> {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-} > {-# LANGUAGE ScopedTypeVariables #-} > {-# LANGUAGE FunctionalDependencies #-} > {-# LANGUAGE UndecidableInstances #-} > module Language.Fibration where(The final solution arrived at by the end of this message does not require -fallow-undecidable-instances above.) Let us consider a module of the following interface (a signature, in ML speak): } module type FN = sig } type a } type b } val app : a -> b } end This is the interface of a regular function. It can be thought of as a compiler stage or network protocol stack that translates one intermediate language or representation (type a) into another (type b). Here are two sample modules of that signature: } module TIF = struct } type a = int } type b = float } let app x = float_of_int x } end } } module TFI = struct } type a = float } type b = int } let app x = truncate x + 1 } end In our Haskell translation, a signature corresponds to a type class, and an implementation (a structure, aka module) to an instance:

> class NFN a b where > napp:: a -> b > > instance NFN Integer Float where > napp x = fromInteger x > > instance NFN Float Integer where > napp x = truncate x + 1Let us write a module that represents a composition appr . appl of two FN-functions: appl: al->bl and appr: ar->br. In order for the composition to be well-formed, the result type of appl must be the argument type of appr: bl = ar (which we will call the intermediate type t). Let us further suppose that we wish to make this intermediate type explicit (e.g., for inspection, to resolve overloading, to invoke the two intermediate functions separately, etc). Thus we arrive at the following interface: } module type NFN1 = sig } type a1 } type t } type b1 } val app1 : a1 -> b1 } end Or, in Haskell

> class NFN1 a t b where > napp1:: t -> a -> bNote that the type of the intermediate result is really needed in Haskell, to resolve the overloading and properly select the instance. The composition of two modules of the signature FN is computed by the following transparent functor: } module NFn1(L: FN)(R: (FN with type a = L.b)) = struct } type a1 = L.a } type t = L.b } type b1 = R.b } let app1 x = R.app (L.app x) } end It takes two modules of the signature FN, labeled L and R. We should note a _sharing constraint_: the type a of module R must be the same as the type b of module L. The result of the NFn1 is a module of the signature NFN1. Here is an example of using the module } module TIFI = NFn1(TIF)(TFI) } let test_tifi = TIFI.app1 7;; (* 8 *) In Haskell, the functor corresponds to an instance with constraints that correspond to the argument signatures. The sharing is expressed by sharing of the names of type variables:

> instance (NFN a t, NFN t b) => NFN1 a t b where > napp1 t x = napp (napp x `asTypeOf` t) > > test_nfn1::Integer > test_nfn1 = napp1 (undefined::Float) (7::Integer) -- 8Suppose we wish to compose two modules NFN1 again. Again, we wish to expose all intermediate types } module type NFN2 = sig } type a2 } type tl type t type tr } type b2 } val app2 : a2 -> b2 } end } } module NFn2(L: NFN1)(R: (NFN1 with type a1 = L.b1)) = struct } type a2 = L.a1 } type tl = L.t } type t = L.b1 } type tr = R.t } type b2 = R.b1 } let app2 x = R.app1 (L.app1 x) } end We should note again that the functor NFn2 imports two modules of the signature NFN1 and re-exports their types, after relabeling them to avoid ambiguity and applying the sharing constraint R.a1 = L.b1. In Haskell:

> class NFN2 a tl t tr b where > napp2:: (tl,t,tr) -> a -> b > > instance (NFN1 a tl t, NFN1 t tr b) => NFN2 a tl t tr b where > napp2 (tl,t,tr) x = napp1 tr $ ((napp1 tl x) `asTypeOf` t)We can do that again: } module type NFN3 = sig } type a3 } type tl type t type tr } type b3 } val app3 : a3 -> b3 } end } } module NFn3(L: NFN2)(R: (NFN2 with type a2 = L.b2)) = struct } type a3 = L.a2 } type tll = L.tl } type tl = L.t } type trl = L.tr } type t = L.b2 } type tlr = R.tl } type tr = R.t } type trr = R.tr } type b3 = R.b2 } let app3 x = R.app2 (L.app2 x) } end In Haskell:

> class NFN3 a tll tl tlr t trl tr trr b where > napp3:: ((tll,tl,tlr),t,(trl,tr,trr)) -> a -> b > > instance (NFN2 a tll tl tlr t, NFN2 t trl tr trr b) > => NFN3 a tll tl tlr t trl tr trr b where > napp3 (tl,t,tr) x = napp2 tr $ ((napp2 tl x) `asTypeOf` t)Here is a usage example } module TII = struct } type a = int } type b = int } let app x = x + 2 } end } } module NM1 = NFn1(TII)(TII) } module NM2 = NFn2(NM1)(NM1) } module NM3 = NFn3(NM2)(NM2) } let test_nm3 = NM3.app3 5;; (* 21 *)

> instance NFN Integer Integer where > napp x = x + 2 > > test_nfn3:: Integer > test_nfn3 = let i = undefined::Integer > i3 = (i,i,i) > in napp3 (i3,i,i3) (5::Integer) -- 21The exponential explosion of the type variables is apparent. The term expressions, the module expressions, and the sharing constraints are all `linear'. That is, if we wish to define another level of composition, NFN4, we write an expression similar to NFn3, which, if we disregard the type variables, has roughly the same size, in characters. It's only when we look at the type variables we see the explosion. The explosion can be overcome if could magically say: import module NFNn as L; import module NFNn as R; make sure that L.bn = R.an; and re-export the rest. Alas, we can't deal with the type variables of a structure 'in bulk'. If we wish to re-export them, we have to enumerate them all. The explosion is particularly apparent in Haskell, where we refer to type parameters of a class by their position rather than by their name. If we wish to write another level of composition, say, NFN4, we merely need the first type variable NFN3 and the last type variable of NFN3. Alas, we have to enumerate all the type variables in-between. It turns out that we _can_ refer to type variables of a module `in bulk', both in OCaml and in Haskell. To do that, we introduce a more structural representation: } module type FN1 = sig } type a type b type t } module ML : (FN with type a = a and type b = t) } module MR : (FN with type b = b and type a = t) } val app : a -> b } end } } module Fn1(L: FN)(R: (FN with type a = L.b)) = struct } type a = L.a type b = R.b type t = L.b } module ML = L module MR = R } let app x = R.app (L.app x) } end } } module type FN2 = sig } type a type b type t } module ML : (FN1 with type a = a and type b = t) } module MR : (FN1 with type b = b and type a = t) } val app : a -> b } end } } module Fn2(L: FN1)(R: (FN1 with type a = L.b)) = struct } type a = L.a type b = R.b type t = L.b } module ML = L module MR = R } let app x = R.app (L.app x) } end The details of the two halves of the composition are stowed away in the submodules ML and MR. We avoid the explosion in Ocaml because we can mention, for example, the type tl in NFN2 above as ML.t in FN2 instead. We can build a chain of functions using source code of size logarithmic in the length of the chain. Let us extend the chain one more time for illustration, and show an example: } module type FN3 = sig } type a type b type t } module ML : (FN2 with type a = a and type b = t) } module MR : (FN2 with type b = b and type a = t) } val app : a -> b } end } } module Fn3(L: FN2)(R: (FN2 with type a = L.b)) = struct } type a = L.a type b = R.b type t = L.b } module ML = L module MR = R } let app x = R.app (L.app x) } end } } module M1 = Fn1(TIF)(TFI) } module M2 = Fn2(M1)(M1) } module M3 = Fn3(M2)(M2) } let test_m3 = M3.app 5;; (* 9 *) With the help of Haskell type classes, we can also stow away the detailed type information of a module. First, we re-define our class representing the signature FN to take an extra type parameter:

> class FN s a b | s -> a, s -> b where > app:: s -> a -> bThe parameter 's' is a `label' that uniquely identifies an instance of the class FN: in other words, the label 's' represents a module of a signature FN. The label is a `proxy' for the module. Here are a few examples of such modules:

> instance FN (Integer->Float) Integer Float where > app _ x = fromInteger x > > instance FN (Float->Integer) Float Integer where > app _ x = truncate x + 1 > > instance FN (Integer->Integer) Integer Integer where > app _ x = x + 2 > > data L = L > instance FN L Integer Integer where > app _ x = x + 2In the last example, we used an `artificial' label `L'. Now we can write the signature and the functor FN1 that `composes' FN once, the signature FN2 and the corresponding functor that compose FN twice, etc. However, because in Haskell an instance can refer to itself, we can create a recursive functor and a signature:

> instance (FN u a t, FN v t b) => FN (u,v) a b where > app s = app (snd s) . app (fst s)The FN instance above subsumes the old classes NFN1, NFN2, NFN3, etc., all under the same FN class:

> fn'1 = undefined :: (Integer->Float, Float->Integer) > fn'2 = (fn'1, fn'1) > fn'3 = (fn'2, fn'2) > test_fn' = app fn'3 5 -- 9If fn'1 is a 2-stage compiler, then fn'2 is a 4-stage compiler and fn'3 is an 8-stage one. The types of fn'1, fn'2, fn'3 above grow exponentially, just as the Ocaml signature FN2 above is twice the size of FN1 when expanded out. But signature definitions in Ocaml and type synonyms in Haskell allow us to avoid the explosion in the source code. Even though the details of these composed modules are stowed away, they are not hidden. Indeed, the label uniquely determines the the module. We can inspect the label or its type to find sub-labels, which uniquely describe the intermediate modules and their internal types. For example, here is a Haskell function that runs the first 3 stages of an 8-stage compiler like fn'3:

> stages123of8 ~((s12,(s3,s4)),s5678) > = app s3 . app s12The type of stages123of8 is inferred to be *Fibration> :t stages123of8 stages123of8 :: forall b a s5678 s4 s3 s12 b1. (FN s12 a b1, FN s3 b1 b) => ((s12, (s3, s4)), s5678) -> a -> b Here ((s12,(s3,s4)),s5678) is a dummy type argument that identifies the module (in other words, resolves the overloading). Note that the term and type above only mentions the parts of the module that are actually used, not the exponentially-sized details of say s5678. We thus avoid exponential blowup and achieve sharing by specification.

> test_stagem3 = stages123of8 fn'3 5*Fibration> :t test_stagem3 test_stagem3 :: Float *Fibration> test_stagem3 6.0 We should point out that we have indeed accessed an intermediate type in fn'3: although the whole compiler "app fn'3" maps Integer to Integer, the first three stages map Integer to an intermediate type Float. We have taken a great advantage of the pattern-matching ability of the Haskell compiler: the ability to unify one type expression with the other and bind type variables. REFERENCES Robert Harper, and Benjamin C. Pierce. 2003. Design issues in advanced module systems. In Advanced topics in types and programming languages, ed. Benjamin C. Pierce. Cambridge: MIT Press. Draft manuscript. Mark P. Jones. 1995. From Hindley-Milner types to first-class structures. In Proceedings of the Haskell workshop, ed. Paul Hudak. Tech. Rep. YALEU/ DCS/RR-1075, New Haven: Department of Computer Science, Yale University. http://www.cse.ogi.edu/~mpj/pubs/haskwork95.pdf Mark P. Jones. 1996. Using parameterized signatures to express modular structure. In POPL '96: Conference record of the annual ACM symposium on principles of programming languages, 68-78. New York: ACM Press. http://www.cse.ogi.edu/~mpj/pubs/paramsig.html http://www.cse.ogi.edu/~mpj/pubs/paramsig.pdf David B. MacQueen. 1986. Using dependent types to express modular structure. In POPL '86: Conference record of the annual ACM symposium on principles of programming languages, 277-286. New York: ACM Press. http://www.cs.bell-labs.com/who/dbm/papers/popl86/paper.ps Benjamin C. Pierce. 2000. Advanced module systems: A guide for the perplexed. ICFP invited talk. http://www.cis.upenn.edu/~bcpierce/papers/modules-icfp.ps Chung-chieh Shan. 2004. Higher-order modules in System F-omega and Haskell. Draft manuscript. http://www.cs.rutgers.edu/~ccshan/xlate/xlate.pdf Zhong Shao. 1999a. Transparent modules with fully syntactic signatures. In ICFP '99: Proceedings of the ACM international conference on functional programming, vol. 34(9) of ACM SIGPLAN Notices, 220-232. New York: ACM Press. http://flint.cs.yale.edu/flint/publications/fullsig.pdf Zhong Shao. 1999b. Transparent modules with fully syntactic signatures. Tech. Rep. YALEU/ DCS/ TR-1181, Department of Computer Science, Yale University, New Haven. http://flint.cs.yale.edu/flint/publications/fullsig-tr.pdf