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Advanced Polymorphism in Simpler-Typed Languages

 

Parametric polymorphism is parameterization of a term by a type, permitting the term to be used in differently-typed contexts. Types of polymorphic terms contain universal quantification. For the sake of inference and ease of type-checking, mainstream functional programming languages restrict where in the type the quantifiers may appear and over what they quantify. Hindley-Milner system, or core ML, limits polymorphism to rank-1 (prenex): all quantifiers in a type must be at the front and quantify over monomorphic types. OCaml and Haskell relax the rank restriction, permitting quantifiers inside certain types such as records. Haskell also permits parameterization by arbitrary type constructors: we can define a tree data type abstracting not only over the type of leaf values but also over the type of the collection to hold node's children.

We do encounter problems that run into the limits of polymorphism in Haskell, let alone in ML. We may need to parameterize by type functions that are not constructors; we may need to quantify over not only types but also kinds. Or we think we need. Some of these demanding problems turn out solvable in Haskell or even ML. On this page we collect such examples. Most of the code is simple OCaml, demonstrating that the plain Hindley-Milner system can be surprisingly expressive.


 

Warm-up: polyvariadic functions

As a warm-up, we show the trivial OCaml code for the problem that may at first blush appear unsolvable within the Hindley-Milner system. The problem is to write a function that first takes the numeral n; The function then takes n other arguments and returns them as the n-element list. It seems the function should have the type n -> ('a ->)n -> 'a list , which is of course not possible in OCaml.

The solution takes three lines; the first two define the building blocks of our numerals -- zero and the successor. The third line defines the desired function p . An indented line after each definition shows the response of the OCaml top-level, with the inferred type for each function.

     let n0 = fun k -> k [];;
       val n0 : ('a list -> 'b) -> 'b = <fun>
     let s n k x = n (fun v -> k (x::v));;
       val s : (('a list -> 'b) -> 'c) -> ('a list -> 'b) -> 'a -> 'c = <fun>
     let p sel = sel (fun x -> x);;
       val p : (('a -> 'a) -> 'b) -> 'b = <fun>
We use the function p as follows. (Again, an indented line underneath each statement is the response of the OCaml top-level.)
     p n0;;
       - : 'a list = []
     
     p (s n0) 1;;
       - : int list = [1]
     
     p (s (s n0)) 1 2;;
       - : int list = [1; 2]
     
     p (s (s (s n0))) 'a' 'b' 'c';;
       - : char list = ['a'; 'b'; 'c']
Version
The current version is September 2008.
References
The problem is a variation of functional unparsing. It is solved using the general approach well explained in
Olivier Danvy: Functional Unparsing. J. Functional Programming, 1998, v8, N6, pp. 621-625

Polyvariadic zipWith and the tautology checker of boolean functions of arbitrary many arguments are described in the Functional Pearl
Daniel Fridlender and Mia Indrika: Do We Need Dependent Types? J. Functional Programming, 2000, v10, N4, pp. 409-415
< http://www.math.chalmers.se/~indrika/jfp.ps.gz >
The concluding section of the paper thoroughly discusses the custom numeral approach.

Polyvariadic functions and keyword arguments
The introduction section of that page describes several other simple encodings of polyvariadic functions, including a simple Forth-like interpreter.

double_generic.ml [6K]
A simple generalization: a generic `function' (or, recipe) that can be instantiated to different data types (sums, products, lists) and to the different number of arguments. In effect, we unify the generic map with the polyvariadic zipWith .

 

Polymorphic selection

Selection from heterogeneous structures such as nested tuples seems to require parameterization of a term by a type function that is not a type constructor. We have run into such selection problems on several occasions; last time, when designing typed printf and scanf sharing the same formatting specification. The problem turns out solvable with mere Hindley-Milner polymorphism, after we exploit the deep connection between polymorphism and inlining.

The problem is to select a component from two tuples. If the tuples are homogeneous, the problem is easy: we define the function f1 that takes a selector as the argument. To select a component, we apply f1 either to fst or to snd. (An indented line below each definition or statement shows the response of the OCaml top-level.)

     let f1 sel = (sel (1,2), sel (3,4));;
       val f1 : (int * int -> 'a) -> 'a * 'a = <fun>
     
     f1 fst;;
       - : int * int = (1, 3)
     f1 snd;;
       - : int * int = (2, 4)

Heterogeneous tuples cause serious trouble, however. For example:

     let f2 sel = (sel (1,'b'), sel (true,"four"));;
     Error: This expression has type bool * string
     	   but an expression was expected of type int * char
The first problem is indicated in the error message. In a Hindley-Milner system, function arguments must be monomorphic. Therefore, it is not possible to apply sel (received as the argument to f2) to the pairs of different types. There is another problem with f2. Let us try to define this function in the less restrictive System F:
     let f2 (sel: forall a b. (a,b) -> a) =
       (sel @ int @ char (1,'b'), sel @ bool @ string (true,"four"))
where @ is the type application. We can apply f2 to fst without trouble: f2 (Fun ta tb -> fun (x:ta,y:tb) -> x) assuming the notation Fun for the type abstraction. Alas, we cannot apply f2 to snd, which has a different polymorphic type forall a b. (a,b) -> b. Even System F is not expressive to write f2. We need an abstraction not only over types but over arbitrary type functions of the kind * -> * -> *.

It turns out the selection from heterogeneous tuples is expressible in a Hindley-Milner system. If we look back at f2, we notice that there are only two choices for sel. Therefore, we can compute the applications f2 fst and f2 snd separately, and later choose between the two results. We use thunks to delay the computations until chosen.

     let f3 =
       (* f2 applied to fst *)
       let f2_fst () = (fst (1,'b'), fst (true,"four")) in
       (* f2 applied to snd *)
       let f2_snd () = (snd (1,'b'), snd (true,"four")) in
       fun sel -> sel (f2_fst, f2_snd) ()
         val f3 : ((unit -> int * bool) * (unit -> char * string) -> unit -> 'a) -> 'a = <fun>
     
     f3 fst;;
       - : int * bool = (1, true)
     f3 snd;;
      - : char * string = ('b', "four")

The trick is a higher-order version of the technique commonly used in partial evaluation. It is related to `narrowing' in functional-logic programming. It is essentially the eta-rule for sums:

     fun x -> C[match x with V1 -> e1 | V2 -> e2] ==>
     fun x -> match x with V1 -> C[e1] | V2 -> C[e2]
where the context C[] does not bind x.
Version
The current version is July 18, 2008.
References
Interpreting types as abstract values
The second part of these lecture notes expounds the deep relation between parametric polymorphism and `inlining'.

 

Higher-order polymorphic selection

This article shows an example of how looking at a problem in a different way turns it from impossible (at least, in System F) to elementary, expressible in the Hindley-Milner system. The problem is a higher-order version of selection from heterogeneous structures, described above. Recall, we wanted to write a function g that takes a selector such as the tuple selector
     fst :: forall a b. (a,b) -> a
     snd :: forall a b. (a,b) -> b
as an argument and applies it to several heterogeneous tuples. For example:
     g sel = (sel (1,'b'), sel (true,"four"))
It is already a problem to type such a function in System F, let alone in the Hindley-Milner system. But we want more: a function that takes functions like g as an argument. The original problem was posed by Vladimir Reshetnikov on the Haskell-Cafe mailing list on June 6, 2009. We describe here an elaborated version: We would like to type check the following definition of the function fs along with several examples of its use.
     fs g = (g snd, (), g fst)
     
     t1 = fs id
     t2 = fs (:[])
     t3 = fs (\sel -> sel (True,False))
     
     t4 = fs (\sel -> sel (True,"False"))
     t5 = fs (\sel -> head $ sel ([True],"False"))
     
     t6 = fs (\sel -> (sel (1,'b'), sel (true,"four")))
The examples t1 through t3 type check in Haskell98; the others are flagged as ill-typed. Indeed, the inferred type of fs is (((a, a) -> a) -> t) -> (t, (), t), betraying the selection from tuples whose components have the same type. In the example t4 the selector is applied to the patently heterogeneous tuple. We saw earlier that the type of a function taking a tuple selector as an argument must be parametrized by a type function of the kind * -> * -> *. Such polymorphism is already not representable in System F. Now we wish to pass such a function as an argument to fs.

First we show a brute-force solution, emulating the necessary higher-rank polymorphism. Then we change the point of view and the problem becomes trivial. The complex solution introduces an extra level of interpretation:

     {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-}
     {-# LANGUAGE FunctionalDependencies, UndecidableInstances #-}
     
     data Fst = Fst
     data Snd = Snd
     fs g = (apply g Snd, (), apply g Fst)
replacing the functions like fst and snd with their `encodings' fstE and sndE, to be interpreted by the extensible interpreter Apply. The base cases of the interpreter are:
     class Apply f x y | f x -> y where apply :: f -> x -> y
     
     instance Apply (x->y) x y    where  apply = ($)
     
     instance Apply Fst (x,y) x   where apply _ = fst
     instance Apply Snd (x,y) y   where apply _ = snd
To write the simple tests t1 through t3 we add the encoding for functions that can be passed as arguments of fs. Representing functions by their encoding lets us get around the too complex polymorphism:
     newtype Sel a w = Sel (((a,a) -> a) -> w)
     
     t1 = fs (Sel id)
     t2 = fs (Sel (:[]))
     t3 = fs (Sel (\sel -> sel (True,False)))
     -- (False,(),True)
     
     instance Apply (Sel a w) Fst w where  apply (Sel f) _ = f fst
     instance Apply (Sel a w) Snd w where  apply (Sel f) _ = f snd
That encoding is too simple for t4. Hence we introduce a more complex encoding, and extend our interpreter:
     newtype PSel obj = PSel obj
     instance Apply Fst obj w => Apply (PSel obj) Fst w where
         apply (PSel obj) _ = apply Fst obj
     instance Apply Snd obj w => Apply (PSel obj) Snd w where
         apply (PSel obj) _ = apply Snd obj
     
     t13 = fs (PSel (True,False))
     t4 = fs (PSel (True,"False"))
     -- ("False",(),True)
Alas, to express t5 and t6 we have to complicate our encoding further still.

Let us pause however. The number of selectors is finite and small; for example, there are only two selectors from tuples, fst and snd. This gives us an idea to represent the function g as a `table', `indexed' by the selector argument. This change of representation is quite like applying the eta-rule for sums. The function fs will receive the table: a pair of the results (g fst, g snd).

     g' = (g_fst, g_snd)
      where
      g_fst = (fst (1,'b'), fst (True,"four")) 
      g_snd = (snd (1,'b'), snd (True,"four"))
     
     fs g' = (snd g', (), fst g')
     
     t1 = fs (id,id)
     t2 = fs ((:[]),(:[]))
     t3 = fs (True,False)
     
     t4 = fs (True,"False")
     -- ("False",(),True)
     t5 = fs (head [True], head "False")
     -- ('F',(),True)
     
     t6 = fs g'
     -- (('b',"four"),(),(1,True))
Thanks to the non-strict evaluation of Haskell, the results are not actually computed unless needed. This code type checks in Haskell98, or any Hindley-Milner system. Changing the representation made the problem trivial.
Version
The current version is 1.2, October 2009.
References
Type-class overloaded functions: second-order typeclass programming with backtracking
More extensive example of using the interpreter Apply

Haskell with only one typeclass
Another extensive example of Apply (called C in that code)

 

Parametric polymorphism over a type class

Bas van Dijk, in a message posted on Haskell-Cafe on 30 September 2008 has posed a problem of parameterizing a function by a type class rather than by a type. The standard Prelude functions
     read        :: Read a => String  -> a
     fromInteger :: Num a  => Integer -> a
are the examples of parameterizing a term by a type, which is constrained to be a member of a type class, Read or Num, respectively. With rank-2 polymorphism, we can write terms that accept polymorphic functions like read and fromInteger as arguments, for example:
     foo :: (Num c, Num d) => (forall b. Num b => a -> b) -> a -> (c, d)
     foo f x = (f x, f x)
     bar :: (Read c, Read d) => (forall b. Read b => a -> b) -> a -> (c, d)
     bar f x = (f x, f x)
     
     testFoo = foo fromInteger 1  :: (Int, Float) -- (1,1.0)
     testBar = bar read       "1" :: (Int, Float) -- (1,1.0)
Since higher-rank types in general cannot be inferred, we must supply signatures for the functions foo and bar . Writing signatures for top-level functions is overall a good habit. If the functions foo and bar were local however, defined in a where clause, we must still write their full signatures -- and that is annoying.

The functions foo and bar illustrate parameterizing terms by polymorphic terms. Bas van Dijk has observed that foo and bar look quite alike. It is tempting to abstract away the difference between them, generalizing foo and bar to a single function baz to be used like

     testBaz1 = baz fromInteger 1  :: (Int, Float)
     testBaz2 = baz read       "1" :: (Int, Float)
The only difference between foo and bar is the type class constraint: Num for foo and Read for bar . The desired term baz must therefore be parameterized by a type class. The article develops the solution without resorting to the recently added Constraint kind, showing that first-class constraints have always been available in Haskell.

It turns out that quantification over a type class is easy, after we add a layer of indirection. We introduce a type class that relates, or interprets, types as type classes. That type class acts as a look-up table from types to type class constraints. We can now use ordinary types to represent type classes. Ordinary polymorphism -- parameterization over types -- becomes in effect parameterization over type classes.

We implement the idea as follows, using these extensions:

     {-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-}
     {-# LANGUAGE FlexibleInstances, TypeSynonymInstances #-}
Although functional dependencies are required, overlapping or undecidable instances are not. We also get by without exotic extensions such as mutually recursive instances, required by `SYB with class.'

We define the ``look-up table'' from types to type classes as the following type class:

     class Apply l a b | l b -> a where
         apply :: l -> a -> b
The type class is the generalization of the ordinary function application. We can even give the following instance for it (although we won't be needing it here).
     instance Apply (a->b) a b where
         apply = ($)
We now add two entries into our look-up table from types to type classes. The types LRead and LFromInt below will act as look-up keys.
     data LRead = LRead
     instance Read b => Apply LRead String b where
         apply _ = read
     
     data LFromInt = LFromInt
     instance Num b => Apply LFromInt Integer b where
         apply _ = fromInteger
The type LRead indeed relates to the type class constraint Read while LFromInt relates to Num . We are done. The function baz is simply
     baz f x = (apply f x, apply f x)
to be used as follows:
     testBaz1 = baz LFromInt  1  :: (Int, Float)
     testBaz2 = baz LRead    "1" :: (Int, Float)
Incidentally, if it were not for the polymorphic numeric literal 1 , we could have defined Apply with only two parameters and without the functional dependency.

Although we have accomplished a seemingly high-ranking polymorphism, rank-2 types emphatically were not needed. Therefore, all types are inferred. We did not have to write the signature for baz . It is still interesting to see what the type checker has inferred:

     *Main> :t baz
     baz :: (Apply l a b, Apply l a b1) => l -> a -> (b, b1)
The type of baz does show it to be a generalization of both foo and bar . The term is polymorphic over the type l of keys in our look-up table of type class constraints. In effect, baz is polymorphic over type classes -- as desired.

Since a type class acts as a type predicate in Haskell, quantification over arbitrary type predicates gives us the unrestricted set comprehension. We should be able to write something like Russell paradox. Indeed we can. The following code understandably needs the UndecidableInstances extension.

     data Delta = Delta
     instance Apply l l b => Apply Delta l b where
         apply _ x = apply x x
The definition of apply is not recursive: the method apply for the instance Apply Delta l b is defined in terms of a generally different apply method, for the instance Apply l l b . However, we may, later on, instantiate l to be Delta . After all, the quantification in the above instance is impredicative, over a class that includes the instance being defined. We introduce
     omega () = apply Delta Delta
whose inferred type
     *Main> :t omega
     omega :: () -> b
leaves no doubts about the behavior of the term: omega () promises to produce a value of any type whatsoever. Of course the evaluation of omega () does not terminate. Since apply is sort of a functional application, the term apply x x above does look quite like the self-application in \x -> x x , which is the `kernel' of fixed-point combinators. The self-application and hence fixed-point combinators are not expressible in the simply-typed lambda-calculus. Our calculus is certainly not simply typed.
Version
The current version is October 1, 2008.
References
Type-class overloaded functions: second-order typeclass programming with backtracking
More extensive example of using the type class Apply

Class-parameterized classes, and the type-level logarithm
An illustration of parameterizing a type class by a type class.

Many faces of the fixed-point combinator

 

Leibniz equality can be made injective

Haskell type families, as it turns out, let us write Leibniz equality witnesses of the injectivity of type constructors. We may use Leibniz equality not only for introduction but also for elimination of type constructors, which has been considered impossible. The key is a realization that type-constructor polymorphism extends to the polymorphism over arbitrary type functions. Leibniz injectivity is also the example of type functions seemingly more expressive than type-class functional dependencies.

The paper on ``Typing Dynamic Typing'' (Baars and Swierstra, ICFP 2002) demonstrated the first implementation and application of Leibniz equality witnesses in Haskell:

     newtype EQU a b = EQU{subst:: forall c. c a -> c b}
A term eq :: EQU a b witnesses the equality of the types a and b: in any context c, the type a can be substituted with the type b. The context is represented by the type constructor c. The witness gives the constructive proof of substitutability: given a term of the type c a, we can always obtain a term of the type c b. Since we do not know anything about the context c, such a substitution may only happen if the two types are indeed equal. Hence the sole non-bottom witness is
     refl :: EQU a a
     refl =  EQU id
testifying that equality is reflexive. Leibniz equality became quite popular since it lets us represent GADTs to a large extent. The power of the Leibniz equality comes from the freedom to choose the context c. The following example illustrates that power, by concisely proving that the equality is transitive. Here is the 1-line witness of the proof:
     tran :: EQU a u -> EQU u b -> EQU a b
     tran au ub = subst ub au
We treat the type EQU a u as (EQU a) u, that is, as an application of the `type constructor' (EQU a) to the type u. We then apply the witness EQU u b to replace the type u with the type b in the context (EQU a), giving us the desired EQU a b.

Implementing type checkers and type inferencers in Haskell along the lines of typing dynamic typing requires the proof that two arrow types are equal if and only if their components are equal. In one direction,

     eq_arr :: EQU a1 a2 -> EQU b1 b2 -> EQU (a1->b1) (a2->b2)
witnesses that if the argument and the result types of two arrow types are equal, the arrow types themselves are equal. Fortunately, for this, commonly used direction, the proof is easy, after we define suitable type-level combinators F1 and F2 to place the types being equated in the right context:
     newtype F1 t b a = F1{unF1:: EQU t (a->b)}
     newtype F2 t a b = F2{unF2:: EQU t (a->b)}
     
     eq_arr a1a2 b1b2 = 
         unF2 . subst b1b2 . F2 . unF1 . subst a1a2 . F1 $ refl
In the type EQU (a1->b1) (a2->b2), the type `constructor' (F1 (a1->b1) b2) represents the context of the type a2, and the type constructor (F2 (a1->b1) a2) is the context of the type b2. Leibniz equality only applies if the context of a type is represented as a type constructor; therefore, we sometimes have to define auxiliary newtypes to put the contexts in the required form.

In the reverse direction, we have to prove that if two arrow types are equal, their components (e.g., the argument types) are equal as well. The proof will witness the injectivity of the arrow type constructor. It was thought that injectivity cannot be proven with Leibniz equality at all. For example, the paper ``Implementing Cut Elimination: A Case Study of Simulating Dependent Types in Haskell'' by Chen, Zhu and Xi, PADL'04, needed the reverse, elimination direction and found it impossible to obtain with Leibniz equality witnesses. We demonstrate that type families help.

Type families let us define ``subtractive contexts'', so that we can view a type a as the type (a->b) placed into the context that removes (->b). Again we use an auxiliary newtype to make the subtractive context appear as a type constructor.

     type family Fst a :: *
     type instance Fst (x->y) = x
     type instance Fst (x,y)  = x
     -- etc.
     
     newtype FstA a b = FstA{unFstA :: EQU (Fst a) (Fst b)}
     
     eq_arr1 :: EQU (a1->b1) (a2->b2) -> EQU a1 a2
     eq_arr1 eq = unFstA . subst eq $ ra
      where
       ra :: FstA (a1->b1) (a1->b1)
       ra = FstA refl

Jeremy Yallop has suggested a simple generalization that lets us write generic injectivity witnesses, for arbitrary type constructors f and f1:

     eq_f  :: EQU (f a)     (f1 b)     -> EQU a b
     eq_f2 :: EQU (f a1 b1) (f1 a2 b2) -> EQU a1 a2
These witnesses testify that Leibniz equality is indeed stronger than mere bijection.
Version
The current version is March 2011.
References
LeibnizInjective.hs [4K]
The complete code for the article, with a simple test of the injectivity. The code implements Jeremy Yallop's suggestion. The first version of the code was posted on the Haskell-Cafe mailing list on Sun, 2 May 2010 18:28:02 -0700 (PDT).

Proving False with impredicativity, injectivity, and type case analysis



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