Table of Contents
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
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.
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
the representation is trivial to inspect and manipulate, in contrast with
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
approach used here is also dependent on OCaml's definition of physical
equality to identify
refcells, and on MLs ability to ensure that the
references are immutable via type abstraction.
The following short examples help illustrate use of the library. For more
extensive examples, see
An ABT for the λ-calculus
Here is a short example showing a naive implementation of the simply typed
lambda calculus using
ABTs representing the syntax of a language are produced by applying the
Abt.Make functor to a module implementing the
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
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
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
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
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)))
Unification over λ-calculus terms
The ABTs produced by applying the
Abt.Make functor to an
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) ]")
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
refcells to track binding scope in
conversation with Callan McGill, and the representation of free and bound
variables was influenced by his post "Locally
with-test & >= "1.10.1"
with-test & >= "0.2.0"
with-test & >= "0.17"