Table of Contents
um-abt is an OCaml library implementing abstract binding trees (ABTs)
exhibiting the properties defined in Robert
Harper's Practical Foundations for
Programming Labguages (PFPL) augmented with
unification modulo ɑ-equivalence.
This library aims for the following qualities:
- It should be correct.
- It should be well tested, to ensure its correctness.
- It should be easy to use.
- It should be well documented.
This ABT library has two distinctive (afaik) features:
-
The library augments the binding functionality of the ABT approach with unification modulo ɑ-equivalence. We 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.
-
The library implements variable binding via (what we might call) binding by reference; i.e., variable binding is recorded in the pointer structure of "immutable" reference cells. This is an experiment: being unaware of any 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.
The latest released version can be installed with opam:
opam install um-abtTo install the head of development
opam pin [email protected]:shonfeder/um-abt.gitThe following short examples help illustrate use of the library. For more extensive examples, see test/example/example.ml.
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, sexp]
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] specifying the ABTs that can be created from the operators **)
include Abt.Make (O)
(* Define some helper constructors, that make construction more concise. *)
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))
endThe 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.tMost 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
could have define 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.
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 nFinally, 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.)
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 -- i.e., the bound [x]
in the definitation of [k] is unfied with the bound [z] in our term on the lefthand side: *)
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) ]")-
Harper explicitly connects binding scope with pointers in PFPL (tho I have not seen another functional implementation that takes this connection literally):
The crucial idea is that any use of an identifier should be understood as a reference, or abstract pointer, to its binding. (PFPL, 2nd ed., p. 6)
-
I discussed the idea of using
refcells to track binding scope in conversation with Callan McGill, and the representation of free and bound variables was influenced by his post "Locally Nameless". -
My initial implementation of an ABT library was informed by Neel Krishnaswami's post on ABTs. There are still some aspects of that approach that show up here.
-
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.