GEXL Lives!! Solving the Expression Problem in C#

GEXL is a general expression library providing core primitives for building and processing expression trees. I had abandoned it awhile ago because I quickly ran into a serious, well-known issue: The Expression Problem.

Basically, the expression problem boils down to the fact that typical programming solutions are extensible in only a single direction: either the data types or the operations defined on them, but not both, can be extended without modifying existing code. It takes some sophisticated type machinery to safely solve the expression problem, and various languages, including OCaml, Haskell, Beta/gBeta, and Scala, have acceptable solutions. Languages supporting multimethods support less type safe extension, so I exclude them here (although Nice might do it properly).

So GEXL, which aims to provide both data and their operations as data and visitors respectively, runs smack into the expression problem. Fortunately, Mads Torgersen's paper, The Expression Problem Revisited, provides 4 new solutions by exploiting the generics of C# and Java. The first 3 solutions are still biased towards data-centric or operation-centric extensions, but the last solution provides full extensibility in both directions in ordinary C# with generics. GEXL is saved! There's just one catch: it's not type safe since it requires 2 casts. Fortunately, these casts are encapsulated in the core data and visitor classes, which are parameterized by the types they must check and cast to. The developer extending the core will also need to be a bit careful to make sure he's parameterizing the visitor and expression types at the same level as the extension he's writing.

These core classes are now implemented in GEXL, together with two interpreters and pretty printers written for the untyped lambda calculus, one implemented as data-centric extension of the core library, and the other as an operation-centric extension; the two total about 300 lines of excessively verbose C#, including comments. The paper provides a mixed example of data and operation extensions if you're interested in extending in both directions.

There is a disadvantage of this solution however: visitors can no longer return the results of their computation, due to a type dependency between the Data and Op; if we were to parameterize IVisitor with a return type, IVisitor<R>, as I described earlier, Op must also be parameterized by R, which forces Data to also be parameterized by R since Data's type is constrained by IVisitor.

This means that the entire expression tree will be parameterized by the return type of the operation that we wish to perform, R, which means it can only be traversed by operations that return R! No more typechecking returning bools and tree transforms returning new trees on the same expression tree. Type safety has backed us into a corner; this is in fact a GADT problem, and as covered previously, C# generics fall just short of GADTs. If C# were to be augmented by full equational constraints, instead of the limited form they have now, this problem would be solvable, since we could simply parameterize the methods themselves with the appropriate constraints.

In fact, full equational constraints aren't needed; there are only two limitations of C#'s type system impeding this solution:

1. Limitations of type constraints to derivations: a constraint placed on a generic parameter must be a non-sealed class or an interface, so a constraint limiting a type parameter to a string is not possible. This should be relaxed so that the type constraint is a full subtype relation. The current design is simply a consequence of #2.
   
2. Lack of type constraint refinements: a constraint cannot be refined to a more specific type when subtyping. For instance, the declaration "class Foo where T : S", cannot be refined to "class Bar : Foo where T : U" where U is a subtype of S.
   

As it stands, the least offensive solution is for each visitor to retain its computed value as a private field.

So now that GEXL lives, perhaps we'll also see the eventual revival of Orc.NET. In the meantime, I'll try porting the interpreter for my language to GEXL and see how painful it turns out to be. :-)

P.S. There's a further interesting post in the above LTU thread: it provides a solution to the expression problem using just sum and recursive types; it requires significant boilerplate, but the extensions are still possible, which is impressive.