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 unification modulo ɑ-equivalence. We might therefore describe this library as an implementation of unifiable abstract binding trees (or UABTs): where ABTs provide a general and reusable system for variable binding, UABTs add a general and reusable system for nominal unification.

   Unification is lovely and not used nearly enough, imo.

2. The library implements variable binding via (what we might call) binding by reference; i.e., variable binding is recorded in the pointer structure of by way of "immutable" reference cells. This is somewhat of an experiment: being unaware of other implementations using this approach, I worked out the details as I went. So far, it seems to have yielded [trivial ɑ-equivalence and substitution algorithms, and enabled nominal unification][], without requiring any bureaucratic fiddling with renaming, variable permutations, or fresh variables.

Caveat emptor: I am not an academic PLT researcher and this library has not yet been used extensively.

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 untyped 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.

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 generated ABT will have the following form:

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.

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 the free variable "M" with the body of the [k] combinator, 
     note that the nominal unification is modulo bound variables: *)
  let unified_term = (lam "z" (v "M") =.= k) |> Result.get_ok in
  assert (to_string unified_term = "(λz.(λy.z))");

  (* 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 functional implementation that takes this connection literally):
• 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 initial implementation of ABTs in OCaml was informed by Neel Krishnaswami's post on ABTs.
• I consulted Christian Urban's paper Nominal Unification Revisited and Urban, Pitts, and Gabby's Nominal Unification for guidance on the harrier bits of unification modulo ɑ-equivalence, tho this library does not currently take advantage of the strategy of nominal permutations described there.