The source for this post is online at 2012-09-03-system-f-hoas-2.rkt.
Last time we implemented the obvious version of a System F interpreter and type-checker. But, it was annoying to have to implement substitution and a type environment, because they are tedious and error-prone (especially w.r.t renaming free variables.) This week, we’ll re-present them using the binding technique, higher-order abstract syntax, or HOAS (which has a bit of an unsightly pronunciation.)
In our last version, we represented terms (types and programs) as first-order, recursive data. This included representing program identifiers as symbols. With HOAS, we use identifiers in the meta-language (Racket) to represent identifiers in the object-language (System F) and functions in the meta-language (Racket) to represent binding in the object-language (System F).
Here are the new terms:
The thing to notice is that we’ve removed the ID constructed and removed the binding names from ABS and TYABS, replacing the body with a function that accepts the substitution that would have been inserted where the name was used.
We do the same thing TYID and TYARR ("for all") in the types:
Here’s an new version of the example, DOUBLE function.
And example use of the function gives it the successor function and calls it with the number 3.
(APP (APP (TYAPP DOUBLE (TYNUM)) (SUCC)) (NUM 3))
When we run this program, we would expect it to return 5.
The interpreter for this language is still mostly the same, except wherever we called substitution before, we now have a function that will do the substitution for us.
The beautiful thing about this is that we can totally ignore substitution, because Racket implemented it for us.
HOAS is an amazing technique because of this convenience. If you want to learn more about HOAS, I suggest starting from Eli Barzilay’s various publications on it.
But, HOAS is not without its problems. The first is that the binding structure of the meta-language and object-language should be similar, or at least binding in the object-language should not have any exotic analysis tied to it that would not be reflected in the meta-language. This makes using HOAS to implement Scheme and Racket macros challenging. Another problem is that HOAS terms are not easy to do induction on, so it is difficult to just "open them up for a look", because the only way to get at the inside is to perform the substitution. For example, let’s look at how we need to change the type-checker.
Recall our type checker examples, the doubling example and this program, which contains a type error:
The first thing about the type-checker is that it doesn’t use an environment or substitution, as it did before. Instead, for type applications, where it used type substitution it simply uses the HOAS substitution. But, the situation is more complicated for the replacement of the environment. The environment was used to map value identifiers to their types during the analysis. But during type-checking, we have no values, so we don’t have anything to substitute in place of the identifier so we can inspect the body of the function.
We could create a new kind of value term that "is" a type that we could substitute in with the expected type. I don’t like this approach because these terms don’t have any meaningful run-time behavior. Instead, I like to write a function that takes a type and creates a value that has that type: type->val.
(define type->val (match-lambda [(TYNUM) (NUM 0)] [(ARR dom rng) (ABS dom (λ (val) (type->val rng)))]))
Once this function is in place, it is simple to write the type-of function (I’ve put the most interesting cases on top):
(define type-of (match-lambda [(ABS ty val->body) (ARR ty (type-of (val->body (type->val ty))))] [(TYAPP rator rand) (match (type-of rator) [(TYARR type->body) (type-of (type->body rand))] [_ #f])] [(APP rator rand) (match (type-of rator) [(ARR dom rng) (and (equal? dom (type-of rand)) rng)] [_ #f])] [(TYABS type->body) (TYARR type->body)] [(SUCC) (ARR (TYNUM) (TYNUM))] [(NUM _) (TYNUM)] [_ #f]))
And now we have another implementation of System F.
One other great thing about HOAS though, that could inspire another implementation, is that it becomes easier to use GADTs to enforce in the meta-language the type-correctness of the object-language terms. My student, Dan Burton, has written about that in this literate Haskell file.
By the way, if you use this code at home, make sure you put the code in this order: