the SKI combinators and see what they look like when printed in the usual notation:

(* [v x] is a free variable named "x" *)
let x, y, z = Syntax.(v "x", v "y", v "z")

let s = Syntax.(lam "x" (lam "y" (lam "z" (app (app x y) (app y z)))))
let k = Syntax.(lam "x" (lam "y" x))
let i = Syntax.(lam "x" x)

let () =
  assert (Syntax.to_string s = "(λx.(λy.(λz.((x y) (y z)))))");
  assert (Syntax.to_string k = "(λx.(λy.x))");
  assert (Syntax.to_string i = "(λx.x)");

Note that equality between ABTs is defined in terms of ɑ-equivalence, so we can define the i using any variable, and it will be equivalent:

let () =
  assert Syntax.(equal i (lam "y" y))

Now let's define reduction, using the API provided by our generated Syntax. We use pattern matching to j

open Syntax

let rec eval : t -> t =
 fun t ->
  match t with
  | Opr (App (m, n)) -> apply (eval m) (eval n)
  (* No other terms can be evaluated *)
  | _                -> t

and apply : t -> t -> t =
 fun m n ->
  match m with
  | Bnd (bnd, t)  -> subst bnd ~value:n t
  | Opr (Lam bnd) -> eval (apply bnd n)
  (* otherwise the application can't be evaluated *)
  | _             -> app m n

Finally, let's illustrate the correctness of our implementation with a few simple evaluations, demonstrating that our SKI combinators behave as expected:

let () =
  (* Let equality be ɑ-equivalence on our syntax for the following examples *)
  let (=) = Syntax.equal in
  let open Syntax in
  assert (eval (app i x)                 = x);
  assert (eval (app (app k x) y)         = x);
  assert (eval (app (app (app s x) y) z) = (app (app x y) (app y z)))

(See https://en.wikipedia.org/wiki/SKI_combinator_calculus#Informal_description for reference.)

Unification over λ-calculus terms

The ABTs produced by applying the Abt.Make functor to an Operator implementation support first-order, syntactic unification modulo ɑ-equivalence.

• Unification is (currently) limited to first-order, because there is no support for variables standing for operators.

• Unification is (currently) syntactic, because we do not perform any evaluation to determine if two ABTs can be unified.

• Unification is modulo ɑ-equivalence, because two ɑ-equivalent ABTs are considered equal during unification.

let () =
  let open Syntax in

  (* The generated [Syntax] module includes a [Unification] submodule

     - the [=?=] operator checks for unifiability
     - the [=.=] operator gives an [Ok] result with the unified term, if its operands unify,
       or else an [Error] indicating why the unification failed
     - the [unify] function is like [=.=], but it also gives the substitution used to produce
       a unified term *)
  let ((=?=), (=.=), unify) = Unification.((=?=), (=.=), unify) in

  (* A free variable will unify with anything *)
  assert (v "X" =?= s);

  (* Again, unification is modulo ɑ-equivalence *)
  assert (lam "y" (lam "x" y) =?= lam "x" (lam "y" x));

  (* Here we unify a the free variable "M" with the body of the [k] combinator *)
  let unified_term = (lam "x" (v "M") =.= k) |> Result.get_ok in
  assert (to_string unified_term = "(λx.(λy.x))");

  (* The substitution allows retrieval the bound values of the free variables *)
  let _, substitution = unify (lam "x" (v "M")) k |> Result.get_ok in
  assert (Unification.Subst.to_string substitution = "[ M -> (λy.x) ]")

Additional References

• Harper explicitly connects binding scope with pointers in PFPL, tho I have not seen another implementation that takes this connection literally to bypass the usual pain of tedious renaming and inscrutable De Bruijn indices.

• I discussed the idea of using ref cells to track binding scope in conversation with Callan McGill, and the representation of free and bound variables was influenced by his post "Locally Nameless".

• The implementation of ABTs in OCaml was informed by Neel Krishnaswami's post on ABTs (but is substantialy different).